Subgroups — C₂×C₄×#14523954 #30853155

Order 17152.

1790 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

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

	subgroup	G/H
center	C₂²×C₄²	C₂×D₆₇
commutator	C₂×C₆₇	C₂×C₄³
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×C₄×#97
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
3. C₄×#14523954	C₂	direct (N × H)
6. C₂×#14523954	C₄	direct (N × H)
8. C₂×C₄×C₆₇⋊C₈	C₄	semidirect
18. #14523954	C₂×C₄	direct (N × H)
19. C₄×C₆₇⋊C₈	C₂×C₄	semidirect
31. C₂×C₆₇⋊C₈	C₄²	semidirect
56. C₂×C₄×C₆₇	#97	semidirect
65. C₆₇⋊C₈	C₂×C₄²	semidirect
74. C₄×C₆₇	C₂×#97	semidirect
77. C₂×C₆₇	C₄×#97	semidirect
85. C₆₇	C₂×C₄×#97	semidirect
137. C₂×C₄	#14523954	direct (N × H)
155. C₄	C₂×#14523954	direct (N × H)
157. C₂	C₄×#14523954	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×#14523954	2
C₂²×C₄×C₆₇⋊C₈	2
C₄×#14523954	2
C₂×C₄³×C₆₇	2
C₂×C₄×#97	67

Subgroups — C2×C4×#14523954 #30853155

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#14523954 #30853155