Subgroups — #31892677

Order 18176.

470 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22598895	C₂	1	1	yes	maximal, characteristic
2	#22597287	C₂	1	1	yes	maximal, characteristic, complemented
3	#22586838	C₂	1	1	yes	maximal, characteristic, complemented
4	#22586739	C₂	1	1	yes	maximal, characteristic
5	#22586181	C₂	1	1	yes	maximal, characteristic, complemented
6	#22586178	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₁×#1980	C₂	1	1	yes	maximal, characteristic
8	#17986556	C₂²	1	1	yes	characteristic
9	#17986395	C₄	1	1	yes	characteristic, complemented
10	#17986371	C₄	1	1	yes	characteristic, complemented
11	#17986351	C₄	1	1	yes	characteristic, complemented
12	#17986345	C₄	1	1	yes	characteristic, complemented
13	#17986226	—	2	2	no	—
14	#17986129	—	2	1	no	—
15	#17986127	—	2	2	no	—
16	C₈×Dic₁₄₂	—	2	1	no	—
17	C₄×#11765853	—	2	2	no	—
18	#17986052	C₂²	1	1	yes	characteristic
19	#17986027	—	2	1	no	—
20	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
21	#17985957	—	2	1	no	—
22	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
23	#17985735	—	2	2	no	—
24	D₄⋊(C₇₁⋊C₈)	—	2	1	no	—
25	D₄⋊(C₇₁⋊C₈)	—	2	1	no	—
26	#532×C₇₁	C₂²	1	1	yes	characteristic
27	C₂×#104×C₇₁	—	2	1	no	—
28	C₂×#104×C₇₁	—	2	1	no	—
29	C₄×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
30	C₄×C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
31	C₂×#11765853	—	2	2	no	—
32	C₂²×C₇₁⋊C₈	—	2	1	no	—
33	C₂²×C₇₁⋊C₈	—	2	1	no	—
34	#14556128	C₂×C₄	1	1	yes	characteristic
35	#14556128	C₂×C₄	1	1	yes	characteristic
36	#14556120	—	2	1	no	—
37	#14556107	—	2	1	no	—
38	#14556107	D₄	1	2	yes	—
39	C₄×Dic₁₄₂	—	2	1	no	—
40	C₄×Dic₁₄₂	D₄	1	2	yes	—
41	#14556087	—	2	1	no	—
42	#14556059	—	2	2	no	—
43	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
44	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
45	C₄⋊(C₇₁⋊C₈)	D₄	1	1	yes	characteristic
46	C₄⋊(C₇₁⋊C₈)	D₄	1	1	yes	characteristic
47	Dic₇₁⋊C₈	—	2	2	no	—
48	Dic₇₁⋊C₈	—	2	2	no	—
49	#14556051	—	2	2	no	—
50	#14556049	—	2	2	no	—
51	C₂²⋊(C₇₁⋊C₈)	—	2	1	no	—
52	#14556038	—	2	2	no	—
53	C₄×C₇₁⋊C₈	—	2	1	no	—
54	C₄×C₇₁⋊C₈	—	2	1	no	—
55	C₄×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
56	C₄×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
57	C₈×Dic₇₁	—	2	2	no	—
58	C₂×#41×C₇₁	—	2	1	no	—
59	C₂×M₁₆×C₇₁	D₄	1	1	yes	characteristic
60	C₂×M₁₆×C₇₁	D₄	1	1	yes	characteristic
61	C₄⋊Q₈×C₇₁	C₂×C₄	1	1	yes	characteristic
62	#127×C₇₁	C₂×C₄	1	1	yes	characteristic
63	#124×C₇₁	—	2	1	no	—
64	#121×C₇₁	—	2	1	no	—
65	#121×C₇₁	—	2	1	no	—
66	C₄×D₄×C₇₁	—	2	1	no	—
67	C₄×D₄×C₇₁	—	2	1	no	—
68	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
69	C₂×C₄²×C₇₁	—	2	1	no	—
70	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
71	#104×C₇₁	—	4	2	no	—
72	#104×C₇₁	—	4	2	no	—
73	#97×C₇₁	—	2	1	no	—
74	C₄×C₈×C₇₁	—	2	1	no	—
75	#95×C₇₁	—	2	1	no	—
76	C₂×Dic₁₄₂	—	2	1	no	—
77	C₂×Dic₁₄₂	C₂²⋊C₄	1	2	yes	—
78	#11765853	—	4	4	no	—
79	C₂×C₇₁⋊C₈	—	2	1	no	—
80	C₂×C₇₁⋊C₈	—	2	1	no	—
81	C₂×C₇₁⋊C₈	—	2	1	no	—
82	C₂×C₇₁⋊C₈	—	2	1	no	—
83	C₂×C₇₁⋊C₈	—	4	2	no	—
84	C₂×C₇₁⋊C₈	—	4	1	no	—
85	C₂×C₇₁⋊C₈	—	4	1	no	—
86	C₂×C₇₁⋊C₈	—	4	1	no	—
87	C₄⋊Dic₇₁	—	2	1	no	—
88	C₄⋊Dic₇₁	—	2	1	no	—
89	C₄⋊Dic₇₁	—	2	1	no	—
90	C₄⋊Dic₇₁	C₂²⋊C₄	1	2	yes	—
91	Dic₇₁⋊C₄	—	2	2	no	—
92	Dic₇₁⋊C₄	—	2	2	no	—
93	C₂²⋊Dic₇₁	—	2	2	no	—
94	C₄×Dic₇₁	—	2	2	no	—
95	#41×C₇₁	—	4	1	no	—
96	#41×C₇₁	—	4	1	no	—
97	#41×C₇₁	—	4	1	no	—
98	#41×C₇₁	—	4	1	no	—
99	C₂×Q₈×C₇₁	—	2	1	no	—
100	C₂×D₄×C₇₁	—	2	1	no	—
101	C₂×D₄×C₇₁	—	2	1	no	—
102	C₂×D₄×C₇₁	—	2	1	no	—
103	C₂×D₄×C₇₁	—	4	1	no	—
104	C₂²×C₄×C₇₁	—	4	1	no	—
105	C₂²×C₄×C₇₁	—	4	1	no	—
106	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
109	M₁₆×C₇₁	—	2	1	no	—
110	M₁₆×C₇₁	—	2	1	no	—
111	M₁₆×C₇₁	—	2	1	no	—
112	M₁₆×C₇₁	SD₁₆	1	2	yes	complemented
113	C₂×C₈×C₇₁	—	2	1	no	—
114	C₂×C₈×C₇₁	—	2	1	no	—
115	C₄⋊C₄×C₇₁	—	2	2	no	—
116	C₄⋊C₄×C₇₁	—	2	1	no	—
117	C₄⋊C₄×C₇₁	—	2	1	no	—
118	C₄⋊C₄×C₇₁	—	4	1	no	—
119	C₂²⋊C₄×C₇₁	—	4	1	no	—
120	C₂²⋊C₄×C₇₁	—	4	1	no	—
121	C₄²×C₇₁	—	4	1	no	—
122	C₄²×C₇₁	—	4	1	no	—
123	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
124	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
125	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
126	C₄²×C₇₁	#41	1	1	yes	characteristic
127	C₂³×C₇₁	—	4	1	no	—
128	C₂³×C₇₁	#121	1	1	yes	characteristic
129	Q₈×C₇₁	—	4	1	no	—
130	Q₈×C₇₁	—	4	1	no	—
131	D₄×C₇₁	—	4	2	no	—
132	D₄×C₇₁	—	4	1	no	—
133	D₄×C₇₁	—	4	1	no	—
134	D₄×C₇₁	—	4	1	no	—
135	D₄×C₇₁	—	4	1	no	—
136	D₄×C₇₁	—	8	1	no	—
137	D₄×C₇₁	—	8	1	no	—
138	C₂×C₄×C₇₁	—	2	1	no	—
139	C₂×C₄×C₇₁	—	2	1	no	—
140	C₂×C₄×C₇₁	—	4	1	no	—
141	C₂×C₄×C₇₁	—	4	2	no	—
142	C₂×C₄×C₇₁	—	4	1	no	—
143	C₂×C₄×C₇₁	—	8	1	no	—
144	C₂×C₄×C₇₁	—	8	1	no	—
145	C₂×C₄×C₇₁	—	8	1	no	—
146	C₂×C₄×C₇₁	—	8	1	no	—
147	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
148	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
149	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
150	C₂×C₄×C₇₁	#120	1	1	yes	characteristic
151	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
152	C₂×C₄×C₇₁	#123	1	1	yes	characteristic
153	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
154	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
155	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
156	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
157	C₂×Dic₇₁	—	2	2	no	—
158	C₂×Dic₇₁	—	2	2	no	—
159	Dic₁₄₂	—	4	1	no	—
160	Dic₁₄₂	—	4	1	no	—
161	Dic₁₄₂	—	4	2	no	—
162	C₈×C₇₁	—	2	2	no	—
163	C₈×C₇₁	—	4	1	no	—
164	C₇₁⋊C₈	—	2	2	no	—
165	C₇₁⋊C₈	—	4	1	no	—
166	C₇₁⋊C₈	—	4	1	no	—
167	C₇₁⋊C₈	—	4	1	no	—
168	C₇₁⋊C₈	—	8	1	no	—
169	C₂²×C₇₁	—	2	1	no	—
170	C₂²×C₇₁	—	2	1	no	—
171	C₂²×C₇₁	—	8	1	no	—
172	C₂²×C₇₁	—	8	1	no	—
173	C₂²×C₇₁	—	8	1	no	—
174	C₂²×C₇₁	#386	1	1	yes	characteristic
175	C₂²×C₇₁	#460	1	1	yes	characteristic
176	C₂²×C₇₁	#464	1	1	yes	characteristic
177	C₄×C₇₁	—	2	1	no	—
178	C₄×C₇₁	—	2	1	no	—
179	C₄×C₇₁	—	2	1	no	—
180	C₄×C₇₁	—	8	2	no	—
181	C₄×C₇₁	—	8	1	no	—
182	C₄×C₇₁	C₄×SD₁₆	1	1	yes	characteristic
183	C₄×C₇₁	#440	1	1	yes	characteristic
184	C₄×C₇₁	#443	1	1	yes	characteristic
185	C₄×C₇₁	#444	1	1	yes	characteristic
186	C₄×C₇₁	#448	1	1	yes	characteristic
187	C₄×C₇₁	#449	1	1	yes	characteristic
188	Dic₇₁	—	4	2	no	—
189	Dic₇₁	—	4	2	no	—
190	#9054	—	71	1	no	maximal, Sylow
191	C₂×C₇₁	—	2	1	no	—
192	C₂×C₇₁	—	2	1	no	—
193	C₂×C₇₁	—	16	1	no	—
194	C₂×C₇₁	#1876	1	1	yes	characteristic
195	C₂×C₇₁	#1880	1	1	yes	characteristic
196	C₂×C₇₁	#1884	1	1	yes	characteristic
197	#3167	—	71	1	no	—
198	#2962	—	71	1	no	—
199	#1980	D₇₁	1	1	yes	characteristic
200	#1503	—	71	1	no	—
201	#1480	—	71	1	no	—
202	#1279	—	71	1	no	—
203	#1278	—	71	1	no	—
204	C₇₁	#9054	1	1	yes	characteristic, complemented, Sylow
205	#533	—	71	1	no	—
206	#532	C₂×D₇₁	1	1	yes	characteristic
207	#498	—	71	1	no	—
208	#492	—	71	1	no	—
209	#487	—	71	1	no	—
210	#486	—	71	1	no	—
211	#465	—	142	2	no	—
212	#446	—	142	1	no	—
213	#446	—	142	2	no	—
214	C₈×Q₈	—	142	1	no	—
215	C₄×SD₁₆	—	142	2	no	—
216	#431	—	71	1	no	—
217	#424	—	142	1	no	—
218	C₂×C₄⋊C₈	—	71	1	no	—
219	C₂×#104	—	2	1	no	—
220	C₂×#104	—	2	1	no	—
221	#405	—	142	1	no	—
222	C₄×M₁₆	C₂×D₇₁	1	1	yes	characteristic
223	C₂×C₄×C₈	—	71	1	no	—
224	C₄×C₄⋊C₄	C₂×D₇₁	1	1	yes	characteristic
225	#328	—	142	2	no	—
226	D₄⋊C₈	—	142	1	no	—
227	D₄⋊C₈	—	142	1	no	—
228	C₂×#41	—	2	1	no	—
229	C₂×SD₁₆	—	142	2	no	—
230	C₂×M₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
231	C₂×M₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
232	C₂²×C₈	—	142	1	no	—
233	C₂²×C₈	—	142	1	no	—
234	C₄⋊Q₈	—	71	1	no	—
235	C₄⋊Q₈	—	71	1	no	—
236	C₄⋊Q₈	C₄×D₇₁	1	1	yes	characteristic
237	#127	C₄×D₇₁	1	1	yes	characteristic
238	#125	—	142	1	no	—
239	#124	—	2	1	no	—
240	C₂²⋊Q₈	—	71	2	no	—
241	C₂²⋊Q₈	—	142	1	no	—
242	#121	—	2	1	no	—
243	#121	—	2	1	no	—
244	C₄×Q₈	—	71	2	no	—
245	C₄×Q₈	—	142	1	no	—
246	C₄×D₄	—	2	1	no	—
247	C₄×D₄	—	2	1	no	—
248	#117	—	142	1	no	—
249	C₂×C₄⋊C₄	—	2	1	no	—
250	C₂×C₄²	—	2	1	no	—
251	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
252	#106	—	142	2	no	—
253	C₄⋊C₈	—	71	1	no	—
254	C₄⋊C₈	—	71	1	no	—
255	C₄⋊C₈	—	142	2	no	—
256	C₄⋊C₈	—	142	2	no	—
257	C₄⋊C₈	—	142	1	no	—
258	C₄⋊C₈	—	142	1	no	—
259	#104	—	4	2	no	—
260	#104	—	4	2	no	—
261	#103	—	142	2	no	—
262	#102	—	142	2	no	—
263	C₂²⋊C₈	—	142	1	no	—
264	#97	—	2	1	no	—
265	#97	—	142	2	no	—
266	C₄×C₈	—	2	1	no	—
267	C₄×C₈	—	71	1	no	—
268	C₄×C₈	—	71	1	no	—
269	C₄×C₈	—	142	2	no	—
270	C₄×C₈	—	142	1	no	—
271	C₄×C₈	—	142	1	no	—
272	#95	—	2	1	no	—
273	#41	—	4	1	no	—
274	#41	—	4	1	no	—
275	#41	—	4	1	no	—
276	#41	—	4	1	no	—
277	C₂×Q₈	—	2	1	no	—
278	C₂×Q₈	—	71	2	no	—
279	C₂×Q₈	—	142	1	no	—
280	C₂×D₄	—	2	1	no	—
281	C₂×D₄	—	2	1	no	—
282	C₂×D₄	—	2	1	no	—
283	C₂×D₄	—	4	1	no	—
284	C₂²×C₄	—	4	1	no	—
285	C₂²×C₄	—	4	1	no	—
286	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
287	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
288	C₂²×C₄	#11765864	1	1	yes	characteristic
289	SD₁₆	—	284	4	no	—
290	M₁₆	—	2	1	no	—
291	M₁₆	—	2	1	no	—
292	M₁₆	—	2	1	no	—
293	M₁₆	#11765853	1	2	yes	complemented
294	C₂×C₈	—	2	1	no	—
295	C₂×C₈	—	2	1	no	—
296	C₂×C₈	—	284	1	no	—
297	C₂×C₈	—	284	1	no	—
298	C₂×C₈	—	284	2	no	—
299	C₂×C₈	—	284	1	no	—
300	C₂×C₈	—	142	1	no	—
301	C₂×C₈	—	142	1	no	—
302	C₂×C₈	—	142	1	no	—
303	C₂×C₈	—	142	1	no	—
304	C₄⋊C₄	—	2	2	no	—
305	C₄⋊C₄	—	2	1	no	—
306	C₄⋊C₄	—	2	1	no	—
307	C₄⋊C₄	—	4	1	no	—
308	C₄⋊C₄	—	71	2	no	—
309	C₄⋊C₄	—	142	1	no	—
310	C₄⋊C₄	—	142	1	no	—
311	C₄⋊C₄	—	142	1	no	—
312	C₄⋊C₄	—	142	2	no	—
313	C₄⋊C₄	—	142	2	no	—
314	C₂²⋊C₄	—	4	1	no	—
315	C₂²⋊C₄	—	4	1	no	—
316	C₂²⋊C₄	—	142	2	no	—
317	C₄²	—	4	1	no	—
318	C₄²	—	4	1	no	—
319	C₄²	—	142	2	no	—
320	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
321	C₄²	D₄×D₇₁	1	1	yes	characteristic
322	C₄²	D₄×D₇₁	1	1	yes	characteristic
323	C₄²	#11765865	1	1	yes	characteristic
324	C₂³	—	4	1	no	—
325	C₂³	#14556099	1	1	yes	characteristic
326	Q₈	—	4	1	no	—
327	Q₈	—	4	1	no	—
328	Q₈	—	284	2	no	—
329	Q₈	—	284	1	no	—
330	Q₈	—	284	1	no	—
331	D₄	—	4	2	no	—
332	D₄	—	4	1	no	—
333	D₄	—	4	1	no	—
334	D₄	—	4	1	no	—
335	D₄	—	4	1	no	—
336	D₄	—	8	1	no	—
337	D₄	—	8	1	no	—
338	C₂×C₄	—	2	1	no	—
339	C₂×C₄	—	2	1	no	—
340	C₂×C₄	—	4	1	no	—
341	C₂×C₄	—	4	2	no	—
342	C₂×C₄	—	4	1	no	—
343	C₂×C₄	—	8	1	no	—
344	C₂×C₄	—	8	1	no	—
345	C₂×C₄	—	8	1	no	—
346	C₂×C₄	—	8	1	no	—
347	C₂×C₄	—	142	2	no	—
348	C₂×C₄	—	142	2	no	—
349	C₂×C₄	C₂²⋊C₄×D₇₁	1	1	yes	characteristic
350	C₂×C₄	(C₂²⋊D₇₁)⋊C₄	1	1	yes	characteristic
351	C₂×C₄	C₄×C₂²⋊D₇₁	1	1	yes	characteristic
352	C₂×C₄	#14556096	1	1	yes	characteristic
353	C₂×C₄	#14556101	1	1	yes	characteristic
354	C₂×C₄	#14556110	1	1	yes	characteristic
355	C₂×C₄	C₂×#11765853	1	1	yes	characteristic
356	C₂×C₄	#14556148	1	1	yes	characteristic
357	C₂×C₄	#14556154	1	1	yes	characteristic
358	C₂×C₄	#14556158	1	1	yes	characteristic
359	C₈	—	2	2	no	—
360	C₈	—	4	1	no	—
361	C₈	—	284	1	no	—
362	C₈	—	284	1	no	—
363	C₈	—	284	1	no	—
364	C₈	—	568	1	no	—
365	C₈	—	142	2	no	—
366	C₂²	—	2	1	no	—
367	C₂²	—	2	1	no	—
368	C₂²	—	8	1	no	—
369	C₂²	—	8	1	no	—
370	C₂²	—	8	1	no	—
371	C₂²	#17985875	1	1	yes	characteristic
372	C₂²	#17986196	1	1	yes	characteristic
373	C₂²	#17986218	1	1	yes	characteristic
374	C₄	—	2	1	no	—
375	C₄	—	2	1	no	—
376	C₄	—	2	1	no	—
377	C₄	—	8	2	no	—
378	C₄	—	8	1	no	—
379	C₄	—	284	2	no	—
380	C₄	—	284	2	no	—
381	C₄	C₄×#11765853	1	1	yes	characteristic
382	C₄	#17986099	1	1	yes	characteristic
383	C₄	#17986113	1	1	yes	characteristic
384	C₄	#17986118	1	1	yes	characteristic
385	C₄	#17986137	1	1	yes	characteristic
386	C₄	#17986142	1	1	yes	characteristic
387	C₂	—	2	1	no	—
388	C₂	—	2	1	no	—
389	C₂	—	16	1	no	—
390	C₂	#22588664	1	1	yes	characteristic
391	C₂	#22588698	1	1	yes	characteristic
392	C₂	#22588720	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14556096
commutator	C₄²×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1980	C₂
socle	C₂²×C₇₁	#386
prime radical	#532×C₇₁	C₂²
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1980	D₇₁
hypocenter	C₇₁	#9054
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1980

Complements

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

N	H ≅ G/N	type
2. #22597287	C₂	semidirect
3. #22586838	C₂	semidirect
5. #22586181	C₂	semidirect
6. #22586178	C₂	semidirect
9. #17986395	C₄	semidirect
10. #17986371	C₄	semidirect
11. #17986351	C₄	semidirect
12. #17986345	C₄	semidirect
112. M₁₆×C₇₁	SD₁₆	semidirect
204. C₇₁	#9054	semidirect
293. M₁₆	#11765853	semidirect

Maximal subgroups

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

subgroup	index
#22598895	2
#22597287	2
#22586838	2
#22586739	2
#22586181	2
#22586178	2
C₇₁×#1980	2
#9054	71