Subgroups — #16855862

Order 3968.

430 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14521599	C₂	1	1	yes	maximal, characteristic, complemented
2	#14521373	C₂	1	2	yes	maximal, complemented
3	#14521334	C₂	1	1	yes	maximal, characteristic, complemented
4	#14521329	C₂	1	1	yes	maximal, characteristic, complemented
5	#14521249	C₂	1	1	yes	maximal, characteristic, complemented
6	#14521244	C₂	1	1	yes	maximal, characteristic, complemented
7	#14521194	C₂	1	2	yes	maximal, complemented
8	#14521159	C₂	1	2	yes	maximal, complemented
9	#14521147	C₂	1	2	yes	maximal, complemented
10	#14521106	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#41×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×#92495	—	2	1	no	—
14	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₄×D₄×C₃₁	—	2	1	no	—
17	C₄×D₄×C₃₁	—	2	1	no	—
18	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
19	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
20	C₃₁×#117	—	2	1	no	—
21	C₃₁×#117	C₂²	1	1	yes	characteristic
22	C₂×C₄²×C₃₁	—	2	1	no	—
23	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
24	C₂×#92481	C₂²	1	2	yes	—
25	C₂×#92479	C₂²	1	2	yes	—
26	#11758507	—	2	1	no	—
27	#11758507	C₂²	1	2	yes	—
28	#11758452	—	2	1	no	—
29	#11758451	C₂²	1	1	yes	characteristic
30	C₄×D₁₂₄	—	2	1	no	—
31	C₄×D₁₂₄	C₂²	1	2	yes	—
32	#11758446	C₂²	1	1	yes	characteristic
33	#11758423	C₂²	1	2	yes	—
34	#11758420	C₂²	1	2	yes	—
35	C₂²⋊(C₃₁⋊C₈)	C₂²	1	2	yes	—
36	#11758397	C₂²	1	2	yes	—
37	#11758397	C₂²	1	2	yes	—
38	#11758396	C₂²	1	2	yes	—
39	#11758395	C₂²	1	2	yes	—
40	C₄⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
41	C₄⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
42	#11758391	C₂²	1	2	yes	—
43	#41×C₃₁	—	2	2	no	—
44	#41×C₃₁	—	2	2	no	—
45	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
46	C₂×D₄×C₃₁	—	2	1	no	—
47	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₃₁	—	2	1	no	—
49	C₂²×C₄×C₃₁	—	2	1	no	—
50	C₂²×C₄×C₃₁	—	2	1	no	—
51	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
52	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
53	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
54	C₂×C₂²⋊D₃₁	—	2	2	no	—
55	#92495	—	4	4	no	—
56	C₂×D₁₂₄	—	2	1	no	—
57	C₂×D₁₂₄	—	2	1	no	—
58	C₂×D₁₂₄	C₂³	1	2	yes	—
59	C₂×C₄×D₃₁	—	2	2	no	—
60	C₂×Dic₆₂	—	2	1	no	—
61	C₄⋊C₄×C₃₁	—	2	1	no	—
62	C₄⋊C₄×C₃₁	—	2	1	no	—
63	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
64	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
65	C₂²⋊C₄×C₃₁	—	2	1	no	—
66	C₂²⋊C₄×C₃₁	—	2	1	no	—
67	C₂²⋊C₄×C₃₁	—	2	1	no	—
68	C₄²×C₃₁	—	2	1	no	—
69	C₄²×C₃₁	—	2	1	no	—
70	C₄²×C₃₁	—	2	1	no	—
71	C₄²×C₃₁	D₄	1	1	yes	characteristic
72	C₄²×C₃₁	D₄	1	1	yes	characteristic
73	C₄²×C₃₁	C₂³	1	1	yes	characteristic
74	C₄²×C₃₁	C₂³	1	1	yes	characteristic
75	#92481	—	2	4	no	—
76	#92479	—	2	4	no	—
77	(C₂×D₃₁)⋊C₄	—	2	2	no	—
78	(C₂×D₃₁)⋊C₄	—	2	2	no	—
79	C₄⋊Dic₃₁	—	2	1	no	—
80	C₄⋊Dic₃₁	C₂³	1	2	yes	—
81	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
82	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
83	C₂³×C₃₁	—	2	1	no	—
84	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
85	C₂²×D₃₁	—	2	2	no	—
86	Q₈×C₃₁	—	2	1	no	—
87	Q₈×C₃₁	#41	1	2	yes	complemented
88	D₄×C₃₁	—	2	4	no	—
89	D₄×C₃₁	—	2	1	no	—
90	D₄×C₃₁	#41	1	2	yes	complemented
91	C₂×C₄×C₃₁	—	2	4	no	—
92	C₂×C₄×C₃₁	—	2	1	no	—
93	C₂×C₄×C₃₁	—	2	2	no	—
94	C₂×C₄×C₃₁	—	2	1	no	—
95	C₂×C₄×C₃₁	—	2	1	no	—
96	C₂×C₄×C₃₁	—	2	1	no	—
97	C₂×C₄×C₃₁	—	2	1	no	—
98	C₂×C₄×C₃₁	—	4	1	no	—
99	C₂×C₄×C₃₁	—	4	2	no	—
100	C₂×C₄×C₃₁	—	4	2	no	—
101	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
107	C₂²⋊D₃₁	—	4	4	no	—
108	C₂×Dic₃₁	—	2	2	no	—
109	D₁₂₄	—	2	2	no	—
110	D₁₂₄	—	2	2	no	—
111	D₁₂₄	—	4	2	no	—
112	D₁₂₄	—	4	2	no	—
113	C₄×D₃₁	—	4	4	no	—
114	Dic₆₂	—	4	2	no	—
115	C₃₁⋊C₈	—	2	2	no	—
116	C₃₁⋊C₈	—	2	2	no	—
117	#3115	—	31	1	no	maximal, Sylow
118	C₂²×C₃₁	—	2	1	no	—
119	C₂²×C₃₁	—	2	2	no	—
120	C₂²×C₃₁	—	2	1	no	—
121	C₂²×C₃₁	—	2	1	no	—
122	C₂²×C₃₁	—	4	1	no	—
123	C₂²×C₃₁	—	4	1	no	—
124	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
125	C₂×D₃₁	—	4	2	no	—
126	C₂×D₃₁	—	4	2	no	—
127	C₂×D₃₁	—	4	2	no	—
128	C₄×C₃₁	—	2	1	no	—
129	C₄×C₃₁	—	2	2	no	—
130	C₄×C₃₁	—	2	1	no	—
131	C₄×C₃₁	—	2	1	no	—
132	C₄×C₃₁	—	2	1	no	—
133	C₄×C₃₁	—	4	1	no	—
134	C₄×C₃₁	—	4	1	no	—
135	C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
136	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
137	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
138	Dic₃₁	—	4	2	no	—
139	#532	—	31	1	no	—
140	C₄×#41	D₃₁	1	1	yes	characteristic, complemented
141	#482	—	31	2	no	—
142	#475	—	31	1	no	—
143	#474	—	31	1	no	—
144	#460	—	31	1	no	—
145	#459	—	31	1	no	—
146	#449	—	31	2	no	—
147	#442	—	31	2	no	—
148	#440	—	31	2	no	—
149	#432	—	31	1	no	—
150	C₂×C₃₁	—	4	1	no	—
151	C₂×C₃₁	—	4	2	no	—
152	C₂×C₃₁	#525	1	1	yes	characteristic
153	C₂×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
154	C₂×C₃₁	#575	1	1	yes	characteristic
155	D₃₁	—	8	2	no	—
156	C₂×#41	—	62	1	no	—
157	C₂×#41	C₂×D₃₁	1	1	yes	characteristic
158	C₂×SD₁₆	—	31	2	no	—
159	C₂×D₈	—	31	2	no	—
160	C₄⋊Q₈	—	31	1	no	—
161	#127	—	31	1	no	—
162	#124	—	62	1	no	—
163	#121	—	31	2	no	—
164	#121	—	62	1	no	—
165	C₄×Q₈	C₂×D₃₁	1	1	yes	characteristic
166	C₄×Q₈	C₂×D₃₁	1	1	yes	characteristic
167	C₄×D₄	—	2	1	no	—
168	C₄×D₄	—	2	1	no	—
169	C₄×D₄	—	31	2	no	—
170	C₄×D₄	—	62	1	no	—
171	C₄×D₄	C₂×D₃₁	1	1	yes	characteristic
172	C₄×D₄	C₂×D₃₁	1	1	yes	characteristic
173	#117	—	2	1	no	—
174	#117	C₂×D₃₁	1	1	yes	characteristic
175	C₂×C₄²	—	2	1	no	—
176	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
177	#107	—	31	2	no	—
178	#106	—	31	2	no	—
179	C₄⋊C₈	—	31	1	no	—
180	C₄⋊C₈	—	31	1	no	—
181	#103	—	31	2	no	—
182	#102	—	31	2	no	—
183	#102	—	31	2	no	—
184	#102	—	31	2	no	—
185	C₂²⋊C₈	—	31	2	no	—
186	#97	—	31	2	no	—
187	C₃₁	#3115	1	1	yes	characteristic, complemented, Sylow
188	#41	—	2	2	no	—
189	#41	—	2	2	no	—
190	#41	—	124	4	no	—
191	C₂×Q₈	—	62	1	no	—
192	C₂×Q₈	C₂²×D₃₁	1	1	yes	characteristic
193	C₂×D₄	—	2	1	no	—
194	C₂×D₄	—	31	2	no	—
195	C₂×D₄	—	62	2	no	—
196	C₂×D₄	—	62	1	no	—
197	C₂×D₄	—	62	1	no	—
198	C₂×D₄	C₂²×D₃₁	1	1	yes	characteristic
199	C₂²×C₄	—	2	1	no	—
200	C₂²×C₄	—	2	1	no	—
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	62	2	no	—
203	C₂²×C₄	C₂²⋊D₃₁	1	1	yes	characteristic
204	C₂²×C₄	C₂²⋊D₃₁	1	1	yes	characteristic
205	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
206	SD₁₆	—	62	4	no	—
207	D₈	—	62	4	no	—
208	C₂×C₈	—	31	2	no	—
209	C₂×C₈	—	31	2	no	—
210	C₄⋊C₄	—	2	1	no	—
211	C₄⋊C₄	—	2	1	no	—
212	C₄⋊C₄	—	31	2	no	—
213	C₄⋊C₄	—	62	1	no	—
214	C₄⋊C₄	C₂²×D₃₁	1	1	yes	characteristic
215	C₄⋊C₄	C₂²×D₃₁	1	1	yes	characteristic
216	C₂²⋊C₄	—	2	1	no	—
217	C₂²⋊C₄	—	2	1	no	—
218	C₂²⋊C₄	—	2	1	no	—
219	C₂²⋊C₄	—	62	2	no	—
220	C₂²⋊C₄	—	62	2	no	—
221	C₄²	—	2	1	no	—
222	C₄²	—	2	1	no	—
223	C₄²	—	2	1	no	—
224	C₄²	C₂²⋊D₃₁	1	1	yes	characteristic
225	C₄²	C₂²⋊D₃₁	1	1	yes	characteristic
226	C₄²	C₂²×D₃₁	1	1	yes	characteristic
227	C₄²	C₂²×D₃₁	1	1	yes	characteristic
228	C₂³	—	2	1	no	—
229	C₂³	—	62	2	no	—
230	C₂³	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
231	Q₈	—	2	1	no	—
232	Q₈	—	124	2	no	—
233	Q₈	#92495	1	2	yes	complemented
234	D₄	—	2	4	no	—
235	D₄	—	2	1	no	—
236	D₄	—	62	2	no	—
237	D₄	—	62	2	no	—
238	D₄	—	124	2	no	—
239	D₄	—	124	4	no	—
240	D₄	—	124	2	no	—
241	D₄	#92495	1	2	yes	complemented
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	62	2	no	—
253	C₂×C₄	—	124	4	no	—
254	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
255	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
256	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
257	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
258	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
259	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
260	C₈	—	62	2	no	—
261	C₈	—	62	2	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	4	1	no	—
268	C₂²	—	124	2	no	—
269	C₂²	—	124	2	no	—
270	C₂²	—	124	2	no	—
271	C₂²	C₂²×C₂²⋊D₃₁	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	2	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	4	1	no	—
278	C₄	—	4	1	no	—
279	C₄	—	124	2	no	—
280	C₄	#11758526	1	2	yes	—
281	C₄	C₂×#92495	1	1	yes	characteristic
282	C₄	C₂×#92495	1	1	yes	characteristic
283	C₂	—	4	1	no	—
284	C₂	—	4	2	no	—
285	C₂	—	248	2	no	—
286	C₂	#14521560	1	1	yes	characteristic
287	C₂	C₂×#11758526	1	1	yes	characteristic
288	C₂	#14521847	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₃₁
commutator, cosocle	C₂×C₄×C₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₄×#41×C₃₁	C₂
socle	C₂²×C₃₁	C₂²×D₄
hypercenter	C₄×#41	D₃₁
hypocenter	C₃₁	#3115
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. #14521599	C₂	semidirect
2. #14521373	C₂	semidirect
3. #14521334	C₂	semidirect
4. #14521329	C₂	semidirect
5. #14521249	C₂	semidirect
6. #14521244	C₂	semidirect
7. #14521194	C₂	semidirect
8. #14521159	C₂	semidirect
9. #14521147	C₂	semidirect
10. #14521106	C₂	semidirect
11. C₄×#41×C₃₁	C₂	semidirect
87. Q₈×C₃₁	#41	semidirect
90. D₄×C₃₁	#41	semidirect
140. C₄×#41	D₃₁	semidirect
187. C₃₁	#3115	semidirect
233. Q₈	#92495	semidirect
241. D₄	#92495	semidirect

Maximal subgroups

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

subgroup	index
#14521599	2
#14521373	2
#14521334	2
#14521329	2
#14521249	2
#14521244	2
#14521194	2
#14521159	2
#14521147	2
#14521106	2
C₄×#41×C₃₁	2
#3115	31