Subgroups — #16855866

Order 3968.

386 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14521601	C₂	1	1	yes	maximal, characteristic, complemented
2	#41×Dic₃₁	C₂	1	1	yes	maximal, characteristic
3	#14521391	C₂	1	1	yes	maximal, characteristic, complemented
4	#14521387	C₂	1	1	yes	maximal, characteristic, complemented
5	#14521342	C₂	1	1	yes	maximal, characteristic, complemented
6	#14521337	C₂	1	1	yes	maximal, characteristic, complemented
7	#14521257	C₂	1	1	yes	maximal, characteristic, complemented
8	#14521252	C₂	1	1	yes	maximal, characteristic, complemented
9	#14521211	C₂	1	1	yes	maximal, characteristic
10	#14521156	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×#452	C₂	1	1	yes	maximal, characteristic
16	C₂×#41×C₃₁	C₂²	1	1	yes	characteristic
17	C₂×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
18	C₂×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
20	Q₈×Dic₃₁	C₂²	1	1	yes	characteristic
21	Q₈×Dic₃₁	C₂²	1	1	yes	characteristic
22	Q₈×Dic₃₁	C₂²	1	1	yes	characteristic
23	#11758521	C₂²	1	1	yes	characteristic
24	C₂×#92482	C₂²	1	1	yes	characteristic
25	D₄×Dic₃₁	—	2	1	no	—
26	D₄×Dic₃₁	—	2	1	no	—
27	D₄×Dic₃₁	C₂²	1	1	yes	characteristic
28	D₄×Dic₃₁	C₂²	1	1	yes	characteristic
29	C₂×#92480	C₂²	1	1	yes	characteristic
30	#11758503	—	2	1	no	—
31	#11758503	C₂²	1	1	yes	characteristic
32	#11758501	C₂²	1	1	yes	characteristic
33	C₂×C₄×Dic₃₁	—	2	1	no	—
34	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
35	#11758466	—	2	1	no	—
36	#11758465	C₂²	1	1	yes	characteristic
37	#11758464	—	2	1	no	—
38	#11758464	C₂²	1	1	yes	characteristic
39	#11758455	—	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	#11758423	C₂²	1	1	yes	characteristic
46	#11758423	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×Q₈×C₃₁	C₂³	1	1	yes	characteristic
63	C₂×D₄×C₃₁	—	2	1	no	—
64	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
65	C₂²×C₄×C₃₁	—	2	1	no	—
66	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
67	C₂²×Dic₃₁	—	2	1	no	—
68	C₂²×Dic₃₁	—	2	1	no	—
69	C₂²×Dic₃₁	D₄	1	1	yes	characteristic
70	C₂²×Dic₃₁	D₄	1	1	yes	characteristic
71	C₂×Dic₆₂	—	2	1	no	—
72	C₂×Dic₆₂	C₂³	1	1	yes	characteristic
73	Q₁₆×C₃₁	—	2	2	no	—
74	SD₁₆×C₃₁	—	2	2	no	—
75	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
76	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
77	C₄⋊C₄×C₃₁	—	2	1	no	—
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	#92482	—	2	2	no	—
85	#92480	—	2	2	no	—
86	C₄⋊Dic₃₁	—	2	1	no	—
87	C₄⋊Dic₃₁	—	2	1	no	—
88	C₄⋊Dic₃₁	—	2	1	no	—
89	C₄⋊Dic₃₁	C₂³	1	1	yes	characteristic
90	C₄⋊Dic₃₁	C₂³	1	1	yes	characteristic
91	C₄⋊Dic₃₁	C₂³	1	1	yes	characteristic
92	Dic₃₁⋊C₄	—	2	1	no	—
93	Dic₃₁⋊C₄	—	2	1	no	—
94	Dic₃₁⋊C₄	—	2	1	no	—
95	Dic₃₁⋊C₄	—	2	1	no	—
96	Dic₃₁⋊C₄	—	2	1	no	—
97	C₄×Dic₃₁	—	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	C₂×D₄	1	1	yes	characteristic
110	Q₈×C₃₁	—	2	1	no	—
111	Q₈×C₃₁	—	2	1	no	—
112	Q₈×C₃₁	—	2	1	no	—
113	Q₈×C₃₁	#41	1	2	yes	—
114	D₄×C₃₁	—	2	4	no	—
115	D₄×C₃₁	—	2	1	no	—
116	D₄×C₃₁	#41	1	2	yes	—
117	C₂×C₄×C₃₁	—	2	4	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	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
125	C₂×Dic₃₁	—	2	2	no	—
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	2	no	—
132	C₂×Dic₃₁	—	4	2	no	—
133	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
134	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
135	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
136	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
137	Dic₆₂	—	2	1	no	—
138	Dic₆₂	—	2	1	no	—
139	Dic₆₂	—	4	2	no	—
140	C₈×C₃₁	—	2	1	no	—
141	C₈×C₃₁	—	2	1	no	—
142	C₃₁⋊C₈	—	2	1	no	—
143	C₃₁⋊C₈	—	2	1	no	—
144	#3116	—	31	1	no	maximal, Sylow
145	C₂²×C₃₁	—	2	1	no	—
146	C₂²×C₃₁	—	2	2	no	—
147	C₂²×C₃₁	—	2	1	no	—
148	C₂²×C₃₁	—	2	1	no	—
149	C₂²×C₃₁	—	4	1	no	—
150	C₂²×C₃₁	—	4	1	no	—
151	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
152	C₄×C₃₁	—	2	1	no	—
153	C₄×C₃₁	—	2	2	no	—
154	C₄×C₃₁	—	4	1	no	—
155	C₄×C₃₁	—	4	1	no	—
156	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
157	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
158	Dic₃₁	—	2	1	no	—
159	Dic₃₁	—	2	1	no	—
160	Dic₃₁	—	2	1	no	—
161	Dic₃₁	—	4	1	no	—
162	Dic₃₁	—	4	1	no	—
163	Dic₃₁	—	4	1	no	—
164	Dic₃₁	—	4	1	no	—
165	Dic₃₁	#137	1	2	yes	—
166	#533	—	31	1	no	—
167	C₄×#41	—	31	1	no	—
168	#485	—	31	1	no	—
169	#485	—	31	1	no	—
170	#477	—	31	1	no	—
171	#476	—	31	1	no	—
172	#462	—	31	1	no	—
173	#461	—	31	1	no	—
174	#452	—	31	1	no	—
175	#452	D₃₁	1	1	yes	characteristic
176	#441	—	31	1	no	—
177	#441	—	31	1	no	—
178	#440	—	31	1	no	—
179	#440	—	31	1	no	—
180	#432	—	31	1	no	—
181	C₂×C₃₁	—	4	1	no	—
182	C₂×C₃₁	—	4	2	no	—
183	C₂×C₃₁	#525	1	1	yes	characteristic
184	C₂×C₃₁	C₂×#137	1	1	yes	characteristic
185	C₂×C₃₁	#575	1	1	yes	characteristic
186	C₂×#41	C₂×D₃₁	1	1	yes	characteristic
187	C₂×Q₁₆	—	31	1	no	—
188	C₂×Q₁₆	C₂×D₃₁	1	1	yes	characteristic
189	C₂×SD₁₆	—	31	1	no	—
190	C₂×SD₁₆	C₂×D₃₁	1	1	yes	characteristic
191	C₄⋊Q₈	—	31	1	no	—
192	C₄⋊Q₈	—	31	1	no	—
193	#125	—	62	1	no	—
194	C₂²⋊Q₈	—	31	1	no	—
195	C₂²⋊Q₈	—	62	1	no	—
196	C₂²⋊Q₈	C₂×D₃₁	1	1	yes	characteristic
197	C₄×Q₈	—	31	1	no	—
198	C₄×Q₈	—	31	1	no	—
199	C₄×Q₈	—	31	1	no	—
200	C₄×Q₈	—	31	1	no	—
201	C₄×Q₈	—	62	1	no	—
202	C₄×D₄	—	31	1	no	—
203	C₄×D₄	—	31	1	no	—
204	C₄×D₄	—	62	1	no	—
205	C₄×D₄	—	62	1	no	—
206	#117	—	31	1	no	—
207	#117	—	62	1	no	—
208	#117	—	62	1	no	—
209	C₂×C₄²	—	31	1	no	—
210	C₂×C₄²	—	62	1	no	—
211	#107	—	31	1	no	—
212	#107	—	31	1	no	—
213	#106	—	31	1	no	—
214	#106	—	31	1	no	—
215	C₄⋊C₈	—	31	1	no	—
216	C₄⋊C₈	—	31	1	no	—
217	#103	—	31	1	no	—
218	#103	—	31	1	no	—
219	#103	—	31	1	no	—
220	#103	—	31	1	no	—
221	#103	—	31	1	no	—
222	#103	C₂×D₃₁	1	1	yes	characteristic
223	#102	—	31	1	no	—
224	#102	C₂×D₃₁	1	1	yes	characteristic
225	C₂²⋊C₈	—	31	1	no	—
226	C₂²⋊C₈	C₂×D₃₁	1	1	yes	characteristic
227	#97	—	31	1	no	—
228	#97	—	31	1	no	—
229	C₃₁	#3116	1	1	yes	characteristic, complemented, Sylow
230	#41	—	2	2	no	—
231	#41	—	2	2	no	—
232	C₂×Q₈	—	31	1	no	—
233	C₂×Q₈	—	62	1	no	—
234	C₂×Q₈	C₂²×D₃₁	1	1	yes	characteristic
235	C₂×Q₈	C₂²×D₃₁	1	1	yes	characteristic
236	C₂×D₄	—	2	1	no	—
237	C₂×D₄	C₂²×D₃₁	1	1	yes	characteristic
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	—	31	1	no	—
240	C₂²×C₄	—	31	1	no	—
241	C₂²×C₄	—	62	1	no	—
242	C₂²×C₄	—	62	1	no	—
243	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
244	Q₁₆	—	2	2	no	—
245	Q₁₆	—	62	2	no	—
246	SD₁₆	—	2	2	no	—
247	SD₁₆	—	62	2	no	—
248	C₂×C₈	—	31	1	no	—
249	C₂×C₈	—	31	1	no	—
250	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
251	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	—	31	1	no	—
254	C₄⋊C₄	—	31	1	no	—
255	C₄⋊C₄	—	31	1	no	—
256	C₄⋊C₄	—	62	1	no	—
257	C₄⋊C₄	—	62	1	no	—
258	C₄⋊C₄	—	62	1	no	—
259	C₄⋊C₄	—	62	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	C₂²×D₃₁	1	1	yes	characteristic
265	C₂²⋊C₄	—	2	1	no	—
266	C₂²⋊C₄	—	62	1	no	—
267	C₂²⋊C₄	—	62	1	no	—
268	C₂²⋊C₄	—	62	1	no	—
269	C₂²⋊C₄	—	62	1	no	—
270	C₄²	—	31	1	no	—
271	C₄²	—	31	1	no	—
272	C₄²	—	31	1	no	—
273	C₄²	—	31	1	no	—
274	C₄²	—	62	1	no	—
275	C₄²	—	62	1	no	—
276	C₄²	—	62	1	no	—
277	C₄²	—	62	1	no	—
278	C₄²	—	62	1	no	—
279	C₂³	—	2	1	no	—
280	C₂³	D₄×D₃₁	1	1	yes	characteristic
281	Q₈	—	2	1	no	—
282	Q₈	—	2	1	no	—
283	Q₈	—	2	1	no	—
284	Q₈	—	62	1	no	—
285	Q₈	—	62	1	no	—
286	Q₈	—	124	2	no	—
287	Q₈	#92497	1	2	yes	—
288	D₄	—	2	4	no	—
289	D₄	—	2	1	no	—
290	D₄	#92497	1	2	yes	—
291	C₂×C₄	—	2	4	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	31	1	no	—
298	C₂×C₄	—	31	1	no	—
299	C₂×C₄	—	31	1	no	—
300	C₂×C₄	—	31	1	no	—
301	C₂×C₄	—	62	2	no	—
302	C₂×C₄	—	62	1	no	—
303	C₂×C₄	—	62	1	no	—
304	C₂×C₄	—	62	1	no	—
305	C₂×C₄	—	62	1	no	—
306	C₂×C₄	—	124	2	no	—
307	C₂×C₄	—	124	1	no	—
308	C₂×C₄	—	124	2	no	—
309	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
310	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
311	C₈	—	2	1	no	—
312	C₈	—	2	1	no	—
313	C₈	—	62	1	no	—
314	C₈	—	62	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	2	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	4	1	no	—
320	C₂²	—	4	1	no	—
321	C₂²	C₂×D₄×D₃₁	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	2	no	—
324	C₄	—	4	1	no	—
325	C₄	—	4	1	no	—
326	C₄	—	31	2	no	—
327	C₄	—	62	1	no	—
328	C₄	—	62	1	no	—
329	C₄	—	62	1	no	—
330	C₄	—	124	1	no	—
331	C₄	—	124	1	no	—
332	C₄	—	124	1	no	—
333	C₄	—	124	1	no	—
334	C₄	C₂×#92497	1	1	yes	characteristic
335	C₄	C₂×#92497	1	1	yes	characteristic
336	C₂	—	4	1	no	—
337	C₂	—	4	2	no	—
338	C₂	#14521555	1	1	yes	characteristic
339	C₂	#137×D₃₁	1	1	yes	characteristic
340	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	C₃₁×#452	C₂
socle	C₂²×C₃₁	C₂²×D₄
prime radical	C₂×#41×C₃₁	C₂²
hypercenter	#452	D₃₁
hypocenter	C₃₁	#3116
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #452

Complements

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

N	H ≅ G/N	type
1. #14521601	C₂	semidirect
3. #14521391	C₂	semidirect
4. #14521387	C₂	semidirect
5. #14521342	C₂	semidirect
6. #14521337	C₂	semidirect
7. #14521257	C₂	semidirect
8. #14521252	C₂	semidirect
10. #14521156	C₂	semidirect
11. #14521153	C₂	semidirect
12. #14521150	C₂	semidirect
13. #14521146	C₂	semidirect
14. #14521103	C₂	semidirect
229. C₃₁	#3116	semidirect

Maximal subgroups

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

subgroup	index
#14521601	2
#41×Dic₃₁	2
#14521391	2
#14521387	2
#14521342	2
#14521337	2
#14521257	2
#14521252	2
#14521211	2
#14521156	2
#14521153	2
#14521150	2
#14521146	2
#14521103	2
C₃₁×#452	2
#3116	31