Subgroups — #16858424

Order 3968.

372 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14521768	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×Dic₆₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#14521408	C₂	1	1	yes	maximal, characteristic, complemented
4	#14521407	C₂	1	1	yes	maximal, characteristic, complemented
5	#14521398	C₂	1	1	yes	maximal, characteristic, complemented
6	#14521345	C₂	1	1	yes	maximal, characteristic, complemented
7	#14521342	C₂	1	1	yes	maximal, characteristic, complemented
8	#14521327	C₂	1	1	yes	maximal, characteristic, complemented
9	#14521262	C₂	1	1	yes	maximal, characteristic
10	#14521252	C₂	1	1	yes	maximal, characteristic, complemented
11	#14521178	C₂	1	1	yes	maximal, characteristic, complemented
12	#14521150	C₂	1	1	yes	maximal, characteristic, complemented
13	#14521149	C₂	1	1	yes	maximal, characteristic, complemented
14	D₈×Dic₃₁	C₂	1	1	yes	maximal, characteristic
15	C₃₁×#463	C₂	1	1	yes	maximal, characteristic
16	C₂²×Dic₆₂	—	2	1	no	—
17	C₂×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
18	C₂×D₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#124	C₂²	1	1	yes	characteristic
20	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
21	Q₈×Dic₃₁	C₂²	1	1	yes	characteristic
22	#11758521	C₂²	1	1	yes	characteristic
23	#11758513	C₂²	1	1	yes	characteristic
24	D₄×Dic₃₁	C₂²	1	1	yes	characteristic
25	D₄×Dic₃₁	C₂²	1	1	yes	characteristic
26	D₄×Dic₃₁	C₂²	1	1	yes	characteristic
27	C₂×#92480	C₂²	1	1	yes	characteristic
28	#11758501	—	2	1	no	—
29	#11758465	C₂²	1	1	yes	characteristic
30	#11758465	C₂²	1	1	yes	characteristic
31	#11758456	—	2	1	no	—
32	#11758456	—	2	1	no	—
33	#11758455	—	2	1	no	—
34	#11758455	—	2	1	no	—
35	#11758455	—	2	1	no	—
36	#11758454	—	2	1	no	—
37	#11758446	C₂²	1	1	yes	characteristic
38	C₄×Dic₆₂	C₂²	1	1	yes	characteristic
39	C₄×Dic₆₂	C₂²	1	1	yes	characteristic
40	C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
41	C₃₁×#103	C₂²	1	1	yes	characteristic
42	C₃₁×#102	C₂²	1	1	yes	characteristic
43	#11758423	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	#11758406	C₂²	1	1	yes	characteristic
49	#11758406	C₂²	1	1	yes	characteristic
50	#11758404	C₂²	1	1	yes	characteristic
51	#11758404	C₂²	1	1	yes	characteristic
52	#11758403	C₂²	1	1	yes	characteristic
53	Dic₃₁⋊C₈	C₂²	1	1	yes	characteristic
54	Dic₃₁⋊C₈	C₂²	1	1	yes	characteristic
55	C₈×Dic₃₁	C₂²	1	1	yes	characteristic
56	#11758398	C₂²	1	1	yes	characteristic
57	#11758395	C₂²	1	1	yes	characteristic
58	#11758391	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
60	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
61	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₃₁	—	2	1	no	—
63	C₂²×Dic₃₁	—	2	1	no	—
64	C₂²×Dic₃₁	—	2	1	no	—
65	C₂²×Dic₃₁	—	2	1	no	—
66	C₂×Dic₆₂	—	2	1	no	—
67	C₂×Dic₆₂	—	2	1	no	—
68	C₂×Dic₆₂	—	2	1	no	—
69	C₂×Dic₆₂	—	4	4	no	—
70	C₂×Dic₆₂	D₄	1	1	yes	characteristic
71	C₂×Dic₆₂	C₂³	1	1	yes	characteristic
72	SD₁₆×C₃₁	—	2	2	no	—
73	D₈×C₃₁	—	2	2	no	—
74	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
75	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
77	C₂²⋊C₄×C₃₁	—	2	1	no	—
78	C₂²⋊C₄×C₃₁	—	2	1	no	—
79	C₂²⋊C₄×C₃₁	—	2	1	no	—
80	C₄²×C₃₁	C₂³	1	1	yes	characteristic
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	#92480	—	2	2	no	—
87	C₄⋊Dic₃₁	—	2	1	no	—
88	C₄⋊Dic₃₁	—	2	1	no	—
89	C₄⋊Dic₃₁	D₄	1	1	yes	characteristic
90	C₄⋊Dic₃₁	C₂³	1	1	yes	characteristic
91	C₄⋊Dic₃₁	C₂³	1	1	yes	characteristic
92	C₄⋊Dic₃₁	C₂³	1	1	yes	characteristic
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	Dic₃₁⋊C₄	—	2	1	no	—
98	Dic₃₁⋊C₄	D₄	1	1	yes	characteristic
99	Dic₃₁⋊C₄	D₄	1	1	yes	characteristic
100	C₄×Dic₃₁	—	2	1	no	—
101	C₄×Dic₃₁	—	2	1	no	—
102	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
103	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
104	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
105	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
106	C₂³×C₃₁	—	2	1	no	—
107	C₂³×C₃₁	—	2	1	no	—
108	Q₈×C₃₁	—	2	1	no	—
109	Q₈×C₃₁	—	2	1	no	—
110	D₄×C₃₁	—	2	1	no	—
111	D₄×C₃₁	—	2	1	no	—
112	D₄×C₃₁	—	2	1	no	—
113	D₄×C₃₁	#41	1	2	yes	—
114	C₂×C₄×C₃₁	—	2	1	no	—
115	C₂×C₄×C₃₁	—	2	1	no	—
116	C₂×C₄×C₃₁	—	4	2	no	—
117	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
120	C₂×Dic₃₁	—	2	1	no	—
121	C₂×Dic₃₁	—	2	1	no	—
122	C₂×Dic₃₁	—	2	1	no	—
123	C₂×Dic₃₁	—	2	1	no	—
124	C₂×Dic₃₁	—	4	2	no	—
125	C₂×Dic₃₁	—	4	2	no	—
126	C₂×Dic₃₁	—	4	2	no	—
127	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
128	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
129	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
131	Dic₆₂	—	2	2	no	—
132	Dic₆₂	—	2	1	no	—
133	Dic₆₂	—	2	1	no	—
134	Dic₆₂	—	4	2	no	—
135	Dic₆₂	—	4	2	no	—
136	Dic₆₂	—	4	2	no	—
137	C₈×C₃₁	—	2	1	no	—
138	C₈×C₃₁	—	2	1	no	—
139	C₃₁⋊C₈	—	2	1	no	—
140	C₃₁⋊C₈	—	2	1	no	—
141	#3367	—	31	1	no	maximal, Sylow
142	C₂²×C₃₁	—	2	2	no	—
143	C₂²×C₃₁	—	4	1	no	—
144	C₂²×C₃₁	—	4	1	no	—
145	C₂²×C₃₁	—	4	1	no	—
146	C₂²×C₃₁	—	4	1	no	—
147	C₂²×C₃₁	—	4	1	no	—
148	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
149	C₄×C₃₁	—	2	1	no	—
150	C₄×C₃₁	—	2	1	no	—
151	C₄×C₃₁	—	4	1	no	—
152	C₄×C₃₁	—	4	1	no	—
153	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
154	C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
155	Dic₃₁	—	2	1	no	—
156	Dic₃₁	—	2	1	no	—
157	Dic₃₁	—	2	1	no	—
158	Dic₃₁	—	4	1	no	—
159	Dic₃₁	—	4	1	no	—
160	Dic₃₁	—	4	1	no	—
161	Dic₃₁	—	4	1	no	—
162	Dic₃₁	AGL(1,ℤ/8)	1	2	yes	—
163	#556	—	31	1	no	—
164	D₄×Q₈	—	31	1	no	—
165	#489	—	31	1	no	—
166	#489	—	31	1	no	—
167	#487	—	31	1	no	—
168	#477	—	31	1	no	—
169	#477	—	31	1	no	—
170	#474	—	31	1	no	—
171	#463	—	31	1	no	—
172	#463	D₃₁	1	1	yes	characteristic
173	#461	—	31	1	no	—
174	#446	—	31	1	no	—
175	#440	—	31	1	no	—
176	#440	—	31	1	no	—
177	C₄×D₈	—	31	1	no	—
178	C₂×C₃₁	—	4	2	no	—
179	C₂×C₃₁	—	8	1	no	—
180	C₂×C₃₁	#548	1	1	yes	characteristic
181	C₂×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
182	C₂×C₃₁	#578	1	1	yes	characteristic
183	C₂²×Q₈	—	62	1	no	—
184	C₂×SD₁₆	—	31	1	no	—
185	C₂×SD₁₆	C₂×D₃₁	1	1	yes	characteristic
186	C₂×D₈	C₂×D₃₁	1	1	yes	characteristic
187	C₄⋊Q₈	—	31	1	no	—
188	C₄⋊Q₈	—	31	1	no	—
189	C₄⋊Q₈	—	31	1	no	—
190	C₄⋊Q₈	—	31	1	no	—
191	#126	—	62	1	no	—
192	#126	—	62	1	no	—
193	#124	—	31	1	no	—
194	#124	C₂×D₃₁	1	1	yes	characteristic
195	C₂²⋊Q₈	—	62	1	no	—
196	C₂²⋊Q₈	—	62	1	no	—
197	C₂²⋊Q₈	—	62	1	no	—
198	C₂²⋊Q₈	—	62	1	no	—
199	C₄×Q₈	—	31	1	no	—
200	C₄×Q₈	—	31	1	no	—
201	C₄×Q₈	—	31	1	no	—
202	C₄×D₄	—	31	1	no	—
203	C₄×D₄	—	31	1	no	—
204	C₄×D₄	—	31	1	no	—
205	C₄×D₄	C₂×D₃₁	1	1	yes	characteristic
206	#117	—	62	1	no	—
207	#107	—	31	1	no	—
208	#107	—	31	1	no	—
209	#107	—	31	1	no	—
210	C₄⋊C₈	—	31	1	no	—
211	C₄⋊C₈	—	31	1	no	—
212	C₄⋊C₈	C₂×D₃₁	1	1	yes	characteristic
213	#103	—	31	1	no	—
214	#103	—	31	1	no	—
215	#103	—	31	1	no	—
216	#103	—	31	1	no	—
217	#103	—	31	1	no	—
218	#103	C₂×D₃₁	1	1	yes	characteristic
219	#102	—	31	1	no	—
220	#102	—	31	1	no	—
221	#102	—	31	1	no	—
222	#102	C₂×D₃₁	1	1	yes	characteristic
223	#97	—	31	1	no	—
224	#97	—	31	1	no	—
225	C₄×C₈	—	31	1	no	—
226	C₃₁	#3367	1	1	yes	characteristic, complemented, Sylow
227	C₂×Q₈	—	31	1	no	—
228	C₂×Q₈	—	31	1	no	—
229	C₂×Q₈	—	62	1	no	—
230	C₂×Q₈	—	62	1	no	—
231	C₂×Q₈	—	62	1	no	—
232	C₂×Q₈	—	124	4	no	—
233	C₂×Q₈	C₂²×D₃₁	1	1	yes	characteristic
234	C₂×D₄	C₂²×D₃₁	1	1	yes	characteristic
235	C₂×D₄	C₂²×D₃₁	1	1	yes	characteristic
236	C₂²×C₄	—	2	1	no	—
237	C₂²×C₄	—	62	1	no	—
238	C₂²×C₄	—	62	1	no	—
239	C₂²×C₄	—	62	1	no	—
240	SD₁₆	—	2	2	no	—
241	SD₁₆	—	62	2	no	—
242	D₈	—	2	2	no	—
243	C₂×C₈	—	31	1	no	—
244	C₂×C₈	—	31	1	no	—
245	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
246	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
247	C₄⋊C₄	—	31	1	no	—
248	C₄⋊C₄	—	31	1	no	—
249	C₄⋊C₄	—	31	1	no	—
250	C₄⋊C₄	—	31	1	no	—
251	C₄⋊C₄	—	31	1	no	—
252	C₄⋊C₄	—	31	1	no	—
253	C₄⋊C₄	—	62	1	no	—
254	C₄⋊C₄	—	62	1	no	—
255	C₄⋊C₄	—	62	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	C₂²×D₃₁	1	1	yes	characteristic
261	C₂²⋊C₄	—	2	1	no	—
262	C₂²⋊C₄	—	2	1	no	—
263	C₂²⋊C₄	—	2	1	no	—
264	C₂²⋊C₄	—	62	1	no	—
265	C₂²⋊C₄	—	62	1	no	—
266	C₂²⋊C₄	—	62	1	no	—
267	C₂²⋊C₄	—	62	1	no	—
268	C₂²⋊C₄	—	62	1	no	—
269	C₄²	—	31	1	no	—
270	C₄²	—	31	1	no	—
271	C₄²	—	62	1	no	—
272	C₄²	—	62	1	no	—
273	C₄²	C₂²×D₃₁	1	1	yes	characteristic
274	C₂³	—	2	1	no	—
275	C₂³	—	2	1	no	—
276	Q₈	—	2	1	no	—
277	Q₈	—	2	1	no	—
278	Q₈	—	62	2	no	—
279	Q₈	—	62	1	no	—
280	Q₈	—	62	1	no	—
281	Q₈	—	124	2	no	—
282	Q₈	—	124	2	no	—
283	Q₈	—	124	2	no	—
284	D₄	—	2	1	no	—
285	D₄	—	2	1	no	—
286	D₄	—	2	1	no	—
287	D₄	#92497	1	2	yes	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	31	1	no	—
292	C₂×C₄	—	31	1	no	—
293	C₂×C₄	—	31	1	no	—
294	C₂×C₄	—	31	1	no	—
295	C₂×C₄	—	62	1	no	—
296	C₂×C₄	—	62	1	no	—
297	C₂×C₄	—	62	1	no	—
298	C₂×C₄	—	62	1	no	—
299	C₂×C₄	—	124	2	no	—
300	C₂×C₄	—	124	2	no	—
301	C₂×C₄	—	124	2	no	—
302	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
303	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
304	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₈	—	62	1	no	—
308	C₈	—	62	1	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	4	1	no	—
311	C₂²	—	4	1	no	—
312	C₂²	—	4	1	no	—
313	C₂²	—	4	1	no	—
314	C₂²	—	4	1	no	—
315	C₂²	C₂×D₄×D₃₁	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	4	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	31	2	no	—
321	C₄	—	62	1	no	—
322	C₄	—	62	1	no	—
323	C₄	—	62	1	no	—
324	C₄	—	124	1	no	—
325	C₄	—	124	1	no	—
326	C₄	—	124	1	no	—
327	C₄	—	124	1	no	—
328	C₄	C₂×#92497	1	1	yes	characteristic
329	C₄	#11758567	1	1	yes	characteristic
330	C₂	—	4	2	no	—
331	C₂	—	8	1	no	—
332	C₂	#14521705	1	1	yes	characteristic
333	C₂	AGL(1,ℤ/8)×D₃₁	1	1	yes	characteristic
334	C₂	#14521863	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×#463	C₂
socle	C₂²×C₃₁	C₂²×D₄
prime radical	C₂×D₈×C₃₁	C₂²
hypercenter	#463	D₃₁
hypocenter	C₃₁	#3367
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #463

Complements

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

N	H ≅ G/N	type
1. #14521768	C₂	semidirect
2. D₄×Dic₆₂	C₂	semidirect
3. #14521408	C₂	semidirect
4. #14521407	C₂	semidirect
5. #14521398	C₂	semidirect
6. #14521345	C₂	semidirect
7. #14521342	C₂	semidirect
8. #14521327	C₂	semidirect
10. #14521252	C₂	semidirect
11. #14521178	C₂	semidirect
12. #14521150	C₂	semidirect
13. #14521149	C₂	semidirect
226. C₃₁	#3367	semidirect

Maximal subgroups

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

subgroup	index
#14521768	2
D₄×Dic₆₂	2
#14521408	2
#14521407	2
#14521398	2
#14521345	2
#14521342	2
#14521327	2
#14521262	2
#14521252	2
#14521178	2
#14521150	2
#14521149	2
D₈×Dic₃₁	2
C₃₁×#463	2
#3367	31