Subgroups — #31915617

Order 18176.

442 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	#22597050	C₂	1	1	yes	maximal, characteristic
2	#22590095	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#17985784	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#17985783	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#17985782	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#17985774	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#350×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14556134	C₂²	1	1	yes	characteristic
9	#17986545	C₄	1	1	yes	characteristic
10	C₂×#14556068	C₂²	1	1	yes	characteristic
11	C₂²×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
12	#17986075	—	2	2	no	—
13	C₈×C₂²⋊D₇₁	—	2	2	no	—
14	#17986022	C₄	1	1	yes	characteristic
15	#17985967	C₄	1	1	yes	characteristic
16	#17985966	C₄	1	1	yes	characteristic
17	C₂×C₈×Dic₇₁	C₂²	1	1	yes	characteristic
18	Dic₇₁⋊C₁₆	—	2	2	no	—
19	#17985784	—	2	2	no	—
20	#17985783	—	2	2	no	—
21	#17985782	—	2	2	no	—
22	#17985774	—	2	2	no	—
23	C₂×#131×C₇₁	C₂²	1	1	yes	characteristic
24	C₂×M₃₂×C₇₁	C₂²	1	1	yes	characteristic
25	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
26	#350×C₇₁	—	2	2	no	—
27	C₂×#11765865	C₂×C₄	1	1	yes	characteristic
28	#14556134	—	2	1	no	—
29	#14556134	—	2	1	no	—
30	#14556134	—	2	1	no	—
31	#14556134	—	2	1	no	—
32	C₂×#11765849	C₂×C₄	1	1	yes	characteristic
33	C₂×#11765848	—	2	1	no	—
34	C₂²×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
35	C₂×C₈×D₇₁	—	2	1	no	—
36	#14556126	C₈	1	1	yes	characteristic
37	#14556124	C₈	1	1	yes	characteristic
38	#14556117	C₈	1	1	yes	characteristic
39	#14556115	C₈	1	1	yes	characteristic
40	#14556100	—	2	2	no	—
41	C₄×C₂²⋊D₇₁	—	2	2	no	—
42	C₂×C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
43	#14556068	—	2	1	no	—
44	#14556068	—	2	1	no	—
45	C₂×C₇₁⋊C₁₆	—	2	1	no	—
46	C₂×C₇₁⋊C₁₆	—	2	1	no	—
47	C₂×C₇₁⋊C₁₆	—	2	1	no	—
48	C₂×C₇₁⋊C₁₆	D₄	1	2	yes	complemented
49	Dic₇₁⋊C₈	—	2	2	no	—
50	C₂²⋊(C₇₁⋊C₈)	—	2	2	no	—
51	(C₂×D₇₁)⋊C₈	—	2	2	no	—
52	C₈×Dic₇₁	D₄	1	2	yes	—
53	C₈×Dic₇₁	D₄	1	2	yes	—
54	C₂×#41×C₇₁	C₂×C₄	1	1	yes	characteristic
55	#131×C₇₁	—	2	1	no	—
56	#131×C₇₁	—	2	1	no	—
57	#131×C₇₁	—	2	1	no	—
58	#131×C₇₁	—	2	1	no	—
59	C₂×M₁₆×C₇₁	—	2	1	no	—
60	C₂×M₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
61	C₂²×C₈×C₇₁	—	2	1	no	—
62	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
63	M₃₂×C₇₁	—	2	1	no	—
64	M₃₂×C₇₁	—	2	1	no	—
65	C₂×C₁₆×C₇₁	—	2	1	no	—
66	C₂×C₁₆×C₇₁	—	2	1	no	—
67	C₂×C₁₆×C₇₁	—	2	1	no	—
68	C₂×C₁₆×C₇₁	D₄	1	2	yes	complemented
69	#11765865	—	2	1	no	—
70	#11765865	—	2	1	no	—
71	#11765865	—	2	1	no	—
72	#11765865	—	2	1	no	—
73	C₂×Dic₁₄₂	C₂×C₈	1	1	yes	characteristic
74	C₂×D₂₈₄	C₂×C₈	1	1	yes	characteristic
75	C₂×C₂²⋊D₇₁	—	2	1	no	—
76	C₂×C₂²⋊D₇₁	—	2	2	no	—
77	C₂²×Dic₇₁	M₁₆	1	2	yes	—
78	C₂×C₄×D₇₁	—	2	1	no	—
79	#11765849	—	2	1	no	—
80	#11765849	—	2	1	no	—
81	#11765848	—	4	1	no	—
82	#11765848	—	4	1	no	—
83	C₂×C₇₁⋊C₈	—	2	1	no	—
84	C₂×C₇₁⋊C₈	—	2	1	no	—
85	C₂×C₇₁⋊C₈	—	2	1	no	—
86	C₂×C₇₁⋊C₈	C₂²⋊C₄	1	2	yes	—
87	C₈×D₇₁	—	4	1	no	—
88	C₈×D₇₁	—	4	1	no	—
89	Dic₇₁⋊C₄	—	2	2	no	—
90	C₂²⋊Dic₇₁	—	2	2	no	—
91	(C₂×D₇₁)⋊C₄	—	2	2	no	—
92	C₄×Dic₇₁	C₂²⋊C₄	1	2	yes	—
93	C₄×Dic₇₁	C₂×C₈	1	1	yes	characteristic
94	C₄×Dic₇₁	C₂×C₈	1	1	yes	characteristic
95	C₇₁⋊C₁₆	—	2	2	no	—
96	C₇₁⋊C₁₆	—	4	1	no	—
97	#41×C₇₁	—	2	1	no	—
98	#41×C₇₁	—	2	1	no	—
99	#41×C₇₁	—	2	1	no	—
100	#41×C₇₁	—	2	1	no	—
101	C₂×Q₈×C₇₁	C₂×C₈	1	1	yes	characteristic
102	C₂×D₄×C₇₁	—	2	1	no	—
103	C₂×D₄×C₇₁	C₂×C₈	1	1	yes	characteristic
104	C₂²×C₄×C₇₁	—	2	1	no	—
105	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
106	M₁₆×C₇₁	—	2	1	no	—
107	M₁₆×C₇₁	—	2	1	no	—
108	M₁₆×C₇₁	—	4	1	no	—
109	M₁₆×C₇₁	—	4	1	no	—
110	C₂×C₈×C₇₁	—	2	1	no	—
111	C₂×C₈×C₇₁	—	4	1	no	—
112	C₂×C₈×C₇₁	—	4	1	no	—
113	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
118	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
119	C₁₆×C₇₁	—	2	2	no	—
120	C₁₆×C₇₁	—	4	1	no	—
121	C₂³×C₇₁	—	2	1	no	—
122	C₂³×C₇₁	C₂×M₁₆	1	1	yes	characteristic
123	C₂²×D₇₁	—	2	1	no	—
124	Q₈×C₇₁	—	2	1	no	—
125	Q₈×C₇₁	—	2	1	no	—
126	D₄×C₇₁	—	2	1	no	—
127	D₄×C₇₁	—	2	1	no	—
128	D₄×C₇₁	—	4	1	no	—
129	D₄×C₇₁	—	4	1	no	—
130	C₂×C₄×C₇₁	—	2	1	no	—
131	C₂×C₄×C₇₁	—	4	1	no	—
132	C₂×C₄×C₇₁	—	4	1	no	—
133	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
135	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
137	C₂×C₄×C₇₁	#131	1	1	yes	characteristic
138	C₂×C₄×C₇₁	#131	1	1	yes	characteristic
139	C₂²⋊D₇₁	—	4	4	no	—
140	C₂²⋊D₇₁	—	4	1	no	—
141	C₂²⋊D₇₁	—	4	1	no	—
142	C₂×Dic₇₁	—	2	1	no	—
143	C₂×Dic₇₁	—	4	2	no	—
144	C₂×Dic₇₁	C₂²⋊C₈	1	2	yes	—
145	C₂×Dic₇₁	C₂²⋊C₈	1	2	yes	—
146	D₂₈₄	—	2	1	no	—
147	D₂₈₄	—	2	1	no	—
148	C₄×D₇₁	—	4	1	no	—
149	C₄×D₇₁	—	4	1	no	—
150	Dic₁₄₂	—	2	1	no	—
151	Dic₁₄₂	—	2	1	no	—
152	C₈×C₇₁	—	4	1	no	—
153	C₈×C₇₁	#121	1	1	yes	characteristic
154	C₈×C₇₁	#121	1	1	yes	characteristic
155	C₈×C₇₁	#124	1	1	yes	characteristic
156	C₈×C₇₁	#127	1	1	yes	characteristic
157	C₇₁⋊C₈	—	2	2	no	—
158	C₇₁⋊C₈	—	4	1	no	—
159	C₂²×C₇₁	—	2	2	no	—
160	C₂²×C₇₁	—	4	2	no	—
161	C₂²×C₇₁	—	4	1	no	—
162	C₂²×C₇₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
163	C₂²×C₇₁	#436	1	1	yes	characteristic
164	C₂²×C₇₁	#436	1	1	yes	characteristic
165	C₂×D₇₁	—	4	2	no	—
166	C₂×D₇₁	—	4	1	no	—
167	C₄×C₇₁	—	4	1	no	—
168	C₄×C₇₁	#390	1	1	yes	characteristic
169	C₄×C₇₁	#408	1	1	yes	characteristic
170	C₄×C₇₁	C₈×D₄	1	1	yes	characteristic
171	C₄×C₇₁	C₈×D₄	1	1	yes	characteristic
172	Dic₇₁	—	2	2	no	—
173	Dic₇₁	—	2	2	no	—
174	Dic₇₁	—	4	1	no	—
175	#13192	—	71	1	no	maximal, Sylow
176	C₂×C₇₁	—	2	1	no	—
177	C₂×C₇₁	—	2	1	no	—
178	C₂×C₇₁	—	8	1	no	—
179	C₂×C₇₁	#1954	1	1	yes	characteristic
180	C₂×C₇₁	#2117	1	2	yes	—
181	D₇₁	—	8	1	no	—
182	#2931	—	71	1	no	—
183	#2150	—	71	1	no	—
184	C₂×#350	—	71	1	no	—
185	C₂×#350	—	71	1	no	—
186	C₂×#350	—	71	1	no	—
187	C₂×#350	D₇₁	1	1	yes	characteristic, complemented
188	C₂×#346	—	71	1	no	—
189	C₇₁	#13192	1	1	yes	characteristic, complemented, Sylow
190	C₂×#131	—	71	1	no	—
191	C₂×#131	C₂×D₇₁	1	1	yes	characteristic
192	#532	—	71	1	no	—
193	C₂×M₃₂	—	71	1	no	—
194	C₂×M₃₂	C₂×D₇₁	1	1	yes	characteristic
195	C₂²×C₁₆	—	71	1	no	—
196	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
197	#436	—	142	2	no	—
198	C₈×D₄	—	142	2	no	—
199	#423	—	71	1	no	—
200	#408	—	71	1	no	—
201	#408	—	71	1	no	—
202	C₂×C₄×C₈	—	71	1	no	—
203	C₄⋊C₁₆	—	142	2	no	—
204	#350	—	2	2	no	—
205	#350	—	142	2	no	—
206	#350	—	142	2	no	—
207	#350	—	142	2	no	—
208	#346	—	142	2	no	—
209	C₂×#41	—	71	1	no	—
210	C₂×#41	C₄×D₇₁	1	1	yes	characteristic
211	#131	—	2	1	no	—
212	#131	—	2	1	no	—
213	#131	—	2	1	no	—
214	#131	—	2	1	no	—
215	#131	—	142	1	no	—
216	#131	—	142	1	no	—
217	#131	—	142	1	no	—
218	#131	—	142	1	no	—
219	C₂×M₁₆	—	2	1	no	—
220	C₂×M₁₆	—	71	1	no	—
221	C₂×M₁₆	—	142	1	no	—
222	C₂×M₁₆	C₄×D₇₁	1	1	yes	characteristic
223	C₂²×C₈	—	2	1	no	—
224	C₂²×C₈	—	71	1	no	—
225	C₂²×C₈	—	142	1	no	—
226	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
227	C₄⋊Q₈	—	71	1	no	—
228	#127	—	71	1	no	—
229	#124	—	71	1	no	—
230	#124	—	71	1	no	—
231	#121	—	142	2	no	—
232	C₄×D₄	—	142	2	no	—
233	C₂×C₄²	—	71	1	no	—
234	M₃₂	—	2	1	no	—
235	M₃₂	—	2	1	no	—
236	M₃₂	—	142	1	no	—
237	M₃₂	—	142	1	no	—
238	C₂×C₁₆	—	2	1	no	—
239	C₂×C₁₆	—	2	1	no	—
240	C₂×C₁₆	—	2	1	no	—
241	C₂×C₁₆	—	71	2	no	—
242	C₂×C₁₆	—	142	1	no	—
243	C₂×C₁₆	—	142	1	no	—
244	C₂×C₁₆	—	142	1	no	—
245	C₂×C₁₆	C₂²⋊D₇₁	1	2	yes	complemented
246	C₄⋊C₈	—	142	2	no	—
247	C₂²⋊C₈	—	142	2	no	—
248	C₂²⋊C₈	—	142	2	no	—
249	C₄×C₈	—	71	2	no	—
250	C₄×C₈	—	71	2	no	—
251	#41	—	2	1	no	—
252	#41	—	2	1	no	—
253	#41	—	2	1	no	—
254	#41	—	2	1	no	—
255	#41	—	142	1	no	—
256	#41	—	142	1	no	—
257	#41	—	142	1	no	—
258	#41	—	142	1	no	—
259	C₂×Q₈	—	71	1	no	—
260	C₂×Q₈	C₈×D₇₁	1	1	yes	characteristic
261	C₂×D₄	—	2	1	no	—
262	C₂×D₄	—	71	1	no	—
263	C₂×D₄	—	142	2	no	—
264	C₂×D₄	—	142	1	no	—
265	C₂×D₄	C₈×D₇₁	1	1	yes	characteristic
266	C₂²×C₄	—	2	1	no	—
267	C₂²×C₄	—	71	2	no	—
268	C₂²×C₄	—	142	1	no	—
269	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
270	M₁₆	—	2	1	no	—
271	M₁₆	—	2	1	no	—
272	M₁₆	—	4	1	no	—
273	M₁₆	—	4	1	no	—
274	M₁₆	—	284	1	no	—
275	M₁₆	—	284	1	no	—
276	M₁₆	—	142	1	no	—
277	M₁₆	—	142	1	no	—
278	C₂×C₈	—	2	1	no	—
279	C₂×C₈	—	4	1	no	—
280	C₂×C₈	—	4	1	no	—
281	C₂×C₈	—	284	1	no	—
282	C₂×C₈	—	284	1	no	—
283	C₂×C₈	—	71	2	no	—
284	C₂×C₈	—	142	1	no	—
285	C₂×C₈	—	142	1	no	—
286	C₂×C₈	—	142	1	no	—
287	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
288	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
289	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
290	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
291	C₂×C₈	#11765864	1	1	yes	characteristic
292	C₂×C₈	#11765865	1	1	yes	characteristic
293	C₄⋊C₄	—	142	2	no	—
294	C₂²⋊C₄	—	142	2	no	—
295	C₂²⋊C₄	—	142	2	no	—
296	C₄²	—	71	2	no	—
297	C₄²	—	71	1	no	—
298	C₄²	—	71	1	no	—
299	C₁₆	—	2	2	no	—
300	C₁₆	—	4	1	no	—
301	C₁₆	—	284	1	no	—
302	C₁₆	—	142	2	no	—
303	C₂³	—	2	1	no	—
304	C₂³	—	142	1	no	—
305	C₂³	M₁₆×D₇₁	1	1	yes	characteristic
306	Q₈	—	2	1	no	—
307	Q₈	—	2	1	no	—
308	Q₈	—	142	1	no	—
309	Q₈	—	142	1	no	—
310	D₄	—	2	1	no	—
311	D₄	—	2	1	no	—
312	D₄	—	4	1	no	—
313	D₄	—	4	1	no	—
314	D₄	—	284	1	no	—
315	D₄	—	284	1	no	—
316	D₄	—	284	4	no	—
317	D₄	—	142	1	no	—
318	D₄	—	142	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	284	1	no	—
323	C₂×C₄	—	284	1	no	—
324	C₂×C₄	—	284	2	no	—
325	C₂×C₄	—	71	2	no	—
326	C₂×C₄	—	71	2	no	—
327	C₂×C₄	—	142	1	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	#14556134	1	1	yes	characteristic
333	C₂×C₄	#14556136	1	1	yes	characteristic
334	C₈	—	4	1	no	—
335	C₈	—	284	1	no	—
336	C₈	—	142	2	no	—
337	C₈	#14556100	1	1	yes	characteristic
338	C₈	#14556101	1	1	yes	characteristic
339	C₈	#14556114	1	1	yes	characteristic
340	C₈	#14556124	1	1	yes	characteristic
341	C₂²	—	2	2	no	—
342	C₂²	—	4	2	no	—
343	C₂²	—	4	1	no	—
344	C₂²	—	284	2	no	—
345	C₂²	—	284	1	no	—
346	C₂²	C₂²⋊C₈×D₇₁	1	1	yes	characteristic
347	C₂²	#17986074	1	1	yes	characteristic
348	C₂²	#17986075	1	1	yes	characteristic
349	C₄	—	4	1	no	—
350	C₄	—	284	1	no	—
351	C₄	—	142	2	no	—
352	C₄	—	142	2	no	—
353	C₄	#17985899	1	1	yes	characteristic
354	C₄	#17985964	1	1	yes	characteristic
355	C₄	(C₂²⋊D₇₁)⋊C₈	1	1	yes	characteristic
356	C₄	C₈×C₂²⋊D₇₁	1	1	yes	characteristic
357	C₂	—	2	1	no	—
358	C₂	—	2	1	no	—
359	C₂	—	8	1	no	—
360	C₂	—	568	1	no	—
361	C₂	#22589132	1	1	yes	characteristic
362	C₂	#22589966	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₂×#350×C₇₁	C₂
socle	C₂²×C₇₁	C₂×C₂²⋊C₈
prime radical	#17986545	C₄
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂×#350	D₇₁
hypocenter	C₇₁	#13192
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#350

Complements

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

N	H ≅ G/N	type
2. #22590095	C₂	semidirect
3. C₂×#17985784	C₂	semidirect
4. C₂×#17985783	C₂	semidirect
5. C₂×#17985782	C₂	semidirect
6. C₂×#17985774	C₂	semidirect
7. C₂×#350×C₇₁	C₂	semidirect
48. C₂×C₇₁⋊C₁₆	D₄	semidirect
68. C₂×C₁₆×C₇₁	D₄	semidirect
187. C₂×#350	D₇₁	semidirect
189. C₇₁	#13192	semidirect
245. C₂×C₁₆	C₂²⋊D₇₁	semidirect

Maximal subgroups

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

subgroup	index
#22597050	2
#22590095	2
C₂×#17985784	2
C₂×#17985783	2
C₂×#17985782	2
C₂×#17985774	2
C₂×#350×C₇₁	2
#13192	71