Subgroups — #16854873

Order 3968.

364 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14521583	C₂	1	1	yes	maximal, characteristic
2	#14521179	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×Dic₆₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#14521128	C₂	1	1	yes	maximal, characteristic, complemented
5	#14521117	C₂	1	2	yes	maximal, complemented
6	C₈×D₁₂₄	C₂	1	1	yes	maximal, characteristic, complemented
7	(C₂²⋊D₃₁)⋊C₈	C₂	1	2	yes	maximal, complemented
8	#14521109	C₂	1	2	yes	maximal, complemented
9	#14521074	C₂	1	2	yes	maximal, complemented
10	C₂×C₄⋊(C₃₁⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#431	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
13	#11758507	C₄	1	1	yes	characteristic
14	#11758506	C₄	1	2	yes	—
15	C₄×C₂²⋊D₃₁	C₂²	1	2	yes	—
16	#11758501	C₄	1	1	yes	characteristic
17	C₂²×C₃₁⋊C₈	C₂²	1	2	yes	—
18	C₂×#92469	C₂²	1	2	yes	—
19	C₂×C₈×D₃₁	C₂²	1	2	yes	—
20	#11758453	C₄	1	2	yes	—
21	#11758452	C₄	1	1	yes	characteristic
22	C₄×D₁₂₄	C₂²	1	1	yes	characteristic
23	#11758449	C₂²	1	2	yes	—
24	#11758447	C₄	1	1	yes	characteristic
25	C₄×Dic₆₂	C₂²	1	1	yes	characteristic
26	C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
27	C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
28	C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
29	C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
30	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
31	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
32	(C₂×D₃₁)⋊C₈	C₂²	1	2	yes	—
33	(C₂×D₃₁)⋊C₈	C₂²	1	2	yes	—
34	#11758403	C₂²	1	2	yes	—
35	Dic₃₁⋊C₈	C₂²	1	2	yes	—
36	Dic₃₁⋊C₈	C₂²	1	2	yes	—
37	C₈×Dic₃₁	C₂²	1	2	yes	—
38	C₄⋊(C₃₁⋊C₈)	C₂²	1	2	yes	—
39	C₄⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
40	C₄⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
41	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
44	C₂×C₂²⋊D₃₁	C₂×C₄	1	2	yes	—
45	C₂×D₁₂₄	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₃₁	C₂³	1	2	yes	—
47	C₂×Dic₆₂	C₂×C₄	1	1	yes	characteristic
48	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
49	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
50	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
51	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
52	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
53	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
54	C₄²×C₃₁	C₂³	1	1	yes	characteristic
55	C₄²×C₃₁	C₂³	1	1	yes	characteristic
56	C₂²⋊Dic₃₁	C₂×C₄	1	2	yes	—
57	(C₂×D₃₁)⋊C₄	C₂×C₄	1	2	yes	—
58	(C₂×D₃₁)⋊C₄	C₂×C₄	1	2	yes	—
59	(C₂×D₃₁)⋊C₄	C₂×C₄	1	2	yes	—
60	(C₂×D₃₁)⋊C₄	C₂×C₄	1	2	yes	—
61	C₄⋊Dic₃₁	C₂×C₄	1	1	yes	characteristic
62	C₄⋊Dic₃₁	C₂×C₄	1	1	yes	characteristic
63	Dic₃₁⋊C₄	C₂×C₄	1	2	yes	—
64	Dic₃₁⋊C₄	C₂×C₄	1	2	yes	—
65	Dic₃₁⋊C₄	C₂×C₄	1	2	yes	—
66	Dic₃₁⋊C₄	C₂×C₄	1	2	yes	—
67	C₄×Dic₃₁	C₂³	1	2	yes	—
68	C₂×C₃₁⋊C₈	—	4	2	no	—
69	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
70	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
71	#92469	—	4	2	no	—
72	C₈×D₃₁	—	4	2	no	—
73	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
74	C₂²×D₃₁	C₂²×C₄	1	2	yes	—
75	C₂×C₄×C₃₁	—	2	2	no	—
76	C₂×C₄×C₃₁	—	2	2	no	—
77	C₂×C₄×C₃₁	—	2	1	no	—
78	C₂×C₄×C₃₁	—	2	1	no	—
79	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
85	C₂²⋊D₃₁	—	2	4	no	—
86	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
87	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
88	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
89	D₁₂₄	—	2	2	no	—
90	C₄×D₃₁	—	2	2	no	—
91	C₄×D₃₁	—	2	2	no	—
92	Dic₆₂	—	2	2	no	—
93	C₈×C₃₁	—	2	1	no	—
94	C₈×C₃₁	—	2	1	no	—
95	C₈×C₃₁	—	2	1	no	—
96	C₈×C₃₁	—	2	1	no	—
97	C₃₁⋊C₈	—	2	2	no	—
98	C₃₁⋊C₈	—	2	2	no	—
99	#2994	—	31	1	no	maximal, Sylow
100	C₂²×C₃₁	—	2	1	no	—
101	C₂²×C₃₁	—	2	1	no	—
102	C₂²×C₃₁	#131	1	2	yes	—
103	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
104	C₂×D₃₁	—	2	2	no	—
105	C₂×D₃₁	—	2	2	no	—
106	C₂×D₃₁	—	2	2	no	—
107	C₄×C₃₁	—	2	1	no	—
108	C₄×C₃₁	—	2	1	no	—
109	C₄×C₃₁	—	2	1	no	—
110	C₄×C₃₁	—	2	1	no	—
111	C₄×C₃₁	#131	1	2	yes	—
112	C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
113	C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
114	Dic₃₁	—	2	2	no	—
115	Dic₃₁	—	2	2	no	—
116	Dic₃₁	—	2	2	no	—
117	#529	—	31	1	no	—
118	#446	—	31	1	no	—
119	C₈×Q₈	—	31	1	no	—
120	#436	—	31	1	no	—
121	#435	—	31	2	no	—
122	C₈×D₄	—	31	2	no	—
123	C₈×D₄	—	31	1	no	—
124	#433	—	31	2	no	—
125	#431	D₃₁	1	1	yes	characteristic, complemented
126	#424	—	31	2	no	—
127	C₂×C₄⋊C₈	—	31	1	no	—
128	C₂×C₃₁	—	2	2	no	—
129	C₂×C₃₁	#520	1	1	yes	characteristic
130	C₂×C₃₁	C₂×#131	1	1	yes	characteristic
131	C₂×C₃₁	C₂×#131	1	1	yes	characteristic
132	D₃₁	—	4	2	no	—
133	C₂×M₁₆	—	31	2	no	—
134	C₂²×C₈	—	31	2	no	—
135	C₂²×C₈	—	31	2	no	—
136	#126	—	31	2	no	—
137	#125	—	31	1	no	—
138	#124	—	31	1	no	—
139	#123	—	31	2	no	—
140	C₂²⋊Q₈	—	31	1	no	—
141	#121	—	31	1	no	—
142	C₄×Q₈	—	31	1	no	—
143	C₄×D₄	—	31	2	no	—
144	C₄×D₄	—	31	1	no	—
145	#117	—	31	2	no	—
146	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
147	C₄⋊C₈	—	31	2	no	—
148	C₄⋊C₈	—	31	2	no	—
149	C₄⋊C₈	—	31	2	no	—
150	C₄⋊C₈	—	31	1	no	—
151	C₄⋊C₈	—	31	1	no	—
152	C₄⋊C₈	C₂×D₃₁	1	1	yes	characteristic
153	C₄⋊C₈	C₂×D₃₁	1	1	yes	characteristic
154	C₂²⋊C₈	—	31	2	no	—
155	C₂²⋊C₈	—	31	2	no	—
156	C₂²⋊C₈	C₂×D₃₁	1	1	yes	characteristic
157	C₂²⋊C₈	C₂×D₃₁	1	1	yes	characteristic
158	#97	—	31	2	no	—
159	C₄×C₈	—	31	2	no	—
160	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic
161	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic
162	C₃₁	#2994	1	1	yes	characteristic, complemented, Sylow
163	C₂×Q₈	—	31	1	no	—
164	C₂×D₄	—	31	2	no	—
165	C₂×D₄	—	31	1	no	—
166	C₂²×C₄	—	31	2	no	—
167	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
168	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
169	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
170	M₁₆	—	124	2	no	—
171	C₂×C₈	—	31	2	no	—
172	C₂×C₈	—	31	2	no	—
173	C₂×C₈	—	124	2	no	—
174	C₂×C₈	—	124	2	no	—
175	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
176	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
177	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
178	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
179	C₄⋊C₄	—	31	2	no	—
180	C₄⋊C₄	—	31	2	no	—
181	C₄⋊C₄	—	31	2	no	—
182	C₄⋊C₄	—	31	2	no	—
183	C₄⋊C₄	—	31	1	no	—
184	C₄⋊C₄	—	31	1	no	—
185	C₂²⋊C₄	—	31	2	no	—
186	C₂²⋊C₄	—	31	2	no	—
187	C₂²⋊C₄	—	31	2	no	—
188	C₂²⋊C₄	—	31	2	no	—
189	C₂²⋊C₄	—	31	2	no	—
190	C₄²	—	31	2	no	—
191	C₄²	C₄×D₃₁	1	1	yes	characteristic
192	C₄²	C₄×D₃₁	1	1	yes	characteristic
193	C₄²	C₂²×D₃₁	1	1	yes	characteristic
194	C₄²	C₂²×D₃₁	1	1	yes	characteristic
195	C₂³	—	31	2	no	—
196	C₂³	C₂×C₄×D₃₁	1	1	yes	characteristic
197	Q₈	—	62	2	no	—
198	D₄	—	62	4	no	—
199	D₄	—	62	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	31	2	no	—
205	C₂×C₄	—	31	2	no	—
206	C₂×C₄	—	31	2	no	—
207	C₂×C₄	—	62	2	no	—
208	C₂×C₄	—	62	2	no	—
209	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
210	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
211	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
212	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
213	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
214	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
215	C₈	—	2	1	no	—
216	C₈	—	2	1	no	—
217	C₈	—	2	1	no	—
218	C₈	—	2	1	no	—
219	C₈	—	62	2	no	—
220	C₈	—	62	2	no	—
221	C₂²	—	2	1	no	—
222	C₂²	—	2	1	no	—
223	C₂²	—	62	2	no	—
224	C₂²	—	62	2	no	—
225	C₂²	—	62	2	no	—
226	C₂²	#11758478	1	2	yes	—
227	C₂²	C₂²×C₄×D₃₁	1	1	yes	characteristic
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	62	2	no	—
233	C₄	—	62	2	no	—
234	C₄	—	62	2	no	—
235	C₄	#11758484	1	2	yes	—
236	C₄	#11758566	1	1	yes	characteristic
237	C₄	#11758567	1	1	yes	characteristic
238	C₂	—	2	2	no	—
239	C₂	—	124	2	no	—
240	C₂	#14521529	1	1	yes	characteristic
241	C₂	C₂×#11758478	1	1	yes	characteristic
242	C₂	C₂×#11758484	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃₁
commutator, socle	C₂²×C₃₁	C₂³×C₄
Fitting	C₃₁×#431	C₂
prime radical	#11758507	C₄
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	#431	D₃₁
hypocenter	C₃₁	#2994
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ #431

Complements

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

N	H ≅ G/N	type
2. #14521179	C₂	semidirect
3. C₈×Dic₆₂	C₂	semidirect
4. #14521128	C₂	semidirect
5. #14521117	C₂	semidirect
6. C₈×D₁₂₄	C₂	semidirect
7. (C₂²⋊D₃₁)⋊C₈	C₂	semidirect
8. #14521109	C₂	semidirect
9. #14521074	C₂	semidirect
10. C₂×C₄⋊(C₃₁⋊C₈)	C₂	semidirect
11. C₃₁×#431	C₂	semidirect
125. #431	D₃₁	semidirect
162. C₃₁	#2994	semidirect

Maximal subgroups

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

subgroup	index
#14521583	2
#14521179	2
C₈×Dic₆₂	2
#14521128	2
#14521117	2
C₈×D₁₂₄	2
(C₂²⋊D₃₁)⋊C₈	2
#14521109	2
#14521074	2
C₂×C₄⋊(C₃₁⋊C₈)	2
C₃₁×#431	2
#2994	31