Subgroups — #32011361

Order 18176.

796 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22597257	C₂	1	1	yes	maximal, characteristic, complemented
2	#22597200	C₂	1	1	yes	maximal, characteristic, complemented
3	#22589245	C₂	1	1	yes	maximal, characteristic, complemented
4	#22589239	C₂	1	1	yes	maximal, characteristic, complemented
5	#22588878	C₂	1	1	yes	maximal, characteristic, complemented
6	#22588873	C₂	1	1	yes	maximal, characteristic, complemented
7	#22588868	C₂	1	1	yes	maximal, characteristic, complemented
8	#22588863	C₂	1	1	yes	maximal, characteristic, complemented
9	#22588836	C₂	1	1	yes	maximal, characteristic, complemented
10	#22588829	C₂	1	1	yes	maximal, characteristic, complemented
11	#22588822	C₂	1	1	yes	maximal, characteristic, complemented
12	#22588815	C₂	1	1	yes	maximal, characteristic, complemented
13	#22588464	C₂	1	1	yes	maximal, characteristic, complemented
14	#22588095	C₂	1	1	yes	maximal, characteristic, complemented
15	C₇₁×#3154	C₂	1	1	yes	maximal, characteristic
16	#17986766	C₂²	1	1	yes	characteristic
17	#17986766	C₂²	1	1	yes	characteristic
18	#17986477	C₂²	1	1	yes	characteristic
19	#17986474	C₂²	1	1	yes	characteristic
20	#17986121	—	2	2	no	—
21	#17986116	—	2	2	no	—
22	#17986100	—	2	2	no	—
23	SD₁₆×Dic₇₁	—	2	2	no	—
24	C₂×#14556064	C₂²	1	1	yes	characteristic
25	#17986037	C₂²	1	1	yes	characteristic
26	#17986016	C₂²	1	1	yes	characteristic
27	#17986016	C₂²	1	1	yes	characteristic
28	#17986016	C₂²	1	1	yes	characteristic
29	#17986016	C₂²	1	1	yes	characteristic
30	#17986008	C₂²	1	1	yes	characteristic
31	#17986008	C₂²	1	1	yes	characteristic
32	#17986004	C₂²	1	1	yes	characteristic
33	#17986004	C₂²	1	1	yes	characteristic
34	C₂×#14556047	C₂²	1	1	yes	characteristic
35	C₂×#14556045	C₂²	1	1	yes	characteristic
36	#17985973	C₂²	1	1	yes	characteristic
37	#17985973	C₂²	1	1	yes	characteristic
38	#17985954	C₂²	1	1	yes	characteristic
39	#17985954	C₂²	1	1	yes	characteristic
40	#17985770	C₂²	1	1	yes	characteristic
41	#17985770	C₂²	1	1	yes	characteristic
42	C₇₁⋊#338	C₂²	1	1	yes	characteristic
43	#17985756	C₂²	1	1	yes	characteristic
44	#17985755	C₂²	1	1	yes	characteristic
45	#17985755	C₂²	1	1	yes	characteristic
46	#17985755	C₂²	1	1	yes	characteristic
47	#17985755	C₂²	1	1	yes	characteristic
48	C₂×#137×C₇₁	C₂²	1	1	yes	characteristic
49	C₂×AGL(1,ℤ/8)×C₇₁	C₂²	1	1	yes	characteristic
50	C₂×#135×C₇₁	C₂²	1	1	yes	characteristic
51	C₂²×SD₁₆×C₇₁	C₂²	1	1	yes	characteristic
52	#536×C₇₁	C₂²	1	1	yes	characteristic
53	#535×C₇₁	C₂²	1	1	yes	characteristic
54	#497×C₇₁	C₄	1	2	yes	complemented
55	#496×C₇₁	C₄	1	2	yes	complemented
56	#495×C₇₁	C₄	1	2	yes	complemented
57	#492×C₇₁	C₄	1	2	yes	complemented
58	#405×C₇₁	C₂²	1	1	yes	characteristic
59	#14556135	—	2	2	no	—
60	#14556135	—	2	2	no	—
61	#14556135	—	4	1	no	—
62	#14556135	—	4	1	no	—
63	C₂×#11765849	—	4	1	no	—
64	C₂×#11765849	—	4	1	no	—
65	C₂×#11765849	C₂³	1	1	yes	characteristic
66	C₂×#11765849	C₂³	1	1	yes	characteristic
67	C₂×#11765849	C₂³	1	1	yes	characteristic
68	C₂×#11765849	C₂³	1	1	yes	characteristic
69	Q₈×Dic₇₁	—	2	2	no	—
70	Q₈×Dic₇₁	—	4	1	no	—
71	D₄×Dic₇₁	—	2	2	no	—
72	D₄×Dic₇₁	—	4	1	no	—
73	#14556087	—	4	1	no	—
74	#14556087	C₂³	1	1	yes	characteristic
75	#14556087	C₂³	1	1	yes	characteristic
76	#14556087	C₂³	1	1	yes	characteristic
77	#14556087	C₂³	1	1	yes	characteristic
78	C₂×C₂²⋊Dic₇₁	—	4	1	no	—
79	#14556064	—	2	1	no	—
80	#14556064	—	2	1	no	—
81	#14556062	—	2	1	no	—
82	#14556060	—	2	1	no	—
83	#14556055	—	2	2	no	—
84	#14556055	—	2	2	no	—
85	#14556055	—	4	1	no	—
86	#14556055	—	4	1	no	—
87	#14556052	—	2	2	no	—
88	#14556052	—	4	1	no	—
89	#14556049	—	2	2	no	—
90	#14556049	—	4	1	no	—
91	#14556047	—	4	1	no	—
92	#14556045	—	4	1	no	—
93	#14556043	—	4	1	no	—
94	#14556043	—	4	1	no	—
95	#14556038	—	2	1	no	—
96	#14556038	D₄	1	2	yes	complemented
97	C₈×Dic₇₁	—	2	1	no	—
98	C₈×Dic₇₁	D₄	1	2	yes	complemented
99	C₂×#41×C₇₁	C₂³	1	1	yes	characteristic
100	C₂×#41×C₇₁	C₂³	1	1	yes	characteristic
101	C₂²×Q₈×C₇₁	C₂³	1	1	yes	characteristic
102	C₂²×D₄×C₇₁	C₂³	1	1	yes	characteristic
103	#137×C₇₁	—	4	1	no	—
104	#137×C₇₁	—	4	1	no	—
105	AGL(1,ℤ/8)×C₇₁	—	4	1	no	—
106	AGL(1,ℤ/8)×C₇₁	—	4	1	no	—
107	#135×C₇₁	—	2	1	no	—
108	#135×C₇₁	—	2	1	no	—
109	#135×C₇₁	—	2	1	no	—
110	#135×C₇₁	—	2	1	no	—
111	C₂×Q₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic, complemented
112	C₂×Q₁₆×C₇₁	C₂×C₄	1	2	yes	complemented
113	C₂×SD₁₆×C₇₁	—	2	1	no	—
114	C₂×SD₁₆×C₇₁	—	2	1	no	—
115	C₂×SD₁₆×C₇₁	—	2	1	no	—
116	C₂×SD₁₆×C₇₁	—	2	1	no	—
117	C₂×SD₁₆×C₇₁	C₂×C₄	1	2	yes	—
118	C₂×SD₁₆×C₇₁	C₂×C₄	1	2	yes	—
119	C₂×SD₁₆×C₇₁	C₂×C₄	1	2	yes	—
120	C₂×SD₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic, complemented
121	C₂×SD₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic, complemented
122	C₂×SD₁₆×C₇₁	C₂×C₄	1	2	yes	complemented
123	C₂×D₈×C₇₁	C₂×C₄	1	2	yes	—
124	C₂×D₈×C₇₁	C₂×C₄	1	1	yes	characteristic, complemented
125	C₂×M₁₆×C₇₁	C₂³	1	1	yes	characteristic
126	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
127	C₄⋊Q₈×C₇₁	C₂×C₄	1	2	yes	complemented
128	#127×C₇₁	C₂×C₄	1	2	yes	—
129	#124×C₇₁	C₂×C₄	1	2	yes	—
130	#124×C₇₁	C₂×C₄	1	2	yes	complemented
131	#123×C₇₁	—	4	1	no	—
132	C₂²⋊Q₈×C₇₁	—	4	1	no	—
133	#121×C₇₁	—	4	1	no	—
134	#120×C₇₁	—	4	1	no	—
135	#117×C₇₁	C₂³	1	1	yes	characteristic
136	#97×C₇₁	C₂×C₄	1	2	yes	complemented
137	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	complemented
138	C₂²×Dic₇₁	—	4	2	no	—
139	#11765849	—	2	1	no	—
140	#11765849	—	2	1	no	—
141	#11765849	—	2	1	no	—
142	#11765849	—	2	1	no	—
143	#11765849	—	2	1	no	—
144	#11765849	—	2	1	no	—
145	#11765849	—	2	1	no	—
146	#11765849	—	2	1	no	—
147	#11765849	—	4	2	no	—
148	#11765849	—	4	2	no	—
149	C₂×C₇₁⋊C₈	—	2	1	no	—
150	C₂×C₇₁⋊C₈	—	2	1	no	—
151	C₂×C₇₁⋊C₈	—	2	1	no	—
152	C₂×C₇₁⋊C₈	—	4	2	no	—
153	C₂×C₇₁⋊C₈	—	4	2	no	—
154	C₂×C₇₁⋊C₈	C₂×D₄	1	2	yes	—
155	C₄⋊Dic₇₁	—	2	1	no	—
156	C₄⋊Dic₇₁	—	2	1	no	—
157	C₄⋊Dic₇₁	—	2	1	no	—
158	C₄⋊Dic₇₁	—	2	1	no	—
159	C₄⋊Dic₇₁	—	4	2	no	—
160	C₂²⋊Dic₇₁	—	2	1	no	—
161	C₂²⋊Dic₇₁	—	2	1	no	—
162	C₂²⋊Dic₇₁	—	2	1	no	—
163	C₂²⋊Dic₇₁	—	2	1	no	—
164	C₂²⋊Dic₇₁	—	4	2	no	—
165	C₄×Dic₇₁	—	2	1	no	—
166	C₄×Dic₇₁	—	2	1	no	—
167	C₄×Dic₇₁	—	2	1	no	—
168	C₄×Dic₇₁	—	4	2	no	—
169	C₄×Dic₇₁	C₂×D₄	1	2	yes	—
170	C₂⁴×C₇₁	—	2	1	no	—
171	#41×C₇₁	—	2	1	no	—
172	#41×C₇₁	—	2	1	no	—
173	#41×C₇₁	—	2	1	no	—
174	#41×C₇₁	—	2	1	no	—
175	#41×C₇₁	—	2	1	no	—
176	#41×C₇₁	—	2	1	no	—
177	#41×C₇₁	—	2	1	no	—
178	#41×C₇₁	—	2	1	no	—
179	C₂×Q₈×C₇₁	—	2	1	no	—
180	C₂×Q₈×C₇₁	—	2	1	no	—
181	C₂×Q₈×C₇₁	—	2	1	no	—
182	C₂×Q₈×C₇₁	—	2	1	no	—
183	C₂×Q₈×C₇₁	—	2	1	no	—
184	C₂×Q₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
185	C₂×Q₈×C₇₁	C₂²×C₄	1	2	yes	—
186	C₂×Q₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
187	C₂×D₄×C₇₁	—	2	1	no	—
188	C₂×D₄×C₇₁	—	2	1	no	—
189	C₂×D₄×C₇₁	—	2	1	no	—
190	C₂×D₄×C₇₁	—	2	1	no	—
191	C₂×D₄×C₇₁	—	2	1	no	—
192	C₂×D₄×C₇₁	—	2	1	no	—
193	C₂×D₄×C₇₁	—	2	1	no	—
194	C₂×D₄×C₇₁	—	4	2	no	—
195	C₂×D₄×C₇₁	C₂²×C₄	1	2	yes	—
196	C₂×D₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
197	C₂×D₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
198	C₂²×C₄×C₇₁	—	2	1	no	—
199	C₂²×C₄×C₇₁	—	2	1	no	—
200	C₂²×C₄×C₇₁	—	2	1	no	—
201	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
202	Q₁₆×C₇₁	—	2	1	no	—
203	Q₁₆×C₇₁	—	2	1	no	—
204	Q₁₆×C₇₁	—	4	2	no	—
205	SD₁₆×C₇₁	—	2	2	no	—
206	SD₁₆×C₇₁	—	2	2	no	—
207	SD₁₆×C₇₁	—	2	2	no	—
208	SD₁₆×C₇₁	—	2	2	no	—
209	SD₁₆×C₇₁	—	2	1	no	—
210	SD₁₆×C₇₁	—	2	1	no	—
211	SD₁₆×C₇₁	—	2	1	no	—
212	SD₁₆×C₇₁	—	2	1	no	—
213	SD₁₆×C₇₁	—	4	2	no	—
214	SD₁₆×C₇₁	—	4	2	no	—
215	D₈×C₇₁	—	2	1	no	—
216	D₈×C₇₁	—	2	1	no	—
217	D₈×C₇₁	—	4	2	no	—
218	M₁₆×C₇₁	—	2	1	no	—
219	M₁₆×C₇₁	—	2	1	no	—
220	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
221	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
222	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
223	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
224	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
225	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
226	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
227	C₄⋊C₄×C₇₁	—	2	1	no	—
228	C₄⋊C₄×C₇₁	—	4	2	no	—
229	C₂²⋊C₄×C₇₁	—	2	1	no	—
230	C₂²⋊C₄×C₇₁	—	4	2	no	—
231	C₂²⋊C₄×C₇₁	—	4	2	no	—
232	C₄²×C₇₁	C₂²×C₄	1	2	yes	—
233	C₂³×C₇₁	—	2	1	no	—
234	C₂³×C₇₁	—	2	2	no	—
235	C₂³×C₇₁	—	2	1	no	—
236	C₂³×C₇₁	—	4	1	no	—
237	C₂³×C₇₁	—	4	1	no	—
238	C₂³×C₇₁	—	16	1	no	—
239	C₂³×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
240	Q₈×C₇₁	—	2	2	no	—
241	Q₈×C₇₁	—	2	1	no	—
242	Q₈×C₇₁	—	2	1	no	—
243	Q₈×C₇₁	—	2	1	no	—
244	Q₈×C₇₁	—	2	2	no	—
245	Q₈×C₇₁	—	2	1	no	—
246	Q₈×C₇₁	—	4	1	no	—
247	Q₈×C₇₁	—	4	1	no	—
248	D₄×C₇₁	—	2	1	no	—
249	D₄×C₇₁	—	2	1	no	—
250	D₄×C₇₁	—	2	2	no	—
251	D₄×C₇₁	—	2	1	no	—
252	D₄×C₇₁	—	2	2	no	—
253	D₄×C₇₁	—	2	1	no	—
254	D₄×C₇₁	—	4	1	no	—
255	D₄×C₇₁	—	4	1	no	—
256	D₄×C₇₁	—	4	1	no	—
257	D₄×C₇₁	—	4	1	no	—
258	D₄×C₇₁	—	4	1	no	—
259	D₄×C₇₁	—	4	1	no	—
260	D₄×C₇₁	—	16	2	no	—
261	C₂×C₄×C₇₁	—	2	2	no	—
262	C₂×C₄×C₇₁	—	2	1	no	—
263	C₂×C₄×C₇₁	—	2	2	no	—
264	C₂×C₄×C₇₁	—	2	1	no	—
265	C₂×C₄×C₇₁	—	4	1	no	—
266	C₂×C₄×C₇₁	—	4	1	no	—
267	C₂×C₄×C₇₁	—	4	1	no	—
268	C₂×C₄×C₇₁	—	4	1	no	—
269	C₂×C₄×C₇₁	—	4	1	no	—
270	C₂×C₄×C₇₁	—	4	1	no	—
271	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
272	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
273	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
274	C₂×C₄×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
275	C₂×C₄×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
276	C₂×C₄×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
277	C₂×Dic₇₁	—	2	2	no	—
278	C₂×Dic₇₁	—	4	1	no	—
279	C₂×Dic₇₁	—	4	1	no	—
280	C₂×Dic₇₁	—	4	1	no	—
281	C₂×Dic₇₁	—	16	2	no	—
282	C₈×C₇₁	—	2	2	no	—
283	C₈×C₇₁	C₄×D₄	1	1	yes	characteristic
284	C₈×C₇₁	C₄×D₄	1	1	yes	characteristic
285	C₈×C₇₁	C₄×D₄	1	1	yes	characteristic
286	C₈×C₇₁	C₄×D₄	1	1	yes	characteristic
287	C₇₁⋊C₈	—	2	2	no	—
288	C₇₁⋊C₈	—	4	1	no	—
289	C₇₁⋊C₈	—	4	1	no	—
290	C₇₁⋊C₈	—	4	1	no	—
291	C₂²×C₇₁	—	4	1	no	—
292	C₂²×C₇₁	—	4	1	no	—
293	C₂²×C₇₁	—	4	1	no	—
294	C₂²×C₇₁	—	4	2	no	—
295	C₂²×C₇₁	—	8	1	no	—
296	C₂²×C₇₁	—	8	2	no	—
297	C₂²×C₇₁	—	8	1	no	—
298	C₂²×C₇₁	—	8	2	no	—
299	C₂²×C₇₁	—	16	1	no	—
300	C₂²×C₇₁	#518	1	1	yes	characteristic
301	C₂²×C₇₁	#535	1	1	yes	characteristic
302	C₂²×C₇₁	#540	1	1	yes	characteristic
303	C₄×C₇₁	—	4	2	no	—
304	C₄×C₇₁	—	4	1	no	—
305	C₄×C₇₁	—	4	1	no	—
306	C₄×C₇₁	—	4	2	no	—
307	C₄×C₇₁	C₂×C₄×D₄	1	1	yes	characteristic
308	C₄×C₇₁	#518	1	1	yes	characteristic
309	C₄×C₇₁	#535	1	1	yes	characteristic
310	C₄×C₇₁	#540	1	1	yes	characteristic
311	Dic₇₁	—	4	2	no	—
312	Dic₇₁	—	8	1	no	—
313	Dic₇₁	—	8	1	no	—
314	Dic₇₁	—	8	1	no	—
315	#22284	—	71	1	no	maximal, Sylow
316	C₂×C₇₁	—	2	1	no	—
317	C₂×C₇₁	—	2	1	no	—
318	C₂×C₇₁	—	2	1	no	—
319	C₂×C₇₁	—	8	1	no	—
320	C₂×C₇₁	—	8	1	no	—
321	C₂×C₇₁	—	8	2	no	—
322	C₂×C₇₁	#2328	1	1	yes	characteristic
323	#3154	D₇₁	1	1	yes	characteristic
324	#2957	—	71	1	no	—
325	#2949	—	71	1	no	—
326	#1977	—	71	1	no	—
327	#1976	—	71	1	no	—
328	#1912	—	71	1	no	—
329	#1911	—	71	1	no	—
330	#1910	—	71	1	no	—
331	#1909	—	71	1	no	—
332	#1905	—	71	1	no	—
333	#1904	—	71	1	no	—
334	#1903	—	71	1	no	—
335	#1902	—	71	1	no	—
336	#1833	—	71	1	no	—
337	#1747	—	71	1	no	—
338	C₇₁	#22284	1	1	yes	characteristic, complemented, Sylow
339	C₂×#137	C₂×D₇₁	1	1	yes	characteristic
340	C₂×AGL(1,ℤ/8)	C₂×D₇₁	1	1	yes	characteristic
341	C₂×#135	C₂×D₇₁	1	1	yes	characteristic
342	C₂²×SD₁₆	C₂×D₇₁	1	1	yes	characteristic
343	#568	—	71	1	no	—
344	#568	—	71	1	no	—
345	#536	C₂×D₇₁	1	1	yes	characteristic
346	#535	C₂×D₇₁	1	1	yes	characteristic
347	#519	—	71	1	no	—
348	#518	—	71	1	no	—
349	#497	Dic₇₁	1	2	yes	complemented
350	#496	Dic₇₁	1	2	yes	complemented
351	#495	Dic₇₁	1	2	yes	complemented
352	#492	Dic₇₁	1	2	yes	complemented
353	#444	—	142	2	no	—
354	#443	—	142	2	no	—
355	#440	—	142	2	no	—
356	C₄×SD₁₆	—	142	2	no	—
357	C₂×#108	—	71	1	no	—
358	#427	—	71	1	no	—
359	#421	—	71	1	no	—
360	#421	—	71	1	no	—
361	#421	—	71	1	no	—
362	#421	—	71	1	no	—
363	#419	—	71	1	no	—
364	#419	—	71	1	no	—
365	#418	—	71	1	no	—
366	#418	—	71	1	no	—
367	C₂×#101	—	71	1	no	—
368	C₂×#100	—	71	1	no	—
369	#410	—	71	1	no	—
370	#410	—	71	1	no	—
371	#405	—	71	1	no	—
372	#405	—	71	1	no	—
373	#405	C₂×D₇₁	1	1	yes	characteristic
374	#344	—	71	1	no	—
375	#344	—	71	1	no	—
376	#338	—	71	1	no	—
377	H(3,ℤ/4)	—	71	1	no	—
378	H(3,ℤ/4)	—	71	1	no	—
379	H(3,ℤ/4)	—	71	1	no	—
380	H(3,ℤ/4)	—	71	1	no	—
381	H(3,ℤ/4)	—	71	1	no	—
382	C₂×#41	C₂²×D₇₁	1	1	yes	characteristic
383	C₂×#41	C₂²×D₇₁	1	1	yes	characteristic
384	C₂²×Q₈	C₂²×D₇₁	1	1	yes	characteristic
385	C₂²×D₄	C₂²×D₇₁	1	1	yes	characteristic
386	#137	—	4	1	no	—
387	#137	—	4	1	no	—
388	AGL(1,ℤ/8)	—	4	1	no	—
389	AGL(1,ℤ/8)	—	4	1	no	—
390	#135	—	2	1	no	—
391	#135	—	2	1	no	—
392	#135	—	2	1	no	—
393	#135	—	2	1	no	—
394	C₂×Q₁₆	C₂×Dic₇₁	1	1	yes	characteristic, complemented
395	C₂×Q₁₆	C₂×Dic₇₁	1	2	yes	complemented
396	C₂×SD₁₆	—	2	1	no	—
397	C₂×SD₁₆	—	2	1	no	—
398	C₂×SD₁₆	—	2	1	no	—
399	C₂×SD₁₆	—	2	1	no	—
400	C₂×SD₁₆	C₂×Dic₇₁	1	2	yes	—
401	C₂×SD₁₆	C₂×Dic₇₁	1	2	yes	—
402	C₂×SD₁₆	C₂×Dic₇₁	1	2	yes	—
403	C₂×SD₁₆	C₂×Dic₇₁	1	1	yes	characteristic, complemented
404	C₂×SD₁₆	C₂×Dic₇₁	1	1	yes	characteristic, complemented
405	C₂×SD₁₆	C₂×Dic₇₁	1	2	yes	complemented
406	C₂×D₈	C₂×Dic₇₁	1	2	yes	—
407	C₂×D₈	C₂×Dic₇₁	1	1	yes	characteristic, complemented
408	#131	—	284	1	no	—
409	#131	—	284	1	no	—
410	#131	—	142	2	no	—
411	#131	—	142	2	no	—
412	C₂×M₁₆	—	284	1	no	—
413	C₂×M₁₆	—	284	1	no	—
414	C₂×M₁₆	—	71	1	no	—
415	C₂×M₁₆	—	71	1	no	—
416	C₂×M₁₆	—	71	1	no	—
417	C₂×M₁₆	—	71	1	no	—
418	C₂×M₁₆	C₂²×D₇₁	1	1	yes	characteristic
419	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
420	C₄⋊Q₈	C₂×Dic₇₁	1	2	yes	complemented
421	#127	C₂×Dic₇₁	1	2	yes	—
422	#124	C₂×Dic₇₁	1	2	yes	—
423	#124	C₂×Dic₇₁	1	2	yes	complemented
424	#123	—	4	1	no	—
425	C₂²⋊Q₈	—	4	1	no	—
426	#121	—	4	1	no	—
427	#120	—	4	1	no	—
428	C₄×Q₈	—	284	1	no	—
429	C₄×Q₈	—	142	2	no	—
430	C₄×D₄	—	284	1	no	—
431	C₄×D₄	—	142	2	no	—
432	#117	—	284	1	no	—
433	#117	—	71	1	no	—
434	#117	—	71	1	no	—
435	#117	—	71	1	no	—
436	#117	—	71	1	no	—
437	#117	C₂²×D₇₁	1	1	yes	characteristic
438	C₂×C₂²⋊C₄	—	284	1	no	—
439	#108	—	142	1	no	—
440	#108	—	142	1	no	—
441	#107	—	142	1	no	—
442	#106	—	142	1	no	—
443	#104	—	284	1	no	—
444	#104	—	284	1	no	—
445	#104	—	142	2	no	—
446	#104	—	142	2	no	—
447	#103	—	284	1	no	—
448	#103	—	142	2	no	—
449	#102	—	284	1	no	—
450	#102	—	142	2	no	—
451	#101	—	284	1	no	—
452	#100	—	284	1	no	—
453	#99	—	284	1	no	—
454	#99	—	284	1	no	—
455	#97	—	71	2	no	—
456	#97	—	142	1	no	—
457	#97	C₂×Dic₇₁	1	2	yes	complemented
458	C₄×C₈	—	71	2	no	—
459	C₄×C₈	—	142	1	no	—
460	C₄×C₈	C₂×Dic₇₁	1	2	yes	complemented
461	C₂⁴	—	2	1	no	—
462	#41	—	2	1	no	—
463	#41	—	2	1	no	—
464	#41	—	2	1	no	—
465	#41	—	2	1	no	—
466	#41	—	2	1	no	—
467	#41	—	2	1	no	—
468	#41	—	2	1	no	—
469	#41	—	2	1	no	—
470	C₂×Q₈	—	2	1	no	—
471	C₂×Q₈	—	2	1	no	—
472	C₂×Q₈	—	2	1	no	—
473	C₂×Q₈	—	2	1	no	—
474	C₂×Q₈	—	2	1	no	—
475	C₂×Q₈	C₂²×Dic₇₁	1	1	yes	characteristic
476	C₂×Q₈	C₂²×Dic₇₁	1	2	yes	—
477	C₂×Q₈	C₂²×Dic₇₁	1	1	yes	characteristic
478	C₂×D₄	—	2	1	no	—
479	C₂×D₄	—	2	1	no	—
480	C₂×D₄	—	2	1	no	—
481	C₂×D₄	—	2	1	no	—
482	C₂×D₄	—	2	1	no	—
483	C₂×D₄	—	2	1	no	—
484	C₂×D₄	—	2	1	no	—
485	C₂×D₄	—	4	2	no	—
486	C₂×D₄	C₂²×Dic₇₁	1	2	yes	—
487	C₂×D₄	C₂²×Dic₇₁	1	1	yes	characteristic
488	C₂×D₄	C₂²×Dic₇₁	1	1	yes	characteristic
489	C₂²×C₄	—	2	1	no	—
490	C₂²×C₄	—	2	1	no	—
491	C₂²×C₄	—	2	1	no	—
492	C₂²×C₄	—	284	2	no	—
493	C₂²×C₄	C₂³×D₇₁	1	1	yes	characteristic
494	Q₁₆	—	2	1	no	—
495	Q₁₆	—	2	1	no	—
496	Q₁₆	—	4	2	no	—
497	SD₁₆	—	2	2	no	—
498	SD₁₆	—	2	2	no	—
499	SD₁₆	—	2	2	no	—
500	SD₁₆	—	2	2	no	—
501	SD₁₆	—	2	1	no	—
502	SD₁₆	—	2	1	no	—
503	SD₁₆	—	2	1	no	—
504	SD₁₆	—	2	1	no	—
505	SD₁₆	—	4	2	no	—
506	SD₁₆	—	4	2	no	—
507	D₈	—	2	1	no	—
508	D₈	—	2	1	no	—
509	D₈	—	4	2	no	—
510	M₁₆	—	2	1	no	—
511	M₁₆	—	2	1	no	—
512	M₁₆	—	284	2	no	—
513	M₁₆	—	284	2	no	—
514	M₁₆	—	142	1	no	—
515	M₁₆	—	142	1	no	—
516	M₁₆	—	142	1	no	—
517	M₁₆	—	142	1	no	—
518	M₁₆	—	142	1	no	—
519	M₁₆	—	142	1	no	—
520	M₁₆	—	142	1	no	—
521	M₁₆	—	142	1	no	—
522	C₂×C₈	—	284	2	no	—
523	C₂×C₈	—	284	2	no	—
524	C₂×C₈	—	71	2	no	—
525	C₂×C₈	—	142	1	no	—
526	C₂×C₈	—	142	1	no	—
527	C₂×C₈	—	142	1	no	—
528	C₂×C₈	C₂²×Dic₇₁	1	2	yes	—
529	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
530	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
531	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
532	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
533	C₂×C₈	#11765864	1	1	yes	characteristic
534	C₂×C₈	#11765864	1	1	yes	characteristic
535	C₄⋊C₄	—	2	1	no	—
536	C₄⋊C₄	—	4	2	no	—
537	C₄⋊C₄	—	284	2	no	—
538	C₄⋊C₄	—	142	1	no	—
539	C₄⋊C₄	—	142	1	no	—
540	C₄⋊C₄	—	142	1	no	—
541	C₄⋊C₄	—	142	1	no	—
542	C₂²⋊C₄	—	2	1	no	—
543	C₂²⋊C₄	—	4	2	no	—
544	C₂²⋊C₄	—	4	2	no	—
545	C₂²⋊C₄	—	284	2	no	—
546	C₂²⋊C₄	—	142	1	no	—
547	C₂²⋊C₄	—	142	1	no	—
548	C₂²⋊C₄	—	142	1	no	—
549	C₂²⋊C₄	—	142	1	no	—
550	C₄²	—	284	2	no	—
551	C₄²	—	71	2	no	—
552	C₄²	—	142	1	no	—
553	C₄²	—	142	1	no	—
554	C₄²	—	142	1	no	—
555	C₄²	C₂²×Dic₇₁	1	2	yes	—
556	C₂³	—	2	1	no	—
557	C₂³	—	2	2	no	—
558	C₂³	—	2	1	no	—
559	C₂³	—	4	1	no	—
560	C₂³	—	4	1	no	—
561	C₂³	—	16	1	no	—
562	C₂³	#14556173	1	1	yes	characteristic
563	Q₈	—	2	2	no	—
564	Q₈	—	2	1	no	—
565	Q₈	—	2	1	no	—
566	Q₈	—	2	1	no	—
567	Q₈	—	2	2	no	—
568	Q₈	—	2	1	no	—
569	Q₈	—	4	1	no	—
570	Q₈	—	4	1	no	—
571	D₄	—	2	1	no	—
572	D₄	—	2	1	no	—
573	D₄	—	2	2	no	—
574	D₄	—	2	1	no	—
575	D₄	—	2	2	no	—
576	D₄	—	2	1	no	—
577	D₄	—	4	1	no	—
578	D₄	—	4	1	no	—
579	D₄	—	4	1	no	—
580	D₄	—	4	1	no	—
581	D₄	—	4	1	no	—
582	D₄	—	4	1	no	—
583	D₄	—	16	2	no	—
584	C₂×C₄	—	2	2	no	—
585	C₂×C₄	—	2	1	no	—
586	C₂×C₄	—	2	2	no	—
587	C₂×C₄	—	2	1	no	—
588	C₂×C₄	—	4	1	no	—
589	C₂×C₄	—	4	1	no	—
590	C₂×C₄	—	4	1	no	—
591	C₂×C₄	—	4	1	no	—
592	C₂×C₄	—	4	1	no	—
593	C₂×C₄	—	4	1	no	—
594	C₂×C₄	—	284	1	no	—
595	C₂×C₄	—	284	1	no	—
596	C₂×C₄	—	284	1	no	—
597	C₂×C₄	—	1136	2	no	—
598	C₂×C₄	—	142	2	no	—
599	C₂×C₄	C₂³×Dic₇₁	1	1	yes	characteristic
600	C₂×C₄	C₂×D₄×D₇₁	1	1	yes	characteristic
601	C₂×C₄	C₂×#11765864	1	1	yes	characteristic
602	C₂×C₄	#14556173	1	1	yes	characteristic
603	C₂×C₄	#14556173	1	1	yes	characteristic
604	C₂×C₄	#14556173	1	1	yes	characteristic
605	C₈	—	2	2	no	—
606	C₈	—	284	1	no	—
607	C₈	—	284	1	no	—
608	C₈	—	284	1	no	—
609	C₈	—	142	2	no	—
610	C₈	D₄×Dic₇₁	1	1	yes	characteristic
611	C₈	D₄×Dic₇₁	1	1	yes	characteristic
612	C₈	D₄×Dic₇₁	1	1	yes	characteristic
613	C₈	D₄×Dic₇₁	1	1	yes	characteristic
614	C₂²	—	4	1	no	—
615	C₂²	—	4	1	no	—
616	C₂²	—	4	1	no	—
617	C₂²	—	4	2	no	—
618	C₂²	—	8	1	no	—
619	C₂²	—	8	2	no	—
620	C₂²	—	8	1	no	—
621	C₂²	—	8	2	no	—
622	C₂²	—	16	1	no	—
623	C₂²	#17986474	1	1	yes	characteristic
624	C₂²	#17986562	1	1	yes	characteristic
625	C₂²	#17986603	1	1	yes	characteristic
626	C₄	—	4	2	no	—
627	C₄	—	4	1	no	—
628	C₄	—	4	1	no	—
629	C₄	—	4	2	no	—
630	C₄	—	284	2	no	—
631	C₄	—	568	1	no	—
632	C₄	—	568	1	no	—
633	C₄	—	568	1	no	—
634	C₄	C₂×D₄×Dic₇₁	1	1	yes	characteristic
635	C₄	#17986474	1	1	yes	characteristic
636	C₄	#17986562	1	1	yes	characteristic
637	C₄	#17986603	1	1	yes	characteristic
638	C₂	—	2	1	no	—
639	C₂	—	2	1	no	—
640	C₂	—	2	1	no	—
641	C₂	—	8	2	no	—
642	C₂	—	8	1	no	—
643	C₂	—	8	1	no	—
644	C₂	#22591066	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22591066
commutator	C₂×C₄×C₇₁	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₁
Fitting, prime radical	C₇₁×#3154	C₂
socle	C₂×C₇₁	#2328
cosocle	C₂²×C₄×C₇₁	C₂⁴
hypercenter	#3154	D₇₁
hypocenter	C₇₁	#22284
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #3154

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #22597257	C₂	semidirect
2. #22597200	C₂	semidirect
3. #22589245	C₂	semidirect
4. #22589239	C₂	semidirect
5. #22588878	C₂	semidirect
6. #22588873	C₂	semidirect
7. #22588868	C₂	semidirect
8. #22588863	C₂	semidirect
9. #22588836	C₂	semidirect
10. #22588829	C₂	semidirect
11. #22588822	C₂	semidirect
12. #22588815	C₂	semidirect
13. #22588464	C₂	semidirect
14. #22588095	C₂	semidirect
54. #497×C₇₁	C₄	semidirect
55. #496×C₇₁	C₄	semidirect
56. #495×C₇₁	C₄	semidirect
57. #492×C₇₁	C₄	semidirect
96. #14556038	D₄	semidirect
98. C₈×Dic₇₁	D₄	semidirect
111. C₂×Q₁₆×C₇₁	C₂×C₄	semidirect
112. C₂×Q₁₆×C₇₁	C₂×C₄	semidirect
120. C₂×SD₁₆×C₇₁	C₂×C₄	semidirect
121. C₂×SD₁₆×C₇₁	C₂×C₄	semidirect
122. C₂×SD₁₆×C₇₁	C₂×C₄	semidirect
124. C₂×D₈×C₇₁	C₂×C₄	semidirect
127. C₄⋊Q₈×C₇₁	C₂×C₄	semidirect
130. #124×C₇₁	C₂×C₄	semidirect
136. #97×C₇₁	C₂×C₄	semidirect
137. C₄×C₈×C₇₁	C₂×C₄	semidirect
338. C₇₁	#22284	semidirect
349. #497	Dic₇₁	semidirect
350. #496	Dic₇₁	semidirect
351. #495	Dic₇₁	semidirect
352. #492	Dic₇₁	semidirect
394. C₂×Q₁₆	C₂×Dic₇₁	semidirect
395. C₂×Q₁₆	C₂×Dic₇₁	semidirect
403. C₂×SD₁₆	C₂×Dic₇₁	semidirect
404. C₂×SD₁₆	C₂×Dic₇₁	semidirect
405. C₂×SD₁₆	C₂×Dic₇₁	semidirect
407. C₂×D₈	C₂×Dic₇₁	semidirect
420. C₄⋊Q₈	C₂×Dic₇₁	semidirect
423. #124	C₂×Dic₇₁	semidirect
457. #97	C₂×Dic₇₁	semidirect
460. C₄×C₈	C₂×Dic₇₁	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22597257	2
#22597200	2
#22589245	2
#22589239	2
#22588878	2
#22588873	2
#22588868	2
#22588863	2
#22588836	2
#22588829	2
#22588822	2
#22588815	2
#22588464	2
#22588095	2
C₇₁×#3154	2
#22284	71