Subgroups — #16855813

Order 3968.

392 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14521580	C₂	1	1	yes	maximal, characteristic, complemented
2	#41×Dic₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#14521391	C₂	1	1	yes	maximal, characteristic, complemented
4	#14521370	C₂	1	1	yes	maximal, characteristic, complemented
5	#14521354	C₂	1	1	yes	maximal, characteristic, complemented
6	#14521347	C₂	1	1	yes	maximal, characteristic, complemented
7	#14521269	C₂	1	1	yes	maximal, characteristic, complemented
8	#14521262	C₂	1	1	yes	maximal, characteristic, complemented
9	#14521211	C₂	1	1	yes	maximal, characteristic, complemented
10	#14521161	C₂	1	1	yes	maximal, characteristic, complemented
11	#14521153	C₂	1	1	yes	maximal, characteristic, complemented
12	#14521150	C₂	1	1	yes	maximal, characteristic, complemented
13	#14521146	C₂	1	1	yes	maximal, characteristic, complemented
14	#14521103	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₁×#449	C₂	1	1	yes	maximal, characteristic
16	C₂×#41×C₃₁	C₂²	1	1	yes	characteristic
17	C₂×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
18	C₂×D₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#121	C₂²	1	1	yes	characteristic
20	Q₈×Dic₃₁	C₂²	1	1	yes	characteristic
21	Q₈×Dic₃₁	C₂²	1	1	yes	characteristic
22	C₂×#92482	C₂²	1	1	yes	characteristic
23	#11758513	C₂²	1	1	yes	characteristic
24	#11758512	—	2	1	no	—
25	D₄×Dic₃₁	—	2	1	no	—
26	D₄×Dic₃₁	—	2	1	no	—
27	D₄×Dic₃₁	—	2	1	no	—
28	D₄×Dic₃₁	C₂²	1	1	yes	characteristic
29	D₄×Dic₃₁	C₂²	1	1	yes	characteristic
30	D₄×Dic₃₁	C₂²	1	1	yes	characteristic
31	C₂×#92480	C₂²	1	1	yes	characteristic
32	#11758503	—	2	1	no	—
33	#11758503	C₂²	1	1	yes	characteristic
34	#11758501	C₂²	1	1	yes	characteristic
35	C₂×C₄×Dic₃₁	—	2	1	no	—
36	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
37	#11758467	C₂²	1	1	yes	characteristic
38	#11758464	C₂²	1	1	yes	characteristic
39	#11758456	—	2	1	no	—
40	#11758454	—	2	1	no	—
41	C₃₁×#103	C₂²	1	1	yes	characteristic
42	C₃₁×#102	C₂²	1	1	yes	characteristic
43	C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
44	#11758423	C₂²	1	1	yes	characteristic
45	#11758420	C₂²	1	1	yes	characteristic
46	#11758420	C₂²	1	1	yes	characteristic
47	#11758420	C₂²	1	1	yes	characteristic
48	C₂²⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
49	#11758406	C₂²	1	1	yes	characteristic
50	#11758405	C₂²	1	1	yes	characteristic
51	#11758404	C₂²	1	1	yes	characteristic
52	#11758404	C₂²	1	1	yes	characteristic
53	#11758403	C₂²	1	1	yes	characteristic
54	#11758403	C₂²	1	1	yes	characteristic
55	Dic₃₁⋊C₈	C₂²	1	1	yes	characteristic
56	Dic₃₁⋊C₈	C₂²	1	1	yes	characteristic
57	#11758396	C₂²	1	1	yes	characteristic
58	#11758395	C₂²	1	1	yes	characteristic
59	#41×C₃₁	—	2	2	no	—
60	#41×C₃₁	—	2	2	no	—
61	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
62	C₂×D₄×C₃₁	—	2	1	no	—
63	C₂×D₄×C₃₁	—	2	1	no	—
64	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
65	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
66	C₂²×C₄×C₃₁	—	2	1	no	—
67	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
68	C₂²×Dic₃₁	—	2	1	no	—
69	C₂²×Dic₃₁	—	2	1	no	—
70	C₂²×Dic₃₁	—	2	1	no	—
71	C₂²×Dic₃₁	D₄	1	1	yes	characteristic
72	C₂²×Dic₃₁	D₄	1	1	yes	characteristic
73	C₂×Dic₆₂	C₂³	1	1	yes	characteristic
74	SD₁₆×C₃₁	—	2	2	no	—
75	D₈×C₃₁	—	2	2	no	—
76	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
77	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
79	C₂²⋊C₄×C₃₁	—	2	1	no	—
80	C₂²⋊Dic₃₁	—	2	1	no	—
81	C₂²⋊Dic₃₁	—	2	1	no	—
82	C₂²⋊Dic₃₁	—	2	1	no	—
83	C₂²⋊Dic₃₁	—	2	1	no	—
84	C₂²⋊Dic₃₁	—	2	1	no	—
85	C₂²⋊Dic₃₁	—	2	1	no	—
86	C₂²⋊Dic₃₁	—	2	1	no	—
87	#92482	—	2	2	no	—
88	#92480	—	2	2	no	—
89	C₄⋊Dic₃₁	—	2	1	no	—
90	C₄⋊Dic₃₁	—	2	1	no	—
91	C₄⋊Dic₃₁	—	2	1	no	—
92	C₄⋊Dic₃₁	C₂³	1	1	yes	characteristic
93	C₄⋊Dic₃₁	C₂³	1	1	yes	characteristic
94	C₄⋊Dic₃₁	C₂³	1	1	yes	characteristic
95	Dic₃₁⋊C₄	—	2	1	no	—
96	Dic₃₁⋊C₄	—	2	1	no	—
97	Dic₃₁⋊C₄	—	2	1	no	—
98	C₄×Dic₃₁	—	2	1	no	—
99	C₄×Dic₃₁	—	2	1	no	—
100	C₄×Dic₃₁	—	2	1	no	—
101	C₄×Dic₃₁	—	2	1	no	—
102	C₄×Dic₃₁	D₄	1	1	yes	characteristic
103	C₄×Dic₃₁	D₄	1	1	yes	characteristic
104	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
105	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
106	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
107	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
108	C₂³×C₃₁	—	2	1	no	—
109	C₂³×C₃₁	—	2	1	no	—
110	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
111	Q₈×C₃₁	—	2	1	no	—
112	Q₈×C₃₁	#41	1	2	yes	—
113	D₄×C₃₁	—	2	4	no	—
114	D₄×C₃₁	—	2	1	no	—
115	D₄×C₃₁	—	2	1	no	—
116	D₄×C₃₁	—	2	1	no	—
117	D₄×C₃₁	—	4	2	no	—
118	D₄×C₃₁	#41	1	2	yes	—
119	C₂×C₄×C₃₁	—	2	4	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	C₂×Dic₃₁	—	2	1	no	—
127	C₂×Dic₃₁	—	2	1	no	—
128	C₂×Dic₃₁	—	2	1	no	—
129	C₂×Dic₃₁	—	2	1	no	—
130	C₂×Dic₃₁	—	4	1	no	—
131	C₂×Dic₃₁	—	4	1	no	—
132	C₂×Dic₃₁	—	4	2	no	—
133	C₂×Dic₃₁	—	4	2	no	—
134	C₂×Dic₃₁	—	4	2	no	—
135	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
136	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
137	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
138	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
139	Dic₆₂	—	2	1	no	—
140	Dic₆₂	—	2	1	no	—
141	C₈×C₃₁	—	2	1	no	—
142	C₈×C₃₁	—	2	1	no	—
143	C₃₁⋊C₈	—	2	1	no	—
144	C₃₁⋊C₈	—	2	1	no	—
145	#3111	—	31	1	no	maximal, Sylow
146	C₂²×C₃₁	—	2	1	no	—
147	C₂²×C₃₁	—	2	2	no	—
148	C₂²×C₃₁	—	2	1	no	—
149	C₂²×C₃₁	—	2	1	no	—
150	C₂²×C₃₁	—	4	1	no	—
151	C₂²×C₃₁	—	4	1	no	—
152	C₂²×C₃₁	—	4	1	no	—
153	C₂²×C₃₁	—	4	1	no	—
154	C₂²×C₃₁	—	4	1	no	—
155	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
156	C₄×C₃₁	—	2	1	no	—
157	C₄×C₃₁	—	2	2	no	—
158	C₄×C₃₁	—	4	1	no	—
159	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
160	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
161	Dic₃₁	—	2	1	no	—
162	Dic₃₁	—	2	1	no	—
163	Dic₃₁	—	2	1	no	—
164	Dic₃₁	—	2	1	no	—
165	Dic₃₁	—	4	1	no	—
166	Dic₃₁	—	4	1	no	—
167	Dic₃₁	—	4	1	no	—
168	Dic₃₁	—	4	1	no	—
169	#529	—	31	1	no	—
170	C₄×#41	—	31	1	no	—
171	#485	—	31	1	no	—
172	#482	—	31	1	no	—
173	#479	—	31	1	no	—
174	#478	—	31	1	no	—
175	#464	—	31	1	no	—
176	#463	—	31	1	no	—
177	#452	—	31	1	no	—
178	#449	D₃₁	1	1	yes	characteristic
179	#442	—	31	1	no	—
180	#441	—	31	1	no	—
181	#440	—	31	1	no	—
182	#440	—	31	1	no	—
183	#432	—	31	1	no	—
184	C₂×C₃₁	—	4	1	no	—
185	C₂×C₃₁	—	4	2	no	—
186	C₂×C₃₁	—	8	1	no	—
187	C₂×C₃₁	#525	1	1	yes	characteristic
188	C₂×C₃₁	#575	1	1	yes	characteristic
189	C₂×C₃₁	#575	1	1	yes	characteristic
190	C₂×#41	C₂×D₃₁	1	1	yes	characteristic
191	C₂×Q₁₆	—	31	1	no	—
192	C₂×SD₁₆	—	31	1	no	—
193	C₂×SD₁₆	C₂×D₃₁	1	1	yes	characteristic
194	C₂×D₈	C₂×D₃₁	1	1	yes	characteristic
195	#126	—	62	1	no	—
196	#125	—	31	1	no	—
197	#124	—	31	1	no	—
198	#123	—	62	1	no	—
199	C₂²⋊Q₈	—	31	1	no	—
200	#121	C₂×D₃₁	1	1	yes	characteristic
201	C₄×Q₈	—	31	1	no	—
202	C₄×Q₈	—	31	1	no	—
203	C₄×Q₈	—	31	1	no	—
204	C₄×D₄	—	31	1	no	—
205	C₄×D₄	—	31	1	no	—
206	C₄×D₄	—	31	1	no	—
207	C₄×D₄	—	62	1	no	—
208	C₄×D₄	—	62	1	no	—
209	C₄×D₄	—	62	1	no	—
210	#117	—	31	1	no	—
211	#117	—	62	1	no	—
212	#117	—	62	1	no	—
213	C₂×C₄²	—	31	1	no	—
214	C₂×C₄²	—	62	1	no	—
215	#107	—	31	1	no	—
216	#107	—	31	1	no	—
217	#106	—	31	1	no	—
218	#106	—	31	1	no	—
219	C₄⋊C₈	—	31	1	no	—
220	C₄⋊C₈	—	31	1	no	—
221	#103	—	31	1	no	—
222	#103	—	31	1	no	—
223	#103	—	31	1	no	—
224	#103	C₂×D₃₁	1	1	yes	characteristic
225	#102	—	31	1	no	—
226	#102	—	31	1	no	—
227	#102	—	31	1	no	—
228	#102	C₂×D₃₁	1	1	yes	characteristic
229	C₂²⋊C₈	—	31	1	no	—
230	C₂²⋊C₈	C₂×D₃₁	1	1	yes	characteristic
231	#97	—	31	1	no	—
232	#97	—	31	1	no	—
233	C₃₁	#3111	1	1	yes	characteristic, complemented, Sylow
234	#41	—	2	2	no	—
235	#41	—	2	2	no	—
236	C₂×Q₈	—	31	1	no	—
237	C₂×Q₈	C₂²×D₃₁	1	1	yes	characteristic
238	C₂×D₄	—	2	1	no	—
239	C₂×D₄	—	2	1	no	—
240	C₂×D₄	C₂²×D₃₁	1	1	yes	characteristic
241	C₂×D₄	C₂²×D₃₁	1	1	yes	characteristic
242	C₂²×C₄	—	2	1	no	—
243	C₂²×C₄	—	31	1	no	—
244	C₂²×C₄	—	31	1	no	—
245	C₂²×C₄	—	62	1	no	—
246	C₂²×C₄	—	62	1	no	—
247	C₂²×C₄	—	62	1	no	—
248	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
249	Q₁₆	—	62	2	no	—
250	SD₁₆	—	2	2	no	—
251	SD₁₆	—	62	2	no	—
252	D₈	—	2	2	no	—
253	C₂×C₈	—	31	1	no	—
254	C₂×C₈	—	31	1	no	—
255	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
256	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
257	C₄⋊C₄	—	31	1	no	—
258	C₄⋊C₄	—	31	1	no	—
259	C₄⋊C₄	—	31	1	no	—
260	C₄⋊C₄	—	62	1	no	—
261	C₄⋊C₄	—	62	1	no	—
262	C₄⋊C₄	—	62	1	no	—
263	C₄⋊C₄	—	62	1	no	—
264	C₄⋊C₄	—	62	1	no	—
265	C₄⋊C₄	—	62	1	no	—
266	C₄⋊C₄	C₂²×D₃₁	1	1	yes	characteristic
267	C₂²⋊C₄	—	2	1	no	—
268	C₂²⋊C₄	—	62	1	no	—
269	C₂²⋊C₄	—	62	1	no	—
270	C₂²⋊C₄	—	62	1	no	—
271	C₂²⋊C₄	—	62	1	no	—
272	C₂²⋊C₄	—	62	1	no	—
273	C₂²⋊C₄	—	62	1	no	—
274	C₂²⋊C₄	—	62	1	no	—
275	C₄²	—	31	1	no	—
276	C₄²	—	31	1	no	—
277	C₄²	—	31	1	no	—
278	C₄²	—	31	1	no	—
279	C₄²	—	62	1	no	—
280	C₄²	—	62	1	no	—
281	C₄²	—	62	1	no	—
282	C₄²	—	62	1	no	—
283	C₂³	—	2	1	no	—
284	C₂³	—	2	1	no	—
285	C₂³	D₄×D₃₁	1	1	yes	characteristic
286	Q₈	—	2	1	no	—
287	Q₈	—	62	1	no	—
288	Q₈	—	62	1	no	—
289	Q₈	#92497	1	2	yes	—
290	D₄	—	2	4	no	—
291	D₄	—	2	1	no	—
292	D₄	—	2	1	no	—
293	D₄	—	2	1	no	—
294	D₄	—	4	2	no	—
295	D₄	#92497	1	2	yes	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	31	1	no	—
302	C₂×C₄	—	31	1	no	—
303	C₂×C₄	—	31	1	no	—
304	C₂×C₄	—	31	1	no	—
305	C₂×C₄	—	62	1	no	—
306	C₂×C₄	—	62	1	no	—
307	C₂×C₄	—	62	1	no	—
308	C₂×C₄	—	62	1	no	—
309	C₂×C₄	—	124	1	no	—
310	C₂×C₄	—	124	1	no	—
311	C₂×C₄	—	124	2	no	—
312	C₂×C₄	—	124	2	no	—
313	C₂×C₄	—	124	2	no	—
314	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
315	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
316	C₈	—	2	1	no	—
317	C₈	—	2	1	no	—
318	C₈	—	62	1	no	—
319	C₈	—	62	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	2	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	4	1	no	—
325	C₂²	—	4	1	no	—
326	C₂²	—	4	1	no	—
327	C₂²	—	4	1	no	—
328	C₂²	—	4	1	no	—
329	C₂²	C₂×D₄×D₃₁	1	1	yes	characteristic
330	C₄	—	2	1	no	—
331	C₄	—	2	2	no	—
332	C₄	—	4	1	no	—
333	C₄	—	62	1	no	—
334	C₄	—	62	1	no	—
335	C₄	—	62	1	no	—
336	C₄	—	62	1	no	—
337	C₄	—	124	1	no	—
338	C₄	—	124	1	no	—
339	C₄	—	124	1	no	—
340	C₄	—	124	1	no	—
341	C₄	C₂×#92497	1	1	yes	characteristic
342	C₄	C₂×#92497	1	1	yes	characteristic
343	C₂	—	4	1	no	—
344	C₂	—	4	2	no	—
345	C₂	—	8	1	no	—
346	C₂	#14521555	1	1	yes	characteristic
347	C₂	#14521845	1	1	yes	characteristic
348	C₂	#14521846	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₃₁
commutator, cosocle	C₂×C₄×C₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₁
Fitting, prime radical	C₃₁×#449	C₂
socle	C₂²×C₃₁	C₂²×D₄
hypercenter	#449	D₃₁
hypocenter	C₃₁	#3111
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #449

Complements

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

N	H ≅ G/N	type
1. #14521580	C₂	semidirect
2. #41×Dic₃₁	C₂	semidirect
3. #14521391	C₂	semidirect
4. #14521370	C₂	semidirect
5. #14521354	C₂	semidirect
6. #14521347	C₂	semidirect
7. #14521269	C₂	semidirect
8. #14521262	C₂	semidirect
9. #14521211	C₂	semidirect
10. #14521161	C₂	semidirect
11. #14521153	C₂	semidirect
12. #14521150	C₂	semidirect
13. #14521146	C₂	semidirect
14. #14521103	C₂	semidirect
233. C₃₁	#3111	semidirect

Maximal subgroups

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

subgroup	index
#14521580	2
#41×Dic₃₁	2
#14521391	2
#14521370	2
#14521354	2
#14521347	2
#14521269	2
#14521262	2
#14521211	2
#14521161	2
#14521153	2
#14521150	2
#14521146	2
#14521103	2
C₃₁×#449	2
#3111	31