Subgroups — C₂×#14521003 #16854122

Order 3968.

592 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×#92469	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₈×D₃₁	C₂	1	1	yes	maximal, characteristic
3	C₂×#11758449	C₂	1	1	yes	maximal, characteristic
4	#14521003	C₂	1	8	yes	maximal, direct product factor
5	C₂×#11758391	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11758403	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₈×Dic₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₄×D₃₁	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×(C₂×D₃₁)⋊C₄	C₄	1	2	yes	—
14	C₂×Dic₃₁⋊C₄	C₄	1	2	yes	—
15	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
16	C₂²×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
17	C₂²×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×#92469	C₄	1	8	yes	complemented
19	C₂×#92469	C₂²	1	4	yes	—
20	C₂×C₈×D₃₁	C₄	1	8	yes	complemented
21	C₂×C₈×D₃₁	C₂²	1	4	yes	—
22	#11758449	C₄	1	4	yes	—
23	#11758449	C₂²	1	4	yes	—
24	C₄×C₈×C₃₁	C₂²	1	4	yes	complemented
25	#11758403	C₂²	1	4	yes	complemented
26	C₈×Dic₃₁	C₂²	1	4	yes	complemented
27	#11758391	C₂²	1	4	yes	complemented
28	C₂³×D₃₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
31	C₂²×Dic₃₁	C₂×C₄	1	2	yes	—
32	C₂²×Dic₃₁	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×D₃₁	C₂×C₄	1	2	yes	—
34	C₂×C₄×D₃₁	C₂×C₄	1	4	yes	—
35	C₂×C₄×D₃₁	C₂×C₄	1	4	yes	—
36	C₂×C₄×D₃₁	C₂³	1	2	yes	—
37	C₂×C₈×C₃₁	C₂×C₄	1	4	yes	—
38	C₂×C₈×C₃₁	C₂×C₄	1	4	yes	—
39	C₂×C₈×C₃₁	C₂³	1	2	yes	—
40	C₂×C₈×C₃₁	C₂³	1	2	yes	—
41	C₄²×C₃₁	C₂×C₄	1	2	yes	—
42	C₄²×C₃₁	C₂³	1	2	yes	—
43	(C₂×D₃₁)⋊C₄	C₂×C₄	1	8	yes	—
44	Dic₃₁⋊C₄	C₂×C₄	1	8	yes	—
45	C₄×Dic₃₁	C₂×C₄	1	2	yes	—
46	C₄×Dic₃₁	C₂³	1	2	yes	—
47	C₂×C₃₁⋊C₈	C₂×C₄	1	4	yes	—
48	C₂×C₃₁⋊C₈	C₂×C₄	1	4	yes	—
49	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
50	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
51	#92469	C₂×C₄	1	16	yes	complemented
52	C₈×D₃₁	C₂×C₄	1	16	yes	complemented
53	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
54	C₂²×D₃₁	—	2	4	no	—
55	C₂²×D₃₁	C₄²	1	4	yes	—
56	C₂²×D₃₁	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₃₁	—	2	4	no	—
58	C₂×C₄×C₃₁	C₂²×C₄	1	4	yes	—
59	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
61	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
62	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
63	C₂×Dic₃₁	—	2	4	no	—
64	C₂×Dic₃₁	C₄²	1	4	yes	—
65	C₂×Dic₃₁	C₂²×C₄	1	4	yes	—
66	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
67	C₄×D₃₁	C₄²	1	8	yes	—
68	C₄×D₃₁	C₂²×C₄	1	4	yes	—
69	C₄×D₃₁	C₂²×C₄	1	4	yes	—
70	C₈×C₃₁	C₂²×C₄	1	4	yes	—
71	C₈×C₃₁	C₂²×C₄	1	4	yes	—
72	C₃₁⋊C₈	C₂²×C₄	1	4	yes	—
73	C₃₁⋊C₈	C₂²×C₄	1	4	yes	—
74	C₂×#405	—	31	1	no	maximal, Sylow
75	C₂²×C₃₁	C₂×C₄²	1	2	yes	—
76	C₂²×C₃₁	#131	1	4	yes	—
77	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
78	C₂×D₃₁	—	2	4	no	—
79	C₂×D₃₁	—	2	8	no	—
80	C₂×D₃₁	C₂×C₄²	1	4	yes	—
81	C₄×C₃₁	—	2	4	no	—
82	C₄×C₃₁	C₂×C₄²	1	2	yes	—
83	C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
84	C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
85	Dic₃₁	—	2	4	no	—
86	Dic₃₁	C₂×C₄²	1	4	yes	—
87	C₂²×M₁₆	—	31	1	no	—
88	C₂³×C₈	—	31	1	no	—
89	C₂×#117	—	31	1	no	—
90	#405	—	31	8	no	—
91	C₂×#97	—	31	1	no	—
92	C₂×#97	—	31	1	no	—
93	C₂×C₄×C₈	—	31	1	no	—
94	C₂×C₄×C₈	D₃₁	1	1	yes	characteristic, complemented
95	C₂×C₃₁	#405	1	4	yes	complemented
96	C₂×C₃₁	C₂²×C₄²	1	1	yes	characteristic
97	C₂×C₃₁	C₂×#131	1	2	yes	—
98	D₃₁	—	2	4	no	—
99	C₂³×C₄	—	31	1	no	—
100	C₂×M₁₆	—	31	8	no	—
101	C₂×M₁₆	—	31	4	no	—
102	C₂²×C₈	—	31	1	no	—
103	C₂²×C₈	—	31	1	no	—
104	C₂²×C₈	—	31	8	no	—
105	C₂²×C₈	—	31	4	no	—
106	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
107	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
108	#117	—	31	4	no	—
109	#117	—	31	4	no	—
110	C₂×C₄⋊C₄	—	31	2	no	—
111	C₂×C₂²⋊C₄	—	31	2	no	—
112	C₂×C₄²	—	31	1	no	—
113	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
114	#97	—	31	4	no	—
115	#97	—	31	4	no	—
116	C₄×C₈	—	31	4	no	—
117	C₄×C₈	C₂×D₃₁	1	4	yes	complemented
118	C₃₁	C₂×#405	1	1	yes	characteristic, complemented, Sylow
119	C₂⁴	—	31	1	no	—
120	C₂²×C₄	—	31	1	no	—
121	C₂²×C₄	—	31	2	no	—
122	C₂²×C₄	—	31	4	no	—
123	C₂²×C₄	—	31	4	no	—
124	C₂²×C₄	—	31	2	no	—
125	C₂²×C₄	—	31	2	no	—
126	C₂²×C₄	C₄×D₃₁	1	2	yes	—
127	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
128	M₁₆	—	31	16	no	—
129	C₂×C₈	—	31	4	no	—
130	C₂×C₈	—	31	4	no	—
131	C₂×C₈	—	31	2	no	—
132	C₂×C₈	—	31	2	no	—
133	C₂×C₈	—	31	16	no	—
134	C₂×C₈	C₄×D₃₁	1	4	yes	—
135	C₂×C₈	C₄×D₃₁	1	4	yes	—
136	C₂×C₈	C₂²×D₃₁	1	2	yes	—
137	C₂×C₈	C₂²×D₃₁	1	2	yes	—
138	C₄⋊C₄	—	31	8	no	—
139	C₂²⋊C₄	—	31	8	no	—
140	C₄²	—	31	2	no	—
141	C₄²	—	31	2	no	—
142	C₄²	C₄×D₃₁	1	2	yes	—
143	C₄²	C₂²×D₃₁	1	2	yes	—
144	C₂³	—	31	4	no	—
145	C₂³	—	31	2	no	—
146	C₂³	—	62	4	no	—
147	C₂³	C₂×C₄×D₃₁	1	1	yes	characteristic
148	C₂×C₄	—	2	4	no	—
149	C₂×C₄	—	31	4	no	—
150	C₂×C₄	—	31	2	no	—
151	C₂×C₄	—	31	8	no	—
152	C₂×C₄	—	31	4	no	—
153	C₂×C₄	—	31	4	no	—
154	C₂×C₄	—	31	4	no	—
155	C₂×C₄	—	62	4	no	—
156	C₂×C₄	C₂×C₄×D₃₁	1	4	yes	—
157	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
158	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
159	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
160	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
161	C₈	—	31	4	no	—
162	C₈	—	31	4	no	—
163	C₈	C₂×C₄×D₃₁	1	4	yes	—
164	C₈	C₂×C₄×D₃₁	1	4	yes	—
165	C₂²	—	31	4	no	—
166	C₂²	—	62	8	no	—
167	C₂²	—	62	4	no	—
168	C₂²	C₄²×D₃₁	1	2	yes	—
169	C₂²	#11758478	1	4	yes	—
170	C₂²	C₂²×C₄×D₃₁	1	1	yes	characteristic
171	C₄	—	2	4	no	—
172	C₄	—	31	4	no	—
173	C₄	—	62	4	no	—
174	C₄	C₄²×D₃₁	1	2	yes	—
175	C₄	C₂²×C₄×D₃₁	1	1	yes	characteristic
176	C₄	C₂²×C₄×D₃₁	1	1	yes	characteristic
177	C₂	—	62	4	no	—
178	C₂	#14521003	1	4	yes	direct product factor
179	C₂	C₂×C₄²×D₃₁	1	1	yes	characteristic
180	C₂	C₂×#11758478	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂×D₃₁
commutator	C₂×C₃₁	C₂²×C₄²
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₂×C₄×C₈×C₃₁	C₂
socle	C₂³×C₃₁	C₂²×C₄
prime radical	C₂³×D₃₁	C₂×C₄
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	C₂×C₄×C₈	D₃₁
hypocenter	C₃₁	C₂×#405
LCS	G ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
4. #14521003	C₂	direct (N × H)
5. C₂×#11758391	C₂	semidirect
6. C₂×#11758403	C₂	semidirect
7. C₂×C₈×Dic₃₁	C₂	semidirect
8. C₂×C₄×C₈×C₃₁	C₂	semidirect
18. C₂×#92469	C₄	semidirect
20. C₂×C₈×D₃₁	C₄	semidirect
24. C₄×C₈×C₃₁	C₂²	semidirect
25. #11758403	C₂²	semidirect
26. C₈×Dic₃₁	C₂²	semidirect
27. #11758391	C₂²	semidirect
51. #92469	C₂×C₄	semidirect
52. C₈×D₃₁	C₂×C₄	semidirect
94. C₂×C₄×C₈	D₃₁	semidirect
95. C₂×C₃₁	#405	semidirect
117. C₄×C₈	C₂×D₃₁	semidirect
118. C₃₁	C₂×#405	semidirect
178. C₂	#14521003	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×#92469	2
C₂²×C₈×D₃₁	2
C₂×#11758449	2
#14521003	2
C₂×#11758391	2
C₂×#11758403	2
C₂×C₈×Dic₃₁	2
C₂×C₄×C₈×C₃₁	2
C₂×#405	31

Subgroups — C2×#14521003 #16854122

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14521003 #16854122