Subgroups — #31888493

Order 18176.

444 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22597315	C₂	1	1	yes	maximal, characteristic
2	#22588026	C₂	1	1	yes	maximal, characteristic, complemented
3	#22586929	C₂	1	1	yes	maximal, characteristic
4	#22586827	C₂	1	1	yes	maximal, characteristic
5	#22586240	C₂	1	1	yes	maximal, characteristic, complemented
6	#22586234	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×M₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#17986533	C₂²	1	1	yes	characteristic
9	#17986129	C₄	1	1	yes	characteristic, complemented
10	C₈×Dic₁₄₂	C₄	1	1	yes	characteristic, complemented
11	#17986078	C₄	1	1	yes	characteristic, complemented
12	#17986070	—	2	1	no	—
13	C₈×D₂₈₄	C₄	1	1	yes	characteristic, complemented
14	C₈×C₂²⋊D₇₁	—	2	1	no	—
15	#17986058	—	2	1	no	—
16	#17986056	C₂²	1	1	yes	characteristic
17	#17985953	—	2	1	no	—
18	C₂×#14556039	C₂²	1	1	yes	characteristic
19	C₂×#14556038	—	2	1	no	—
20	C₂×C₈×Dic₇₁	—	2	1	no	—
21	#17985845	C₂²	1	1	yes	characteristic
22	#17985740	C₄	1	2	yes	complemented
23	#17985738	C₄	1	2	yes	complemented
24	#17985737	C₄	1	2	yes	—
25	#17985734	C₄	1	2	yes	—
26	#431×C₇₁	C₂²	1	1	yes	characteristic
27	C₄×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
29	#322×C₇₁	—	2	1	no	—
30	C₈²×C₇₁	—	2	1	no	—
31	C₂×#11765848	—	2	1	no	—
32	C₂²×C₇₁⋊C₈	—	2	1	no	—
33	C₂×C₈×D₇₁	—	2	1	no	—
34	#14556122	—	2	1	no	—
35	#14556120	C₂×C₄	1	1	yes	characteristic
36	#14556116	C₂×C₄	1	1	yes	characteristic
37	#14556111	—	2	1	no	—
38	#14556107	C₂×C₄	1	1	yes	characteristic
39	#14556102	C₂×C₄	1	1	yes	characteristic
40	C₄×Dic₁₄₂	C₂×C₄	1	1	yes	characteristic
41	C₄×D₂₈₄	C₂×C₄	1	1	yes	characteristic
42	C₄×C₂²⋊D₇₁	—	2	1	no	—
43	#14556085	—	2	1	no	—
44	C₂×C₄×Dic₇₁	—	2	1	no	—
45	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	1	yes	characteristic
46	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	1	yes	characteristic
47	Dic₇₁⋊C₈	—	2	1	no	—
48	Dic₇₁⋊C₈	—	2	1	no	—
49	C₂²⋊(C₇₁⋊C₈)	—	2	1	no	—
50	(C₂×D₇₁)⋊C₈	—	2	1	no	—
51	(C₂×D₇₁)⋊C₈	—	2	1	no	—
52	#14556039	—	2	1	no	—
53	#14556039	—	2	1	no	—
54	#14556039	C₂×C₄	1	2	yes	—
55	#14556038	—	2	4	no	—
56	#14556038	—	2	1	no	—
57	C₈×Dic₇₁	—	2	4	no	—
58	C₈×Dic₇₁	—	2	1	no	—
59	C₇₁⋊#95	—	2	1	no	—
60	C₇₁⋊#95	C₂×C₄	1	2	yes	—
61	C₂×M₁₆×C₇₁	—	2	1	no	—
62	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
63	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
64	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
65	C₄⋊C₈×C₇₁	—	2	1	no	—
66	C₂²⋊C₈×C₇₁	C₂×C₄	1	2	yes	—
67	#97×C₇₁	C₂×C₄	1	2	yes	—
68	C₄×C₈×C₇₁	—	2	1	no	—
69	C₄×C₈×C₇₁	—	2	1	no	—
70	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
71	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
72	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
73	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
74	C₂×Dic₁₄₂	C₄²	1	1	yes	characteristic
75	C₂×D₂₈₄	C₄²	1	1	yes	characteristic
76	C₂×C₂²⋊D₇₁	—	2	1	no	—
77	C₂²×Dic₇₁	—	2	2	no	—
78	C₂×C₄×D₇₁	—	2	1	no	—
79	#11765848	—	4	2	no	—
80	C₂×C₇₁⋊C₈	—	2	2	no	—
81	C₂×C₇₁⋊C₈	—	2	2	no	—
82	C₂×C₇₁⋊C₈	—	2	2	no	—
83	C₂×C₇₁⋊C₈	—	2	1	no	—
84	C₂×C₇₁⋊C₈	—	2	1	no	—
85	C₈×D₇₁	—	4	2	no	—
86	C₄⋊Dic₇₁	C₄²	1	1	yes	characteristic
87	C₄⋊Dic₇₁	C₄²	1	1	yes	characteristic
88	Dic₇₁⋊C₄	—	2	1	no	—
89	Dic₇₁⋊C₄	—	2	1	no	—
90	Dic₇₁⋊C₄	—	2	1	no	—
91	Dic₇₁⋊C₄	—	2	1	no	—
92	C₂²⋊Dic₇₁	—	2	1	no	—
93	(C₂×D₇₁)⋊C₄	—	2	1	no	—
94	(C₂×D₇₁)⋊C₄	—	2	1	no	—
95	(C₂×D₇₁)⋊C₄	—	2	1	no	—
96	(C₂×D₇₁)⋊C₄	—	2	1	no	—
97	C₄×Dic₇₁	—	2	2	no	—
98	C₄×Dic₇₁	—	2	2	no	—
99	C₄×Dic₇₁	—	2	1	no	—
100	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
101	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
103	M₁₆×C₇₁	—	2	4	no	—
104	C₂×C₈×C₇₁	—	2	2	no	—
105	C₂×C₈×C₇₁	—	2	2	no	—
106	C₂×C₈×C₇₁	—	2	2	no	—
107	C₂×C₈×C₇₁	—	2	2	no	—
108	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
112	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
113	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₇₁	#41	1	1	yes	characteristic
116	C₂³×C₇₁	C₄×D₄	1	1	yes	characteristic
117	C₂²×D₇₁	—	2	1	no	—
118	C₂×C₄×C₇₁	—	2	2	no	—
119	C₂×C₄×C₇₁	—	2	1	no	—
120	C₂×C₄×C₇₁	—	2	1	no	—
121	C₂×C₄×C₇₁	—	4	1	no	—
122	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
123	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
125	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
126	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
127	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
128	C₂²⋊D₇₁	—	4	2	no	—
129	C₂×Dic₇₁	—	2	4	no	—
130	C₂×Dic₇₁	—	2	1	no	—
131	C₂×Dic₇₁	—	2	1	no	—
132	C₂×Dic₇₁	—	2	1	no	—
133	C₂×Dic₇₁	—	4	2	no	—
134	C₂×Dic₇₁	—	4	2	no	—
135	D₂₈₄	—	4	1	no	—
136	C₄×D₇₁	—	4	1	no	—
137	C₄×D₇₁	—	4	1	no	—
138	Dic₁₄₂	—	4	1	no	—
139	C₈×C₇₁	—	2	4	no	—
140	C₈×C₇₁	—	2	1	no	—
141	C₈×C₇₁	—	2	1	no	—
142	C₈×C₇₁	—	4	2	no	—
143	C₈×C₇₁	#104	1	2	yes	—
144	C₈×C₇₁	#104	1	2	yes	—
145	C₇₁⋊C₈	—	2	4	no	—
146	C₇₁⋊C₈	—	4	1	no	—
147	C₇₁⋊C₈	—	4	1	no	—
148	C₂²×C₇₁	—	2	1	no	—
149	C₂²×C₇₁	—	2	1	no	—
150	C₂²×C₇₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
151	C₂²×C₇₁	#443	1	2	yes	—
152	C₂×D₇₁	—	4	1	no	—
153	C₂×D₇₁	—	4	1	no	—
154	C₂×D₇₁	—	4	1	no	—
155	C₄×C₇₁	—	2	1	no	—
156	C₄×C₇₁	—	2	1	no	—
157	C₄×C₇₁	—	2	1	no	—
158	C₄×C₇₁	—	2	1	no	—
159	C₄×C₇₁	—	2	1	no	—
160	C₄×C₇₁	C₂×#104	1	1	yes	characteristic
161	C₄×C₇₁	C₂×#104	1	1	yes	characteristic
162	Dic₇₁	—	4	4	no	—
163	Dic₇₁	—	4	1	no	—
164	Dic₇₁	—	4	1	no	—
165	Dic₇₁	—	4	1	no	—
166	#8247	—	71	1	no	maximal, Sylow
167	C₂×C₇₁	—	2	2	no	—
168	C₂×C₇₁	#1757	1	1	yes	characteristic
169	C₂×C₇₁	C₄×#104	1	1	yes	characteristic
170	C₂×C₇₁	#1764	1	1	yes	characteristic
171	D₇₁	—	8	1	no	—
172	#2966	—	71	1	no	—
173	#1726	—	71	1	no	—
174	#1524	—	71	1	no	—
175	#1500	—	71	1	no	—
176	C₈×M₁₆	D₇₁	1	1	yes	characteristic, complemented
177	#1302	—	71	1	no	—
178	#1300	—	71	1	no	—
179	C₇₁	#8247	1	1	yes	characteristic, complemented, Sylow
180	#529	—	71	1	no	—
181	#446	—	71	1	no	—
182	C₈×Q₈	—	71	1	no	—
183	#436	—	71	1	no	—
184	#435	—	142	1	no	—
185	C₈×D₄	—	71	1	no	—
186	C₈×D₄	—	142	1	no	—
187	#433	—	142	1	no	—
188	#432	—	71	1	no	—
189	#431	C₂×D₇₁	1	1	yes	characteristic
190	#405	—	142	1	no	—
191	C₄×M₁₆	C₂×D₇₁	1	1	yes	characteristic
192	C₂×#97	—	71	1	no	—
193	C₂×#97	—	142	1	no	—
194	C₂×C₄×C₈	—	142	1	no	—
195	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
196	#376	—	71	1	no	—
197	#330	—	71	2	no	—
198	#329	—	71	2	no	—
199	#328	—	71	2	no	—
200	#327	—	71	2	no	—
201	#322	—	2	1	no	—
202	C₈²	—	2	1	no	—
203	C₂×M₁₆	—	2	1	no	—
204	C₂×M₁₆	—	142	1	no	—
205	C₂²×C₈	—	142	1	no	—
206	C₂²×C₈	—	142	1	no	—
207	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
208	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
209	#126	—	142	1	no	—
210	#125	—	71	1	no	—
211	#124	—	71	1	no	—
212	#123	—	142	1	no	—
213	C₂²⋊Q₈	—	71	1	no	—
214	#121	—	71	1	no	—
215	C₄×Q₈	—	71	1	no	—
216	C₄×D₄	—	71	1	no	—
217	C₄×D₄	—	142	1	no	—
218	#117	—	142	1	no	—
219	C₂×C₄²	—	142	1	no	—
220	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
221	C₄⋊C₈	—	2	1	no	—
222	C₄⋊C₈	—	71	1	no	—
223	C₄⋊C₈	—	71	1	no	—
224	C₄⋊C₈	—	142	1	no	—
225	C₄⋊C₈	—	142	1	no	—
226	C₂²⋊C₈	—	142	1	no	—
227	C₂²⋊C₈	—	142	1	no	—
228	C₂²⋊C₈	—	142	1	no	—
229	C₂²⋊C₈	C₄×D₇₁	1	2	yes	—
230	#97	—	71	2	no	—
231	#97	—	142	4	no	—
232	#97	—	142	1	no	—
233	#97	—	142	1	no	—
234	#97	—	142	1	no	—
235	#97	C₄×D₇₁	1	2	yes	—
236	C₄×C₈	—	2	1	no	—
237	C₄×C₈	—	2	1	no	—
238	C₄×C₈	—	142	4	no	—
239	C₄×C₈	—	142	1	no	—
240	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
241	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
242	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
243	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
244	#95	—	71	2	no	—
245	#95	—	142	1	no	—
246	C₂×Q₈	—	71	1	no	—
247	C₂×D₄	—	71	1	no	—
248	C₂×D₄	—	142	1	no	—
249	C₂²×C₄	—	142	2	no	—
250	C₂²×C₄	—	142	1	no	—
251	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
252	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
253	C₂²×C₄	#11765865	1	1	yes	characteristic
254	M₁₆	—	2	4	no	—
255	M₁₆	—	284	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	284	2	no	—
261	C₂×C₈	—	142	1	no	—
262	C₂×C₈	—	142	1	no	—
263	C₂×C₈	—	142	2	no	—
264	C₂×C₈	—	142	2	no	—
265	C₂×C₈	—	142	2	no	—
266	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
267	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
268	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
269	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
270	C₄⋊C₄	—	71	1	no	—
271	C₄⋊C₄	—	71	1	no	—
272	C₄⋊C₄	—	142	1	no	—
273	C₄⋊C₄	—	142	1	no	—
274	C₄⋊C₄	—	142	1	no	—
275	C₄⋊C₄	—	142	1	no	—
276	C₂²⋊C₄	—	142	1	no	—
277	C₂²⋊C₄	—	142	1	no	—
278	C₂²⋊C₄	—	142	1	no	—
279	C₂²⋊C₄	—	142	1	no	—
280	C₂²⋊C₄	—	142	1	no	—
281	C₄²	—	142	2	no	—
282	C₄²	—	142	2	no	—
283	C₄²	—	142	1	no	—
284	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
285	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
286	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
287	C₄²	#11765865	1	1	yes	characteristic
288	C₂³	—	142	1	no	—
289	C₂³	C₄×D₂₈₄	1	1	yes	characteristic
290	Q₈	—	284	1	no	—
291	D₄	—	284	1	no	—
292	D₄	—	284	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	284	2	no	—
298	C₂×C₄	—	284	2	no	—
299	C₂×C₄	—	284	1	no	—
300	C₂×C₄	—	284	1	no	—
301	C₂×C₄	—	142	4	no	—
302	C₂×C₄	—	142	1	no	—
303	C₂×C₄	—	142	1	no	—
304	C₂×C₄	—	142	1	no	—
305	C₂×C₄	C₄²×D₇₁	1	1	yes	characteristic
306	C₂×C₄	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
307	C₂×C₄	#14556085	1	1	yes	characteristic
308	C₂×C₄	C₄×C₂²⋊D₇₁	1	1	yes	characteristic
309	C₂×C₄	C₄×C₂²⋊D₇₁	1	1	yes	characteristic
310	C₂×C₄	C₄×D₂₈₄	1	1	yes	characteristic
311	C₈	—	2	4	no	—
312	C₈	—	2	1	no	—
313	C₈	—	2	1	no	—
314	C₈	—	4	2	no	—
315	C₈	—	284	1	no	—
316	C₈	—	284	1	no	—
317	C₈	—	142	4	no	—
318	C₈	#14556054	1	2	yes	—
319	C₈	#14556054	1	2	yes	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	284	1	no	—
323	C₂²	—	284	1	no	—
324	C₂²	—	284	1	no	—
325	C₂²	C₄×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
326	C₂²	#17986117	1	2	yes	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	284	4	no	—
333	C₄	—	284	1	no	—
334	C₄	—	284	1	no	—
335	C₄	—	284	1	no	—
336	C₄	C₂×#14556054	1	1	yes	characteristic
337	C₄	C₂×#14556054	1	1	yes	characteristic
338	C₂	—	2	2	no	—
339	C₂	—	568	1	no	—
340	C₂	#22588141	1	1	yes	characteristic
341	C₂	C₄×#14556054	1	1	yes	characteristic
342	C₂	#22588177	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	(C₂×D₇₁)⋊C₄
commutator	C₂×C₄×C₇₁	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₈×M₁₆×C₇₁	C₂
socle	C₂²×C₇₁	C₄×C₂²⋊C₄
prime radical	#14556102	C₂×C₄
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	C₈×M₁₆	D₇₁
hypocenter	C₇₁	#8247
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
2. #22588026	C₂	semidirect
5. #22586240	C₂	semidirect
6. #22586234	C₂	semidirect
7. C₈×M₁₆×C₇₁	C₂	semidirect
9. #17986129	C₄	semidirect
10. C₈×Dic₁₄₂	C₄	semidirect
11. #17986078	C₄	semidirect
13. C₈×D₂₈₄	C₄	semidirect
22. #17985740	C₄	semidirect
23. #17985738	C₄	semidirect
176. C₈×M₁₆	D₇₁	semidirect
179. C₇₁	#8247	semidirect

Maximal subgroups

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

subgroup	index
#22597315	2
#22588026	2
#22586929	2
#22586827	2
#22586240	2
#22586234	2
C₈×M₁₆×C₇₁	2
#8247	71