Subgroups — #16857623

Order 3968.

386 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14521770	C₂	1	1	yes	maximal, characteristic, complemented
2	#14521701	C₂	1	1	yes	maximal, characteristic
3	#14521398	C₂	1	1	yes	maximal, characteristic, complemented
4	#14521392	C₂	1	1	yes	maximal, characteristic, complemented
5	#14521387	C₂	1	1	yes	maximal, characteristic, complemented
6	#14521328	C₂	1	1	yes	maximal, characteristic, complemented
7	#14521289	C₂	1	1	yes	maximal, characteristic, complemented
8	#14521257	C₂	1	1	yes	maximal, characteristic, complemented
9	#14521210	C₂	1	1	yes	maximal, characteristic, complemented
10	#14521209	C₂	1	1	yes	maximal, characteristic, complemented
11	#14521134	C₂	1	1	yes	maximal, characteristic, complemented
12	C₈×C₂²⋊D₃₁	C₂	1	1	yes	maximal, characteristic, complemented
13	#14521046	C₂	1	1	yes	maximal, characteristic, complemented
14	#14521044	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₁×#467	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#92499	C₂²	1	1	yes	characteristic
17	C₂×#92495	C₂²	1	1	yes	characteristic
18	C₂×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
19	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
20	C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
21	C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
22	#11758523	C₂²	1	1	yes	characteristic
23	Q₈×Dic₃₁	C₂²	1	1	yes	characteristic
24	#11758521	C₂²	1	1	yes	characteristic
25	C₂×#92482	C₂²	1	1	yes	characteristic
26	C₂×#92481	C₂²	1	1	yes	characteristic
27	C₄×C₂²⋊D₃₁	C₂²	1	1	yes	characteristic
28	#11758503	C₂²	1	1	yes	characteristic
29	C₂×C₈×D₃₁	C₂²	1	1	yes	characteristic
30	#11758475	—	2	1	no	—
31	#11758474	C₂²	1	1	yes	characteristic
32	#11758471	—	2	1	no	—
33	#11758470	C₂²	1	1	yes	characteristic
34	#11758469	C₂²	1	1	yes	characteristic
35	C₄⋊C₄×D₃₁	—	2	1	no	—
36	#11758465	C₂²	1	1	yes	characteristic
37	#11758464	—	2	1	no	—
38	#11758462	—	2	1	no	—
39	#11758461	—	2	1	no	—
40	#11758460	—	2	1	no	—
41	#11758456	—	2	1	no	—
42	C₃₁×#107	C₂²	1	1	yes	characteristic
43	C₃₁×#103	C₂²	1	1	yes	characteristic
44	C₃₁×#103	C₂²	1	1	yes	characteristic
45	#11758423	C₂²	1	1	yes	characteristic
46	#11758423	C₂²	1	1	yes	characteristic
47	C₂²⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
48	#11758409	C₂²	1	1	yes	characteristic
49	#11758409	C₂²	1	1	yes	characteristic
50	(C₂×D₃₁)⋊C₈	C₂²	1	1	yes	characteristic
51	#11758404	C₂²	1	1	yes	characteristic
52	#11758404	C₂²	1	1	yes	characteristic
53	Dic₃₁⋊C₈	C₂²	1	1	yes	characteristic
54	C₈×Dic₃₁	C₂²	1	1	yes	characteristic
55	#11758398	C₂²	1	1	yes	characteristic
56	#11758397	C₂²	1	1	yes	characteristic
57	#11758396	C₂²	1	1	yes	characteristic
58	#11758395	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
60	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₃₁	—	2	1	no	—
63	C₂×C₂²⋊D₃₁	D₄	1	1	yes	characteristic
64	#92499	—	2	2	no	—
65	#92499	—	4	1	no	—
66	#92495	—	2	2	no	—
67	#92495	—	4	1	no	—
68	C₂×D₁₂₄	—	2	1	no	—
69	C₂×D₁₂₄	C₂³	1	1	yes	characteristic
70	C₂×C₄×D₃₁	—	2	1	no	—
71	C₂×C₄×D₃₁	—	2	1	no	—
72	C₂×C₄×D₃₁	—	2	1	no	—
73	C₂×C₄×D₃₁	C₂³	1	1	yes	characteristic
74	C₂×Dic₆₂	—	2	1	no	—
75	C₂×Dic₆₂	C₂³	1	1	yes	characteristic
76	Q₁₆×C₃₁	—	4	1	no	—
77	C₂×C₈×C₃₁	—	4	1	no	—
78	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
79	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
80	C₄⋊C₄×C₃₁	—	2	1	no	—
81	C₄⋊C₄×C₃₁	—	2	1	no	—
82	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
83	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
84	C₂²⋊C₄×C₃₁	—	2	1	no	—
85	C₂²⋊C₄×C₃₁	—	2	1	no	—
86	C₂²⋊Dic₃₁	—	2	1	no	—
87	C₂²⋊Dic₃₁	D₄	1	1	yes	characteristic
88	#92482	—	4	1	no	—
89	#92481	—	2	2	no	—
90	(C₂×D₃₁)⋊C₄	—	2	1	no	—
91	(C₂×D₃₁)⋊C₄	—	2	1	no	—
92	(C₂×D₃₁)⋊C₄	—	2	1	no	—
93	(C₂×D₃₁)⋊C₄	—	2	1	no	—
94	(C₂×D₃₁)⋊C₄	D₄	1	1	yes	characteristic
95	C₄⋊Dic₃₁	—	2	1	no	—
96	C₄⋊Dic₃₁	—	2	1	no	—
97	C₄⋊Dic₃₁	C₂³	1	1	yes	characteristic
98	C₄⋊Dic₃₁	C₂³	1	1	yes	characteristic
99	Dic₃₁⋊C₄	—	2	1	no	—
100	Dic₃₁⋊C₄	—	2	1	no	—
101	Dic₃₁⋊C₄	—	2	1	no	—
102	Dic₃₁⋊C₄	—	2	1	no	—
103	Dic₃₁⋊C₄	D₄	1	1	yes	characteristic
104	C₄×Dic₃₁	—	2	1	no	—
105	C₄×Dic₃₁	—	2	1	no	—
106	C₄×Dic₃₁	—	2	1	no	—
107	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
108	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
109	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
110	C₈×D₃₁	—	4	1	no	—
111	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
112	C₂²×D₃₁	—	2	1	no	—
113	C₂²×D₃₁	C₂×D₄	1	1	yes	characteristic
114	Q₈×C₃₁	—	2	1	no	—
115	Q₈×C₃₁	—	2	1	no	—
116	Q₈×C₃₁	—	2	1	no	—
117	Q₈×C₃₁	—	2	1	no	—
118	C₂×C₄×C₃₁	—	2	1	no	—
119	C₂×C₄×C₃₁	—	2	1	no	—
120	C₂×C₄×C₃₁	—	2	1	no	—
121	C₂×C₄×C₃₁	—	2	1	no	—
122	C₂×C₄×C₃₁	—	2	1	no	—
123	C₂×C₄×C₃₁	—	2	1	no	—
124	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
126	C₂²⋊D₃₁	—	2	2	no	—
127	C₂²⋊D₃₁	—	4	2	no	—
128	C₂×Dic₃₁	—	2	1	no	—
129	C₂×Dic₃₁	—	2	1	no	—
130	C₂×Dic₃₁	—	2	1	no	—
131	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
132	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
133	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
134	D₁₂₄	—	2	1	no	—
135	D₁₂₄	—	4	2	no	—
136	D₁₂₄	#41	1	2	yes	—
137	C₄×D₃₁	—	2	1	no	—
138	C₄×D₃₁	—	2	1	no	—
139	C₄×D₃₁	—	4	2	no	—
140	C₄×D₃₁	—	4	2	no	—
141	C₄×D₃₁	—	4	2	no	—
142	Dic₆₂	—	2	1	no	—
143	Dic₆₂	—	2	1	no	—
144	Dic₆₂	—	4	2	no	—
145	C₈×C₃₁	—	2	1	no	—
146	C₈×C₃₁	—	2	1	no	—
147	C₃₁⋊C₈	—	2	1	no	—
148	C₃₁⋊C₈	—	2	1	no	—
149	#3302	—	31	1	no	maximal, Sylow
150	C₂²×C₃₁	—	2	1	no	—
151	C₂²×C₃₁	—	2	1	no	—
152	C₂²×C₃₁	—	2	1	no	—
153	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
154	C₂×D₃₁	—	2	1	no	—
155	C₂×D₃₁	—	2	2	no	—
156	C₂×D₃₁	—	2	1	no	—
157	C₂×D₃₁	—	2	1	no	—
158	C₂×D₃₁	—	4	1	no	—
159	C₂×D₃₁	—	4	1	no	—
160	C₄×C₃₁	—	2	1	no	—
161	C₄×C₃₁	—	4	1	no	—
162	C₄×C₃₁	—	4	1	no	—
163	C₄×C₃₁	—	4	1	no	—
164	C₄×C₃₁	—	4	1	no	—
165	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
166	C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
167	Dic₃₁	—	2	1	no	—
168	Dic₃₁	—	2	1	no	—
169	Dic₃₁	—	4	1	no	—
170	Dic₃₁	—	4	1	no	—
171	Dic₃₁	—	4	1	no	—
172	Dic₃₁	#135	1	2	yes	—
173	#556	—	31	1	no	—
174	#547	—	31	1	no	—
175	#487	—	31	1	no	—
176	#485	—	31	1	no	—
177	#485	—	31	1	no	—
178	#474	—	31	1	no	—
179	#467	—	31	1	no	—
180	#467	D₃₁	1	1	yes	characteristic, complemented
181	#462	—	31	1	no	—
182	#452	—	31	1	no	—
183	#452	—	31	1	no	—
184	C₄×SD₁₆	—	31	1	no	—
185	C₈×D₄	—	31	1	no	—
186	#416	—	31	1	no	—
187	#416	—	31	1	no	—
188	C₂×C₃₁	—	4	1	no	—
189	C₂×C₃₁	#546	1	1	yes	characteristic
190	C₂×C₃₁	C₂×#135	1	1	yes	characteristic
191	C₂×C₃₁	#578	1	1	yes	characteristic
192	D₃₁	—	4	1	no	—
193	D₃₁	—	4	2	no	—
194	C₂×#41	—	31	1	no	—
195	C₂×#41	—	31	1	no	—
196	C₂×Q₁₆	—	31	1	no	—
197	C₂×Q₁₆	C₂×D₃₁	1	1	yes	characteristic
198	C₂×SD₁₆	—	31	1	no	—
199	C₂²×C₈	—	31	1	no	—
200	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
201	C₄⋊Q₈	—	31	1	no	—
202	C₄⋊Q₈	—	31	1	no	—
203	#126	—	62	1	no	—
204	#126	—	62	1	no	—
205	#124	—	62	1	no	—
206	#123	—	62	1	no	—
207	#123	—	62	1	no	—
208	C₂²⋊Q₈	—	31	1	no	—
209	C₂²⋊Q₈	—	31	1	no	—
210	C₂²⋊Q₈	C₂×D₃₁	1	1	yes	characteristic
211	C₂²⋊Q₈	C₂×D₃₁	1	1	yes	characteristic
212	#121	—	62	1	no	—
213	C₄×Q₈	—	31	1	no	—
214	C₄×Q₈	—	62	1	no	—
215	C₄×D₄	—	31	1	no	—
216	C₄×D₄	—	31	1	no	—
217	#117	—	31	1	no	—
218	#117	—	31	1	no	—
219	C₂×C₄⋊C₄	—	62	1	no	—
220	#107	—	31	1	no	—
221	#107	C₂×D₃₁	1	1	yes	characteristic
222	#106	—	31	1	no	—
223	C₄⋊C₈	—	31	1	no	—
224	#103	—	31	1	no	—
225	#103	—	31	1	no	—
226	#103	—	31	1	no	—
227	#103	—	31	1	no	—
228	#103	—	31	1	no	—
229	#103	C₂×D₃₁	1	1	yes	characteristic
230	#103	C₂×D₃₁	1	1	yes	characteristic
231	#102	—	31	1	no	—
232	#102	—	31	1	no	—
233	#102	—	31	1	no	—
234	C₂²⋊C₈	—	31	1	no	—
235	C₂²⋊C₈	—	31	1	no	—
236	C₄×C₈	—	31	1	no	—
237	C₃₁	#3302	1	1	yes	characteristic, complemented, Sylow
238	#41	—	62	2	no	—
239	#41	—	62	2	no	—
240	#41	—	124	1	no	—
241	#41	—	124	1	no	—
242	C₂×Q₈	—	31	1	no	—
243	C₂×Q₈	—	62	1	no	—
244	C₂×Q₈	C₂²×D₃₁	1	1	yes	characteristic
245	C₂×Q₈	C₂²×D₃₁	1	1	yes	characteristic
246	C₂×D₄	—	31	1	no	—
247	C₂×D₄	—	31	1	no	—
248	C₂×D₄	—	62	1	no	—
249	C₂×D₄	—	62	1	no	—
250	C₂²×C₄	—	31	1	no	—
251	C₂²×C₄	—	62	1	no	—
252	C₂²×C₄	—	62	1	no	—
253	C₂²×C₄	—	62	1	no	—
254	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
255	Q₁₆	—	4	1	no	—
256	Q₁₆	—	124	1	no	—
257	SD₁₆	—	62	2	no	—
258	C₂×C₈	—	4	1	no	—
259	C₂×C₈	—	31	1	no	—
260	C₂×C₈	—	31	1	no	—
261	C₂×C₈	—	124	1	no	—
262	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
263	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
264	C₄⋊C₄	—	2	1	no	—
265	C₄⋊C₄	—	2	1	no	—
266	C₄⋊C₄	—	31	1	no	—
267	C₄⋊C₄	—	31	1	no	—
268	C₄⋊C₄	—	31	1	no	—
269	C₄⋊C₄	—	62	1	no	—
270	C₄⋊C₄	—	62	1	no	—
271	C₄⋊C₄	—	62	1	no	—
272	C₄⋊C₄	—	62	1	no	—
273	C₄⋊C₄	—	62	1	no	—
274	C₄⋊C₄	—	62	1	no	—
275	C₄⋊C₄	C₂²×D₃₁	1	1	yes	characteristic
276	C₄⋊C₄	C₂²×D₃₁	1	1	yes	characteristic
277	C₂²⋊C₄	—	2	1	no	—
278	C₂²⋊C₄	—	2	1	no	—
279	C₂²⋊C₄	—	31	1	no	—
280	C₂²⋊C₄	—	31	1	no	—
281	C₂²⋊C₄	—	62	1	no	—
282	C₂²⋊C₄	—	62	1	no	—
283	C₂²⋊C₄	—	62	1	no	—
284	C₂²⋊C₄	—	62	1	no	—
285	C₂²⋊C₄	—	62	1	no	—
286	C₄²	—	31	1	no	—
287	C₄²	—	62	1	no	—
288	C₄²	—	62	1	no	—
289	C₄²	—	62	1	no	—
290	C₂³	—	31	1	no	—
291	C₂³	—	62	1	no	—
292	C₂³	D₄×D₃₁	1	1	yes	characteristic
293	Q₈	—	2	1	no	—
294	Q₈	—	2	1	no	—
295	Q₈	—	2	1	no	—
296	Q₈	—	2	1	no	—
297	Q₈	—	62	1	no	—
298	Q₈	—	62	1	no	—
299	Q₈	—	124	2	no	—
300	D₄	—	31	2	no	—
301	D₄	—	62	2	no	—
302	D₄	—	62	1	no	—
303	D₄	—	124	2	no	—
304	D₄	—	124	2	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	31	1	no	—
312	C₂×C₄	—	31	1	no	—
313	C₂×C₄	—	31	1	no	—
314	C₂×C₄	—	62	1	no	—
315	C₂×C₄	—	62	1	no	—
316	C₂×C₄	—	62	1	no	—
317	C₂×C₄	—	62	1	no	—
318	C₂×C₄	—	62	1	no	—
319	C₂×C₄	—	124	2	no	—
320	C₂×C₄	—	124	2	no	—
321	C₂×C₄	—	124	2	no	—
322	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
323	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
324	C₈	—	2	1	no	—
325	C₈	—	2	1	no	—
326	C₈	—	62	1	no	—
327	C₈	—	62	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	62	1	no	—
332	C₂²	—	62	1	no	—
333	C₂²	—	62	2	no	—
334	C₂²	—	62	1	no	—
335	C₂²	—	124	1	no	—
336	C₂²	—	124	1	no	—
337	C₂²	C₂×D₄×D₃₁	1	1	yes	characteristic
338	C₄	—	2	1	no	—
339	C₄	—	4	1	no	—
340	C₄	—	4	1	no	—
341	C₄	—	4	1	no	—
342	C₄	—	4	1	no	—
343	C₄	—	31	2	no	—
344	C₄	—	62	1	no	—
345	C₄	—	62	1	no	—
346	C₄	—	124	1	no	—
347	C₄	—	124	1	no	—
348	C₄	—	124	1	no	—
349	C₄	#11758561	1	1	yes	characteristic
350	C₄	#41×D₃₁	1	1	yes	characteristic
351	C₂	—	4	1	no	—
352	C₂	—	124	1	no	—
353	C₂	—	124	2	no	—
354	C₂	#14521688	1	1	yes	characteristic
355	C₂	#135×D₃₁	1	1	yes	characteristic
356	C₂	#14521862	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₃₁
commutator, cosocle	C₂×C₄×C₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₃₁×#467	C₂
socle	C₂²×C₃₁	C₂²×D₄
prime radical	#14521701	C₂
hypercenter	#467	D₃₁
hypocenter	C₃₁	#3302
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #467

Complements

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

N	H ≅ G/N	type
1. #14521770	C₂	semidirect
3. #14521398	C₂	semidirect
4. #14521392	C₂	semidirect
5. #14521387	C₂	semidirect
6. #14521328	C₂	semidirect
7. #14521289	C₂	semidirect
8. #14521257	C₂	semidirect
9. #14521210	C₂	semidirect
10. #14521209	C₂	semidirect
11. #14521134	C₂	semidirect
12. C₈×C₂²⋊D₃₁	C₂	semidirect
13. #14521046	C₂	semidirect
14. #14521044	C₂	semidirect
15. C₃₁×#467	C₂	semidirect
180. #467	D₃₁	semidirect
237. C₃₁	#3302	semidirect

Maximal subgroups

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

subgroup	index
#14521770	2
#14521701	2
#14521398	2
#14521392	2
#14521387	2
#14521328	2
#14521289	2
#14521257	2
#14521210	2
#14521209	2
#14521134	2
C₈×C₂²⋊D₃₁	2
#14521046	2
#14521044	2
C₃₁×#467	2
#3302	31