Subgroups — C₂×#22558406 #30776320

Order 17152.

214 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	C₂×#16863230	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×Dic₆₇	C₂	1	1	yes	maximal, characteristic
3	#22558406	C₂	1	4	yes	maximal, direct product factor
4	C₂×M₆₄×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₆₇⋊C₁₆	C₄	1	1	yes	characteristic
6	C₂×C₈×Dic₆₇	C₄	1	1	yes	characteristic
7	#16863230	—	2	1	no	—
8	#16863230	C₂²	1	2	yes	—
9	C₂×C₆₇⋊C₃₂	—	2	1	no	—
10	C₁₆×Dic₆₇	C₄	1	2	yes	—
11	C₁₆×Dic₆₇	C₂²	1	2	yes	—
12	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
13	M₆₄×C₆₇	—	2	1	no	—
14	M₆₄×C₆₇	C₂²	1	2	yes	—
15	C₂×C₃₂×C₆₇	—	2	1	no	—
16	C₂²×C₆₇⋊C₈	C₈	1	1	yes	characteristic
17	C₂×C₄×Dic₆₇	C₈	1	1	yes	characteristic
18	C₂×C₆₇⋊C₁₆	—	2	1	no	—
19	C₂×C₆₇⋊C₁₆	—	2	1	no	—
20	C₂×C₆₇⋊C₁₆	C₂×C₄	1	2	yes	—
21	C₈×Dic₆₇	C₈	1	2	yes	—
22	C₈×Dic₆₇	C₂×C₄	1	2	yes	—
23	C₆₇⋊C₃₂	—	2	2	no	—
24	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
27	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
28	C₂×C₁₆×C₆₇	Q₈	1	1	yes	characteristic
29	C₂×C₁₆×C₆₇	Q₈	1	1	yes	characteristic
30	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
31	C₃₂×C₆₇	—	2	2	no	—
32	C₂²×Dic₆₇	—	2	1	no	—
33	C₂×C₆₇⋊C₈	C₁₆	1	2	yes	—
34	C₂×C₆₇⋊C₈	C₁₆	1	2	yes	—
35	C₂×C₆₇⋊C₈	C₂×C₈	1	2	yes	—
36	C₄×Dic₆₇	—	2	1	no	—
37	C₄×Dic₆₇	C₂×C₈	1	2	yes	—
38	C₆₇⋊C₁₆	—	2	2	no	—
39	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
40	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₆₇	C₂×C₈	1	1	yes	characteristic
43	C₂×C₈×C₆₇	M₁₆	1	1	yes	characteristic
44	C₂×C₈×C₆₇	M₁₆	1	1	yes	characteristic
45	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
46	C₁₆×C₆₇	C₄⋊C₄	1	1	yes	characteristic
47	C₁₆×C₆₇	C₄⋊C₄	1	1	yes	characteristic
48	C₁₆×C₆₇	C₂×D₄	1	1	yes	characteristic
49	C₁₆×C₆₇	C₂×Q₈	1	1	yes	characteristic
50	C₂³×C₆₇	M₃₂	1	1	yes	characteristic
51	C₂×C₄×C₆₇	C₄⋊C₈	1	1	yes	characteristic
52	C₂×C₄×C₆₇	C₄⋊C₈	1	1	yes	characteristic
53	C₂×C₄×C₆₇	C₂×C₁₆	1	1	yes	characteristic
54	C₂×C₄×C₆₇	C₂×C₁₆	1	1	yes	characteristic
55	C₂×C₄×C₆₇	M₃₂	1	1	yes	characteristic
56	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
57	C₂×Dic₆₇	—	2	4	no	—
58	C₂×Dic₆₇	—	2	2	no	—
59	C₈×C₆₇	C₄⋊C₈	1	1	yes	characteristic
60	C₈×C₆₇	C₄⋊C₈	1	1	yes	characteristic
61	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₈×C₆₇	C₂×M₁₆	1	1	yes	characteristic
63	C₆₇⋊C₈	C₂×C₁₆	1	4	yes	—
64	C₂²×C₆₇	—	2	2	no	—
65	C₂²×C₆₇	C₄⋊C₁₆	1	1	yes	characteristic
66	C₂²×C₆₇	C₄⋊C₁₆	1	1	yes	characteristic
67	C₂²×C₆₇	C₂×M₃₂	1	1	yes	characteristic
68	C₄×C₆₇	C₄⋊C₁₆	1	1	yes	characteristic
69	C₄×C₆₇	C₄⋊C₁₆	1	1	yes	characteristic
70	C₄×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
71	C₄×C₆₇	C₂²×C₁₆	1	1	yes	characteristic
72	Dic₆₇	—	2	4	no	—
73	C₂×#1422	—	67	1	no	maximal, Sylow
74	C₂×C₆₇	—	2	1	no	—
75	C₂×C₆₇	—	2	1	no	—
76	C₂×C₆₇	#1422	1	2	yes	complemented
77	C₂×C₆₇	C₂×C₄⋊C₁₆	1	1	yes	characteristic
78	C₂×M₆₄	—	67	1	no	—
79	C₂×M₆₄	D₆₇	1	1	yes	characteristic
80	C₂×C₄×C₁₆	—	67	1	no	—
81	#1422	—	67	4	no	—
82	C₆₇	C₂×#1422	1	1	yes	characteristic, complemented, Sylow
83	C₂²×C₁₆	—	67	1	no	—
84	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
85	C₂×C₄×C₈	—	67	1	no	—
86	M₆₄	—	2	1	no	—
87	M₆₄	—	67	2	no	—
88	M₆₄	—	134	1	no	—
89	M₆₄	C₂×D₆₇	1	2	yes	—
90	C₂×C₃₂	—	2	1	no	—
91	C₂×C₃₂	—	134	1	no	—
92	C₄×C₁₆	—	67	2	no	—
93	C₄×C₁₆	—	67	2	no	—
94	C₂²×C₈	—	67	1	no	—
95	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
96	C₂×C₄²	—	67	1	no	—
97	C₂×C₁₆	—	67	2	no	—
98	C₂×C₁₆	—	134	1	no	—
99	C₂×C₁₆	—	134	1	no	—
100	C₂×C₁₆	Dic₁₃₄	1	1	yes	characteristic
101	C₂×C₁₆	Dic₁₃₄	1	1	yes	characteristic
102	C₂×C₁₆	C₄×D₆₇	1	1	yes	characteristic
103	C₂×C₁₆	C₂²⋊D₆₇	1	1	yes	characteristic
104	C₂×C₁₆	C₂²⋊D₆₇	1	1	yes	characteristic
105	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
106	C₄×C₈	—	67	2	no	—
107	C₄×C₈	—	67	2	no	—
108	C₃₂	—	2	2	no	—
109	C₃₂	—	134	2	no	—
110	C₂²×C₄	—	134	1	no	—
111	C₂²×C₄	C₈×D₆₇	1	1	yes	characteristic
112	C₂×C₈	—	67	2	no	—
113	C₂×C₈	—	67	2	no	—
114	C₂×C₈	—	67	2	no	—
115	C₂×C₈	Dic₆₇⋊C₄	1	1	yes	characteristic
116	C₂×C₈	Dic₆₇⋊C₄	1	1	yes	characteristic
117	C₂×C₈	C₈×D₆₇	1	1	yes	characteristic
118	C₂×C₈	#11761611	1	1	yes	characteristic
119	C₂×C₈	#11761611	1	1	yes	characteristic
120	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
121	C₄²	—	67	2	no	—
122	C₄²	—	134	1	no	—
123	C₁₆	—	134	2	no	—
124	C₁₆	Dic₆₇⋊C₄	1	1	yes	characteristic
125	C₁₆	Dic₆₇⋊C₄	1	1	yes	characteristic
126	C₁₆	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
127	C₁₆	C₂×Dic₁₃₄	1	1	yes	characteristic
128	C₂³	#14523982	1	1	yes	characteristic
129	C₂×C₄	—	134	4	no	—
130	C₂×C₄	—	134	2	no	—
131	C₂×C₄	Dic₆₇⋊C₈	1	1	yes	characteristic
132	C₂×C₄	Dic₆₇⋊C₈	1	1	yes	characteristic
133	C₂×C₄	C₁₆×D₆₇	1	1	yes	characteristic
134	C₂×C₄	C₁₆×D₆₇	1	1	yes	characteristic
135	C₂×C₄	#14523982	1	1	yes	characteristic
136	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
137	C₈	—	67	4	no	—
138	C₈	Dic₆₇⋊C₈	1	1	yes	characteristic
139	C₈	Dic₆₇⋊C₈	1	1	yes	characteristic
140	C₈	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
141	C₈	C₂×#11761611	1	1	yes	characteristic
142	C₂²	—	2	2	no	—
143	C₂²	Dic₆₇⋊C₁₆	1	1	yes	characteristic
144	C₂²	Dic₆₇⋊C₁₆	1	1	yes	characteristic
145	C₂²	C₂×#14523982	1	1	yes	characteristic
146	C₄	—	134	4	no	—
147	C₄	Dic₆₇⋊C₁₆	1	1	yes	characteristic
148	C₄	Dic₆₇⋊C₁₆	1	1	yes	characteristic
149	C₄	C₂×Dic₆₇⋊C₈	1	1	yes	characteristic
150	C₄	C₂×C₁₆×D₆₇	1	1	yes	characteristic
151	C₂	—	2	1	no	—
152	C₂	—	2	1	no	—
153	C₂	#22558406	1	2	yes	direct product factor
154	C₂	C₂×Dic₆₇⋊C₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆	C₂²⋊D₆₇
commutator	C₄×C₆₇	C₂²×C₁₆
Frattini	C₂×C₁₆	C₂²×D₆₇
Fitting	C₂×M₆₄×C₆₇	C₂
socle	C₂²×C₆₇	C₄⋊C₁₆
prime radical	C₂³×C₆₇	M₃₂
cosocle	C₂×C₁₆×C₆₇	C₂³
hypercenter	C₂×M₆₄	D₆₇
hypocenter	C₆₇	C₂×#1422
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₁₆ ◁ C₂×M₆₄

Complements

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

N	H ≅ G/N	type
3. #22558406	C₂	direct (N × H)
76. C₂×C₆₇	#1422	semidirect
82. C₆₇	C₂×#1422	semidirect
153. C₂	#22558406	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16863230	2
C₂×C₁₆×Dic₆₇	2
#22558406	2
C₂×M₆₄×C₆₇	2
C₂×#1422	67

Subgroups — C2×#22558406 #30776320

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22558406 #30776320