Subgroups — C₂×#22559955 #30853293

Order 17152.

1438 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₆₇⋊C₈	C₂	1	2	yes	maximal
2	#22559955	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16863151	C₂	1	4	yes	maximal
4	C₂×C₄³×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₆₇⋊C₈	C₂²	1	4	yes	—
6	C₂×C₄×C₆₇⋊C₈	C₄	1	16	yes	complemented
7	C₂×C₄×C₆₇⋊C₈	C₂²	1	8	yes	—
8	#16863151	C₂²	1	16	yes	—
9	C₂²×C₄²×C₆₇	C₄	1	2	yes	—
10	C₂²×C₄²×C₆₇	C₄	1	2	yes	—
11	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
12	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
13	C₄³×C₆₇	C₄	1	4	yes	—
14	C₄³×C₆₇	C₂²	1	4	yes	—
15	C₂²×C₆₇⋊C₈	—	2	16	no	—
16	C₂²×C₆₇⋊C₈	C₂×C₄	1	16	yes	—
17	C₂²×C₆₇⋊C₈	C₂³	1	8	yes	—
18	C₄×C₆₇⋊C₈	C₂×C₄	1	32	yes	complemented
19	C₂³×C₄×C₆₇	C₂×C₄	1	4	yes	—
20	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
23	C₂×C₄²×C₆₇	—	2	8	no	—
24	C₂×C₄²×C₆₇	—	2	8	no	—
25	C₂×C₄²×C₆₇	C₈	1	16	yes	complemented
26	C₂×C₄²×C₆₇	C₂×C₄	1	8	yes	—
27	C₂×C₄²×C₆₇	C₂×C₄	1	8	yes	—
28	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₆₇	C₂×C₄	1	8	yes	—
31	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
32	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
33	C₂×C₆₇⋊C₈	—	2	32	no	—
34	C₂×C₆₇⋊C₈	—	2	16	no	—
35	C₂×C₆₇⋊C₈	C₂²×C₄	1	16	yes	—
36	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₆₇	—	2	8	no	—
38	C₂²×C₄×C₆₇	—	2	8	no	—
39	C₂²×C₄×C₆₇	C₄²	1	4	yes	—
40	C₂²×C₄×C₆₇	C₄²	1	4	yes	—
41	C₂²×C₄×C₆₇	C₂×C₈	1	16	yes	—
42	C₂²×C₄×C₆₇	C₂×C₈	1	8	yes	—
43	C₂²×C₄×C₆₇	M₁₆	1	8	yes	—
44	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
45	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
46	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
48	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
49	C₂²×C₄×C₆₇	#41	1	8	yes	—
50	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
51	C₄²×C₆₇	—	2	8	no	—
52	C₄²×C₆₇	—	2	16	no	—
53	C₄²×C₆₇	—	2	8	no	—
54	C₄²×C₆₇	C₄²	1	8	yes	—
55	C₄²×C₆₇	C₂×C₈	1	32	yes	complemented
56	C₄²×C₆₇	C₂²×C₄	1	8	yes	—
57	C₂³×C₆₇	C₂×C₄²	1	2	yes	—
58	C₂³×C₆₇	#117	1	4	yes	—
59	C₂³×C₆₇	C₂²×C₈	1	4	yes	—
60	C₂³×C₆₇	C₂×M₁₆	1	4	yes	—
61	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
62	C₂×C₄×C₆₇	—	2	8	no	—
63	C₂×C₄×C₆₇	—	2	8	no	—
64	C₂×C₄×C₆₇	—	2	32	no	—
65	C₂×C₄×C₆₇	C₄×C₈	1	16	yes	—
66	C₂×C₄×C₆₇	#97	1	16	yes	—
67	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
68	C₂×C₄×C₆₇	C₂×C₄²	1	4	yes	—
69	C₂×C₄×C₆₇	C₂×C₄²	1	4	yes	—
70	C₂×C₄×C₆₇	#117	1	16	yes	—
71	C₂×C₄×C₆₇	#117	1	4	yes	—
72	C₂×C₄×C₆₇	C₂²×C₈	1	16	yes	—
73	C₂×C₄×C₆₇	C₂²×C₈	1	8	yes	—
74	C₂×C₄×C₆₇	C₂×M₁₆	1	8	yes	—
75	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
76	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
77	C₆₇⋊C₈	—	2	16	no	—
78	C₂²×C₆₇	#376	1	4	yes	—
79	C₂²×C₆₇	C₂×C₄×C₈	1	4	yes	—
80	C₂²×C₆₇	C₂×#97	1	4	yes	—
81	C₂²×C₆₇	#431	1	16	yes	—
82	C₂²×C₆₇	C₂²×C₄²	1	1	yes	characteristic
83	C₂²×C₆₇	C₂×#117	1	2	yes	—
84	C₂²×C₆₇	C₂³×C₈	1	2	yes	—
85	C₂²×C₆₇	C₂²×M₁₆	1	2	yes	—
86	C₄×C₆₇	—	2	16	no	—
87	C₄×C₆₇	#376	1	4	yes	—
88	C₄×C₆₇	C₂×C₄×C₈	1	8	yes	—
89	C₄×C₆₇	C₂×#97	1	8	yes	—
90	C₄×C₆₇	C₂×#117	1	4	yes	—
91	C₂×#1744	—	67	1	no	maximal, Sylow
92	C₂×C₆₇	#1744	1	8	yes	complemented
93	C₂×C₆₇	C₂×#376	1	1	yes	characteristic
94	C₂×C₆₇	C₂²×C₄×C₈	1	1	yes	characteristic
95	C₂×C₆₇	C₂²×#97	1	1	yes	characteristic
96	C₂×C₆₇	C₂×#431	1	4	yes	—
97	C₂²×C₄×C₈	—	67	2	no	—
98	C₂×C₄³	D₆₇	1	1	yes	characteristic
99	#1744	—	67	8	no	—
100	C₂×#336	—	67	4	no	—
101	C₆₇	C₂×#1744	1	1	yes	characteristic, complemented, Sylow
102	C₂³×C₈	—	67	4	no	—
103	C₂²×C₄²	Dic₆₇	1	2	yes	—
104	C₂²×C₄²	Dic₆₇	1	2	yes	—
105	C₂²×C₄²	C₂×D₆₇	1	2	yes	—
106	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
107	C₂×C₄×C₈	—	67	16	no	—
108	C₂×C₄×C₈	—	67	8	no	—
109	C₄³	Dic₆₇	1	4	yes	—
110	C₄³	C₂×D₆₇	1	4	yes	—
111	#336	—	67	16	no	—
112	C₂³×C₄	C₂×Dic₆₇	1	4	yes	—
113	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
114	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
115	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
116	C₂²×C₈	—	67	16	no	—
117	C₂²×C₈	—	67	8	no	—
118	C₂²×C₈	—	134	16	no	—
119	C₂×C₄²	—	2	8	no	—
120	C₂×C₄²	—	2	8	no	—
121	C₂×C₄²	C₆₇⋊C₈	1	16	yes	complemented
122	C₂×C₄²	C₄×D₆₇	1	8	yes	—
123	C₂×C₄²	C₂×Dic₆₇	1	4	yes	—
124	C₂×C₄²	C₂×Dic₆₇	1	8	yes	—
125	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
126	C₂×C₄²	C₂×Dic₆₇	1	8	yes	—
127	C₂×C₄²	C₂²×D₆₇	1	4	yes	—
128	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
129	C₄×C₈	—	67	32	no	—
130	C₂⁴	C₂²×Dic₆₇	1	1	yes	characteristic
131	C₂²×C₄	—	2	8	no	—
132	C₂²×C₄	—	2	8	no	—
133	C₂²×C₄	C₄×Dic₆₇	1	4	yes	—
134	C₂²×C₄	C₄×Dic₆₇	1	4	yes	—
135	C₂²×C₄	C₂×C₆₇⋊C₈	1	16	yes	—
136	C₂²×C₄	C₂×C₆₇⋊C₈	1	8	yes	—
137	C₂²×C₄	#11761612	1	8	yes	—
138	C₂²×C₄	C₂×C₄×D₆₇	1	4	yes	—
139	C₂²×C₄	C₂²×Dic₆₇	1	8	yes	—
140	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
141	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
142	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
143	C₂²×C₄	#11761628	1	8	yes	—
144	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
145	C₂×C₈	—	67	16	no	—
146	C₂×C₈	—	134	32	no	—
147	C₂×C₈	—	134	16	no	—
148	C₄²	—	2	8	no	—
149	C₄²	—	2	16	no	—
150	C₄²	—	2	8	no	—
151	C₄²	C₄×Dic₆₇	1	8	yes	—
152	C₄²	C₂×C₆₇⋊C₈	1	32	yes	complemented
153	C₄²	C₂×C₄×D₆₇	1	8	yes	—
154	C₂³	C₂×C₄×Dic₆₇	1	2	yes	—
155	C₂³	#14524002	1	4	yes	—
156	C₂³	C₂²×C₆₇⋊C₈	1	4	yes	—
157	C₂³	C₂×#11761612	1	4	yes	—
158	C₂³	C₂³×Dic₆₇	1	1	yes	characteristic
159	C₂×C₄	—	2	8	no	—
160	C₂×C₄	—	2	8	no	—
161	C₂×C₄	—	2	32	no	—
162	C₂×C₄	C₄×C₆₇⋊C₈	1	16	yes	—
163	C₂×C₄	#14523954	1	16	yes	—
164	C₂×C₄	C₂×C₄×Dic₆₇	1	2	yes	—
165	C₂×C₄	C₂×C₄×Dic₆₇	1	4	yes	—
166	C₂×C₄	C₂×C₄×Dic₆₇	1	4	yes	—
167	C₂×C₄	#14524000	1	16	yes	—
168	C₂×C₄	#14524002	1	4	yes	—
169	C₂×C₄	C₂²×C₆₇⋊C₈	1	16	yes	—
170	C₂×C₄	C₂²×C₆₇⋊C₈	1	8	yes	—
171	C₂×C₄	C₂×#11761612	1	8	yes	—
172	C₂×C₄	C₂²×C₄×D₆₇	1	2	yes	—
173	C₂×C₄	C₂×#11761628	1	4	yes	—
174	C₈	—	134	16	no	—
175	C₂²	#16863242	1	4	yes	—
176	C₂²	C₂×C₄×C₆₇⋊C₈	1	4	yes	—
177	C₂²	C₂×#14523954	1	4	yes	—
178	C₂²	#16863449	1	16	yes	—
179	C₂²	C₂²×C₄×Dic₆₇	1	1	yes	characteristic
180	C₂²	C₂×#14524002	1	2	yes	—
181	C₂²	C₂³×C₆₇⋊C₈	1	2	yes	—
182	C₂²	C₂²×#11761612	1	2	yes	—
183	C₄	—	2	16	no	—
184	C₄	#16863242	1	4	yes	—
185	C₄	C₂×C₄×C₆₇⋊C₈	1	8	yes	—
186	C₄	C₂×#14523954	1	8	yes	—
187	C₄	C₂×#14524000	1	4	yes	—
188	C₂	#22559955	1	8	yes	direct product factor
189	C₂	C₂×#16863242	1	1	yes	characteristic
190	C₂	C₂²×C₄×C₆₇⋊C₈	1	1	yes	characteristic
191	C₂	C₂²×#14523954	1	1	yes	characteristic
192	C₂	C₂×#16863449	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄²	C₂×D₆₇
commutator	C₂×C₆₇	C₂²×C₄×C₈
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂×C₄³×C₆₇	C₂
socle, prime radical	C₂⁴×C₆₇	C₂²×C₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂×C₄³	D₆₇
hypocenter	C₆₇	C₂×#1744
LCS	G ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄² ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
2. #22559955	C₂	direct (N × H)
6. C₂×C₄×C₆₇⋊C₈	C₄	semidirect
18. C₄×C₆₇⋊C₈	C₂×C₄	semidirect
25. C₂×C₄²×C₆₇	C₈	semidirect
55. C₄²×C₆₇	C₂×C₈	semidirect
92. C₂×C₆₇	#1744	semidirect
101. C₆₇	C₂×#1744	semidirect
121. C₂×C₄²	C₆₇⋊C₈	semidirect
152. C₄²	C₂×C₆₇⋊C₈	semidirect
188. C₂	#22559955	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×C₄×C₆₇⋊C₈	2
#22559955	2
C₂×#16863151	2
C₂×C₄³×C₆₇	2
C₂×#1744	67

Subgroups — C2×#22559955 #30853293

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22559955 #30853293