Subgroups — C₂×#14521099 #16854324

Order 3968.

528 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×Dic₃₁	C₂	1	1	yes	maximal, characteristic
2	#14521099	C₂	1	8	yes	maximal, direct product factor
3	C₂×Dic₃₁⋊C₈	C₂	1	2	yes	maximal, complemented
4	C₂×C₂²⋊(C₃₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
5	C₂×C₈×Dic₃₁	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₁×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
8	C₂³×Dic₃₁	C₄	1	1	yes	characteristic
9	C₂³×Dic₃₁	C₄	1	1	yes	characteristic
10	C₂²×C₈×C₃₁	C₂²	1	2	yes	—
11	C₂×C₄×Dic₃₁	C₄	1	1	yes	characteristic
12	C₂×C₄×Dic₃₁	C₄	1	4	yes	—
13	C₂×C₄×Dic₃₁	C₄	1	1	yes	characteristic
14	C₂×C₄×Dic₃₁	C₂²	1	4	yes	—
15	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
16	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
17	C₂²×C₃₁⋊C₈	C₂²	1	2	yes	—
18	C₃₁×C₂²⋊C₈	C₂²	1	4	yes	—
19	C₂²⋊(C₃₁⋊C₈)	C₂²	1	4	yes	—
20	Dic₃₁⋊C₈	C₂²	1	8	yes	complemented
21	C₈×Dic₃₁	C₂²	1	8	yes	complemented
22	C₂⁴×C₃₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₃₁	—	2	4	no	—
24	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
27	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
28	C₂²×Dic₃₁	—	2	4	no	—
29	C₂²×Dic₃₁	C₈	1	8	yes	complemented
30	C₂²×Dic₃₁	C₂×C₄	1	1	yes	characteristic
31	C₂²×Dic₃₁	C₂×C₄	1	4	yes	—
32	C₂²×Dic₃₁	C₂×C₄	1	1	yes	characteristic
33	C₂²×Dic₃₁	C₂×C₄	1	1	yes	characteristic
34	C₂²×Dic₃₁	C₂×C₄	1	4	yes	—
35	C₂²×Dic₃₁	C₂×C₄	1	1	yes	characteristic
36	C₂×C₈×C₃₁	—	2	4	no	—
37	C₂×C₈×C₃₁	C₂³	1	4	yes	—
38	C₄×Dic₃₁	C₂×C₄	1	4	yes	—
39	C₄×Dic₃₁	C₂×C₄	1	2	yes	—
40	C₄×Dic₃₁	C₂×C₄	1	4	yes	—
41	C₄×Dic₃₁	C₂×C₄	1	2	yes	—
42	C₄×Dic₃₁	C₂³	1	2	yes	—
43	C₄×Dic₃₁	C₂³	1	2	yes	—
44	C₂×C₃₁⋊C₈	—	2	4	no	—
45	C₂×C₃₁⋊C₈	C₂³	1	4	yes	—
46	C₂³×C₃₁	—	2	2	no	—
47	C₂³×C₃₁	—	2	2	no	—
48	C₂³×C₃₁	C₂×C₈	1	4	yes	—
49	C₂³×C₃₁	C₂²×C₄	1	2	yes	—
50	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₃₁	—	2	2	no	—
52	C₂×C₄×C₃₁	—	2	4	no	—
53	C₂×C₄×C₃₁	—	2	4	no	—
54	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
55	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₃₁	#41	1	4	yes	—
57	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
58	C₂×Dic₃₁	—	2	2	no	—
59	C₂×Dic₃₁	—	2	8	no	—
60	C₂×Dic₃₁	C₂×C₈	1	4	yes	—
61	C₂×Dic₃₁	C₂×C₈	1	4	yes	—
62	C₂×Dic₃₁	C₂×C₈	1	16	yes	complemented
63	C₂×Dic₃₁	M₁₆	1	4	yes	—
64	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
65	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
66	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
67	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
68	C₈×C₃₁	—	2	4	no	—
69	C₃₁⋊C₈	—	2	4	no	—
70	C₂×#431	—	31	1	no	maximal, Sylow
71	C₂²×C₃₁	—	2	2	no	—
72	C₂²×C₃₁	—	2	2	no	—
73	C₂²×C₃₁	—	2	8	no	—
74	C₂²×C₃₁	#117	1	2	yes	—
75	C₂²×C₃₁	C₂²×C₈	1	4	yes	—
76	C₂²×C₃₁	C₂²×C₈	1	2	yes	—
77	C₂²×C₃₁	C₂×M₁₆	1	2	yes	—
78	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
79	C₄×C₃₁	—	2	2	no	—
80	C₄×C₃₁	#117	1	2	yes	—
81	C₄×C₃₁	C₂×#41	1	2	yes	—
82	Dic₃₁	—	2	2	no	—
83	Dic₃₁	C₂²×C₈	1	4	yes	—
84	Dic₃₁	C₂²×C₈	1	4	yes	—
85	Dic₃₁	C₂×M₁₆	1	4	yes	—
86	C₂²×C₄²	—	31	1	no	—
87	#431	—	31	8	no	—
88	C₂×C₄⋊C₈	—	31	2	no	—
89	C₂×C₂²⋊C₈	—	31	1	no	—
90	C₂×C₂²⋊C₈	D₃₁	1	1	yes	characteristic
91	C₂×C₄×C₈	—	31	2	no	—
92	C₂×C₃₁	—	2	4	no	—
93	C₂×C₃₁	#431	1	4	yes	complemented
94	C₂×C₃₁	C₂×#117	1	1	yes	characteristic
95	C₂×C₃₁	C₂³×C₈	1	1	yes	characteristic
96	C₂×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
97	C₂³×C₄	—	31	1	no	—
98	C₂³×C₄	—	31	1	no	—
99	C₂³×C₄	C₂×D₃₁	1	1	yes	characteristic
100	C₂²×C₈	—	31	2	no	—
101	C₂²×C₈	C₂×D₃₁	1	2	yes	—
102	C₂×C₄²	—	31	1	no	—
103	C₂×C₄²	—	31	4	no	—
104	C₂×C₄²	—	31	1	no	—
105	C₂×C₄²	—	31	4	no	—
106	C₂×C₄²	—	31	1	no	—
107	C₂×C₄²	—	31	1	no	—
108	C₄⋊C₈	—	31	8	no	—
109	C₂²⋊C₈	—	31	4	no	—
110	C₂²⋊C₈	C₂×D₃₁	1	4	yes	—
111	C₄×C₈	—	31	8	no	—
112	C₃₁	C₂×#431	1	1	yes	characteristic, complemented, Sylow
113	C₂⁴	C₄×D₃₁	1	1	yes	characteristic
114	C₂²×C₄	—	2	4	no	—
115	C₂²×C₄	—	31	8	no	—
116	C₂²×C₄	—	31	1	no	—
117	C₂²×C₄	—	31	1	no	—
118	C₂²×C₄	—	31	4	no	—
119	C₂²×C₄	—	31	1	no	—
120	C₂²×C₄	—	31	4	no	—
121	C₂²×C₄	—	31	1	no	—
122	C₂²×C₄	—	62	4	no	—
123	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
124	C₂²×C₄	C₄×D₃₁	1	2	yes	—
125	C₂²×C₄	C₂²×D₃₁	1	2	yes	—
126	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
127	C₂×C₈	—	2	4	no	—
128	C₂×C₈	—	31	4	no	—
129	C₂×C₈	—	62	4	no	—
130	C₂×C₈	C₂²×D₃₁	1	4	yes	—
131	C₄²	—	31	4	no	—
132	C₄²	—	31	2	no	—
133	C₄²	—	31	4	no	—
134	C₄²	—	31	2	no	—
135	C₄²	—	31	2	no	—
136	C₄²	—	31	2	no	—
137	C₂³	—	2	2	no	—
138	C₂³	—	2	2	no	—
139	C₂³	C₈×D₃₁	1	4	yes	—
140	C₂³	C₂×C₄×D₃₁	1	2	yes	—
141	C₂³	C₂×C₄×D₃₁	1	1	yes	characteristic
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	31	4	no	—
146	C₂×C₄	—	31	4	no	—
147	C₂×C₄	—	31	16	no	—
148	C₂×C₄	—	31	4	no	—
149	C₂×C₄	—	31	2	no	—
150	C₂×C₄	—	31	2	no	—
151	C₂×C₄	—	31	2	no	—
152	C₂×C₄	—	31	2	no	—
153	C₂×C₄	—	62	8	no	—
154	C₂×C₄	—	62	2	no	—
155	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
156	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
157	C₂×C₄	#92497	1	4	yes	—
158	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
159	C₈	—	2	4	no	—
160	C₈	—	62	4	no	—
161	C₂²	—	2	8	no	—
162	C₂²	—	2	2	no	—
163	C₂²	—	2	2	no	—
164	C₂²	#11758454	1	2	yes	—
165	C₂²	C₂×C₈×D₃₁	1	4	yes	—
166	C₂²	C₂×C₈×D₃₁	1	2	yes	—
167	C₂²	M₁₆×D₃₁	1	2	yes	—
168	C₂²	C₂²×C₄×D₃₁	1	1	yes	characteristic
169	C₄	—	2	2	no	—
170	C₄	—	31	4	no	—
171	C₄	—	31	4	no	—
172	C₄	—	31	4	no	—
173	C₄	—	62	2	no	—
174	C₄	#11758454	1	2	yes	—
175	C₄	C₂×#92497	1	2	yes	—
176	C₂	—	2	4	no	—
177	C₂	#14521099	1	4	yes	direct product factor
178	C₂	C₂×#11758454	1	1	yes	characteristic
179	C₂	C₂²×C₈×D₃₁	1	1	yes	characteristic
180	C₂	C₂×M₁₆×D₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₃₁
commutator	C₂×C₃₁	C₂³×C₈
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₂×C₃₁×C₂²⋊C₈	C₂
socle	C₂³×C₃₁	C₂²×C₄
prime radical	C₂⁴×C₃₁	C₂×C₄
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	C₂×C₂²⋊C₈	D₃₁
hypocenter	C₃₁	C₂×#431
LCS	G ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
2. #14521099	C₂	direct (N × H)
3. C₂×Dic₃₁⋊C₈	C₂	semidirect
5. C₂×C₈×Dic₃₁	C₂	semidirect
20. Dic₃₁⋊C₈	C₂²	semidirect
21. C₈×Dic₃₁	C₂²	semidirect
29. C₂²×Dic₃₁	C₈	semidirect
62. C₂×Dic₃₁	C₂×C₈	semidirect
93. C₂×C₃₁	#431	semidirect
112. C₃₁	C₂×#431	semidirect
177. C₂	#14521099	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×Dic₃₁	2
#14521099	2
C₂×Dic₃₁⋊C₈	2
C₂×C₂²⋊(C₃₁⋊C₈)	2
C₂×C₈×Dic₃₁	2
C₂×C₃₁×C₂²⋊C₈	2
C₂×#431	31

Subgroups — C2×#14521099 #16854324

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14521099 #16854324