Subgroups — C₂×#14521166 #16854601

Order 3968.

472 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11758525	C₂	1	2	yes	maximal, complemented
2	#14521166	C₂	1	8	yes	maximal, direct product factor
3	C₂×#11758407	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#11758425	C₂	1	2	yes	maximal, complemented
5	C₂×C₈×Dic₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃₁×#135	C₂	1	1	yes	maximal, characteristic
7	C₂×#41×C₃₁	C₂²	1	2	yes	—
8	C₃₁×#135	C₄	1	4	yes	complemented
9	C₃₁×#135	C₂²	1	4	yes	—
10	C₂×Q₁₆×C₃₁	C₄	1	1	yes	characteristic, complemented
11	C₂×SD₁₆×C₃₁	C₄	1	2	yes	complemented
12	C₂×D₈×C₃₁	C₄	1	1	yes	characteristic, complemented
13	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
14	#11758525	—	2	4	no	—
15	#11758525	C₂²	1	8	yes	complemented
16	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
17	C₂×#92474	—	2	2	no	—
18	C₂×#92474	C₂²	1	2	yes	—
19	C₂²×C₃₁⋊C₈	—	2	2	no	—
20	C₂²×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
21	#11758425	C₂²	1	8	yes	complemented
22	#11758407	C₂²	1	4	yes	complemented
23	C₈×Dic₃₁	C₂²	1	4	yes	complemented
24	#41×C₃₁	—	2	2	no	—
25	#41×C₃₁	—	2	2	no	—
26	#41×C₃₁	C₂×C₄	1	4	yes	—
27	#41×C₃₁	C₂³	1	4	yes	—
28	C₂×Q₈×C₃₁	C₂×C₄	1	2	yes	—
29	C₂×D₄×C₃₁	—	2	2	no	—
30	C₂×D₄×C₃₁	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₃₁	—	2	2	no	—
32	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
33	C₂²×Dic₃₁	—	2	1	no	—
34	Q₁₆×C₃₁	C₂×C₄	1	4	yes	complemented
35	SD₁₆×C₃₁	C₂×C₄	1	8	yes	complemented
36	D₈×C₃₁	C₂×C₄	1	4	yes	complemented
37	C₂×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
38	C₂×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
39	C₂×C₈×C₃₁	C₂×C₄	1	2	yes	—
40	C₂×C₈×C₃₁	C₂³	1	2	yes	—
41	C₄×Dic₃₁	—	2	1	no	—
42	C₄×Dic₃₁	C₂³	1	2	yes	—
43	#92474	—	2	2	no	—
44	#92474	—	2	8	no	—
45	#92474	C₂³	1	4	yes	—
46	C₂×C₃₁⋊C₈	—	2	2	no	—
47	C₂×C₃₁⋊C₈	—	2	8	no	—
48	C₂×C₃₁⋊C₈	D₄	1	1	yes	characteristic
49	C₂×C₃₁⋊C₈	D₄	1	1	yes	characteristic
50	C₂×C₃₁⋊C₈	D₄	1	2	yes	—
51	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
52	C₂³×C₃₁	—	2	2	no	—
53	C₂³×C₃₁	#41	1	1	yes	characteristic
54	Q₈×C₃₁	—	2	2	no	—
55	Q₈×C₃₁	C₂²×C₄	1	4	yes	—
56	D₄×C₃₁	—	2	4	no	—
57	D₄×C₃₁	—	2	4	no	—
58	D₄×C₃₁	—	2	2	no	—
59	D₄×C₃₁	C₂²×C₄	1	4	yes	—
60	C₂×C₄×C₃₁	—	2	2	no	—
61	C₂×C₄×C₃₁	—	2	4	no	—
62	C₂×C₄×C₃₁	—	2	4	no	—
63	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
67	C₂×C₄×C₃₁	#41	1	1	yes	characteristic
68	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
69	C₂×Dic₃₁	—	2	4	no	—
70	C₂×Dic₃₁	—	2	2	no	—
71	C₈×C₃₁	C₂²×C₄	1	2	yes	—
72	C₈×C₃₁	C₂²×C₄	1	2	yes	—
73	C₃₁⋊C₈	—	2	4	no	—
74	C₃₁⋊C₈	C₂×D₄	1	2	yes	—
75	C₃₁⋊C₈	C₂×D₄	1	2	yes	—
76	C₂×#443	—	31	1	no	maximal, Sylow
77	C₂²×C₃₁	—	2	2	no	—
78	C₂²×C₃₁	—	2	4	no	—
79	C₂²×C₃₁	—	4	4	no	—
80	C₂²×C₃₁	C₄×D₄	1	1	yes	characteristic
81	C₂²×C₃₁	C₄×D₄	1	1	yes	characteristic
82	C₂²×C₃₁	C₂×#41	1	1	yes	characteristic
83	C₄×C₃₁	—	2	4	no	—
84	C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
85	C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
86	C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
87	C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
88	Dic₃₁	—	2	4	no	—
89	C₂×#135	D₃₁	1	1	yes	characteristic
90	C₂×#131	—	31	2	no	—
91	#443	—	31	8	no	—
92	C₂×#108	—	31	1	no	—
93	C₂×#104	—	31	2	no	—
94	C₂×C₄×C₈	—	31	1	no	—
95	C₂×C₃₁	—	2	1	no	—
96	C₂×C₃₁	—	2	1	no	—
97	C₂×C₃₁	—	4	4	no	—
98	C₂×C₃₁	#443	1	2	yes	complemented
99	C₂×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
100	C₂×#41	C₂×D₃₁	1	2	yes	—
101	#135	Dic₃₁	1	4	yes	complemented
102	#135	C₂×D₃₁	1	4	yes	—
103	C₂×Q₁₆	Dic₃₁	1	1	yes	characteristic, complemented
104	C₂×SD₁₆	Dic₃₁	1	2	yes	complemented
105	C₂×D₈	Dic₃₁	1	1	yes	characteristic, complemented
106	#131	—	31	8	no	—
107	#131	—	62	4	no	—
108	C₂×M₁₆	—	31	2	no	—
109	C₂×M₁₆	—	62	2	no	—
110	C₂²×C₈	—	31	1	no	—
111	C₂²×C₈	—	62	2	no	—
112	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
113	C₂×C₄²	—	31	1	no	—
114	#108	—	31	4	no	—
115	#104	—	31	8	no	—
116	C₄×C₈	—	31	4	no	—
117	C₃₁	C₂×#443	1	1	yes	characteristic, complemented, Sylow
118	#41	—	2	2	no	—
119	#41	—	2	2	no	—
120	#41	C₂×Dic₃₁	1	4	yes	—
121	#41	C₂²×D₃₁	1	4	yes	—
122	C₂×Q₈	C₂×Dic₃₁	1	2	yes	—
123	C₂×D₄	—	2	2	no	—
124	C₂×D₄	C₂×Dic₃₁	1	2	yes	—
125	C₂²×C₄	—	2	2	no	—
126	C₂²×C₄	—	62	1	no	—
127	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
128	Q₁₆	C₂×Dic₃₁	1	4	yes	complemented
129	SD₁₆	C₂×Dic₃₁	1	8	yes	complemented
130	D₈	C₂×Dic₃₁	1	4	yes	complemented
131	M₁₆	—	31	4	no	—
132	M₁₆	—	62	8	no	—
133	M₁₆	—	62	2	no	—
134	C₂×C₈	—	31	2	no	—
135	C₂×C₈	—	31	1	no	—
136	C₂×C₈	—	31	1	no	—
137	C₂×C₈	—	31	2	no	—
138	C₂×C₈	—	62	8	no	—
139	C₂×C₈	—	62	2	no	—
140	C₂×C₈	C₂×Dic₃₁	1	1	yes	characteristic
141	C₂×C₈	C₂×Dic₃₁	1	1	yes	characteristic
142	C₂×C₈	C₂×Dic₃₁	1	2	yes	—
143	C₂×C₈	C₂²×D₃₁	1	2	yes	—
144	C₄²	—	31	2	no	—
145	C₄²	—	62	1	no	—
146	C₂³	—	2	2	no	—
147	C₂³	#92497	1	1	yes	characteristic
148	Q₈	—	2	2	no	—
149	Q₈	C₂²×Dic₃₁	1	4	yes	—
150	D₄	—	2	4	no	—
151	D₄	—	2	4	no	—
152	D₄	—	2	2	no	—
153	D₄	C₂²×Dic₃₁	1	4	yes	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	62	4	no	—
158	C₂×C₄	—	62	2	no	—
159	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
160	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
161	C₂×C₄	#92497	1	1	yes	characteristic
162	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
163	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
164	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
165	C₈	—	31	2	no	—
166	C₈	—	31	2	no	—
167	C₈	—	62	4	no	—
168	C₈	C₂²×Dic₃₁	1	2	yes	—
169	C₈	C₂²×Dic₃₁	1	2	yes	—
170	C₂²	—	2	2	no	—
171	C₂²	—	2	4	no	—
172	C₂²	—	4	4	no	—
173	C₂²	D₄×Dic₃₁	1	1	yes	characteristic
174	C₂²	D₄×Dic₃₁	1	1	yes	characteristic
175	C₂²	C₂×#92497	1	1	yes	characteristic
176	C₄	—	2	4	no	—
177	C₄	—	62	4	no	—
178	C₄	D₄×Dic₃₁	1	1	yes	characteristic
179	C₄	D₄×Dic₃₁	1	1	yes	characteristic
180	C₄	C₂×D₄×D₃₁	1	1	yes	characteristic
181	C₄	C₂³×Dic₃₁	1	1	yes	characteristic
182	C₂	—	2	1	no	—
183	C₂	—	2	1	no	—
184	C₂	—	4	4	no	—
185	C₂	#14521166	1	2	yes	direct product factor
186	C₂	C₂×D₄×Dic₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₃₁
commutator	C₄×C₃₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃₁
Fitting, prime radical	C₂×C₃₁×#135	C₂
socle	C₂²×C₃₁	C₄×D₄
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	C₂×#135	D₃₁
hypocenter	C₃₁	C₂×#443
LCS	G ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#135

Complements

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

N	H ≅ G/N	type
1. C₂×#11758525	C₂	semidirect
2. #14521166	C₂	direct (N × H)
3. C₂×#11758407	C₂	semidirect
4. C₂×#11758425	C₂	semidirect
5. C₂×C₈×Dic₃₁	C₂	semidirect
8. C₃₁×#135	C₄	semidirect
10. C₂×Q₁₆×C₃₁	C₄	semidirect
11. C₂×SD₁₆×C₃₁	C₄	semidirect
12. C₂×D₈×C₃₁	C₄	semidirect
15. #11758525	C₂²	semidirect
21. #11758425	C₂²	semidirect
22. #11758407	C₂²	semidirect
23. C₈×Dic₃₁	C₂²	semidirect
34. Q₁₆×C₃₁	C₂×C₄	semidirect
35. SD₁₆×C₃₁	C₂×C₄	semidirect
36. D₈×C₃₁	C₂×C₄	semidirect
98. C₂×C₃₁	#443	semidirect
101. #135	Dic₃₁	semidirect
103. C₂×Q₁₆	Dic₃₁	semidirect
104. C₂×SD₁₆	Dic₃₁	semidirect
105. C₂×D₈	Dic₃₁	semidirect
117. C₃₁	C₂×#443	semidirect
128. Q₁₆	C₂×Dic₃₁	semidirect
129. SD₁₆	C₂×Dic₃₁	semidirect
130. D₈	C₂×Dic₃₁	semidirect
185. C₂	#14521166	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#11758525	2
#14521166	2
C₂×#11758407	2
C₂×#11758425	2
C₂×C₈×Dic₃₁	2
C₂×C₃₁×#135	2
C₂×#443	31

Subgroups — C2×#14521166 #16854601

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14521166 #16854601