Subgroups — C₂×#22569290 #31720770

Order 17152.

2234 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569290	C₂	1	16	yes	maximal, direct product factor
2	C₂×Q₈×C₆₇⋊C₈	C₂	1	8	yes	maximal, complemented
3	C₂×#16863449	C₂	1	6	yes	maximal
4	C₂²×C₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
5	Q₈×C₆₇⋊C₈	C₂²	1	64	yes	complemented
6	#16863449	C₂²	1	48	yes	—
7	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	12	yes	—
8	C₂×C₂²⋊(C₆₇⋊C₈)	C₂²	1	4	yes	—
9	C₂×C₄×C₆₇⋊C₈	C₂²	1	12	yes	—
10	C₂³×Q₈×C₆₇	C₄	1	1	yes	characteristic
11	C₂×C₄×Q₈×C₆₇	C₄	1	4	yes	—
12	C₂×C₄×Q₈×C₆₇	C₄	1	8	yes	—
13	C₂×C₄×Q₈×C₆₇	C₂²	1	8	yes	—
14	C₂×C₄×Q₈×C₆₇	C₂²	1	4	yes	—
15	C₂²×C₄⋊C₄×C₆₇	C₄	1	3	yes	—
16	C₂²×C₄²×C₆₇	C₂²	1	3	yes	—
17	C₂²×C₆₇⋊C₈	C₂³	1	8	yes	—
18	C₄⋊(C₆₇⋊C₈)	C₂³	1	48	yes	—
19	C₂²⋊(C₆₇⋊C₈)	C₂³	1	16	yes	—
20	C₄×C₆₇⋊C₈	C₂³	1	48	yes	—
21	C₂²×Q₈×C₆₇	C₈	1	16	yes	complemented
22	C₂²×Q₈×C₆₇	C₂×C₄	1	4	yes	—
23	C₂²×Q₈×C₆₇	C₂×C₄	1	8	yes	—
24	C₂³×C₄×C₆₇	C₂×C₄	1	3	yes	—
25	C₂³×C₄×C₆₇	C₂×C₄	1	3	yes	—
26	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
27	C₄×Q₈×C₆₇	C₂×C₄	1	32	yes	—
28	C₄×Q₈×C₆₇	C₂×C₄	1	16	yes	—
29	C₄×Q₈×C₆₇	C₂³	1	16	yes	—
30	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	12	yes	—
31	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	24	yes	—
32	C₂×C₄²×C₆₇	C₂×C₄	1	6	yes	—
33	C₂×C₄²×C₆₇	C₂×C₄	1	12	yes	—
34	C₂×C₄²×C₆₇	C₂³	1	6	yes	—
35	C₂×C₄²×C₆₇	C₂³	1	12	yes	—
36	C₂×C₆₇⋊C₈	—	2	16	no	—
37	C₂×C₆₇⋊C₈	C₂⁴	1	16	yes	—
38	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
39	C₂×Q₈×C₆₇	C₂×C₈	1	32	yes	—
40	C₂×Q₈×C₆₇	C₂×C₈	1	64	yes	complemented
41	C₂×Q₈×C₆₇	C₂²×C₄	1	16	yes	—
42	C₂²×C₄×C₆₇	—	2	4	no	—
43	C₂²×C₄×C₆₇	—	4	6	no	—
44	C₂²×C₄×C₆₇	C₂×C₈	1	24	yes	—
45	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
47	C₂²×C₄×C₆₇	C₂²×C₄	1	6	yes	—
48	C₂²×C₄×C₆₇	C₂²×C₄	1	12	yes	—
49	C₂²×C₄×C₆₇	C₂²×C₄	1	6	yes	—
50	C₂²×C₄×C₆₇	C₂²×C₄	1	12	yes	—
51	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
52	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
53	C₄⋊C₄×C₆₇	C₂²×C₄	1	48	yes	—
54	C₄²×C₆₇	C₂²×C₄	1	24	yes	—
55	C₄²×C₆₇	C₂²×C₄	1	12	yes	—
56	C₄²×C₆₇	C₂⁴	1	12	yes	—
57	C₂³×C₆₇	—	2	2	no	—
58	C₂³×C₆₇	—	2	2	no	—
59	C₂³×C₆₇	C₂²×C₈	1	4	yes	—
60	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
61	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
62	Q₈×C₆₇	C₂²×C₈	1	64	yes	—
63	C₂×C₄×C₆₇	—	2	2	no	—
64	C₂×C₄×C₆₇	—	2	12	no	—
65	C₂×C₄×C₆₇	—	2	4	no	—
66	C₂×C₄×C₆₇	—	2	4	no	—
67	C₂×C₄×C₆₇	—	4	12	no	—
68	C₂×C₄×C₆₇	C₂²×C₈	1	24	yes	—
69	C₂×C₄×C₆₇	C₂²×C₈	1	48	yes	—
70	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
71	C₂×C₄×C₆₇	C₂³×C₄	1	12	yes	—
72	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
73	C₂×C₄×C₆₇	C₂³×C₄	1	12	yes	—
74	C₂×C₄×C₆₇	2₋¹⁺⁴	1	4	yes	—
75	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
76	C₆₇⋊C₈	—	2	16	no	—
77	C₂²×C₆₇	—	2	2	no	—
78	C₂²×C₆₇	—	2	8	no	—
79	C₂²×C₆₇	—	2	2	no	—
80	C₂²×C₆₇	#519	1	2	yes	—
81	C₂²×C₆₇	C₂³×C₈	1	4	yes	—
82	C₂²×C₆₇	C₂³×C₈	1	2	yes	—
83	C₂²×C₆₇	#568	1	2	yes	—
84	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
85	C₄×C₆₇	—	2	2	no	—
86	C₄×C₆₇	—	2	12	no	—
87	C₄×C₆₇	#519	1	2	yes	—
88	C₄×C₆₇	C₂³×C₈	1	24	yes	—
89	C₄×C₆₇	C₂×2₋¹⁺⁴	1	2	yes	—
90	C₂×#2982	—	67	1	no	maximal, Sylow
91	C₂×C₆₇	—	2	4	no	—
92	C₂×C₆₇	#2982	1	4	yes	complemented
93	C₂×C₆₇	C₂×#519	1	1	yes	characteristic
94	C₂×C₆₇	C₂⁴×C₈	1	1	yes	characteristic
95	C₂×C₆₇	C₂×#568	1	1	yes	characteristic
96	C₂²×C₄×Q₈	D₆₇	1	1	yes	characteristic
97	#2982	—	67	16	no	—
98	C₂×C₈×Q₈	—	67	8	no	—
99	C₂×#431	—	67	6	no	—
100	C₆₇	C₂×#2982	1	1	yes	characteristic, complemented, Sylow
101	C₂³×Q₈	Dic₆₇	1	1	yes	characteristic
102	C₂×C₄×Q₈	Dic₆₇	1	4	yes	—
103	C₂×C₄×Q₈	Dic₆₇	1	8	yes	—
104	C₂×C₄×Q₈	C₂×D₆₇	1	4	yes	—
105	C₂×C₄×Q₈	C₂×D₆₇	1	8	yes	—
106	C₂²×C₄⋊C₄	Dic₆₇	1	3	yes	—
107	C₂²×C₄²	C₂×D₆₇	1	3	yes	—
108	C₈×Q₈	—	67	64	no	—
109	#431	—	67	48	no	—
110	C₂×C₄⋊C₈	—	67	12	no	—
111	C₂×C₂²⋊C₈	—	67	4	no	—
112	C₂×C₄×C₈	—	67	12	no	—
113	C₂²×Q₈	C₆₇⋊C₈	1	16	yes	complemented
114	C₂²×Q₈	C₂×Dic₆₇	1	4	yes	—
115	C₂²×Q₈	C₂×Dic₆₇	1	8	yes	—
116	C₂³×C₄	C₂×Dic₆₇	1	3	yes	—
117	C₂³×C₄	C₂×Dic₆₇	1	3	yes	—
118	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
119	C₂²×C₈	—	67	8	no	—
120	C₄×Q₈	C₂×Dic₆₇	1	16	yes	—
121	C₄×Q₈	C₂×Dic₆₇	1	32	yes	—
122	C₄×Q₈	C₂²×D₆₇	1	16	yes	—
123	C₂×C₄⋊C₄	C₂×Dic₆₇	1	12	yes	—
124	C₂×C₄⋊C₄	C₂×Dic₆₇	1	24	yes	—
125	C₂×C₄²	C₂×Dic₆₇	1	6	yes	—
126	C₂×C₄²	C₂×Dic₆₇	1	12	yes	—
127	C₂×C₄²	C₂²×D₆₇	1	12	yes	—
128	C₂×C₄²	C₂²×D₆₇	1	6	yes	—
129	C₄⋊C₈	—	67	48	no	—
130	C₂²⋊C₈	—	67	16	no	—
131	C₄×C₈	—	67	48	no	—
132	C₂⁴	C₂²×Dic₆₇	1	1	yes	characteristic
133	C₂×Q₈	C₂×C₆₇⋊C₈	1	32	yes	—
134	C₂×Q₈	C₂×C₆₇⋊C₈	1	64	yes	complemented
135	C₂×Q₈	C₂²×Dic₆₇	1	16	yes	—
136	C₂²×C₄	—	2	4	no	—
137	C₂²×C₄	—	4	6	no	—
138	C₂²×C₄	C₂×C₆₇⋊C₈	1	24	yes	—
139	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
140	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
141	C₂²×C₄	C₂²×Dic₆₇	1	6	yes	—
142	C₂²×C₄	C₂²×Dic₆₇	1	12	yes	—
143	C₂²×C₄	C₂²×Dic₆₇	1	6	yes	—
144	C₂²×C₄	C₂²×Dic₆₇	1	12	yes	—
145	C₂²×C₄	C₂³×D₆₇	1	2	yes	—
146	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
147	C₂×C₈	—	67	16	no	—
148	C₂×C₈	—	134	16	no	—
149	C₄⋊C₄	C₂²×Dic₆₇	1	48	yes	—
150	C₄²	C₂²×Dic₆₇	1	24	yes	—
151	C₄²	C₂²×Dic₆₇	1	12	yes	—
152	C₄²	C₂³×D₆₇	1	12	yes	—
153	C₂³	—	2	2	no	—
154	C₂³	—	2	2	no	—
155	C₂³	C₂²×C₆₇⋊C₈	1	4	yes	—
156	C₂³	C₂³×Dic₆₇	1	2	yes	—
157	C₂³	C₂³×Dic₆₇	1	1	yes	characteristic
158	Q₈	C₂²×C₆₇⋊C₈	1	64	yes	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	12	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	4	12	no	—
164	C₂×C₄	C₂²×C₆₇⋊C₈	1	24	yes	—
165	C₂×C₄	C₂²×C₆₇⋊C₈	1	48	yes	—
166	C₂×C₄	C₂³×Dic₆₇	1	2	yes	—
167	C₂×C₄	C₂³×Dic₆₇	1	12	yes	—
168	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
169	C₂×C₄	C₂³×Dic₆₇	1	12	yes	—
170	C₂×C₄	#14524091	1	4	yes	—
171	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
172	C₈	—	134	16	no	—
173	C₂²	—	2	2	no	—
174	C₂²	—	2	8	no	—
175	C₂²	—	2	2	no	—
176	C₂²	#16863874	1	2	yes	—
177	C₂²	C₂³×C₆₇⋊C₈	1	4	yes	—
178	C₂²	C₂³×C₆₇⋊C₈	1	2	yes	—
179	C₂²	#16864163	1	2	yes	—
180	C₂²	C₂⁴×Dic₆₇	1	1	yes	characteristic
181	C₄	—	2	12	no	—
182	C₄	—	2	2	no	—
183	C₄	#16863874	1	2	yes	—
184	C₄	C₂³×C₆₇⋊C₈	1	24	yes	—
185	C₄	C₂×#14524091	1	2	yes	—
186	C₂	—	2	4	no	—
187	C₂	#22569290	1	4	yes	direct product factor
188	C₂	C₂×#16863874	1	1	yes	characteristic
189	C₂	C₂⁴×C₆₇⋊C₈	1	1	yes	characteristic
190	C₂	C₂×#16864163	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂³×D₆₇
commutator	C₂×C₆₇	C₂⁴×C₈
Frattini	C₂×C₄	C₂⁴×D₆₇
Fitting	C₂²×C₄×Q₈×C₆₇	C₂
socle	C₂³×C₆₇	C₂³×C₄
prime radical	C₂⁴×C₆₇	C₂²×C₄
cosocle	C₂×C₄×C₆₇	C₂⁵
hypercenter	C₂²×C₄×Q₈	D₆₇
hypocenter	C₆₇	C₂×#2982
LCS	G ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. #22569290	C₂	direct (N × H)
2. C₂×Q₈×C₆₇⋊C₈	C₂	semidirect
5. Q₈×C₆₇⋊C₈	C₂²	semidirect
21. C₂²×Q₈×C₆₇	C₈	semidirect
40. C₂×Q₈×C₆₇	C₂×C₈	semidirect
92. C₂×C₆₇	#2982	semidirect
100. C₆₇	C₂×#2982	semidirect
113. C₂²×Q₈	C₆₇⋊C₈	semidirect
134. C₂×Q₈	C₂×C₆₇⋊C₈	semidirect
187. C₂	#22569290	direct (N × H)

Maximal subgroups

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

subgroup	index
#22569290	2
C₂×Q₈×C₆₇⋊C₈	2
C₂×#16863449	2
C₂²×C₄×Q₈×C₆₇	2
C₂×#2982	67

Subgroups — C2×#22569290 #31720770

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22569290 #31720770