Subgroups — #16854414

Order 3968.

492 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11758478	C₂	1	1	yes	maximal, characteristic, complemented
2	#14521560	C₂	1	1	yes	maximal, characteristic
3	#14521125	C₂	1	4	yes	maximal, complemented
4	#14521118	C₂	1	4	yes	maximal, complemented
5	#14521075	C₂	1	1	yes	maximal, characteristic, complemented
6	#14521015	C₂	1	1	yes	maximal, characteristic, complemented
7	#14521013	C₂	1	1	yes	maximal, characteristic, complemented
8	#14521004	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
11	C₂×#92495	C₂²	1	1	yes	characteristic
12	C₂×M₁₆×C₃₁	—	2	4	no	—
13	C₂×M₁₆×C₃₁	C₂²	1	2	yes	—
14	C₂×M₁₆×C₃₁	C₂²	1	2	yes	—
15	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
17	#11758530	C₄	1	2	yes	—
18	#11758507	C₄	1	2	yes	—
19	#11758506	C₄	1	2	yes	—
20	C₄×C₂²⋊D₃₁	C₂²	1	4	yes	—
21	#11758503	C₂²	1	1	yes	characteristic
22	#11758501	C₄	1	2	yes	—
23	C₂×#92474	C₂²	1	1	yes	characteristic
24	C₂×#92474	C₂²	1	1	yes	characteristic
25	#11758478	—	2	4	no	—
26	C₂×#92469	C₂²	1	2	yes	complemented
27	C₂×C₈×D₃₁	C₂²	1	2	yes	complemented
28	C₂²⋊(C₃₁⋊C₈)	C₂²	1	4	yes	—
29	(C₂×D₃₁)⋊C₈	C₂²	1	4	yes	complemented
30	#11758403	C₂²	1	2	yes	complemented
31	Dic₃₁⋊C₈	C₂²	1	4	yes	complemented
32	C₈×Dic₃₁	C₂²	1	2	yes	complemented
33	C₂⁴×C₃₁	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₃₁	—	2	2	no	—
35	C₂²×C₄×C₃₁	—	2	2	no	—
36	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
37	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
39	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
40	C₂×C₂²⋊D₃₁	C₂×C₄	1	2	yes	—
41	C₂×C₂²⋊D₃₁	C₂×C₄	1	4	yes	—
42	#92495	—	2	2	no	—
43	#92495	—	2	2	no	—
44	C₂×D₁₂₄	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₃₁	C₂³	1	2	yes	—
46	C₂×Dic₆₂	C₂×C₄	1	1	yes	characteristic
47	M₁₆×C₃₁	—	2	4	no	—
48	M₁₆×C₃₁	—	2	4	no	—
49	C₂×C₈×C₃₁	—	2	2	no	—
50	C₂×C₈×C₃₁	D₄	1	4	yes	complemented
51	C₂×C₈×C₃₁	C₂³	1	2	yes	—
52	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
53	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
54	C₂²⋊Dic₃₁	C₂×C₄	1	4	yes	—
55	C₂²⋊Dic₃₁	C₂×C₄	1	2	yes	—
56	(C₂×D₃₁)⋊C₄	C₂×C₄	1	4	yes	—
57	C₄⋊Dic₃₁	C₂×C₄	1	2	yes	—
58	Dic₃₁⋊C₄	C₂×C₄	1	4	yes	—
59	C₄×Dic₃₁	C₂³	1	2	yes	—
60	#92474	—	2	2	no	—
61	#92474	—	2	2	no	—
62	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
63	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
64	#92469	—	2	4	no	—
65	C₈×D₃₁	—	2	4	no	—
66	C₂³×C₃₁	—	2	2	no	—
67	C₂³×C₃₁	—	4	2	no	—
68	C₂³×C₃₁	C₂²×C₄	1	2	yes	—
69	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
70	C₂²×D₃₁	C₂²×C₄	1	2	yes	—
71	C₂×C₄×C₃₁	—	2	2	no	—
72	C₂×C₄×C₃₁	—	2	4	no	—
73	C₂×C₄×C₃₁	—	2	2	no	—
74	C₂×C₄×C₃₁	—	2	2	no	—
75	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
78	C₂×C₄×C₃₁	#41	1	2	yes	—
79	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
80	C₂²⋊D₃₁	—	2	4	no	—
81	C₂²⋊D₃₁	—	4	4	no	—
82	C₂×Dic₃₁	C₂²×C₄	1	4	yes	—
83	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
84	D₁₂₄	—	2	2	no	—
85	C₄×D₃₁	—	2	2	no	—
86	C₄×D₃₁	—	2	2	no	—
87	Dic₆₂	—	2	2	no	—
88	C₈×C₃₁	—	2	2	no	—
89	C₈×C₃₁	C₂×D₄	1	2	yes	—
90	C₈×C₃₁	C₂×D₄	1	2	yes	—
91	C₃₁⋊C₈	—	2	2	no	—
92	C₃₁⋊C₈	—	2	2	no	—
93	#2930	—	31	1	no	maximal, Sylow
94	C₂²×C₃₁	—	2	2	no	—
95	C₂²×C₃₁	—	2	2	no	—
96	C₂²×C₃₁	—	2	2	no	—
97	C₂²×C₃₁	—	2	2	no	—
98	C₂²×C₃₁	—	4	4	no	—
99	C₂²×C₃₁	C₄×D₄	1	2	yes	—
100	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
101	C₂×D₃₁	—	2	2	no	—
102	C₂×D₃₁	—	2	4	no	—
103	C₄×C₃₁	—	2	2	no	—
104	C₄×C₃₁	C₄×D₄	1	2	yes	—
105	C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
106	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
107	Dic₃₁	—	2	2	no	—
108	Dic₃₁	—	2	4	no	—
109	C₂×#131	—	31	1	no	—
110	C₂²×M₁₆	D₃₁	1	1	yes	characteristic, complemented
111	#525	—	31	1	no	—
112	#436	—	31	4	no	—
113	#435	—	31	4	no	—
114	#424	—	31	1	no	—
115	#408	—	31	1	no	—
116	#407	—	31	1	no	—
117	#405	—	31	1	no	—
118	C₂×C₃₁	—	2	2	no	—
119	C₂×C₃₁	—	4	2	no	—
120	C₂×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
121	C₂×C₃₁	#568	1	2	yes	—
122	D₃₁	—	4	2	no	—
123	C₂×#41	—	31	1	no	—
124	C₂³×C₄	C₂×D₃₁	1	1	yes	characteristic
125	#131	—	62	4	no	—
126	C₂×M₁₆	—	2	4	no	—
127	C₂×M₁₆	—	31	1	no	—
128	C₂×M₁₆	—	31	1	no	—
129	C₂×M₁₆	—	31	2	no	—
130	C₂×M₁₆	C₂×D₃₁	1	2	yes	—
131	C₂×M₁₆	C₂×D₃₁	1	2	yes	—
132	C₂²×C₈	—	31	2	no	—
133	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
134	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
135	#123	—	31	2	no	—
136	C₂²⋊Q₈	—	31	2	no	—
137	#121	—	31	2	no	—
138	#120	—	31	2	no	—
139	C₄×D₄	—	31	4	no	—
140	#117	—	31	1	no	—
141	C₄⋊C₈	—	31	4	no	—
142	C₂²⋊C₈	—	31	4	no	—
143	C₂²⋊C₈	—	31	4	no	—
144	#97	—	31	2	no	—
145	C₄×C₈	—	31	2	no	—
146	C₃₁	#2930	1	1	yes	characteristic, complemented, Sylow
147	C₂⁴	C₄×D₃₁	1	1	yes	characteristic
148	#41	—	62	2	no	—
149	#41	—	62	2	no	—
150	C₂×Q₈	—	31	1	no	—
151	C₂×D₄	—	31	2	no	—
152	C₂×D₄	—	31	4	no	—
153	C₂×D₄	—	31	1	no	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	—	31	2	no	—
157	C₂²×C₄	C₄×D₃₁	1	2	yes	—
158	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
159	C₂²×C₄	C₂²×D₃₁	1	2	yes	—
160	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
161	M₁₆	—	2	4	no	—
162	M₁₆	—	2	4	no	—
163	M₁₆	—	62	2	no	—
164	M₁₆	—	62	2	no	—
165	M₁₆	—	62	4	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	31	2	no	—
168	C₂×C₈	—	31	2	no	—
169	C₂×C₈	—	62	4	no	—
170	C₂×C₈	C₂²⋊D₃₁	1	4	yes	complemented
171	C₂×C₈	C₂²×D₃₁	1	2	yes	—
172	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
173	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
174	C₄⋊C₄	—	31	4	no	—
175	C₄⋊C₄	—	31	2	no	—
176	C₂²⋊C₄	—	31	2	no	—
177	C₂²⋊C₄	—	31	4	no	—
178	C₂²⋊C₄	—	31	4	no	—
179	C₄²	—	31	2	no	—
180	C₂³	—	2	2	no	—
181	C₂³	—	4	2	no	—
182	C₂³	—	31	2	no	—
183	C₂³	C₂×C₄×D₃₁	1	2	yes	—
184	C₂³	C₂×C₄×D₃₁	1	1	yes	characteristic
185	Q₈	—	62	2	no	—
186	D₄	—	62	4	no	—
187	D₄	—	62	2	no	—
188	D₄	—	124	4	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	31	4	no	—
194	C₂×C₄	—	31	2	no	—
195	C₂×C₄	—	62	2	no	—
196	C₂×C₄	—	62	2	no	—
197	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
198	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
199	C₂×C₄	#92495	1	2	yes	—
200	C₂×C₄	C₂×C₂²⋊D₃₁	1	2	yes	—
201	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
202	C₈	—	2	2	no	—
203	C₈	—	62	2	no	—
204	C₈	—	62	2	no	—
205	C₈	C₂×C₂²⋊D₃₁	1	2	yes	—
206	C₈	C₂×C₂²⋊D₃₁	1	2	yes	—
207	C₂²	—	2	2	no	—
208	C₂²	—	2	2	no	—
209	C₂²	—	2	2	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	4	4	no	—
212	C₂²	—	62	2	no	—
213	C₂²	—	62	4	no	—
214	C₂²	C₄×C₂²⋊D₃₁	1	2	yes	—
215	C₂²	C₂²×C₄×D₃₁	1	1	yes	characteristic
216	C₄	—	2	2	no	—
217	C₄	—	62	2	no	—
218	C₄	—	62	4	no	—
219	C₄	C₄×C₂²⋊D₃₁	1	2	yes	—
220	C₄	C₂×#92495	1	1	yes	characteristic
221	C₄	C₂²×C₂²⋊D₃₁	1	1	yes	characteristic
222	C₂	—	2	2	no	—
223	C₂	—	4	2	no	—
224	C₂	—	124	2	no	—
225	C₂	C₂×C₄×C₂²⋊D₃₁	1	1	yes	characteristic
226	C₂	#14521815	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃₁
commutator, socle	C₂²×C₃₁	C₂³×C₄
Fitting	C₂²×M₁₆×C₃₁	C₂
prime radical	#14521560	C₂
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	C₂²×M₁₆	D₃₁
hypocenter	C₃₁	#2930
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#11758478	C₂	semidirect
3. #14521125	C₂	semidirect
4. #14521118	C₂	semidirect
5. #14521075	C₂	semidirect
6. #14521015	C₂	semidirect
7. #14521013	C₂	semidirect
8. #14521004	C₂	semidirect
9. C₂²×M₁₆×C₃₁	C₂	semidirect
26. C₂×#92469	C₂²	semidirect
27. C₂×C₈×D₃₁	C₂²	semidirect
29. (C₂×D₃₁)⋊C₈	C₂²	semidirect
30. #11758403	C₂²	semidirect
31. Dic₃₁⋊C₈	C₂²	semidirect
32. C₈×Dic₃₁	C₂²	semidirect
50. C₂×C₈×C₃₁	D₄	semidirect
110. C₂²×M₁₆	D₃₁	semidirect
146. C₃₁	#2930	semidirect
170. C₂×C₈	C₂²⋊D₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11758478	2
#14521560	2
#14521125	2
#14521118	2
#14521075	2
#14521015	2
#14521013	2
#14521004	2
C₂²×M₁₆×C₃₁	2
#2930	31