Subgroups — #16854838

Order 3968.

398 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#92495	C₂	1	1	yes	maximal, characteristic
2	#14521177	C₂	1	2	yes	maximal, complemented
3	#14521122	C₂	1	2	yes	maximal, complemented
4	#14521117	C₂	1	2	yes	maximal, complemented
5	(C₂²⋊D₃₁)⋊C₈	C₂	1	2	yes	maximal, complemented
6	#14521105	C₂	1	1	yes	maximal, characteristic, complemented
7	#14521101	C₂	1	1	yes	maximal, characteristic, complemented
8	#14521071	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄⋊(C₃₁⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄⋊C₈×D₃₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#432	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#92495	C₄	1	1	yes	characteristic
13	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
14	C₄×C₂²⋊D₃₁	C₄	1	1	yes	characteristic
15	C₄×C₂²⋊D₃₁	C₄	1	1	yes	characteristic
16	C₄×C₂²⋊D₃₁	C₂²	1	1	yes	characteristic
17	C₄×C₂²⋊D₃₁	C₂²	1	1	yes	characteristic
18	#11758503	C₄	1	1	yes	characteristic
19	C₂²×C₃₁⋊C₈	C₂²	1	2	yes	—
20	C₂×#92469	C₂²	1	2	yes	—
21	C₂×C₈×D₃₁	C₂²	1	2	yes	—
22	C₄×D₁₂₄	C₄	1	1	yes	characteristic
23	C₄×D₁₂₄	C₂²	1	1	yes	characteristic
24	#11758449	C₄	1	1	yes	characteristic
25	#11758449	C₄	1	1	yes	characteristic
26	C₄²×D₃₁	C₂²	1	1	yes	characteristic
27	C₄²×D₃₁	C₂²	1	1	yes	characteristic
28	C₄×Dic₆₂	C₄	1	1	yes	characteristic
29	C₄×Dic₆₂	C₂²	1	1	yes	characteristic
30	C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
31	C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
32	C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
33	C₃₁×#97	C₂²	1	2	yes	—
34	(C₂×D₃₁)⋊C₈	C₂²	1	2	yes	—
35	(C₂×D₃₁)⋊C₈	C₂²	1	2	yes	—
36	#11758403	C₂²	1	2	yes	—
37	Dic₃₁⋊C₈	C₂²	1	2	yes	—
38	Dic₃₁⋊C₈	C₂²	1	2	yes	—
39	C₈×Dic₃₁	C₂²	1	2	yes	—
40	C₄⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
41	C₄⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
42	C₄⋊(C₃₁⋊C₈)	C₂²	1	2	yes	—
43	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
46	C₂×C₂²⋊D₃₁	C₂×C₄	1	1	yes	characteristic
47	C₂×C₂²⋊D₃₁	C₂×C₄	1	1	yes	characteristic
48	#92495	—	2	4	no	—
49	C₂×D₁₂₄	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×D₃₁	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₃₁	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₃₁	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄×D₃₁	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×D₃₁	C₂³	1	1	yes	characteristic
55	C₂×C₄×D₃₁	C₂³	1	1	yes	characteristic
56	C₂×Dic₆₂	C₂×C₄	1	1	yes	characteristic
57	C₂×C₈×C₃₁	C₂³	1	2	yes	—
58	C₂×C₈×C₃₁	C₂³	1	2	yes	—
59	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
60	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
61	C₄²×C₃₁	C₂³	1	1	yes	characteristic
62	C₄²×C₃₁	C₂³	1	1	yes	characteristic
63	C₂²⋊Dic₃₁	C₂×C₄	1	1	yes	characteristic
64	C₂²⋊Dic₃₁	C₂×C₄	1	1	yes	characteristic
65	(C₂×D₃₁)⋊C₄	C₂×C₄	1	1	yes	characteristic
66	(C₂×D₃₁)⋊C₄	C₂×C₄	1	1	yes	characteristic
67	(C₂×D₃₁)⋊C₄	C₂×C₄	1	1	yes	characteristic
68	(C₂×D₃₁)⋊C₄	C₂×C₄	1	1	yes	characteristic
69	C₄⋊Dic₃₁	C₂×C₄	1	1	yes	characteristic
70	C₄⋊Dic₃₁	C₂×C₄	1	1	yes	characteristic
71	Dic₃₁⋊C₄	C₂×C₄	1	1	yes	characteristic
72	Dic₃₁⋊C₄	C₂×C₄	1	1	yes	characteristic
73	Dic₃₁⋊C₄	C₂×C₄	1	1	yes	characteristic
74	Dic₃₁⋊C₄	C₂×C₄	1	1	yes	characteristic
75	C₄×Dic₃₁	C₂×C₄	1	1	yes	characteristic
76	C₄×Dic₃₁	C₂×C₄	1	1	yes	characteristic
77	C₄×Dic₃₁	C₂×C₄	1	1	yes	characteristic
78	C₄×Dic₃₁	C₂×C₄	1	1	yes	characteristic
79	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
80	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
81	C₂×C₃₁⋊C₈	—	4	2	no	—
82	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
83	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
84	#92469	—	4	2	no	—
85	C₈×D₃₁	—	4	2	no	—
86	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
87	C₂²×D₃₁	C₂²×C₄	1	1	yes	characteristic
88	C₂²×D₃₁	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₃₁	—	2	2	no	—
90	C₂×C₄×C₃₁	—	2	2	no	—
91	C₂×C₄×C₃₁	—	2	1	no	—
92	C₂×C₄×C₃₁	—	2	1	no	—
93	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
99	C₂²⋊D₃₁	—	2	2	no	—
100	C₂²⋊D₃₁	M₁₆	1	4	yes	complemented
101	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
102	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
103	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
104	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
105	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
106	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
107	D₁₂₄	—	2	2	no	—
108	C₄×D₃₁	—	2	2	no	—
109	C₄×D₃₁	—	2	2	no	—
110	C₄×D₃₁	—	2	2	no	—
111	C₄×D₃₁	—	2	1	no	—
112	C₄×D₃₁	—	2	1	no	—
113	C₄×D₃₁	—	2	1	no	—
114	C₄×D₃₁	—	2	1	no	—
115	C₄×D₃₁	—	2	2	no	—
116	Dic₆₂	—	2	2	no	—
117	C₈×C₃₁	—	2	2	no	—
118	C₈×C₃₁	—	2	2	no	—
119	C₃₁⋊C₈	—	2	2	no	—
120	C₃₁⋊C₈	—	2	2	no	—
121	#2990	—	31	1	no	maximal, Sylow
122	C₂²×C₃₁	—	2	1	no	—
123	C₂²×C₃₁	—	2	1	no	—
124	C₂²×C₃₁	C₂×M₁₆	1	2	yes	—
125	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
126	C₂×D₃₁	—	2	1	no	—
127	C₂×D₃₁	—	2	1	no	—
128	C₂×D₃₁	—	2	1	no	—
129	C₂×D₃₁	—	2	1	no	—
130	C₂×D₃₁	—	2	1	no	—
131	C₂×D₃₁	C₂×M₁₆	1	2	yes	—
132	C₄×C₃₁	—	2	1	no	—
133	C₄×C₃₁	—	2	1	no	—
134	C₄×C₃₁	—	2	1	no	—
135	C₄×C₃₁	—	2	1	no	—
136	C₄×C₃₁	#131	1	2	yes	—
137	C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
138	C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
139	Dic₃₁	—	2	1	no	—
140	Dic₃₁	—	2	1	no	—
141	Dic₃₁	—	2	1	no	—
142	Dic₃₁	—	2	1	no	—
143	Dic₃₁	—	2	1	no	—
144	Dic₃₁	C₂×M₁₆	1	2	yes	—
145	C₄×#41	—	31	1	no	—
146	#446	—	31	2	no	—
147	#435	—	31	2	no	—
148	#435	—	31	2	no	—
149	C₈×D₄	—	31	2	no	—
150	#432	—	31	1	no	—
151	#432	D₃₁	1	1	yes	characteristic, complemented
152	#431	—	31	1	no	—
153	#423	—	31	1	no	—
154	C₂×C₄⋊C₈	—	31	1	no	—
155	C₂×C₄⋊C₈	—	31	1	no	—
156	C₂×C₃₁	—	2	2	no	—
157	C₂×C₃₁	#520	1	1	yes	characteristic
158	C₂×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
159	C₂×C₃₁	C₂×#131	1	1	yes	characteristic
160	D₃₁	—	2	2	no	—
161	D₃₁	—	4	1	no	—
162	C₂×#41	—	31	1	no	—
163	C₂×M₁₆	—	31	2	no	—
164	C₂²×C₈	—	31	2	no	—
165	C₂²×C₈	—	31	2	no	—
166	C₄×Q₈	—	31	1	no	—
167	C₄×Q₈	—	31	1	no	—
168	C₄×D₄	—	31	1	no	—
169	C₄×D₄	—	31	1	no	—
170	C₄×D₄	—	31	1	no	—
171	C₄×D₄	—	31	1	no	—
172	C₄×D₄	—	31	1	no	—
173	C₄×D₄	—	31	1	no	—
174	#117	—	31	1	no	—
175	#117	—	31	1	no	—
176	#117	—	31	1	no	—
177	C₂×C₄²	—	31	1	no	—
178	C₂×C₄²	—	31	1	no	—
179	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
180	C₄⋊C₈	—	31	2	no	—
181	C₄⋊C₈	—	31	2	no	—
182	C₄⋊C₈	—	31	1	no	—
183	C₄⋊C₈	—	31	1	no	—
184	C₄⋊C₈	—	31	2	no	—
185	C₄⋊C₈	C₂×D₃₁	1	1	yes	characteristic
186	C₄⋊C₈	C₂×D₃₁	1	1	yes	characteristic
187	C₂²⋊C₈	—	31	2	no	—
188	C₂²⋊C₈	—	31	2	no	—
189	C₂²⋊C₈	C₂×D₃₁	1	2	yes	—
190	#97	—	31	2	no	—
191	#97	C₂×D₃₁	1	2	yes	—
192	C₄×C₈	—	31	2	no	—
193	C₃₁	#2990	1	1	yes	characteristic, complemented, Sylow
194	#41	—	62	4	no	—
195	C₂×Q₈	—	31	1	no	—
196	C₂×D₄	—	31	1	no	—
197	C₂×D₄	—	31	1	no	—
198	C₂×D₄	—	31	1	no	—
199	C₂²×C₄	—	31	1	no	—
200	C₂²×C₄	—	31	1	no	—
201	C₂²×C₄	—	31	1	no	—
202	C₂²×C₄	—	31	1	no	—
203	C₂²×C₄	—	31	1	no	—
204	C₂²×C₄	—	31	1	no	—
205	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
206	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
207	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
208	M₁₆	—	124	2	no	—
209	C₂×C₈	—	31	2	no	—
210	C₂×C₈	—	31	2	no	—
211	C₂×C₈	—	124	2	no	—
212	C₂×C₈	—	124	2	no	—
213	C₂×C₈	C₂²×D₃₁	1	2	yes	—
214	C₂×C₈	C₂²×D₃₁	1	2	yes	—
215	C₄⋊C₄	—	31	1	no	—
216	C₄⋊C₄	—	31	1	no	—
217	C₄⋊C₄	—	31	1	no	—
218	C₄⋊C₄	—	31	1	no	—
219	C₄⋊C₄	—	31	1	no	—
220	C₄⋊C₄	—	31	1	no	—
221	C₂²⋊C₄	—	31	1	no	—
222	C₂²⋊C₄	—	31	1	no	—
223	C₂²⋊C₄	—	31	1	no	—
224	C₂²⋊C₄	—	31	1	no	—
225	C₂²⋊C₄	—	31	1	no	—
226	C₂²⋊C₄	—	31	1	no	—
227	C₄²	—	31	1	no	—
228	C₄²	—	31	1	no	—
229	C₄²	—	31	1	no	—
230	C₄²	—	31	1	no	—
231	C₄²	—	31	1	no	—
232	C₄²	—	31	1	no	—
233	C₄²	C₄×D₃₁	1	1	yes	characteristic
234	C₄²	C₄×D₃₁	1	1	yes	characteristic
235	C₄²	C₂²×D₃₁	1	1	yes	characteristic
236	C₄²	C₂²×D₃₁	1	1	yes	characteristic
237	C₂³	—	31	1	no	—
238	C₂³	—	31	1	no	—
239	C₂³	C₂×C₄×D₃₁	1	1	yes	characteristic
240	Q₈	—	62	2	no	—
241	D₄	—	31	4	no	—
242	D₄	—	62	2	no	—
243	D₄	—	62	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	31	1	no	—
249	C₂×C₄	—	31	1	no	—
250	C₂×C₄	—	31	1	no	—
251	C₂×C₄	—	31	1	no	—
252	C₂×C₄	—	31	1	no	—
253	C₂×C₄	—	31	1	no	—
254	C₂×C₄	—	62	2	no	—
255	C₂×C₄	—	62	2	no	—
256	C₂×C₄	—	62	2	no	—
257	C₂×C₄	—	62	2	no	—
258	C₂×C₄	—	62	1	no	—
259	C₂×C₄	—	62	1	no	—
260	C₂×C₄	—	62	1	no	—
261	C₂×C₄	—	62	1	no	—
262	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
263	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
264	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
265	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
266	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
267	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
268	C₈	—	2	2	no	—
269	C₈	—	2	2	no	—
270	C₈	—	62	2	no	—
271	C₈	—	62	2	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	31	2	no	—
275	C₂²	—	62	1	no	—
276	C₂²	—	62	1	no	—
277	C₂²	—	62	1	no	—
278	C₂²	—	62	1	no	—
279	C₂²	—	62	1	no	—
280	C₂²	M₁₆×D₃₁	1	2	yes	—
281	C₂²	C₂²×C₄×D₃₁	1	1	yes	characteristic
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	31	2	no	—
287	C₄	—	62	1	no	—
288	C₄	—	62	1	no	—
289	C₄	—	62	1	no	—
290	C₄	—	62	1	no	—
291	C₄	—	62	1	no	—
292	C₄	#11758484	1	2	yes	—
293	C₄	#11758566	1	1	yes	characteristic
294	C₄	#11758567	1	1	yes	characteristic
295	C₂	—	2	2	no	—
296	C₂	—	62	2	no	—
297	C₂	—	124	1	no	—
298	C₂	#14521529	1	1	yes	characteristic
299	C₂	C₂×M₁₆×D₃₁	1	1	yes	characteristic
300	C₂	C₂×#11758484	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃₁
commutator, socle	C₂²×C₃₁	C₂³×C₄
Fitting	C₃₁×#432	C₂
prime radical	C₂×#92495	C₄
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	#432	D₃₁
hypocenter	C₃₁	#2990
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ #432

Complements

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

N	H ≅ G/N	type
2. #14521177	C₂	semidirect
3. #14521122	C₂	semidirect
4. #14521117	C₂	semidirect
5. (C₂²⋊D₃₁)⋊C₈	C₂	semidirect
6. #14521105	C₂	semidirect
7. #14521101	C₂	semidirect
8. #14521071	C₂	semidirect
9. C₂×C₄⋊(C₃₁⋊C₈)	C₂	semidirect
10. C₄⋊C₈×D₃₁	C₂	semidirect
11. C₃₁×#432	C₂	semidirect
100. C₂²⋊D₃₁	M₁₆	semidirect
151. #432	D₃₁	semidirect
193. C₃₁	#2990	semidirect

Maximal subgroups

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

subgroup	index
C₄×#92495	2
#14521177	2
#14521122	2
#14521117	2
(C₂²⋊D₃₁)⋊C₈	2
#14521105	2
#14521101	2
#14521071	2
C₂×C₄⋊(C₃₁⋊C₈)	2
C₄⋊C₈×D₃₁	2
C₃₁×#432	2
#2990	31