Subgroups — C₂×#14521097 #16854317

Order 3968.

460 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#92474	C₂	1	1	yes	maximal, characteristic
2	C₂²×#92474	C₂	1	1	yes	maximal, characteristic
3	#14521097	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11758410	C₂	1	4	yes	maximal, complemented
5	C₂²×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
7	C₂×M₁₆×C₃₁	—	2	1	no	—
8	C₂×M₁₆×C₃₁	—	2	2	no	—
9	C₂×M₁₆×C₃₁	C₂²	1	2	yes	—
10	C₂×M₁₆×C₃₁	C₂²	1	4	yes	—
11	C₂²×C₈×C₃₁	—	2	1	no	—
12	C₂×#92474	—	2	1	no	—
13	C₂×#92474	—	2	2	no	—
14	C₂×#92474	C₄	1	2	yes	—
15	C₂×#92474	C₄	1	2	yes	—
16	C₂×#92474	C₄	1	4	yes	—
17	C₂×#92474	C₂²	1	2	yes	—
18	C₂×#92474	C₂²	1	4	yes	—
19	C₂×#92474	C₂²	1	4	yes	—
20	C₂²×C₃₁⋊C₈	—	2	1	no	—
21	C₂²×C₃₁⋊C₈	C₂²	1	2	yes	—
22	#11758410	C₂²	1	16	yes	complemented
23	C₂⁴×C₃₁	Q₈	1	1	yes	characteristic
24	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₃₁	D₄	1	2	yes	—
30	C₂²×C₄×C₃₁	Q₈	1	1	yes	characteristic
31	C₂²×C₄×C₃₁	Q₈	1	2	yes	—
32	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
34	M₁₆×C₃₁	—	2	4	no	—
35	M₁₆×C₃₁	—	2	2	no	—
36	M₁₆×C₃₁	C₂³	1	4	yes	—
37	C₂×C₈×C₃₁	—	2	4	no	—
38	C₂×C₈×C₃₁	—	2	2	no	—
39	#92474	—	2	4	no	—
40	#92474	—	2	2	no	—
41	#92474	C₂×C₄	1	8	yes	—
42	#92474	C₂×C₄	1	8	yes	—
43	#92474	C₂³	1	4	yes	—
44	C₂×C₃₁⋊C₈	—	2	4	no	—
45	C₂×C₃₁⋊C₈	—	2	2	no	—
46	C₂×C₃₁⋊C₈	C₂×C₄	1	4	yes	—
47	C₂×C₃₁⋊C₈	C₂×C₄	1	4	yes	—
48	C₂×C₃₁⋊C₈	C₂³	1	4	yes	—
49	C₂³×C₃₁	—	4	2	no	—
50	C₂³×C₃₁	C₄⋊C₄	1	1	yes	characteristic
51	C₂³×C₃₁	C₄⋊C₄	1	1	yes	characteristic
52	C₂³×C₃₁	C₄⋊C₄	1	2	yes	—
53	C₂³×C₃₁	C₂×Q₈	1	1	yes	characteristic
54	C₂³×C₃₁	C₂×Q₈	1	2	yes	—
55	C₂×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
56	C₂×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
57	C₂×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
58	C₂×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
59	C₂×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
60	C₂×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
65	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
66	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
67	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
68	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
70	C₂×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
72	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
73	C₈×C₃₁	—	2	4	no	—
74	C₃₁⋊C₈	—	2	4	no	—
75	C₃₁⋊C₈	C₂²×C₄	1	8	yes	—
76	C₂×#430	—	31	1	no	maximal, Sylow
77	C₂²×C₃₁	—	2	2	no	—
78	C₂²×C₃₁	—	2	2	no	—
79	C₂²×C₃₁	—	2	2	no	—
80	C₂²×C₃₁	—	4	2	no	—
81	C₂²×C₃₁	—	4	2	no	—
82	C₂²×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
84	C₂²×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
85	C₂²×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
86	C₂²×C₃₁	C₂²×Q₈	1	1	yes	characteristic
87	C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
89	C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
90	C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
92	C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
93	C₂²×M₁₆	—	31	1	no	—
94	C₂²×M₁₆	—	31	1	no	—
95	C₂²×M₁₆	D₃₁	1	1	yes	characteristic
96	#430	—	31	8	no	—
97	C₂×#108	—	31	4	no	—
98	C₂×C₃₁	—	2	2	no	—
99	C₂×C₃₁	—	2	2	no	—
100	C₂×C₃₁	—	2	1	no	—
101	C₂×C₃₁	—	2	1	no	—
102	C₂×C₃₁	#430	1	2	yes	complemented
103	C₂×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
104	C₂³×C₄	C₂×D₃₁	1	1	yes	characteristic
105	C₂×M₁₆	—	2	2	no	—
106	C₂×M₁₆	—	2	1	no	—
107	C₂×M₁₆	—	31	2	no	—
108	C₂×M₁₆	—	31	2	no	—
109	C₂×M₁₆	—	31	4	no	—
110	C₂×M₁₆	—	31	4	no	—
111	C₂×M₁₆	—	31	2	no	—
112	C₂×M₁₆	—	31	4	no	—
113	C₂×M₁₆	—	62	2	no	—
114	C₂×M₁₆	—	62	1	no	—
115	C₂×M₁₆	C₂×D₃₁	1	2	yes	—
116	C₂×M₁₆	C₂×D₃₁	1	4	yes	—
117	C₂²×C₈	—	2	1	no	—
118	C₂²×C₈	—	31	2	no	—
119	C₂²×C₈	—	62	1	no	—
120	#108	—	31	16	no	—
121	C₃₁	C₂×#430	1	1	yes	characteristic, complemented, Sylow
122	C₂⁴	Dic₆₂	1	1	yes	characteristic
123	C₂²×C₄	Dic₆₂	1	1	yes	characteristic
124	C₂²×C₄	Dic₆₂	1	2	yes	—
125	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
126	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
127	C₂²×C₄	C₄×D₃₁	1	2	yes	—
128	C₂²×C₄	C₂²⋊D₃₁	1	2	yes	—
129	C₂²×C₄	C₂²⋊D₃₁	1	1	yes	characteristic
130	C₂²×C₄	C₂²⋊D₃₁	1	1	yes	characteristic
131	C₂²×C₄	C₂²×D₃₁	1	2	yes	—
132	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
133	M₁₆	—	2	4	no	—
134	M₁₆	—	2	2	no	—
135	M₁₆	—	31	8	no	—
136	M₁₆	—	31	8	no	—
137	M₁₆	—	31	4	no	—
138	M₁₆	—	62	4	no	—
139	M₁₆	—	62	2	no	—
140	M₁₆	C₂²×D₃₁	1	4	yes	—
141	C₂×C₈	—	2	4	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	31	4	no	—
144	C₂×C₈	—	31	4	no	—
145	C₂×C₈	—	31	4	no	—
146	C₂×C₈	—	62	4	no	—
147	C₂×C₈	—	62	2	no	—
148	C₂³	—	4	2	no	—
149	C₂³	Dic₃₁⋊C₄	1	1	yes	characteristic
150	C₂³	Dic₃₁⋊C₄	1	1	yes	characteristic
151	C₂³	Dic₃₁⋊C₄	1	2	yes	—
152	C₂³	C₂×Dic₆₂	1	1	yes	characteristic
153	C₂³	C₂×Dic₆₂	1	2	yes	—
154	C₂×C₄	Dic₃₁⋊C₄	1	2	yes	—
155	C₂×C₄	Dic₃₁⋊C₄	1	2	yes	—
156	C₂×C₄	Dic₃₁⋊C₄	1	2	yes	—
157	C₂×C₄	Dic₃₁⋊C₄	1	1	yes	characteristic
158	C₂×C₄	Dic₃₁⋊C₄	1	1	yes	characteristic
159	C₂×C₄	Dic₃₁⋊C₄	1	2	yes	—
160	C₂×C₄	Dic₃₁⋊C₄	1	2	yes	—
161	C₂×C₄	C₂×Dic₆₂	1	2	yes	—
162	C₂×C₄	C₂×Dic₆₂	1	1	yes	characteristic
163	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
164	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
165	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
166	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
167	C₂×C₄	C₂×C₂²⋊D₃₁	1	2	yes	—
168	C₂×C₄	C₂×C₂²⋊D₃₁	1	2	yes	—
169	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
170	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
171	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
172	C₈	—	2	4	no	—
173	C₈	—	31	8	no	—
174	C₈	—	62	4	no	—
175	C₂²	—	2	2	no	—
176	C₂²	—	2	2	no	—
177	C₂²	—	2	2	no	—
178	C₂²	—	4	2	no	—
179	C₂²	—	4	2	no	—
180	C₂²	C₂×Dic₃₁⋊C₄	1	1	yes	characteristic
181	C₂²	C₂×Dic₃₁⋊C₄	1	2	yes	—
182	C₂²	C₂×Dic₃₁⋊C₄	1	2	yes	—
183	C₂²	C₂×Dic₃₁⋊C₄	1	1	yes	characteristic
184	C₂²	C₂²×Dic₆₂	1	1	yes	characteristic
185	C₄	C₂×Dic₃₁⋊C₄	1	1	yes	characteristic
186	C₄	C₂×Dic₃₁⋊C₄	1	2	yes	—
187	C₄	C₂×Dic₃₁⋊C₄	1	2	yes	—
188	C₄	C₂×Dic₃₁⋊C₄	1	1	yes	characteristic
189	C₄	C₂²×C₄×D₃₁	1	1	yes	characteristic
190	C₄	C₂²×C₂²⋊D₃₁	1	1	yes	characteristic
191	C₂	—	2	2	no	—
192	C₂	—	2	2	no	—
193	C₂	—	2	1	no	—
194	C₂	—	2	1	no	—
195	C₂	#14521097	1	2	yes	direct product factor
196	C₂	C₂²×Dic₃₁⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₃₁
commutator	C₄×C₃₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₂²×M₁₆×C₃₁	C₂
socle	C₂²×C₃₁	C₂×C₄⋊C₄
prime radical	C₂⁴×C₃₁	Q₈
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	C₂²×M₁₆	D₃₁
hypocenter	C₃₁	C₂×#430
LCS	G ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
3. #14521097	C₂	direct (N × H)
4. C₂×#11758410	C₂	semidirect
22. #11758410	C₂²	semidirect
102. C₂×C₃₁	#430	semidirect
121. C₃₁	C₂×#430	semidirect
195. C₂	#14521097	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×#92474	2
#14521097	2
C₂×#11758410	2
C₂²×M₁₆×C₃₁	2
C₂×#430	31

Subgroups — C2×#14521097 #16854317

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14521097 #16854317