Subgroups — C₂×#14521773 #16859261

Order 3968.

1112 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14521773	C₂	1	16	yes	maximal, direct product factor
2	C₂×#11758475	C₂	1	4	yes	maximal, complemented
3	C₂×#11758524	C₂	1	2	yes	maximal
4	C₂×#11758474	C₂	1	2	yes	maximal, complemented
5	C₂×Q₈×Dic₃₁	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×Dic₆₂	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×D₁₂₄	C₂	1	1	yes	maximal, characteristic
8	C₂×#11758469	C₂	1	2	yes	maximal, complemented
9	C₂×C₃₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Q₈×C₃₁	C₂²	1	2	yes	—
11	C₂²×D₁₂₄	C₂²	1	1	yes	characteristic
12	C₂²×C₄×D₃₁	C₂²	1	2	yes	—
13	C₂²×Dic₆₂	C₂²	1	1	yes	characteristic
14	C₃₁×C₄⋊Q₈	C₂²	1	8	yes	complemented
15	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
17	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
18	#11758524	C₂²	1	16	yes	—
19	Q₈×Dic₃₁	C₂²	1	16	yes	complemented
20	C₂×(C₂×D₃₁)⋊C₄	C₂²	1	4	yes	—
21	C₂×(C₂×D₃₁)⋊C₄	C₂²	1	2	yes	—
22	C₂×(C₂×D₃₁)⋊C₄	C₂²	1	4	yes	—
23	C₂×C₄⋊Dic₃₁	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊Dic₃₁	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊Dic₃₁	C₂²	1	4	yes	—
26	C₂×Dic₃₁⋊C₄	C₂²	1	2	yes	—
27	C₂×C₄×Dic₃₁	C₂²	1	4	yes	—
28	C₂×C₄×Dic₃₁	C₂²	1	2	yes	—
29	#11758475	C₂²	1	32	yes	complemented
30	#11758474	C₂²	1	16	yes	complemented
31	#11758469	C₂²	1	16	yes	complemented
32	C₄×D₁₂₄	C₂²	1	8	yes	—
33	C₄×Dic₆₂	C₂²	1	8	yes	complemented
34	C₂³×D₃₁	C₂³	1	2	yes	—
35	C₂×Q₈×C₃₁	—	2	8	no	—
36	C₂×Q₈×C₃₁	C₂³	1	8	yes	—
37	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
38	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
41	C₂²×Dic₃₁	C₂³	1	2	yes	—
42	C₂²×Dic₃₁	C₂³	1	2	yes	—
43	C₂²×Dic₃₁	C₂³	1	2	yes	—
44	C₂×D₁₂₄	—	2	4	no	—
45	C₂×D₁₂₄	C₂³	1	4	yes	—
46	C₂×C₄×D₃₁	—	2	8	no	—
47	C₂×C₄×D₃₁	C₂³	1	8	yes	—
48	C₂×Dic₆₂	—	2	4	no	—
49	C₂×Dic₆₂	C₂³	1	4	yes	—
50	C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
51	C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
52	C₄²×C₃₁	C₂³	1	4	yes	—
53	(C₂×D₃₁)⋊C₄	C₂³	1	8	yes	—
54	(C₂×D₃₁)⋊C₄	C₂³	1	16	yes	—
55	(C₂×D₃₁)⋊C₄	C₂³	1	16	yes	—
56	C₄⋊Dic₃₁	C₂³	1	4	yes	—
57	C₄⋊Dic₃₁	C₂³	1	4	yes	—
58	C₄⋊Dic₃₁	C₂³	1	16	yes	—
59	Dic₃₁⋊C₄	C₂³	1	8	yes	—
60	C₄×Dic₃₁	C₂³	1	8	yes	—
61	C₄×Dic₃₁	C₂³	1	16	yes	—
62	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
63	C₂²×D₃₁	—	2	4	no	—
64	C₂²×D₃₁	—	2	4	no	—
65	C₂²×D₃₁	—	2	4	no	—
66	C₂²×D₃₁	C₂⁴	1	4	yes	—
67	Q₈×C₃₁	—	2	16	no	—
68	C₂×C₄×C₃₁	—	2	8	no	—
69	C₂×C₄×C₃₁	—	2	2	no	—
70	C₂×C₄×C₃₁	#41	1	4	yes	—
71	C₂×C₄×C₃₁	#41	1	4	yes	—
72	C₂×C₄×C₃₁	C₂⁴	1	8	yes	—
73	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
74	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
75	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
76	C₂×Dic₃₁	—	2	4	no	—
77	C₂×Dic₃₁	—	2	4	no	—
78	C₂×Dic₃₁	—	2	4	no	—
79	C₂×Dic₃₁	C₂⁴	1	4	yes	—
80	C₂×Dic₃₁	C₂⁴	1	4	yes	—
81	C₂×Dic₃₁	C₂⁴	1	4	yes	—
82	D₁₂₄	—	2	8	no	—
83	C₄×D₃₁	—	2	16	no	—
84	Dic₆₂	—	2	8	no	—
85	C₂×#556	—	31	1	no	maximal, Sylow
86	C₂²×C₃₁	C₂×#41	1	2	yes	—
87	C₂²×C₃₁	C₂×#41	1	2	yes	—
88	C₂²×C₃₁	2₋¹⁺⁴	1	2	yes	—
89	C₂²×C₃₁	C₂⁵	1	1	yes	characteristic
90	C₂×D₃₁	—	2	4	no	—
91	C₂×D₃₁	—	2	4	no	—
92	C₂×D₃₁	—	2	4	no	—
93	C₂×D₃₁	—	4	8	no	—
94	C₄×C₃₁	—	2	8	no	—
95	C₄×C₃₁	—	2	2	no	—
96	C₄×C₃₁	C₂×#41	1	4	yes	—
97	C₄×C₃₁	C₂×#41	1	4	yes	—
98	Dic₃₁	—	2	4	no	—
99	Dic₃₁	—	2	4	no	—
100	Dic₃₁	—	2	4	no	—
101	#556	—	31	16	no	—
102	C₂×C₄⋊Q₈	D₃₁	1	1	yes	characteristic, complemented
103	C₂×#126	—	31	4	no	—
104	C₂×#124	—	31	2	no	—
105	C₂×C₂²⋊Q₈	—	31	2	no	—
106	C₂×C₄×Q₈	—	31	2	no	—
107	C₂×C₄×Q₈	—	31	1	no	—
108	C₂×C₄×D₄	—	31	1	no	—
109	C₂×#117	—	31	2	no	—
110	C₂×C₃₁	#556	1	4	yes	complemented
111	C₂×C₃₁	C₂²×#41	1	1	yes	characteristic
112	C₂×C₃₁	C₂²×#41	1	1	yes	characteristic
113	C₂×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
114	D₃₁	—	4	4	no	—
115	C₂²×Q₈	—	31	1	no	—
116	C₂²×Q₈	C₂×D₃₁	1	2	yes	—
117	C₂²×D₄	—	31	1	no	—
118	C₂³×C₄	—	31	2	no	—
119	C₄⋊Q₈	C₂×D₃₁	1	8	yes	complemented
120	#126	—	31	32	no	—
121	#124	—	31	16	no	—
122	C₂²⋊Q₈	—	31	16	no	—
123	C₄×Q₈	—	31	16	no	—
124	C₄×Q₈	—	31	8	no	—
125	C₄×D₄	—	31	8	no	—
126	#117	—	31	16	no	—
127	C₂×C₄⋊C₄	—	31	1	no	—
128	C₂×C₄⋊C₄	—	31	1	no	—
129	C₂×C₄⋊C₄	—	31	4	no	—
130	C₂×C₄⋊C₄	—	31	2	no	—
131	C₂×C₄⋊C₄	C₂×D₃₁	1	2	yes	—
132	C₂×C₄⋊C₄	C₂×D₃₁	1	2	yes	—
133	C₂×C₂²⋊C₄	—	31	4	no	—
134	C₂×C₂²⋊C₄	—	31	2	no	—
135	C₂×C₂²⋊C₄	—	31	4	no	—
136	C₂×C₄²	—	31	4	no	—
137	C₂×C₄²	—	31	2	no	—
138	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
139	C₃₁	C₂×#556	1	1	yes	characteristic, complemented, Sylow
140	C₂⁴	—	31	2	no	—
141	C₂×Q₈	—	2	8	no	—
142	C₂×Q₈	—	31	4	no	—
143	C₂×Q₈	—	62	4	no	—
144	C₂×Q₈	C₂²×D₃₁	1	8	yes	—
145	C₂×D₄	—	31	4	no	—
146	C₂×D₄	—	62	4	no	—
147	C₂²×C₄	—	31	2	no	—
148	C₂²×C₄	—	31	2	no	—
149	C₂²×C₄	—	31	2	no	—
150	C₂²×C₄	—	31	8	no	—
151	C₂²×C₄	—	62	8	no	—
152	C₂²×C₄	C₂²×D₃₁	1	4	yes	—
153	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
154	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
155	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
156	C₄⋊C₄	—	31	8	no	—
157	C₄⋊C₄	—	31	4	no	—
158	C₄⋊C₄	—	31	4	no	—
159	C₄⋊C₄	—	31	16	no	—
160	C₄⋊C₄	C₂²×D₃₁	1	8	yes	—
161	C₄⋊C₄	C₂²×D₃₁	1	8	yes	—
162	C₂²⋊C₄	—	31	16	no	—
163	C₂²⋊C₄	—	31	8	no	—
164	C₂²⋊C₄	—	31	16	no	—
165	C₄²	—	31	16	no	—
166	C₄²	—	31	8	no	—
167	C₄²	C₂²×D₃₁	1	4	yes	—
168	C₂³	—	31	4	no	—
169	C₂³	—	62	4	no	—
170	C₂³	—	62	4	no	—
171	C₂³	—	62	4	no	—
172	C₂³	C₂³×D₃₁	1	1	yes	characteristic
173	Q₈	—	2	16	no	—
174	Q₈	—	62	8	no	—
175	D₄	—	62	8	no	—
176	C₂×C₄	—	2	8	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	31	4	no	—
179	C₂×C₄	—	31	4	no	—
180	C₂×C₄	—	31	4	no	—
181	C₂×C₄	—	62	4	no	—
182	C₂×C₄	—	62	4	no	—
183	C₂×C₄	—	62	4	no	—
184	C₂×C₄	—	62	16	no	—
185	C₂×C₄	#92497	1	4	yes	—
186	C₂×C₄	#92499	1	4	yes	—
187	C₂×C₄	C₂³×D₃₁	1	8	yes	—
188	C₂×C₄	C₂³×D₃₁	1	2	yes	—
189	C₂×C₄	C₂³×D₃₁	1	2	yes	—
190	C₂×C₄	C₂³×D₃₁	1	2	yes	—
191	C₂²	—	62	4	no	—
192	C₂²	—	62	4	no	—
193	C₂²	—	62	4	no	—
194	C₂²	—	124	8	no	—
195	C₂²	C₂×#92497	1	2	yes	—
196	C₂²	C₂×#92499	1	2	yes	—
197	C₂²	#11758564	1	2	yes	—
198	C₂²	C₂⁴×D₃₁	1	1	yes	characteristic
199	C₄	—	2	8	no	—
200	C₄	—	2	2	no	—
201	C₄	—	62	4	no	—
202	C₄	—	62	4	no	—
203	C₄	—	62	4	no	—
204	C₄	C₂×#92497	1	4	yes	—
205	C₄	C₂×#92499	1	4	yes	—
206	C₂	—	124	4	no	—
207	C₂	#14521773	1	4	yes	direct product factor
208	C₂	C₂²×#92497	1	1	yes	characteristic
209	C₂	C₂²×#92499	1	1	yes	characteristic
210	C₂	C₂×#11758564	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₃₁
commutator, cosocle	C₂²×C₃₁	C₂⁵
Frattini	C₂²	C₂⁴×D₃₁
Fitting	C₂×C₃₁×C₄⋊Q₈	C₂
socle	C₂³×C₃₁	C₂⁴
prime radical	C₂²×D₁₂₄	C₂²
hypercenter	C₂×C₄⋊Q₈	D₃₁
hypocenter	C₃₁	C₂×#556
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #14521773	C₂	direct (N × H)
2. C₂×#11758475	C₂	semidirect
4. C₂×#11758474	C₂	semidirect
5. C₂×Q₈×Dic₃₁	C₂	semidirect
6. C₂×C₄×Dic₆₂	C₂	semidirect
8. C₂×#11758469	C₂	semidirect
9. C₂×C₃₁×C₄⋊Q₈	C₂	semidirect
14. C₃₁×C₄⋊Q₈	C₂²	semidirect
19. Q₈×Dic₃₁	C₂²	semidirect
29. #11758475	C₂²	semidirect
30. #11758474	C₂²	semidirect
31. #11758469	C₂²	semidirect
33. C₄×Dic₆₂	C₂²	semidirect
102. C₂×C₄⋊Q₈	D₃₁	semidirect
110. C₂×C₃₁	#556	semidirect
119. C₄⋊Q₈	C₂×D₃₁	semidirect
139. C₃₁	C₂×#556	semidirect
207. C₂	#14521773	direct (N × H)

Maximal subgroups

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

subgroup	index
#14521773	2
C₂×#11758475	2
C₂×#11758524	2
C₂×#11758474	2
C₂×Q₈×Dic₃₁	2
C₂×C₄×Dic₆₂	2
C₂×C₄×D₁₂₄	2
C₂×#11758469	2
C₂×C₃₁×C₄⋊Q₈	2
C₂×#556	31

Subgroups — C2×#14521773 #16859261

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14521773 #16859261