Subgroups — #31890247

Order 18176.

516 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14556128	C₂	1	1	yes	maximal, characteristic
2	#22588492	C₂	1	1	yes	maximal, characteristic
3	C₂×#17985747	C₂	1	2	yes	maximal
4	#22586224	C₂	1	2	yes	maximal
5	C₇₁×#1842	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556128	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₁₄₂	C₄	1	2	yes	—
8	C₂×C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
9	C₂×C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
10	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
11	#17985905	C₄	1	2	yes	—
12	#17985897	—	2	1	no	—
13	#17985897	—	2	1	no	—
14	#17985869	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊Dic₇₁	—	2	1	no	—
16	C₄×C₄⋊Dic₇₁	—	2	1	no	—
17	#17985747	—	2	4	no	—
18	C₂×C₄⋊C₈×C₇₁	—	2	1	no	—
19	C₂×C₄⋊C₈×C₇₁	—	2	1	no	—
20	C₂×C₄⋊C₈×C₇₁	C₂²	1	2	yes	—
21	C₄³×C₇₁	C₂²	1	1	yes	characteristic
22	C₂²×Dic₁₄₂	C₂×C₄	1	2	yes	—
23	C₂²×C₇₁⋊C₈	—	2	2	no	—
24	#14556128	—	2	1	no	—
25	#14556128	—	2	1	no	—
26	#14556128	D₄	1	2	yes	—
27	#14556128	Q₈	1	2	yes	—
28	C₄×Dic₁₄₂	—	4	4	no	—
29	C₂×C₄⋊Dic₇₁	—	2	1	no	—
30	C₂×C₄⋊Dic₇₁	—	2	1	no	—
31	C₂×C₄⋊Dic₇₁	—	2	1	no	—
32	C₂×C₄⋊Dic₇₁	—	2	1	no	—
33	C₂×C₄⋊Dic₇₁	C₂×C₄	1	2	yes	—
34	C₂×Dic₇₁⋊C₄	—	2	2	no	—
35	C₂×C₄×Dic₇₁	—	2	2	no	—
36	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
37	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
38	C₄⋊(C₇₁⋊C₈)	—	4	2	no	—
39	C₄⋊(C₇₁⋊C₈)	—	4	2	no	—
40	C₇₁⋊#95	—	2	2	no	—
41	C₇₁⋊#95	—	2	2	no	—
42	C₂²×C₈×C₇₁	—	2	2	no	—
43	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
50	C₄⋊C₈×C₇₁	—	2	2	no	—
51	C₄⋊C₈×C₇₁	—	2	2	no	—
52	C₄⋊C₈×C₇₁	—	4	2	no	—
53	C₄⋊C₈×C₇₁	—	4	2	no	—
54	C₂×Dic₁₄₂	—	2	2	no	—
55	C₂×Dic₁₄₂	—	4	2	no	—
56	C₂×Dic₁₄₂	—	4	2	no	—
57	C₂×Dic₁₄₂	C₄⋊C₄	1	4	yes	—
58	C₂²×Dic₇₁	—	2	2	no	—
59	C₂²×Dic₇₁	—	2	2	no	—
60	C₂×C₇₁⋊C₈	—	4	2	no	—
61	C₂×C₇₁⋊C₈	—	4	2	no	—
62	C₂×C₇₁⋊C₈	—	4	2	no	—
63	C₄⋊Dic₇₁	—	2	2	no	—
64	C₄⋊Dic₇₁	—	2	2	no	—
65	C₄⋊Dic₇₁	—	2	2	no	—
66	C₄⋊Dic₇₁	—	2	2	no	—
67	C₄⋊Dic₇₁	—	4	1	no	—
68	C₄⋊Dic₇₁	—	4	1	no	—
69	C₄⋊Dic₇₁	—	4	1	no	—
70	C₄⋊Dic₇₁	—	4	1	no	—
71	C₄⋊Dic₇₁	—	4	2	no	—
72	Dic₇₁⋊C₄	—	4	4	no	—
73	C₄×Dic₇₁	—	4	4	no	—
74	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₇₁	—	4	2	no	—
82	C₂×C₈×C₇₁	—	4	2	no	—
83	C₂×C₈×C₇₁	—	4	2	no	—
84	C₄²×C₇₁	—	2	2	no	—
85	C₄²×C₇₁	—	2	1	no	—
86	C₄²×C₇₁	—	2	1	no	—
87	C₄²×C₇₁	—	2	1	no	—
88	C₄²×C₇₁	—	2	1	no	—
89	C₄²×C₇₁	—	4	1	no	—
90	C₄²×C₇₁	SD₁₆	1	2	yes	—
91	C₄²×C₇₁	SD₁₆	1	2	yes	—
92	C₄²×C₇₁	Q₁₆	1	2	yes	—
93	C₄²×C₇₁	Q₁₆	1	2	yes	—
94	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
96	C₄²×C₇₁	#41	1	1	yes	characteristic
97	C₄²×C₇₁	#41	1	1	yes	characteristic
98	C₂³×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₇₁	—	2	1	no	—
100	C₂×C₄×C₇₁	—	2	1	no	—
101	C₂×C₄×C₇₁	—	2	1	no	—
102	C₂×C₄×C₇₁	—	2	1	no	—
103	C₂×C₄×C₇₁	—	2	1	no	—
104	C₂×C₄×C₇₁	—	2	1	no	—
105	C₂×C₄×C₇₁	—	2	1	no	—
106	C₂×C₄×C₇₁	—	2	1	no	—
107	C₂×C₄×C₇₁	—	2	1	no	—
108	C₂×C₄×C₇₁	#100	1	2	yes	—
109	C₂×C₄×C₇₁	#103	1	2	yes	—
110	C₂×C₄×C₇₁	#103	1	2	yes	—
111	C₂×C₄×C₇₁	#103	1	2	yes	—
112	C₂×C₄×C₇₁	#103	1	2	yes	—
113	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
115	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
116	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
117	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
118	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
119	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
120	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
121	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
122	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
123	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
124	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
125	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
126	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
127	C₂×Dic₇₁	—	2	4	no	—
128	C₂×Dic₇₁	—	4	2	no	—
129	C₂×Dic₇₁	—	4	2	no	—
130	C₂×Dic₇₁	—	4	2	no	—
131	C₂×Dic₇₁	—	4	2	no	—
132	C₂×Dic₇₁	—	4	2	no	—
133	Dic₁₄₂	—	4	4	no	—
134	Dic₁₄₂	—	4	4	no	—
135	C₈×C₇₁	—	8	2	no	—
136	C₇₁⋊C₈	—	8	2	no	—
137	C₂²×C₇₁	#380	1	1	yes	characteristic
138	C₂²×C₇₁	C₂×#100	1	1	yes	characteristic
139	C₂²×C₇₁	#413	1	1	yes	characteristic
140	C₂²×C₇₁	C₂×#103	1	1	yes	characteristic
141	C₂²×C₇₁	C₂×#103	1	1	yes	characteristic
142	C₂²×C₇₁	#417	1	1	yes	characteristic
143	C₂²×C₇₁	#417	1	1	yes	characteristic
144	C₄×C₇₁	—	2	1	no	—
145	C₄×C₇₁	—	2	1	no	—
146	C₄×C₇₁	—	2	2	no	—
147	C₄×C₇₁	—	2	2	no	—
148	C₄×C₇₁	—	4	1	no	—
149	C₄×C₇₁	#333	1	4	yes	complemented
150	C₄×C₇₁	#460	1	1	yes	characteristic
151	C₄×C₇₁	#460	1	1	yes	characteristic
152	C₄×C₇₁	#462	1	1	yes	characteristic
153	C₄×C₇₁	#462	1	1	yes	characteristic
154	C₄×C₇₁	#475	1	1	yes	characteristic
155	C₄×C₇₁	#475	1	1	yes	characteristic
156	C₄×C₇₁	#477	1	1	yes	characteristic
157	C₄×C₇₁	#477	1	1	yes	characteristic
158	Dic₇₁	—	4	4	no	—
159	Dic₇₁	—	8	2	no	—
160	#8587	—	71	1	no	maximal, Sylow
161	C₂×C₇₁	C₂×#333	1	1	yes	characteristic
162	C₂×C₇₁	#1550	1	1	yes	characteristic
163	C₂×C₇₁	#1550	1	1	yes	characteristic
164	C₂×C₇₁	#1560	1	1	yes	characteristic
165	C₂×C₇₁	#1800	1	1	yes	characteristic
166	C₂×C₇₁	#1803	1	1	yes	characteristic
167	C₂×C₇₁	#1803	1	1	yes	characteristic
168	C₄×C₄⋊Q₈	—	71	1	no	—
169	#1842	—	71	1	no	—
170	#1842	D₇₁	1	1	yes	characteristic
171	C₂×#333	—	71	2	no	—
172	#1296	—	71	2	no	—
173	C₇₁	#8587	1	1	yes	characteristic, complemented, Sylow
174	C₂×C₄⋊Q₈	—	71	1	no	—
175	C₂×C₄×Q₈	—	71	2	no	—
176	C₂×C₄⋊C₈	—	2	1	no	—
177	C₂×C₄⋊C₈	—	2	1	no	—
178	C₂×C₄⋊C₈	—	71	2	no	—
179	C₂×C₄⋊C₈	—	142	1	no	—
180	C₂×C₄⋊C₈	—	142	1	no	—
181	C₂×C₄⋊C₈	C₂×D₇₁	1	2	yes	—
182	#391	—	71	2	no	—
183	#389	—	142	1	no	—
184	#389	—	142	1	no	—
185	#384	—	71	1	no	—
186	C₄×C₄⋊C₄	—	142	1	no	—
187	C₄×C₄⋊C₄	—	142	1	no	—
188	C₄³	C₂×D₇₁	1	1	yes	characteristic
189	#333	—	142	4	no	—
190	C₂²×Q₈	—	71	2	no	—
191	C₂²×C₈	—	2	2	no	—
192	C₂²×C₈	—	142	2	no	—
193	C₄⋊Q₈	—	71	2	no	—
194	C₄⋊Q₈	—	71	2	no	—
195	C₄⋊Q₈	—	142	1	no	—
196	C₄⋊Q₈	—	142	1	no	—
197	C₄×Q₈	—	284	4	no	—
198	C₂×C₄⋊C₄	—	71	2	no	—
199	C₂×C₄⋊C₄	—	142	2	no	—
200	C₂×C₄⋊C₄	—	142	1	no	—
201	C₂×C₄⋊C₄	—	142	1	no	—
202	C₂×C₄⋊C₄	—	142	1	no	—
203	C₂×C₄⋊C₄	—	142	1	no	—
204	C₂×C₄²	—	142	2	no	—
205	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
206	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
207	C₂×C₄²	D₂₈₄	1	1	yes	characteristic
208	C₂×C₄²	D₂₈₄	1	1	yes	characteristic
209	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
210	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
211	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
212	C₄⋊C₈	—	2	2	no	—
213	C₄⋊C₈	—	2	2	no	—
214	C₄⋊C₈	—	4	2	no	—
215	C₄⋊C₈	—	4	2	no	—
216	C₄⋊C₈	—	284	2	no	—
217	C₄⋊C₈	—	284	2	no	—
218	C₄⋊C₈	—	142	2	no	—
219	C₄⋊C₈	—	142	2	no	—
220	#95	—	142	2	no	—
221	#95	—	142	2	no	—
222	C₂×Q₈	—	284	2	no	—
223	C₂×Q₈	—	284	2	no	—
224	C₂×Q₈	—	71	4	no	—
225	C₂×Q₈	—	142	2	no	—
226	C₂²×C₄	—	142	2	no	—
227	C₂²×C₄	—	142	2	no	—
228	C₂²×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
229	C₂²×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
230	C₂²×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
231	C₂²×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
232	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
233	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
234	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
235	C₂×C₈	—	4	2	no	—
236	C₂×C₈	—	4	2	no	—
237	C₂×C₈	—	4	2	no	—
238	C₂×C₈	—	284	2	no	—
239	C₂×C₈	—	284	2	no	—
240	C₂×C₈	—	284	2	no	—
241	C₄⋊C₄	—	284	2	no	—
242	C₄⋊C₄	—	284	4	no	—
243	C₄⋊C₄	—	284	1	no	—
244	C₄⋊C₄	—	284	1	no	—
245	C₄⋊C₄	—	284	1	no	—
246	C₄⋊C₄	—	284	1	no	—
247	C₄⋊C₄	—	142	2	no	—
248	C₄⋊C₄	—	142	2	no	—
249	C₄⋊C₄	—	142	2	no	—
250	C₄⋊C₄	—	142	2	no	—
251	C₄²	—	2	1	no	—
252	C₄²	—	2	2	no	—
253	C₄²	—	2	1	no	—
254	C₄²	—	2	1	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	4	1	no	—
257	C₄²	—	284	4	no	—
258	C₄²	#11765853	1	2	yes	—
259	C₄²	#11765854	1	2	yes	—
260	C₄²	#11765855	1	2	yes	—
261	C₄²	Dic₂₈₄	1	2	yes	—
262	C₄²	D₄×D₇₁	1	1	yes	characteristic
263	C₄²	Q₈×D₇₁	1	1	yes	characteristic
264	C₄²	#11765864	1	1	yes	characteristic
265	C₄²	#11765866	1	1	yes	characteristic
266	C₂³	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
267	Q₈	—	284	4	no	—
268	Q₈	—	284	4	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	284	2	no	—
279	C₂×C₄	—	284	2	no	—
280	C₂×C₄	—	284	2	no	—
281	C₂×C₄	—	284	2	no	—
282	C₂×C₄	—	284	2	no	—
283	C₂×C₄	—	142	4	no	—
284	C₂×C₄	#14556044	1	2	yes	—
285	C₂×C₄	#14556051	1	2	yes	—
286	C₂×C₄	#14556051	1	2	yes	—
287	C₂×C₄	#14556053	1	2	yes	—
288	C₂×C₄	#14556053	1	2	yes	—
289	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
290	C₂×C₄	#14556086	1	1	yes	characteristic
291	C₂×C₄	#14556099	1	1	yes	characteristic
292	C₂×C₄	#14556103	1	1	yes	characteristic
293	C₂×C₄	#14556104	1	1	yes	characteristic
294	C₂×C₄	#14556108	1	1	yes	characteristic
295	C₂×C₄	C₂×#11765853	1	1	yes	characteristic
296	C₂×C₄	C₂×#11765854	1	1	yes	characteristic
297	C₂×C₄	C₂×#11765855	1	1	yes	characteristic
298	C₂×C₄	C₂×Dic₂₈₄	1	1	yes	characteristic
299	C₂×C₄	#14556154	1	1	yes	characteristic
300	C₂×C₄	#14556156	1	1	yes	characteristic
301	C₂×C₄	#14556159	1	1	yes	characteristic
302	C₂×C₄	#14556161	1	1	yes	characteristic
303	C₈	—	8	2	no	—
304	C₈	—	568	2	no	—
305	C₂²	#17985856	1	1	yes	characteristic
306	C₂²	C₂×#14556044	1	1	yes	characteristic
307	C₂²	#17985981	1	1	yes	characteristic
308	C₂²	C₂×#14556051	1	1	yes	characteristic
309	C₂²	C₂×#14556053	1	1	yes	characteristic
310	C₂²	#17985998	1	1	yes	characteristic
311	C₂²	#17986001	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	—	2	2	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	4	1	no	—
317	C₄	—	284	4	no	—
318	C₄	—	568	2	no	—
319	C₄	#17985747	1	4	yes	complemented
320	C₄	#17986196	1	1	yes	characteristic
321	C₄	#17986200	1	1	yes	characteristic
322	C₄	#17986206	1	1	yes	characteristic
323	C₄	#17986210	1	1	yes	characteristic
324	C₄	#17986281	1	1	yes	characteristic
325	C₄	#17986285	1	1	yes	characteristic
326	C₄	#17986291	1	1	yes	characteristic
327	C₄	#17986295	1	1	yes	characteristic
328	C₂	C₂×#17985747	1	1	yes	characteristic
329	C₂	#22587028	1	1	yes	characteristic
330	C₂	#22587031	1	1	yes	characteristic
331	C₂	#22587073	1	1	yes	characteristic
332	C₂	#22588329	1	1	yes	characteristic
333	C₂	#22588342	1	1	yes	characteristic
334	C₂	#22588346	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₇₁)⋊C₄
commutator	C₂²×C₄×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1842	C₂
socle, prime radical	C₂³×C₇₁	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1842	D₇₁
hypocenter	C₇₁	#8587
LCS	G ▷ C₂²×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1842

Complements

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

N	H ≅ G/N	type
149. C₄×C₇₁	#333	semidirect
173. C₇₁	#8587	semidirect
319. C₄	#17985747	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14556128	2
#22588492	2
C₂×#17985747	2
#22586224	2
C₇₁×#1842	2
#8587	71