Subgroups — #31891766

Order 18176.

438 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22588360	C₂	1	1	yes	maximal, characteristic
2	#22587110	C₂	1	1	yes	maximal, characteristic, complemented
3	#22587095	C₂	1	1	yes	maximal, characteristic, complemented
4	#22586722	C₂	1	1	yes	maximal, characteristic
5	#22586241	C₂	1	1	yes	maximal, characteristic, complemented
6	#22586179	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₁×#2906	C₂	1	1	yes	maximal, characteristic
8	#17986052	C₂²	1	1	yes	characteristic
9	#17986023	C₂²	1	1	yes	characteristic
10	C₂×#14556055	—	2	1	no	—
11	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
12	C₄×C₄⋊Dic₇₁	C₂²	1	1	yes	characteristic
13	#17985758	—	2	1	no	—
14	#17985749	—	2	2	no	—
15	#17985749	—	2	1	no	—
16	#17985749	—	2	1	no	—
17	Q₈⋊(C₇₁⋊C₈)	C₄	1	1	yes	characteristic, complemented
18	Q₈⋊(C₇₁⋊C₈)	C₄	1	1	yes	characteristic, complemented
19	D₄⋊(C₇₁⋊C₈)	C₄	1	1	yes	characteristic, complemented
20	D₄⋊(C₇₁⋊C₈)	C₄	1	1	yes	characteristic, complemented
21	C₂×#131×C₇₁	—	2	1	no	—
22	C₄×#41×C₇₁	C₂²	1	1	yes	characteristic
23	#424×C₇₁	—	2	1	no	—
24	#424×C₇₁	—	2	2	no	—
25	#423×C₇₁	—	2	1	no	—
26	#423×C₇₁	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₈×C₇₁	—	2	1	no	—
28	C₂×C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
29	C₂×#11765849	—	2	1	no	—
30	C₂²×C₇₁⋊C₈	—	2	1	no	—
31	C₂×C₄⋊Dic₇₁	—	2	1	no	—
32	C₂×C₄×Dic₇₁	—	2	1	no	—
33	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
34	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
35	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	1	yes	characteristic
36	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	1	yes	characteristic
37	#14556055	—	4	2	no	—
38	C₂²⋊(C₇₁⋊C₈)	—	2	1	no	—
39	C₄×C₇₁⋊C₈	—	2	1	no	—
40	C₄×C₇₁⋊C₈	—	2	1	no	—
41	C₄×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
42	C₄×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
43	C₇₁⋊#95	—	2	1	no	—
44	C₂×#41×C₇₁	D₄	1	1	yes	characteristic
45	#131×C₇₁	—	4	2	no	—
46	#131×C₇₁	—	4	2	no	—
47	C₂×M₁₆×C₇₁	—	2	2	no	—
48	C₂×M₁₆×C₇₁	—	2	1	no	—
49	C₂²×C₈×C₇₁	—	2	2	no	—
50	C₂²×C₈×C₇₁	—	2	1	no	—
51	C₄×Q₈×C₇₁	C₂×C₄	1	1	yes	characteristic
52	C₄×Q₈×C₇₁	Q₈	1	1	yes	characteristic
53	C₄×D₄×C₇₁	—	2	1	no	—
54	C₄×D₄×C₇₁	—	2	1	no	—
55	C₄×D₄×C₇₁	C₂×C₄	1	1	yes	characteristic
56	C₄×D₄×C₇₁	Q₈	1	1	yes	characteristic
57	#117×C₇₁	—	2	1	no	—
58	#117×C₇₁	D₄	1	1	yes	characteristic
59	C₂×C₄²×C₇₁	—	2	1	no	—
60	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
61	C₄⋊C₈×C₇₁	—	2	2	no	—
62	C₄⋊C₈×C₇₁	—	2	2	no	—
63	C₄⋊C₈×C₇₁	—	2	1	no	—
64	C₄⋊C₈×C₇₁	—	2	1	no	—
65	C₄⋊C₈×C₇₁	D₄	1	1	yes	characteristic
66	C₄⋊C₈×C₇₁	D₄	1	1	yes	characteristic
67	C₄⋊C₈×C₇₁	D₄	1	2	yes	—
68	C₂²×Dic₇₁	—	4	1	no	—
69	#11765849	—	4	1	no	—
70	#11765849	—	4	1	no	—
71	C₂×C₇₁⋊C₈	—	2	1	no	—
72	C₂×C₇₁⋊C₈	—	2	1	no	—
73	C₂×C₇₁⋊C₈	—	2	1	no	—
74	C₂×C₇₁⋊C₈	—	2	1	no	—
75	C₂×C₇₁⋊C₈	—	4	2	no	—
76	C₂×C₇₁⋊C₈	—	4	1	no	—
77	C₄⋊Dic₇₁	—	2	2	no	—
78	C₄⋊Dic₇₁	—	4	1	no	—
79	C₄×Dic₇₁	—	4	1	no	—
80	C₄×Dic₇₁	—	4	1	no	—
81	#41×C₇₁	—	2	1	no	—
82	#41×C₇₁	—	2	1	no	—
83	#41×C₇₁	SD₁₆	1	2	yes	—
84	#41×C₇₁	SD₁₆	1	2	yes	—
85	C₂×Q₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
86	C₂×D₄×C₇₁	—	2	1	no	—
87	C₂×D₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
88	C₂²×C₄×C₇₁	—	2	1	no	—
89	C₂²×C₄×C₇₁	—	2	1	no	—
90	C₂²×C₄×C₇₁	—	2	1	no	—
91	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
93	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
94	M₁₆×C₇₁	—	4	4	no	—
95	M₁₆×C₇₁	—	4	1	no	—
96	M₁₆×C₇₁	—	4	1	no	—
97	C₂×C₈×C₇₁	—	2	1	no	—
98	C₂×C₈×C₇₁	—	2	2	no	—
99	C₂×C₈×C₇₁	—	2	1	no	—
100	C₂×C₈×C₇₁	—	4	4	no	—
101	C₂×C₈×C₇₁	—	4	1	no	—
102	C₂×C₈×C₇₁	—	4	1	no	—
103	C₄⋊C₄×C₇₁	—	2	1	no	—
104	C₄⋊C₄×C₇₁	—	2	1	no	—
105	C₄⋊C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
106	C₄⋊C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
107	C₂²⋊C₄×C₇₁	—	2	1	no	—
108	C₂²⋊C₄×C₇₁	—	2	1	no	—
109	C₂²⋊C₄×C₇₁	—	2	1	no	—
110	C₄²×C₇₁	—	2	1	no	—
111	C₄²×C₇₁	—	2	1	no	—
112	C₄²×C₇₁	—	2	1	no	—
113	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
117	C₂³×C₇₁	—	2	1	no	—
118	C₂³×C₇₁	#123	1	1	yes	characteristic
119	Q₈×C₇₁	—	2	1	no	—
120	Q₈×C₇₁	#106	1	2	yes	—
121	D₄×C₇₁	—	2	2	no	—
122	D₄×C₇₁	—	2	2	no	—
123	D₄×C₇₁	—	2	1	no	—
124	D₄×C₇₁	#106	1	2	yes	—
125	C₂×C₄×C₇₁	—	2	1	no	—
126	C₂×C₄×C₇₁	—	2	1	no	—
127	C₂×C₄×C₇₁	—	2	1	no	—
128	C₂×C₄×C₇₁	—	2	2	no	—
129	C₂×C₄×C₇₁	—	2	2	no	—
130	C₂×C₄×C₇₁	—	2	1	no	—
131	C₂×C₄×C₇₁	—	2	1	no	—
132	C₂×C₄×C₇₁	—	4	1	no	—
133	C₂×C₄×C₇₁	—	4	1	no	—
134	C₂×C₄×C₇₁	—	4	1	no	—
135	C₂×C₄×C₇₁	—	8	1	no	—
136	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
139	C₂×C₄×C₇₁	#120	1	1	yes	characteristic
140	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
141	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
142	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
143	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
144	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
145	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
146	C₂×Dic₇₁	—	4	2	no	—
147	C₂×Dic₇₁	—	8	1	no	—
148	C₂×Dic₇₁	—	8	1	no	—
149	C₈×C₇₁	—	4	1	no	—
150	C₈×C₇₁	—	4	2	no	—
151	C₈×C₇₁	—	4	1	no	—
152	C₇₁⋊C₈	—	2	2	no	—
153	C₇₁⋊C₈	—	4	1	no	—
154	C₇₁⋊C₈	—	4	1	no	—
155	C₇₁⋊C₈	—	4	1	no	—
156	C₇₁⋊C₈	—	8	1	no	—
157	C₂²×C₇₁	—	2	1	no	—
158	C₂²×C₇₁	—	2	1	no	—
159	C₂²×C₇₁	—	2	2	no	—
160	C₂²×C₇₁	—	4	1	no	—
161	C₂²×C₇₁	—	4	1	no	—
162	C₂²×C₇₁	#385	1	1	yes	characteristic
163	C₂²×C₇₁	#481	1	1	yes	characteristic
164	C₂²×C₇₁	#483	1	1	yes	characteristic
165	C₄×C₇₁	—	2	2	no	—
166	C₄×C₇₁	—	2	1	no	—
167	C₄×C₇₁	—	2	1	no	—
168	C₄×C₇₁	—	2	1	no	—
169	C₄×C₇₁	—	4	1	no	—
170	C₄×C₇₁	—	4	1	no	—
171	C₄×C₇₁	C₂×#106	1	1	yes	characteristic
172	C₄×C₇₁	#428	1	1	yes	characteristic
173	C₄×C₇₁	#443	1	1	yes	characteristic
174	C₄×C₇₁	#444	1	1	yes	characteristic
175	C₄×C₇₁	#448	1	1	yes	characteristic
176	C₄×C₇₁	#450	1	1	yes	characteristic
177	Dic₇₁	—	8	2	no	—
178	#8904	—	71	1	no	maximal, Sylow
179	C₂×C₇₁	—	2	1	no	—
180	C₂×C₇₁	—	2	1	no	—
181	C₂×C₇₁	—	4	2	no	—
182	C₂×C₇₁	#1853	1	1	yes	characteristic
183	C₂×C₇₁	#1861	1	1	yes	characteristic
184	C₂×C₇₁	#1866	1	1	yes	characteristic
185	#2906	D₇₁	1	1	yes	characteristic
186	#1806	—	71	1	no	—
187	#1570	—	71	1	no	—
188	#1566	—	71	1	no	—
189	#1476	—	71	1	no	—
190	#1303	—	71	1	no	—
191	#1279	—	71	1	no	—
192	C₇₁	#8904	1	1	yes	characteristic, complemented, Sylow
193	C₂×#131	—	2	1	no	—
194	C₄×#41	C₂×D₇₁	1	1	yes	characteristic
195	#431	—	71	1	no	—
196	#424	—	2	1	no	—
197	#424	—	2	2	no	—
198	#423	—	2	1	no	—
199	#423	—	71	1	no	—
200	#423	C₂×D₇₁	1	1	yes	characteristic
201	C₂×C₄⋊C₈	—	2	1	no	—
202	C₂×C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
203	C₂×#104	—	142	1	no	—
204	C₂×C₄×C₈	—	71	1	no	—
205	C₄×C₄⋊C₄	—	71	1	no	—
206	#339	—	142	1	no	—
207	#334	—	142	2	no	—
208	#334	—	142	1	no	—
209	#334	—	142	1	no	—
210	Q₈⋊C₈	—	71	1	no	—
211	Q₈⋊C₈	—	71	1	no	—
212	D₄⋊C₈	—	71	1	no	—
213	D₄⋊C₈	—	71	1	no	—
214	C₂×#41	C₂²⋊D₇₁	1	1	yes	characteristic
215	#131	—	4	2	no	—
216	#131	—	4	2	no	—
217	C₂×M₁₆	—	2	2	no	—
218	C₂×M₁₆	—	2	1	no	—
219	C₂×M₁₆	—	142	1	no	—
220	C₂²×C₈	—	2	2	no	—
221	C₂²×C₈	—	2	1	no	—
222	C₂²×C₈	—	142	1	no	—
223	C₄×Q₈	Dic₁₄₂	1	1	yes	characteristic
224	C₄×Q₈	C₄×D₇₁	1	1	yes	characteristic
225	C₄×D₄	—	2	1	no	—
226	C₄×D₄	—	2	1	no	—
227	C₄×D₄	Dic₁₄₂	1	1	yes	characteristic
228	C₄×D₄	C₄×D₇₁	1	1	yes	characteristic
229	#117	—	2	1	no	—
230	#117	C₂²⋊D₇₁	1	1	yes	characteristic
231	C₂×C₄⋊C₄	—	142	1	no	—
232	C₂×C₄²	—	2	1	no	—
233	C₂×C₄²	—	142	1	no	—
234	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
235	C₄⋊C₈	—	2	2	no	—
236	C₄⋊C₈	—	2	2	no	—
237	C₄⋊C₈	—	2	1	no	—
238	C₄⋊C₈	—	2	1	no	—
239	C₄⋊C₈	—	71	1	no	—
240	C₄⋊C₈	—	71	1	no	—
241	C₄⋊C₈	—	142	1	no	—
242	C₄⋊C₈	—	142	1	no	—
243	C₄⋊C₈	C₂²⋊D₇₁	1	1	yes	characteristic
244	C₄⋊C₈	C₂²⋊D₇₁	1	1	yes	characteristic
245	C₄⋊C₈	C₂²⋊D₇₁	1	2	yes	—
246	#104	—	284	2	no	—
247	C₂²⋊C₈	—	142	1	no	—
248	C₄×C₈	—	71	1	no	—
249	C₄×C₈	—	71	1	no	—
250	C₄×C₈	—	142	1	no	—
251	C₄×C₈	—	142	1	no	—
252	#95	—	142	1	no	—
253	#41	—	2	1	no	—
254	#41	—	2	1	no	—
255	#41	#11765852	1	2	yes	—
256	#41	#11765853	1	2	yes	—
257	C₂×Q₈	Dic₇₁⋊C₄	1	1	yes	characteristic
258	C₂×D₄	—	2	1	no	—
259	C₂×D₄	Dic₇₁⋊C₄	1	1	yes	characteristic
260	C₂²×C₄	—	2	1	no	—
261	C₂²×C₄	—	2	1	no	—
262	C₂²×C₄	—	2	1	no	—
263	C₂²×C₄	—	284	1	no	—
264	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
265	C₂²×C₄	#11765865	1	1	yes	characteristic
266	C₂²×C₄	#11765865	1	1	yes	characteristic
267	M₁₆	—	4	4	no	—
268	M₁₆	—	4	1	no	—
269	M₁₆	—	4	1	no	—
270	M₁₆	—	284	1	no	—
271	M₁₆	—	284	1	no	—
272	C₂×C₈	—	2	1	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	2	1	no	—
275	C₂×C₈	—	4	4	no	—
276	C₂×C₈	—	4	1	no	—
277	C₂×C₈	—	4	1	no	—
278	C₂×C₈	—	284	2	no	—
279	C₂×C₈	—	284	1	no	—
280	C₂×C₈	—	142	1	no	—
281	C₂×C₈	—	142	1	no	—
282	C₂×C₈	—	142	1	no	—
283	C₂×C₈	—	142	1	no	—
284	C₄⋊C₄	—	2	1	no	—
285	C₄⋊C₄	—	2	1	no	—
286	C₄⋊C₄	—	284	1	no	—
287	C₄⋊C₄	—	142	2	no	—
288	C₄⋊C₄	Dic₇₁⋊C₄	1	1	yes	characteristic
289	C₄⋊C₄	Dic₇₁⋊C₄	1	1	yes	characteristic
290	C₂²⋊C₄	—	2	1	no	—
291	C₂²⋊C₄	—	2	1	no	—
292	C₂²⋊C₄	—	2	1	no	—
293	C₄²	—	2	1	no	—
294	C₄²	—	2	1	no	—
295	C₄²	—	2	1	no	—
296	C₄²	—	284	1	no	—
297	C₄²	—	284	1	no	—
298	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
299	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
300	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
301	C₄²	C₂×Dic₁₄₂	1	1	yes	characteristic
302	C₂³	—	2	1	no	—
303	C₂³	#14556111	1	1	yes	characteristic
304	Q₈	—	2	1	no	—
305	Q₈	#14556059	1	2	yes	—
306	D₄	—	2	2	no	—
307	D₄	—	2	2	no	—
308	D₄	—	2	1	no	—
309	D₄	#14556059	1	2	yes	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	8	1	no	—
321	C₂×C₄	—	284	2	no	—
322	C₂×C₄	—	568	1	no	—
323	C₂×C₄	—	568	1	no	—
324	C₂×C₄	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
325	C₂×C₄	C₄×C₂²⋊D₇₁	1	1	yes	characteristic
326	C₂×C₄	C₄×C₂²⋊D₇₁	1	1	yes	characteristic
327	C₂×C₄	#14556097	1	1	yes	characteristic
328	C₂×C₄	#14556107	1	1	yes	characteristic
329	C₂×C₄	#14556107	1	1	yes	characteristic
330	C₂×C₄	C₂×#11765852	1	1	yes	characteristic
331	C₂×C₄	C₂×#11765853	1	1	yes	characteristic
332	C₂×C₄	#14556154	1	1	yes	characteristic
333	C₂×C₄	#14556159	1	1	yes	characteristic
334	C₈	—	4	1	no	—
335	C₈	—	4	2	no	—
336	C₈	—	4	1	no	—
337	C₈	—	284	1	no	—
338	C₈	—	284	1	no	—
339	C₈	—	284	1	no	—
340	C₈	—	568	1	no	—
341	C₈	—	142	2	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	2	2	no	—
345	C₂²	—	4	1	no	—
346	C₂²	—	4	1	no	—
347	C₂²	#17985872	1	1	yes	characteristic
348	C₂²	#17986317	1	1	yes	characteristic
349	C₂²	#17986329	1	1	yes	characteristic
350	C₄	—	2	2	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	4	1	no	—
355	C₄	—	4	1	no	—
356	C₄	—	568	2	no	—
357	C₄	C₂×#14556059	1	1	yes	characteristic
358	C₄	#17986039	1	1	yes	characteristic
359	C₄	#17986114	1	1	yes	characteristic
360	C₄	#17986119	1	1	yes	characteristic
361	C₄	#17986138	1	1	yes	characteristic
362	C₄	#17986150	1	1	yes	characteristic
363	C₂	—	2	1	no	—
364	C₂	—	2	1	no	—
365	C₂	—	4	2	no	—
366	C₂	#22588533	1	1	yes	characteristic
367	C₂	#22588580	1	1	yes	characteristic
368	C₂	#22588615	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14556097
commutator	C₄²×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#2906	C₂
socle	C₂²×C₇₁	#385
prime radical	C₂×#41×C₇₁	D₄
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#2906	D₇₁
hypocenter	C₇₁	#8904
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #2906

Complements

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

N	H ≅ G/N	type
2. #22587110	C₂	semidirect
3. #22587095	C₂	semidirect
5. #22586241	C₂	semidirect
6. #22586179	C₂	semidirect
17. Q₈⋊(C₇₁⋊C₈)	C₄	semidirect
18. Q₈⋊(C₇₁⋊C₈)	C₄	semidirect
19. D₄⋊(C₇₁⋊C₈)	C₄	semidirect
20. D₄⋊(C₇₁⋊C₈)	C₄	semidirect
192. C₇₁	#8904	semidirect

Maximal subgroups

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

subgroup	index
#22588360	2
#22587110	2
#22587095	2
#22586722	2
#22586241	2
#22586179	2
C₇₁×#2906	2
#8904	71