Subgroups — #31914364

Order 18176.

456 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22601442	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#17986119	C₂	1	1	yes	maximal, characteristic
3	#22589969	C₂	1	1	yes	maximal, characteristic, complemented
4	#22589968	C₂	1	1	yes	maximal, characteristic, complemented
5	#22586423	C₂	1	1	yes	maximal, characteristic, complemented
6	#22586419	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₁×#2115	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14556148	C₄	1	1	yes	characteristic
9	C₂×#14556134	C₂²	1	1	yes	characteristic
10	#17986417	—	2	2	no	—
11	#17986415	—	2	2	no	—
12	C₂×#14556068	C₂²	1	1	yes	characteristic
13	C₂×#14556068	C₂²	1	1	yes	characteristic
14	M₃₂×D₇₁	—	2	2	no	—
15	#17986119	—	2	2	no	—
16	#17986119	—	2	2	no	—
17	C₂×#14556063	C₄	1	1	yes	characteristic
18	C₂×#14556055	C₄	1	1	yes	characteristic
19	C₂×#14556054	C₄	1	1	yes	characteristic
20	C₂×#14556038	C₂²	1	1	yes	characteristic
21	#17985784	—	2	2	no	—
22	#17985783	—	2	2	no	—
23	C₂×#131×C₇₁	C₂²	1	1	yes	characteristic
24	C₂×M₃₂×C₇₁	C₂²	1	1	yes	characteristic
25	C₂×M₃₂×C₇₁	C₂²	1	1	yes	characteristic
26	#349×C₇₁	—	2	2	no	—
27	C₂×#11765865	C₂×C₄	1	1	yes	characteristic
28	#14556148	—	2	2	no	—
29	#14556148	—	2	2	no	—
30	C₂×#11765855	C₈	1	1	yes	characteristic
31	C₂×#11765853	C₈	1	1	yes	characteristic
32	C₂×#11765852	C₈	1	1	yes	characteristic
33	C₂×#11765850	C₈	1	1	yes	characteristic
34	#14556134	—	2	1	no	—
35	#14556134	—	2	1	no	—
36	#14556134	D₄	1	2	yes	—
37	#14556134	D₄	1	2	yes	—
38	C₂×#11765849	C₂×C₄	1	1	yes	characteristic
39	C₂×#11765848	—	2	1	no	—
40	C₂²×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
41	C₂×C₈×D₇₁	—	2	1	no	—
42	C₂×C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
43	#14556068	—	2	1	no	—
44	#14556068	—	2	1	no	—
45	#14556068	—	2	1	no	—
46	#14556068	—	2	1	no	—
47	#14556067	—	2	4	no	—
48	C₂×C₇₁⋊C₁₆	—	2	1	no	—
49	C₂×C₇₁⋊C₁₆	D₄	1	2	yes	—
50	C₁₆×D₇₁	—	2	4	no	—
51	#14556063	—	2	2	no	—
52	#14556055	—	2	2	no	—
53	#14556054	—	2	2	no	—
54	#14556038	—	2	1	no	—
55	#14556038	—	2	1	no	—
56	C₂×#41×C₇₁	C₂×C₄	1	1	yes	characteristic
57	#131×C₇₁	—	2	1	no	—
58	#131×C₇₁	—	2	1	no	—
59	#131×C₇₁	—	2	1	no	—
60	#131×C₇₁	—	2	1	no	—
61	C₂×M₁₆×C₇₁	—	2	1	no	—
62	C₂×M₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
63	C₂²×C₈×C₇₁	—	2	1	no	—
64	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
65	M₃₂×C₇₁	—	2	1	no	—
66	M₃₂×C₇₁	—	2	1	no	—
67	M₃₂×C₇₁	—	2	1	no	—
68	M₃₂×C₇₁	D₄	1	2	yes	—
69	C₂×C₁₆×C₇₁	—	2	1	no	—
70	C₂×C₁₆×C₇₁	—	2	1	no	—
71	#11765865	—	2	1	no	—
72	#11765865	—	2	1	no	—
73	#11765865	C₂²⋊C₄	1	2	yes	—
74	#11765865	M₁₆	1	2	yes	—
75	C₂×Dic₁₄₂	C₂×C₈	1	1	yes	characteristic
76	C₂×D₂₈₄	C₂×C₈	1	1	yes	characteristic
77	C₂×C₂²⋊D₇₁	—	2	1	no	—
78	C₂²×Dic₇₁	—	2	1	no	—
79	C₂×C₄×D₇₁	—	2	1	no	—
80	#11765855	—	2	2	no	—
81	#11765853	—	2	2	no	—
82	#11765852	—	2	2	no	—
83	#11765850	—	2	2	no	—
84	#11765849	—	2	1	no	—
85	#11765849	C₂²⋊C₄	1	2	yes	—
86	#11765848	—	2	2	no	—
87	#11765848	—	2	2	no	—
88	C₂×C₇₁⋊C₈	—	2	1	no	—
89	C₂×C₇₁⋊C₈	—	2	1	no	—
90	C₂×C₇₁⋊C₈	—	2	1	no	—
91	C₂×C₇₁⋊C₈	C₂×C₈	1	1	yes	characteristic
92	C₂×C₇₁⋊C₈	C₂×C₈	1	1	yes	characteristic
93	C₈×D₇₁	—	2	2	no	—
94	C₈×D₇₁	—	2	2	no	—
95	C₄×Dic₇₁	—	2	1	no	—
96	C₄×Dic₇₁	—	2	1	no	—
97	C₇₁⋊C₁₆	—	2	2	no	—
98	C₇₁⋊C₁₆	—	4	1	no	—
99	#41×C₇₁	—	2	1	no	—
100	#41×C₇₁	—	2	1	no	—
101	#41×C₇₁	—	2	1	no	—
102	#41×C₇₁	—	2	1	no	—
103	C₂×Q₈×C₇₁	C₂×C₈	1	1	yes	characteristic
104	C₂×D₄×C₇₁	—	2	1	no	—
105	C₂×D₄×C₇₁	C₂×C₈	1	1	yes	characteristic
106	C₂²×C₄×C₇₁	—	2	1	no	—
107	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
108	M₁₆×C₇₁	—	2	1	no	—
109	M₁₆×C₇₁	—	2	1	no	—
110	M₁₆×C₇₁	—	4	1	no	—
111	M₁₆×C₇₁	—	4	1	no	—
112	C₂×C₈×C₇₁	—	2	1	no	—
113	C₂×C₈×C₇₁	—	4	1	no	—
114	C₂×C₈×C₇₁	—	4	1	no	—
115	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
120	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
121	C₁₆×C₇₁	—	2	2	no	—
122	C₁₆×C₇₁	—	4	1	no	—
123	C₂³×C₇₁	—	2	1	no	—
124	C₂³×C₇₁	#131	1	1	yes	characteristic
125	C₂²×D₇₁	—	2	1	no	—
126	Q₈×C₇₁	—	2	1	no	—
127	Q₈×C₇₁	—	2	1	no	—
128	D₄×C₇₁	—	2	1	no	—
129	D₄×C₇₁	—	2	1	no	—
130	D₄×C₇₁	—	4	1	no	—
131	D₄×C₇₁	—	4	1	no	—
132	C₂×C₄×C₇₁	—	2	1	no	—
133	C₂×C₄×C₇₁	—	4	1	no	—
134	C₂×C₄×C₇₁	—	4	1	no	—
135	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
136	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
139	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
140	C₂×C₄×C₇₁	#131	1	1	yes	characteristic
141	C₂²⋊D₇₁	—	2	2	no	—
142	C₂²⋊D₇₁	—	2	2	no	—
143	C₂×Dic₇₁	—	2	1	no	—
144	C₂×Dic₇₁	—	4	1	no	—
145	C₂×Dic₇₁	—	8	1	no	—
146	D₂₈₄	—	2	1	no	—
147	D₂₈₄	C₂²⋊C₈	1	2	yes	—
148	C₄×D₇₁	—	2	2	no	—
149	C₄×D₇₁	—	2	2	no	—
150	Dic₁₄₂	—	2	1	no	—
151	Dic₁₄₂	C₂²⋊C₈	1	2	yes	—
152	C₈×C₇₁	—	4	1	no	—
153	C₈×C₇₁	#120	1	1	yes	characteristic
154	C₈×C₇₁	#121	1	1	yes	characteristic
155	C₈×C₇₁	#121	1	1	yes	characteristic
156	C₈×C₇₁	#123	1	1	yes	characteristic
157	C₇₁⋊C₈	—	2	2	no	—
158	C₇₁⋊C₈	—	2	1	no	—
159	C₇₁⋊C₈	—	2	1	no	—
160	C₂²×C₇₁	—	4	1	no	—
161	C₂²×C₇₁	—	4	1	no	—
162	C₂²×C₇₁	—	8	1	no	—
163	C₂²×C₇₁	#408	1	1	yes	characteristic
164	C₂²×C₇₁	C₈×D₄	1	1	yes	characteristic
165	C₂²×C₇₁	#435	1	1	yes	characteristic
166	C₂×D₇₁	—	2	2	no	—
167	C₂×D₇₁	—	8	1	no	—
168	C₄×C₇₁	—	4	1	no	—
169	C₄×C₇₁	#386	1	1	yes	characteristic
170	C₄×C₇₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
171	C₄×C₇₁	C₈×D₄	1	1	yes	characteristic
172	C₄×C₇₁	#435	1	1	yes	characteristic
173	Dic₇₁	—	2	2	no	—
174	Dic₇₁	—	8	1	no	—
175	#12986	—	71	1	no	maximal, Sylow
176	C₂×C₇₁	—	2	1	no	—
177	C₂×C₇₁	—	2	1	no	—
178	C₂×C₇₁	—	2	1	no	—
179	C₂×C₇₁	—	8	1	no	—
180	C₂×C₇₁	#1869	1	1	yes	characteristic
181	D₇₁	—	4	2	no	—
182	#3407	—	71	1	no	—
183	C₂×#444	—	71	1	no	—
184	#2117	—	71	1	no	—
185	#2117	—	71	1	no	—
186	#2115	D₇₁	1	1	yes	characteristic, complemented
187	#1377	—	71	1	no	—
188	#1375	—	71	1	no	—
189	C₇₁	#12986	1	1	yes	characteristic, complemented, Sylow
190	C₂×#135	—	71	1	no	—
191	C₂×#131	—	71	1	no	—
192	C₂×#131	C₂×D₇₁	1	1	yes	characteristic
193	#504	—	142	2	no	—
194	#504	—	142	2	no	—
195	C₂×M₃₂	—	71	1	no	—
196	C₂×M₃₂	—	71	1	no	—
197	C₂×M₃₂	—	142	2	no	—
198	C₂×M₃₂	C₂×D₇₁	1	1	yes	characteristic
199	C₂×M₃₂	C₂×D₇₁	1	1	yes	characteristic
200	#444	—	142	2	no	—
201	#444	—	142	2	no	—
202	C₂×#108	—	71	1	no	—
203	C₂×#104	—	71	1	no	—
204	C₂×#104	—	71	1	no	—
205	C₂×#97	—	71	1	no	—
206	#350	—	142	2	no	—
207	#350	—	142	2	no	—
208	#349	—	2	2	no	—
209	C₂×#41	—	71	1	no	—
210	C₂×#41	C₄×D₇₁	1	1	yes	characteristic
211	#135	—	142	2	no	—
212	#135	—	142	2	no	—
213	C₂×Q₁₆	—	71	1	no	—
214	C₂×SD₁₆	—	71	1	no	—
215	C₂×SD₁₆	—	71	1	no	—
216	C₂×D₈	—	71	1	no	—
217	#131	—	2	1	no	—
218	#131	—	2	1	no	—
219	#131	—	2	1	no	—
220	#131	—	2	1	no	—
221	#131	—	71	2	no	—
222	#131	—	71	2	no	—
223	#131	—	142	1	no	—
224	#131	—	142	1	no	—
225	C₂×M₁₆	—	2	1	no	—
226	C₂×M₁₆	—	71	1	no	—
227	C₂×M₁₆	—	142	1	no	—
228	C₂×M₁₆	C₄×D₇₁	1	1	yes	characteristic
229	C₂²×C₈	—	2	1	no	—
230	C₂²×C₈	—	71	1	no	—
231	C₂²×C₈	—	142	1	no	—
232	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
233	C₂×C₄²	—	71	1	no	—
234	M₃₂	—	2	1	no	—
235	M₃₂	—	2	1	no	—
236	M₃₂	—	2	1	no	—
237	M₃₂	—	142	4	no	—
238	M₃₂	—	142	1	no	—
239	M₃₂	—	142	1	no	—
240	M₃₂	—	142	1	no	—
241	M₃₂	—	142	1	no	—
242	M₃₂	C₂²⋊D₇₁	1	2	yes	—
243	C₂×C₁₆	—	2	1	no	—
244	C₂×C₁₆	—	2	1	no	—
245	C₂×C₁₆	—	71	2	no	—
246	C₂×C₁₆	—	142	4	no	—
247	C₂×C₁₆	—	142	1	no	—
248	#108	—	142	2	no	—
249	#104	—	142	2	no	—
250	#104	—	142	2	no	—
251	#97	—	142	1	no	—
252	#97	—	142	1	no	—
253	#41	—	2	1	no	—
254	#41	—	2	1	no	—
255	#41	—	2	1	no	—
256	#41	—	2	1	no	—
257	#41	—	71	2	no	—
258	#41	—	71	2	no	—
259	#41	—	142	1	no	—
260	#41	—	142	1	no	—
261	C₂×Q₈	—	71	1	no	—
262	C₂×Q₈	C₈×D₇₁	1	1	yes	characteristic
263	C₂×D₄	—	2	1	no	—
264	C₂×D₄	—	71	1	no	—
265	C₂×D₄	—	142	1	no	—
266	C₂×D₄	C₈×D₇₁	1	1	yes	characteristic
267	C₂²×C₄	—	2	1	no	—
268	C₂²×C₄	—	142	1	no	—
269	C₂²×C₄	—	142	1	no	—
270	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
271	Q₁₆	—	142	2	no	—
272	SD₁₆	—	142	2	no	—
273	SD₁₆	—	142	2	no	—
274	D₈	—	142	2	no	—
275	M₁₆	—	2	1	no	—
276	M₁₆	—	2	1	no	—
277	M₁₆	—	4	1	no	—
278	M₁₆	—	4	1	no	—
279	M₁₆	—	71	2	no	—
280	M₁₆	—	142	1	no	—
281	M₁₆	—	142	2	no	—
282	M₁₆	—	142	2	no	—
283	C₂×C₈	—	2	1	no	—
284	C₂×C₈	—	4	1	no	—
285	C₂×C₈	—	4	1	no	—
286	C₂×C₈	—	71	1	no	—
287	C₂×C₈	—	71	1	no	—
288	C₂×C₈	—	142	1	no	—
289	C₂×C₈	—	142	1	no	—
290	C₂×C₈	—	142	1	no	—
291	C₂×C₈	—	142	2	no	—
292	C₂×C₈	—	142	2	no	—
293	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
294	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
295	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
296	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
297	C₂×C₈	#11765864	1	1	yes	characteristic
298	C₂×C₈	#11765865	1	1	yes	characteristic
299	C₄²	—	142	1	no	—
300	C₄²	—	142	1	no	—
301	C₁₆	—	2	2	no	—
302	C₁₆	—	4	1	no	—
303	C₁₆	—	284	1	no	—
304	C₁₆	—	142	2	no	—
305	C₂³	—	2	1	no	—
306	C₂³	—	142	1	no	—
307	C₂³	#14556134	1	1	yes	characteristic
308	Q₈	—	2	1	no	—
309	Q₈	—	2	1	no	—
310	Q₈	—	71	2	no	—
311	Q₈	—	142	1	no	—
312	D₄	—	2	1	no	—
313	D₄	—	2	1	no	—
314	D₄	—	4	1	no	—
315	D₄	—	4	1	no	—
316	D₄	—	71	2	no	—
317	D₄	—	142	2	no	—
318	D₄	—	142	2	no	—
319	D₄	—	142	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	—	284	1	no	—
324	C₂×C₄	—	568	1	no	—
325	C₂×C₄	—	142	1	no	—
326	C₂×C₄	—	142	2	no	—
327	C₂×C₄	—	142	2	no	—
328	C₂×C₄	C₂²⋊C₄×D₇₁	1	1	yes	characteristic
329	C₂×C₄	(C₂²⋊D₇₁)⋊C₄	1	1	yes	characteristic
330	C₂×C₄	C₄×C₂²⋊D₇₁	1	1	yes	characteristic
331	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
332	C₂×C₄	M₁₆×D₇₁	1	1	yes	characteristic
333	C₂×C₄	#14556136	1	1	yes	characteristic
334	C₈	—	4	1	no	—
335	C₈	—	142	2	no	—
336	C₈	—	142	1	no	—
337	C₈	—	142	1	no	—
338	C₈	#14556096	1	1	yes	characteristic
339	C₈	#14556099	1	1	yes	characteristic
340	C₈	#14556101	1	1	yes	characteristic
341	C₈	#14556110	1	1	yes	characteristic
342	C₂²	—	4	1	no	—
343	C₂²	—	4	1	no	—
344	C₂²	—	8	1	no	—
345	C₂²	—	568	1	no	—
346	C₂²	—	142	2	no	—
347	C₂²	#17985964	1	1	yes	characteristic
348	C₂²	C₈×C₂²⋊D₇₁	1	1	yes	characteristic
349	C₂²	#17986073	1	1	yes	characteristic
350	C₄	—	4	1	no	—
351	C₄	—	568	1	no	—
352	C₄	—	142	2	no	—
353	C₄	#17985875	1	1	yes	characteristic
354	C₄	C₂²⋊C₈×D₇₁	1	1	yes	characteristic
355	C₄	(C₂²⋊D₇₁)⋊C₈	1	1	yes	characteristic
356	C₄	#17986068	1	1	yes	characteristic
357	C₂	—	2	1	no	—
358	C₂	—	2	1	no	—
359	C₂	—	2	1	no	—
360	C₂	—	8	1	no	—
361	C₂	—	284	2	no	—
362	C₂	#22588629	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#17985875
commutator	C₂×C₄×C₇₁	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#2115	C₂
socle	C₂×C₇₁	#1869
prime radical	C₂×#14556148	C₄
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#2115	D₇₁
hypocenter	C₇₁	#12986
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ #2115

Complements

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

N	H ≅ G/N	type
1. #22601442	C₂	semidirect
3. #22589969	C₂	semidirect
4. #22589968	C₂	semidirect
5. #22586423	C₂	semidirect
6. #22586419	C₂	semidirect
7. C₇₁×#2115	C₂	semidirect
186. #2115	D₇₁	semidirect
189. C₇₁	#12986	semidirect

Maximal subgroups

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

subgroup	index
#22601442	2
C₂×#17986119	2
#22589969	2
#22589968	2
#22586423	2
#22586419	2
C₇₁×#2115	2
#12986	71