Subgroups — #31890622

Order 18176.

454 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#11765865	C₂	1	1	yes	maximal, characteristic
2	#22588361	C₂	1	2	yes	maximal
3	#22587099	C₂	1	1	yes	maximal, characteristic, complemented
4	#22586243	C₂	1	2	yes	maximal, complemented
5	C₄×#107×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#11765865	C₂²	1	1	yes	characteristic
7	C₈×Dic₁₄₂	C₄	1	1	yes	characteristic, complemented
8	C₈×Dic₁₄₂	C₄	1	1	yes	characteristic, complemented
9	C₈×D₂₈₄	C₄	1	1	yes	characteristic, complemented
10	C₈×D₂₈₄	C₄	1	1	yes	characteristic, complemented
11	C₈×C₂²⋊D₇₁	—	2	1	no	—
12	C₈×C₂²⋊D₇₁	—	2	1	no	—
13	#17986052	C₂²	1	1	yes	characteristic
14	#17986050	—	2	1	no	—
15	#17986023	C₂²	1	2	yes	—
16	C₂×#14556056	—	2	2	no	—
17	C₄×C₈×D₇₁	—	2	1	no	—
18	#17985759	—	2	2	no	—
19	#17985752	—	2	1	no	—
20	#17985752	—	2	1	no	—
21	C₂×#107×C₇₁	—	2	1	no	—
22	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
23	C₄×C₄⋊C₄×C₇₁	C₂²	1	2	yes	—
24	#340×C₇₁	—	2	1	no	—
25	C₂×#11765865	D₄	1	1	yes	characteristic
26	C₂×#11765849	—	2	2	no	—
27	C₂×C₈×D₇₁	—	2	1	no	—
28	C₂×C₈×D₇₁	—	2	1	no	—
29	C₄×Dic₁₄₂	C₂×C₄	1	1	yes	characteristic
30	C₄×Dic₁₄₂	Q₈	1	1	yes	characteristic
31	C₄×D₂₈₄	C₂×C₄	1	1	yes	characteristic
32	C₄×D₂₈₄	Q₈	1	1	yes	characteristic
33	C₄×C₂²⋊D₇₁	—	2	1	no	—
34	C₄×C₂²⋊D₇₁	—	2	1	no	—
35	#14556087	D₄	1	1	yes	characteristic
36	#14556085	—	2	1	no	—
37	C₄²×D₇₁	—	2	1	no	—
38	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
39	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
40	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	1	yes	characteristic
41	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	1	yes	characteristic
42	Dic₇₁⋊C₈	—	2	1	no	—
43	Dic₇₁⋊C₈	—	2	1	no	—
44	#14556056	—	4	4	no	—
45	C₂²⋊(C₇₁⋊C₈)	—	2	1	no	—
46	(C₂×D₇₁)⋊C₈	—	2	1	no	—
47	(C₂×D₇₁)⋊C₈	—	2	1	no	—
48	C₄×C₇₁⋊C₈	—	2	1	no	—
49	C₈×Dic₇₁	—	2	1	no	—
50	C₈×Dic₇₁	—	2	1	no	—
51	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
52	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
54	C₂×C₄²×C₇₁	—	2	2	no	—
55	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
56	#107×C₇₁	—	2	4	no	—
57	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
60	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
61	#95×C₇₁	—	2	2	no	—
62	#11765865	—	2	1	no	—
63	#11765865	—	2	1	no	—
64	#11765865	D₈	1	2	yes	—
65	#11765865	Q₁₆	1	2	yes	—
66	C₂×Dic₁₄₂	C₄⋊C₄	1	1	yes	characteristic
67	C₂×D₂₈₄	C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₂²⋊D₇₁	—	2	1	no	—
69	C₂×C₄×D₇₁	—	2	1	no	—
70	C₂×C₄×D₇₁	—	2	1	no	—
71	C₂×C₄×D₇₁	—	2	1	no	—
72	#11765849	—	4	2	no	—
73	#11765849	—	4	2	no	—
74	C₂×C₇₁⋊C₈	—	2	1	no	—
75	C₂×C₇₁⋊C₈	—	2	1	no	—
76	C₂×C₇₁⋊C₈	—	4	2	no	—
77	C₈×D₇₁	—	4	2	no	—
78	C₈×D₇₁	—	4	2	no	—
79	C₄⋊Dic₇₁	C₄⋊C₄	1	1	yes	characteristic
80	C₄⋊Dic₇₁	C₄⋊C₄	1	1	yes	characteristic
81	Dic₇₁⋊C₄	—	2	1	no	—
82	Dic₇₁⋊C₄	—	2	1	no	—
83	C₂²⋊Dic₇₁	—	2	1	no	—
84	(C₂×D₇₁)⋊C₄	—	2	1	no	—
85	(C₂×D₇₁)⋊C₄	—	2	1	no	—
86	C₄×Dic₇₁	—	2	1	no	—
87	C₄×Dic₇₁	—	2	1	no	—
88	C₄×Dic₇₁	—	2	1	no	—
89	C₂²×C₄×C₇₁	—	4	2	no	—
90	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
93	C₂×C₈×C₇₁	—	2	2	no	—
94	C₂×C₈×C₇₁	—	4	1	no	—
95	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
99	C₄⋊C₄×C₇₁	—	2	4	no	—
100	C₄⋊C₄×C₇₁	—	4	2	no	—
101	C₄²×C₇₁	—	4	2	no	—
102	C₄²×C₇₁	—	4	2	no	—
103	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
104	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
106	C₄²×C₇₁	#41	1	1	yes	characteristic
107	C₂³×C₇₁	#121	1	1	yes	characteristic
108	C₂²×D₇₁	—	2	1	no	—
109	C₂×C₄×C₇₁	—	2	1	no	—
110	C₂×C₄×C₇₁	—	2	1	no	—
111	C₂×C₄×C₇₁	—	4	1	no	—
112	C₂×C₄×C₇₁	—	4	4	no	—
113	C₂×C₄×C₇₁	—	8	2	no	—
114	C₂×C₄×C₇₁	—	8	2	no	—
115	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
118	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
119	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
121	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
123	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
124	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
125	C₂²⋊D₇₁	—	2	2	no	—
126	C₂²⋊D₇₁	—	2	2	no	—
127	C₂×Dic₇₁	—	2	1	no	—
128	C₂×Dic₇₁	—	2	1	no	—
129	C₂×Dic₇₁	—	2	1	no	—
130	D₂₈₄	—	2	1	no	—
131	D₂₈₄	#107	1	2	yes	complemented
132	C₄×D₇₁	—	2	2	no	—
133	C₄×D₇₁	—	2	2	no	—
134	C₄×D₇₁	—	4	2	no	—
135	C₄×D₇₁	—	4	1	no	—
136	C₄×D₇₁	—	4	1	no	—
137	Dic₁₄₂	—	2	1	no	—
138	Dic₁₄₂	#107	1	2	yes	complemented
139	C₈×C₇₁	—	2	1	no	—
140	C₈×C₇₁	—	2	1	no	—
141	C₈×C₇₁	—	2	1	no	—
142	C₈×C₇₁	#104	1	2	yes	complemented
143	C₇₁⋊C₈	—	4	1	no	—
144	C₇₁⋊C₈	—	4	1	no	—
145	C₇₁⋊C₈	—	8	2	no	—
146	C₂²×C₇₁	—	2	1	no	—
147	C₂²×C₇₁	—	2	1	no	—
148	C₂²×C₇₁	#380	1	1	yes	characteristic
149	C₂²×C₇₁	#466	1	1	yes	characteristic
150	C₂²×C₇₁	#467	1	1	yes	characteristic
151	C₂×D₇₁	—	2	2	no	—
152	C₂×D₇₁	—	4	1	no	—
153	C₂×D₇₁	—	4	1	no	—
154	C₄×C₇₁	—	2	1	no	—
155	C₄×C₇₁	—	2	1	no	—
156	C₄×C₇₁	—	2	1	no	—
157	C₄×C₇₁	—	8	4	no	—
158	C₄×C₇₁	C₂×#104	1	1	yes	characteristic
159	C₄×C₇₁	#421	1	1	yes	characteristic
160	C₄×C₇₁	C₂×#107	1	1	yes	characteristic
161	C₄×C₇₁	#427	1	1	yes	characteristic
162	C₄×C₇₁	#466	1	1	yes	characteristic
163	C₄×C₇₁	#467	1	1	yes	characteristic
164	Dic₇₁	—	2	2	no	—
165	Dic₇₁	—	4	1	no	—
166	Dic₇₁	—	4	1	no	—
167	#8667	—	71	1	no	maximal, Sylow
168	C₂×C₇₁	—	2	1	no	—
169	C₂×C₇₁	—	2	1	no	—
170	C₂×C₇₁	#1806	1	1	yes	characteristic
171	C₂×C₇₁	#1808	1	1	yes	characteristic
172	C₂×C₇₁	#1814	1	1	yes	characteristic
173	D₇₁	—	4	2	no	—
174	C₈×#41	—	71	1	no	—
175	#1806	—	71	2	no	—
176	C₄×#107	D₇₁	1	1	yes	characteristic, complemented
177	#1567	—	71	1	no	—
178	#1303	—	71	2	no	—
179	C₇₁	#8667	1	1	yes	characteristic, complemented, Sylow
180	C₄×#41	—	71	1	no	—
181	C₈×Q₈	—	71	1	no	—
182	C₈×Q₈	—	71	1	no	—
183	C₈×D₄	—	71	1	no	—
184	C₈×D₄	—	71	1	no	—
185	C₈×D₄	—	142	1	no	—
186	C₈×D₄	—	142	1	no	—
187	#431	—	71	1	no	—
188	#431	—	142	1	no	—
189	C₂×#107	—	2	1	no	—
190	#423	—	71	2	no	—
191	C₂×#104	—	142	2	no	—
192	C₂×C₄×C₈	—	142	1	no	—
193	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
194	C₄×C₄⋊C₄	C₂×D₇₁	1	2	yes	—
195	#340	—	2	1	no	—
196	#339	—	142	2	no	—
197	#335	—	142	1	no	—
198	#335	—	142	1	no	—
199	C₂×#41	—	71	1	no	—
200	C₂×M₁₆	—	142	2	no	—
201	C₂²×C₈	—	142	1	no	—
202	C₂²×C₈	—	142	1	no	—
203	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
204	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
205	C₄×Q₈	—	71	1	no	—
206	C₄×Q₈	—	71	1	no	—
207	C₄×D₄	—	71	1	no	—
208	C₄×D₄	—	71	1	no	—
209	C₄×D₄	—	142	1	no	—
210	C₄×D₄	—	142	1	no	—
211	#117	—	71	1	no	—
212	#117	—	142	1	no	—
213	C₂×C₄⋊C₄	—	2	2	no	—
214	C₂×C₄²	—	2	2	no	—
215	C₂×C₄²	—	142	1	no	—
216	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
217	#107	—	2	4	no	—
218	C₄⋊C₈	—	71	1	no	—
219	C₄⋊C₈	—	71	1	no	—
220	C₄⋊C₈	—	142	1	no	—
221	C₄⋊C₈	—	142	1	no	—
222	C₄⋊C₈	—	142	2	no	—
223	C₄⋊C₈	—	142	2	no	—
224	#104	—	284	4	no	—
225	C₂²⋊C₈	—	142	1	no	—
226	C₂²⋊C₈	—	142	1	no	—
227	C₂²⋊C₈	—	142	1	no	—
228	C₄×C₈	—	142	1	no	—
229	C₄×C₈	—	142	1	no	—
230	C₄×C₈	—	142	1	no	—
231	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
232	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
233	C₄×C₈	D₂₈₄	1	1	yes	characteristic
234	C₄×C₈	D₂₈₄	1	1	yes	characteristic
235	#95	—	2	2	no	—
236	#41	—	71	2	no	—
237	#41	—	71	2	no	—
238	#41	—	142	1	no	—
239	#41	—	142	1	no	—
240	C₂×Q₈	—	71	1	no	—
241	C₂×D₄	—	71	1	no	—
242	C₂×D₄	—	142	1	no	—
243	C₂²×C₄	—	4	2	no	—
244	C₂²×C₄	—	142	1	no	—
245	C₂²×C₄	—	142	1	no	—
246	C₂²×C₄	—	142	1	no	—
247	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
248	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
249	C₂²×C₄	#11765864	1	1	yes	characteristic
250	M₁₆	—	284	2	no	—
251	M₁₆	—	284	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	4	1	no	—
254	C₂×C₈	—	284	2	no	—
255	C₂×C₈	—	284	2	no	—
256	C₂×C₈	—	284	2	no	—
257	C₂×C₈	—	142	1	no	—
258	C₂×C₈	—	142	1	no	—
259	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
260	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
261	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
262	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
263	C₄⋊C₄	—	2	4	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₄⋊C₄	—	71	1	no	—
266	C₄⋊C₄	—	71	1	no	—
267	C₄⋊C₄	—	142	1	no	—
268	C₄⋊C₄	—	142	1	no	—
269	C₂²⋊C₄	—	142	1	no	—
270	C₂²⋊C₄	—	142	1	no	—
271	C₂²⋊C₄	—	142	1	no	—
272	C₄²	—	4	2	no	—
273	C₄²	—	4	2	no	—
274	C₄²	—	142	1	no	—
275	C₄²	—	142	1	no	—
276	C₄²	—	142	1	no	—
277	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
278	C₄²	C₂×D₂₈₄	1	1	yes	characteristic
279	C₄²	Q₈×D₇₁	1	1	yes	characteristic
280	C₄²	#11765866	1	1	yes	characteristic
281	C₂³	—	142	1	no	—
282	C₂³	#14556099	1	1	yes	characteristic
283	Q₈	—	71	2	no	—
284	Q₈	—	142	1	no	—
285	D₄	—	71	2	no	—
286	D₄	—	142	2	no	—
287	D₄	—	142	2	no	—
288	D₄	—	142	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	4	4	no	—
293	C₂×C₄	—	8	2	no	—
294	C₂×C₄	—	8	2	no	—
295	C₂×C₄	—	284	2	no	—
296	C₂×C₄	—	284	1	no	—
297	C₂×C₄	—	284	1	no	—
298	C₂×C₄	—	142	1	no	—
299	C₂×C₄	—	142	1	no	—
300	C₂×C₄	—	142	1	no	—
301	C₂×C₄	—	142	2	no	—
302	C₂×C₄	—	142	2	no	—
303	C₂×C₄	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
304	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
305	C₂×C₄	#14556086	1	1	yes	characteristic
306	C₂×C₄	#14556103	1	1	yes	characteristic
307	C₂×C₄	#14556104	1	1	yes	characteristic
308	C₂×C₄	#14556108	1	1	yes	characteristic
309	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
310	C₂×C₄	Q₁₆×D₇₁	1	1	yes	characteristic
311	C₂×C₄	#14556147	1	1	yes	characteristic
312	C₂×C₄	#14556151	1	1	yes	characteristic
313	C₈	—	2	1	no	—
314	C₈	—	2	1	no	—
315	C₈	—	2	1	no	—
316	C₈	—	284	1	no	—
317	C₈	—	284	1	no	—
318	C₈	—	568	2	no	—
319	C₈	#14556056	1	2	yes	complemented
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	284	1	no	—
323	C₂²	—	284	1	no	—
324	C₂²	—	142	2	no	—
325	C₂²	#17985856	1	1	yes	characteristic
326	C₂²	#17986232	1	1	yes	characteristic
327	C₂²	#17986237	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	8	4	no	—
332	C₄	—	284	1	no	—
333	C₄	—	284	1	no	—
334	C₄	—	142	2	no	—
335	C₄	C₂×#14556056	1	1	yes	characteristic
336	C₄	#17986017	1	1	yes	characteristic
337	C₄	#107×D₇₁	1	1	yes	characteristic
338	C₄	#17986036	1	1	yes	characteristic
339	C₄	#17986236	1	1	yes	characteristic
340	C₄	#17986241	1	1	yes	characteristic
341	C₂	—	2	1	no	—
342	C₂	—	2	1	no	—
343	C₂	—	284	2	no	—
344	C₂	#22588361	1	1	yes	characteristic
345	C₂	#22588366	1	1	yes	characteristic
346	C₂	#22588391	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14556103
commutator	C₄²×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₄×#107×C₇₁	C₂
socle	C₂²×C₇₁	#380
prime radical	C₂×#11765865	D₄
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	C₄×#107	D₇₁
hypocenter	C₇₁	#8667
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#107

Complements

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

N	H ≅ G/N	type
3. #22587099	C₂	semidirect
4. #22586243	C₂	semidirect
5. C₄×#107×C₇₁	C₂	semidirect
7. C₈×Dic₁₄₂	C₄	semidirect
8. C₈×Dic₁₄₂	C₄	semidirect
9. C₈×D₂₈₄	C₄	semidirect
10. C₈×D₂₈₄	C₄	semidirect
131. D₂₈₄	#107	semidirect
138. Dic₁₄₂	#107	semidirect
142. C₈×C₇₁	#104	semidirect
176. C₄×#107	D₇₁	semidirect
179. C₇₁	#8667	semidirect
319. C₈	#14556056	semidirect

Maximal subgroups

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

subgroup	index
C₈×#11765865	2
#22588361	2
#22587099	2
#22586243	2
C₄×#107×C₇₁	2
#8667	71