Subgroups — C₆₇×#33410 #30712661

Order 17152.

658 conjugacy classes of subgroups.

The subgroup lattice has 160 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#504×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2913	C₂	1	2	yes	maximal
3	C₆₇×#2159	C₂	1	6	yes	maximal, complemented
4	C₆₇×#2158	C₂	1	3	yes	maximal, complemented
5	C₂×#366×C₆₇	C₂	1	3	yes	maximal, complemented
6	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
7	#520×C₆₇	—	2	2	no	—
8	#504×C₆₇	C₄	1	8	yes	complemented
9	C₂×M₃₂×C₆₇	C₂²	1	3	yes	—
10	C₂²×C₁₆×C₆₇	C₂²	1	3	yes	—
11	#428×C₆₇	—	2	6	no	—
12	#427×C₆₇	C₂²	1	6	yes	—
13	C₂×#107×C₆₇	C₂²	1	6	yes	—
14	#367×C₆₇	C₂²	1	6	yes	—
15	#366×C₆₇	C₂²	1	3	yes	—
16	#366×C₆₇	C₂²	1	6	yes	—
17	#366×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×#41×C₆₇	D₄	1	1	yes	characteristic
19	#131×C₆₇	C₂×C₄	1	4	yes	—
20	#131×C₆₇	Q₈	1	4	yes	—
21	C₂×M₁₆×C₆₇	D₄	1	3	yes	—
22	C₂²×C₈×C₆₇	C₂³	1	3	yes	—
23	C₄×Q₈×C₆₇	—	4	2	no	—
24	C₄×D₄×C₆₇	—	4	6	no	—
25	#117×C₆₇	—	2	6	no	—
26	C₂×C₄⋊C₄×C₆₇	—	2	6	no	—
27	M₃₂×C₆₇	C₂×C₄	1	12	yes	complemented
28	C₂×C₁₆×C₆₇	C₂×C₄	1	12	yes	complemented
29	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
30	C₂×C₁₆×C₆₇	C₂³	1	3	yes	—
31	#107×C₆₇	—	2	6	no	—
32	#107×C₆₇	C₂³	1	6	yes	—
33	#107×C₆₇	C₂³	1	2	yes	—
34	#106×C₆₇	—	2	6	no	—
35	#41×C₆₇	C₄⋊C₄	1	4	yes	—
36	#41×C₆₇	Q₁₆	1	4	yes	—
37	C₂×Q₈×C₆₇	D₈	1	1	yes	characteristic
38	C₂×D₄×C₆₇	D₈	1	3	yes	—
39	C₂²×C₄×C₆₇	—	4	6	no	—
40	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
41	M₁₆×C₆₇	C₄⋊C₄	1	12	yes	—
42	C₂×C₈×C₆₇	C₂²×C₄	1	6	yes	—
43	C₂×C₈×C₆₇	C₂×D₄	1	3	yes	—
44	C₂×C₈×C₆₇	C₂×Q₈	1	6	yes	—
45	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
46	C₄⋊C₄×C₆₇	—	2	2	no	—
47	C₄⋊C₄×C₆₇	—	2	6	no	—
48	C₄⋊C₄×C₆₇	—	4	6	no	—
49	C₂²⋊C₄×C₆₇	—	4	6	no	—
50	C₄²×C₆₇	—	4	6	no	—
51	C₁₆×C₆₇	C₂²×C₄	1	2	yes	—
52	C₁₆×C₆₇	C₂²×C₄	1	6	yes	—
53	C₂³×C₆₇	C₂×D₈	1	3	yes	—
54	Q₈×C₆₇	#107	1	4	yes	—
55	D₄×C₆₇	#107	1	12	yes	—
56	C₂×C₄×C₆₇	—	4	2	no	—
57	C₂×C₄×C₆₇	—	4	6	no	—
58	C₂×C₄×C₆₇	—	16	6	no	—
59	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	6	yes	—
60	C₂×C₄×C₆₇	C₂×D₈	1	3	yes	—
61	C₂×C₄×C₆₇	C₂×Q₁₆	1	6	yes	—
62	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
63	C₈×C₆₇	C₂×C₄⋊C₄	1	6	yes	—
64	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
65	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
66	C₂²×C₆₇	—	2	3	no	—
67	C₂²×C₆₇	—	2	3	no	—
68	C₂²×C₆₇	C₂×#107	1	6	yes	—
69	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
70	C₄×C₆₇	—	8	6	no	—
71	C₄×C₆₇	—	8	2	no	—
72	C₄×C₆₇	C₂×#107	1	6	yes	—
73	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
74	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
75	#33410	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂×C₆₇	—	2	6	no	—
77	C₂×C₆₇	C₂²×#107	1	1	yes	characteristic
78	C₂×C₆₇	#3415	1	1	yes	characteristic
79	C₂×C₆₇	#3417	1	1	yes	characteristic
80	C₂×#504	C₂×C₆₇	1	1	yes	characteristic
81	#2913	C₂×C₆₇	1	2	yes	—
82	#2159	C₂×C₆₇	1	6	yes	complemented
83	#2158	C₂×C₆₇	1	3	yes	complemented
84	C₂×#366	C₂×C₆₇	1	3	yes	complemented
85	C₆₇	#33410	1	1	yes	characteristic, direct product factor, Sylow
86	C₂×#131	C₂²×C₆₇	1	1	yes	characteristic
87	#520	—	2	2	no	—
88	#504	C₄×C₆₇	1	8	yes	complemented
89	C₂×M₃₂	C₂²×C₆₇	1	3	yes	—
90	C₂²×C₁₆	C₂²×C₆₇	1	3	yes	—
91	#428	—	2	6	no	—
92	#427	C₂²×C₆₇	1	6	yes	—
93	C₂×#107	C₂²×C₆₇	1	6	yes	—
94	#367	C₂²×C₆₇	1	6	yes	—
95	#366	C₂²×C₆₇	1	3	yes	—
96	#366	C₂²×C₆₇	1	6	yes	—
97	#366	C₂²×C₆₇	1	1	yes	characteristic
98	C₂×#41	D₄×C₆₇	1	1	yes	characteristic
99	#131	C₂×C₄×C₆₇	1	4	yes	—
100	#131	Q₈×C₆₇	1	4	yes	—
101	C₂×M₁₆	D₄×C₆₇	1	3	yes	—
102	C₂²×C₈	C₂³×C₆₇	1	3	yes	—
103	C₄×Q₈	—	4	2	no	—
104	C₄×D₄	—	4	6	no	—
105	#117	—	2	6	no	—
106	C₂×C₄⋊C₄	—	2	6	no	—
107	M₃₂	C₂×C₄×C₆₇	1	12	yes	complemented
108	C₂×C₁₆	C₂×C₄×C₆₇	1	12	yes	complemented
109	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
110	C₂×C₁₆	C₂³×C₆₇	1	3	yes	—
111	#107	—	2	6	no	—
112	#107	C₂³×C₆₇	1	6	yes	—
113	#107	C₂³×C₆₇	1	2	yes	—
114	#106	—	2	6	no	—
115	#41	C₄⋊C₄×C₆₇	1	4	yes	—
116	#41	Q₁₆×C₆₇	1	4	yes	—
117	C₂×Q₈	D₈×C₆₇	1	1	yes	characteristic
118	C₂×D₄	D₈×C₆₇	1	3	yes	—
119	C₂²×C₄	—	4	6	no	—
120	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
121	M₁₆	C₄⋊C₄×C₆₇	1	12	yes	—
122	C₂×C₈	C₂²×C₄×C₆₇	1	6	yes	—
123	C₂×C₈	C₂×D₄×C₆₇	1	3	yes	—
124	C₂×C₈	C₂×Q₈×C₆₇	1	6	yes	—
125	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
126	C₄⋊C₄	—	2	2	no	—
127	C₄⋊C₄	—	2	6	no	—
128	C₄⋊C₄	—	4	6	no	—
129	C₂²⋊C₄	—	4	6	no	—
130	C₄²	—	4	6	no	—
131	C₁₆	C₂²×C₄×C₆₇	1	2	yes	—
132	C₁₆	C₂²×C₄×C₆₇	1	6	yes	—
133	C₂³	C₂×D₈×C₆₇	1	3	yes	—
134	Q₈	#107×C₆₇	1	4	yes	—
135	D₄	#107×C₆₇	1	12	yes	—
136	C₂×C₄	—	4	2	no	—
137	C₂×C₄	—	4	6	no	—
138	C₂×C₄	—	16	6	no	—
139	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	6	yes	—
140	C₂×C₄	C₂×D₈×C₆₇	1	3	yes	—
141	C₂×C₄	C₂×Q₁₆×C₆₇	1	6	yes	—
142	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
143	C₈	C₂×C₄⋊C₄×C₆₇	1	6	yes	—
144	C₈	C₂³×C₄×C₆₇	1	1	yes	characteristic
145	C₈	C₂²×Q₈×C₆₇	1	1	yes	characteristic
146	C₂²	—	2	3	no	—
147	C₂²	—	2	3	no	—
148	C₂²	C₂×#107×C₆₇	1	6	yes	—
149	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
150	C₄	—	8	6	no	—
151	C₄	—	8	2	no	—
152	C₄	C₂×#107×C₆₇	1	6	yes	—
153	C₄	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
154	C₄	C₂²×Q₁₆×C₆₇	1	1	yes	characteristic
155	C₂	—	2	6	no	—
156	C₂	C₂²×#107×C₆₇	1	1	yes	characteristic
157	C₂	C₆₇×#3415	1	1	yes	characteristic
158	C₂	C₆₇×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂²×D₈
commutator	C₈	C₂³×C₄×C₆₇
Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
prime radical	C₂×#41×C₆₇	D₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×#41×C₆₇ ◁ C₂×#131×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₇×#2159	C₂	semidirect
4. C₆₇×#2158	C₂	semidirect
5. C₂×#366×C₆₇	C₂	semidirect
8. #504×C₆₇	C₄	semidirect
27. M₃₂×C₆₇	C₂×C₄	semidirect
28. C₂×C₁₆×C₆₇	C₂×C₄	semidirect
75. #33410	C₆₇	direct (N × H)
82. #2159	C₂×C₆₇	semidirect
83. #2158	C₂×C₆₇	semidirect
84. C₂×#366	C₂×C₆₇	semidirect
85. C₆₇	#33410	direct (N × H)
88. #504	C₄×C₆₇	semidirect
107. M₃₂	C₂×C₄×C₆₇	semidirect
108. C₂×C₁₆	C₂×C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#504×C₆₇	2
C₆₇×#2913	2
C₆₇×#2159	2
C₆₇×#2158	2
C₂×#366×C₆₇	2
#33410	67

Subgroups — C67×#33410 #30712661

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33410 #30712661