Subgroups — C₂×#22560361 #30856210

Order 17152.

1470 conjugacy classes of subgroups.

The subgroup lattice has 156 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	6	yes	maximal
2	#22560361	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄³×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₆₇⋊C₈	C₂²	1	4	yes	—
5	C₂×C₄⋊(C₆₇⋊C₈)	—	2	24	no	—
6	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	24	yes	—
7	C₂²×C₄²×C₆₇	C₄	1	3	yes	—
8	C₂²×C₄²×C₆₇	C₄	1	1	yes	characteristic
9	C₂²×C₄²×C₆₇	C₂²	1	3	yes	—
10	C₄³×C₆₇	C₄	1	4	yes	—
11	C₄³×C₆₇	C₂²	1	4	yes	—
12	C₂²×C₆₇⋊C₈	—	2	24	no	—
13	C₂²×C₆₇⋊C₈	C₂³	1	8	yes	—
14	C₄⋊(C₆₇⋊C₈)	—	2	48	no	—
15	C₂³×C₄×C₆₇	C₂×C₄	1	3	yes	—
16	C₂³×C₄×C₆₇	C₂×C₄	1	3	yes	—
17	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
18	C₂×C₄²×C₆₇	—	2	12	no	—
19	C₂×C₄²×C₆₇	C₈	1	8	yes	complemented
20	C₂×C₄²×C₆₇	C₂×C₄	1	12	yes	—
21	C₂×C₄²×C₆₇	C₂×C₄	1	6	yes	—
22	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
23	C₂×C₄²×C₆₇	D₄	1	12	yes	—
24	C₂×C₄²×C₆₇	Q₈	1	12	yes	—
25	C₂×C₄²×C₆₇	C₂³	1	6	yes	—
26	C₂×C₆₇⋊C₈	—	2	24	no	—
27	C₂×C₆₇⋊C₈	—	4	16	no	—
28	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₆₇	—	2	12	no	—
30	C₂²×C₄×C₆₇	C₄⋊C₄	1	12	yes	—
31	C₂²×C₄×C₆₇	C₄⋊C₄	1	12	yes	—
32	C₂²×C₄×C₆₇	C₂×C₈	1	12	yes	—
33	C₂²×C₄×C₆₇	M₁₆	1	12	yes	—
34	C₂²×C₄×C₆₇	C₂²×C₄	1	6	yes	—
35	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₆₇	C₂²×C₄	1	6	yes	—
37	C₂²×C₄×C₆₇	C₂×D₄	1	6	yes	—
38	C₂²×C₄×C₆₇	C₂×Q₈	1	6	yes	—
39	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
40	C₄²×C₆₇	—	2	24	no	—
41	C₄²×C₆₇	C₄⋊C₄	1	24	yes	—
42	C₄²×C₆₇	C₂×C₈	1	16	yes	complemented
43	C₄²×C₆₇	C₂×D₄	1	12	yes	—
44	C₄²×C₆₇	C₂×Q₈	1	12	yes	—
45	C₂³×C₆₇	C₂×C₄⋊C₄	1	6	yes	—
46	C₂³×C₆₇	C₂²×C₈	1	2	yes	—
47	C₂³×C₆₇	C₂×M₁₆	1	6	yes	—
48	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
49	C₂×C₄×C₆₇	—	2	12	no	—
50	C₂×C₄×C₆₇	—	2	24	no	—
51	C₂×C₄×C₆₇	C₄⋊C₈	1	48	yes	complemented
52	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	12	yes	—
53	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	6	yes	—
54	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	12	yes	—
55	C₂×C₄×C₆₇	#127	1	4	yes	—
56	C₂×C₄×C₆₇	C₄⋊Q₈	1	12	yes	—
57	C₂×C₄×C₆₇	C₂²×C₈	1	12	yes	—
58	C₂×C₄×C₆₇	C₂×M₁₆	1	12	yes	—
59	C₂×C₄×C₆₇	C₂²×D₄	1	3	yes	—
60	C₂×C₄×C₆₇	C₂²×Q₈	1	3	yes	—
61	C₆₇⋊C₈	—	4	8	no	—
62	C₂²×C₆₇	#384	1	4	yes	—
63	C₂²×C₆₇	C₂×C₄⋊C₈	1	12	yes	—
64	C₂²×C₆₇	#423	1	12	yes	—
65	C₂²×C₆₇	C₂²×C₄⋊C₄	1	3	yes	—
66	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
67	C₂²×C₆₇	C₂²×M₁₆	1	3	yes	—
68	C₄×C₆₇	—	2	12	no	—
69	C₄×C₆₇	#384	1	4	yes	—
70	C₄×C₆₇	C₂×C₄⋊C₈	1	24	yes	complemented
71	C₄×C₆₇	C₂×#127	1	1	yes	characteristic
72	C₄×C₆₇	C₂×C₄⋊Q₈	1	3	yes	—
73	C₂×#1842	—	67	1	no	maximal, Sylow
74	C₂×C₆₇	#1842	1	8	yes	complemented
75	C₂×C₆₇	C₂×#384	1	1	yes	characteristic
76	C₂×C₆₇	C₂²×C₄⋊C₈	1	3	yes	—
77	C₂×C₆₇	C₂×#423	1	3	yes	—
78	C₂²×C₄⋊C₈	—	67	6	no	—
79	C₂×C₄³	D₆₇	1	1	yes	characteristic
80	#1842	—	67	8	no	—
81	C₆₇	C₂×#1842	1	1	yes	characteristic, complemented, Sylow
82	C₂³×C₈	—	67	4	no	—
83	C₂²×C₄²	Dic₆₇	1	3	yes	—
84	C₂²×C₄²	Dic₆₇	1	1	yes	characteristic
85	C₂²×C₄²	C₂×D₆₇	1	3	yes	—
86	C₂×C₄⋊C₈	—	67	24	no	—
87	C₂×C₄⋊C₈	—	134	24	no	—
88	C₄³	Dic₆₇	1	4	yes	—
89	C₄³	C₂×D₆₇	1	4	yes	—
90	C₂³×C₄	C₂×Dic₆₇	1	3	yes	—
91	C₂³×C₄	C₂×Dic₆₇	1	3	yes	—
92	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
93	C₂²×C₈	—	67	8	no	—
94	C₂²×C₈	—	134	24	no	—
95	C₂×C₄²	—	2	12	no	—
96	C₂×C₄²	C₆₇⋊C₈	1	8	yes	complemented
97	C₂×C₄²	Dic₁₃₄	1	12	yes	—
98	C₂×C₄²	D₂₆₈	1	12	yes	—
99	C₂×C₄²	C₂×Dic₆₇	1	12	yes	—
100	C₂×C₄²	C₂×Dic₆₇	1	6	yes	—
101	C₂×C₄²	C₂×Dic₆₇	1	4	yes	—
102	C₂×C₄²	C₂²×D₆₇	1	6	yes	—
103	C₄⋊C₈	—	134	48	no	—
104	C₂⁴	C₂²×Dic₆₇	1	1	yes	characteristic
105	C₂²×C₄	—	2	12	no	—
106	C₂²×C₄	C₄⋊Dic₆₇	1	12	yes	—
107	C₂²×C₄	C₄⋊Dic₆₇	1	12	yes	—
108	C₂²×C₄	C₂×C₆₇⋊C₈	1	12	yes	—
109	C₂²×C₄	#11761612	1	12	yes	—
110	C₂²×C₄	C₂²×Dic₆₇	1	6	yes	—
111	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
112	C₂²×C₄	C₂²×Dic₆₇	1	6	yes	—
113	C₂²×C₄	C₂×D₂₆₈	1	6	yes	—
114	C₂²×C₄	C₂×Dic₁₃₄	1	6	yes	—
115	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
116	C₂×C₈	—	268	16	no	—
117	C₂×C₈	—	134	24	no	—
118	C₄²	—	2	24	no	—
119	C₄²	C₄⋊Dic₆₇	1	24	yes	—
120	C₄²	C₂×C₆₇⋊C₈	1	16	yes	complemented
121	C₄²	C₂×D₂₆₈	1	12	yes	—
122	C₄²	C₂×Dic₁₃₄	1	12	yes	—
123	C₂³	C₂×C₄⋊Dic₆₇	1	6	yes	—
124	C₂³	C₂²×C₆₇⋊C₈	1	2	yes	—
125	C₂³	C₂×#11761612	1	6	yes	—
126	C₂³	C₂³×Dic₆₇	1	1	yes	characteristic
127	C₂×C₄	—	2	12	no	—
128	C₂×C₄	—	2	24	no	—
129	C₂×C₄	C₄⋊(C₆₇⋊C₈)	1	48	yes	complemented
130	C₂×C₄	C₂×C₄⋊Dic₆₇	1	12	yes	—
131	C₂×C₄	C₂×C₄⋊Dic₆₇	1	6	yes	—
132	C₂×C₄	C₂×C₄⋊Dic₆₇	1	12	yes	—
133	C₂×C₄	#14524040	1	4	yes	—
134	C₂×C₄	#14524043	1	12	yes	—
135	C₂×C₄	C₂²×C₆₇⋊C₈	1	12	yes	—
136	C₂×C₄	C₂×#11761612	1	12	yes	—
137	C₂×C₄	C₂²×D₂₆₈	1	3	yes	—
138	C₂×C₄	C₂²×Dic₁₃₄	1	3	yes	—
139	C₈	—	268	8	no	—
140	C₂²	#16863266	1	4	yes	—
141	C₂²	C₂×C₄⋊(C₆₇⋊C₈)	1	12	yes	—
142	C₂²	#16863420	1	12	yes	—
143	C₂²	C₂²×C₄⋊Dic₆₇	1	3	yes	—
144	C₂²	C₂³×C₆₇⋊C₈	1	1	yes	characteristic
145	C₂²	C₂²×#11761612	1	3	yes	—
146	C₄	—	2	12	no	—
147	C₄	#16863266	1	4	yes	—
148	C₄	C₂×C₄⋊(C₆₇⋊C₈)	1	24	yes	complemented
149	C₄	C₂×#14524040	1	1	yes	characteristic
150	C₄	C₂×#14524043	1	3	yes	—
151	C₂	#22560361	1	8	yes	direct product factor
152	C₂	C₂×#16863266	1	1	yes	characteristic
153	C₂	C₂²×C₄⋊(C₆₇⋊C₈)	1	3	yes	—
154	C₂	C₂×#16863420	1	3	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₆₇
commutator	C₂²×C₆₇	C₂³×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×#1842
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
2. #22560361	C₂	direct (N × H)
19. C₂×C₄²×C₆₇	C₈	semidirect
42. C₄²×C₆₇	C₂×C₈	semidirect
51. C₂×C₄×C₆₇	C₄⋊C₈	semidirect
70. C₄×C₆₇	C₂×C₄⋊C₈	semidirect
74. C₂×C₆₇	#1842	semidirect
81. C₆₇	C₂×#1842	semidirect
96. C₂×C₄²	C₆₇⋊C₈	semidirect
120. C₄²	C₂×C₆₇⋊C₈	semidirect
129. C₂×C₄	C₄⋊(C₆₇⋊C₈)	semidirect
148. C₄	C₂×C₄⋊(C₆₇⋊C₈)	semidirect
151. C₂	#22560361	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
#22560361	2
C₂×C₄³×C₆₇	2
C₂×#1842	67

Subgroups — C2×#22560361 #30856210

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22560361 #30856210