Subgroups — C₆₇×#60113 #30739364

Order 17152.

2564 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#568×C₆₇	C₂	1	2	yes	maximal
2	C₂×#568×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2916	C₂	1	16	yes	maximal, complemented
4	C₂×#430×C₆₇	C₂	1	12	yes	maximal, complemented
5	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
6	#568×C₆₇	—	2	8	no	—
7	#568×C₆₇	C₄	1	16	yes	—
8	#568×C₆₇	C₂²	1	16	yes	—
9	C₂×#131×C₆₇	—	2	8	no	—
10	C₂×#131×C₆₇	C₂²	1	8	yes	—
11	C₂²×M₁₆×C₆₇	C₂²	1	12	yes	—
12	C₂²×M₁₆×C₆₇	C₂²	1	6	yes	—
13	#430×C₆₇	C₂²	1	16	yes	—
14	#430×C₆₇	C₂²	1	48	yes	complemented
15	C₂×#108×C₆₇	C₂²	1	48	yes	complemented
16	C₂×#41×C₆₇	C₂×C₄	1	8	yes	—
17	C₂×#41×C₆₇	D₄	1	8	yes	—
18	C₂×#41×C₆₇	Q₈	1	4	yes	—
19	C₂×#41×C₆₇	C₂³	1	4	yes	—
20	C₂²×Q₈×C₆₇	Q₈	1	1	yes	characteristic
21	C₂²×D₄×C₆₇	Q₈	1	3	yes	—
22	C₂³×C₄×C₆₇	C₂³	1	3	yes	—
23	#131×C₆₇	—	2	64	no	—
24	#131×C₆₇	C₂×C₄	1	64	yes	—
25	C₂×M₁₆×C₆₇	—	2	12	no	—
26	C₂×M₁₆×C₆₇	—	2	24	no	—
27	C₂×M₁₆×C₆₇	C₂×C₄	1	48	yes	—
28	C₂×M₁₆×C₆₇	C₂³	1	8	yes	—
29	C₂×M₁₆×C₆₇	C₂³	1	48	yes	—
30	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
31	C₂×M₁₆×C₆₇	C₂³	1	12	yes	—
32	C₂²×C₈×C₆₇	—	2	12	no	—
33	C₂²×C₈×C₆₇	C₂³	1	12	yes	—
34	#108×C₆₇	C₂³	1	64	yes	—
35	C₂⁴×C₆₇	C₂×Q₈	1	3	yes	—
36	#41×C₆₇	C₄⋊C₄	1	32	yes	—
37	#41×C₆₇	C₂²×C₄	1	16	yes	—
38	#41×C₆₇	C₂×D₄	1	16	yes	—
39	C₂×Q₈×C₆₇	C₄⋊C₄	1	8	yes	—
40	C₂×Q₈×C₆₇	C₂×Q₈	1	4	yes	—
41	C₂×D₄×C₆₇	C₄⋊C₄	1	24	yes	—
42	C₂×D₄×C₆₇	C₂×Q₈	1	12	yes	—
43	C₂²×C₄×C₆₇	C₂²×C₄	1	12	yes	—
44	C₂²×C₄×C₆₇	C₂×D₄	1	12	yes	—
45	C₂²×C₄×C₆₇	C₂×Q₈	1	3	yes	—
46	C₂²×C₄×C₆₇	C₂×Q₈	1	6	yes	—
47	C₂²×C₄×C₆₇	C₂⁴	1	6	yes	—
48	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
49	M₁₆×C₆₇	—	2	8	no	—
50	M₁₆×C₆₇	—	2	48	no	—
51	M₁₆×C₆₇	C₂²×C₄	1	16	yes	—
52	M₁₆×C₆₇	C₂²×C₄	1	48	yes	—
53	M₁₆×C₆₇	C₂⁴	1	16	yes	—
54	C₂×C₈×C₆₇	—	2	8	no	—
55	C₂×C₈×C₆₇	—	2	48	no	—
56	C₂×C₈×C₆₇	C₂²×C₄	1	48	yes	—
57	C₂×C₈×C₆₇	C₂⁴	1	8	yes	—
58	C₂³×C₆₇	—	8	3	no	—
59	C₂³×C₆₇	C₂×C₄⋊C₄	1	12	yes	—
60	C₂³×C₆₇	C₂²×Q₈	1	1	yes	characteristic
61	C₂³×C₆₇	C₂²×Q₈	1	6	yes	—
62	Q₈×C₆₇	C₂×C₄⋊C₄	1	16	yes	—
63	D₄×C₆₇	C₂×C₄⋊C₄	1	48	yes	—
64	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	12	yes	—
65	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	24	yes	—
66	C₂×C₄×C₆₇	C₂³×C₄	1	12	yes	—
67	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
68	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
69	C₂×C₄×C₆₇	C₂²×D₄	1	12	yes	—
70	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₆₇	C₂²×Q₈	1	6	yes	—
72	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
73	C₈×C₆₇	—	2	16	no	—
74	C₈×C₆₇	C₂³×C₄	1	16	yes	—
75	C₂²×C₆₇	—	4	1	no	—
76	C₂²×C₆₇	—	4	12	no	—
77	C₂²×C₆₇	—	4	6	no	—
78	C₂²×C₆₇	C₂²×C₄⋊C₄	1	2	yes	—
79	C₂²×C₆₇	C₂²×C₄⋊C₄	1	12	yes	—
80	C₂²×C₆₇	C₂³×Q₈	1	1	yes	characteristic
81	C₄×C₆₇	C₂²×C₄⋊C₄	1	2	yes	—
82	C₄×C₆₇	C₂²×C₄⋊C₄	1	12	yes	—
83	C₄×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
84	C₄×C₆₇	C₂³×D₄	1	1	yes	characteristic
85	#60113	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₂×C₆₇	—	2	2	no	—
87	C₂×C₆₇	—	2	1	no	—
88	C₂×C₆₇	—	2	12	no	—
89	C₂×C₆₇	C₂³×C₄⋊C₄	1	1	yes	characteristic
90	C₂×#568	C₂×C₆₇	1	2	yes	—
91	C₂×#568	C₂×C₆₇	1	1	yes	characteristic
92	#2916	C₂×C₆₇	1	16	yes	complemented
93	C₂×#430	C₂×C₆₇	1	12	yes	complemented
94	C₆₇	#60113	1	1	yes	characteristic, direct product factor, Sylow
95	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
96	#568	—	2	8	no	—
97	#568	C₄×C₆₇	1	16	yes	—
98	#568	C₂²×C₆₇	1	16	yes	—
99	C₂×#131	—	2	8	no	—
100	C₂×#131	C₂²×C₆₇	1	8	yes	—
101	C₂²×M₁₆	C₂²×C₆₇	1	12	yes	—
102	C₂²×M₁₆	C₂²×C₆₇	1	6	yes	—
103	#430	C₂²×C₆₇	1	16	yes	—
104	#430	C₂²×C₆₇	1	48	yes	complemented
105	C₂×#108	C₂²×C₆₇	1	48	yes	complemented
106	C₂×#41	C₂×C₄×C₆₇	1	8	yes	—
107	C₂×#41	D₄×C₆₇	1	8	yes	—
108	C₂×#41	Q₈×C₆₇	1	4	yes	—
109	C₂×#41	C₂³×C₆₇	1	4	yes	—
110	C₂²×Q₈	Q₈×C₆₇	1	1	yes	characteristic
111	C₂²×D₄	Q₈×C₆₇	1	3	yes	—
112	C₂³×C₄	C₂³×C₆₇	1	3	yes	—
113	#131	—	2	64	no	—
114	#131	C₂×C₄×C₆₇	1	64	yes	—
115	C₂×M₁₆	—	2	12	no	—
116	C₂×M₁₆	—	2	24	no	—
117	C₂×M₁₆	C₂×C₄×C₆₇	1	48	yes	—
118	C₂×M₁₆	C₂³×C₆₇	1	8	yes	—
119	C₂×M₁₆	C₂³×C₆₇	1	48	yes	—
120	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
121	C₂×M₁₆	C₂³×C₆₇	1	12	yes	—
122	C₂²×C₈	—	2	12	no	—
123	C₂²×C₈	C₂³×C₆₇	1	12	yes	—
124	#108	C₂³×C₆₇	1	64	yes	—
125	C₂⁴	C₂×Q₈×C₆₇	1	3	yes	—
126	#41	C₄⋊C₄×C₆₇	1	32	yes	—
127	#41	C₂²×C₄×C₆₇	1	16	yes	—
128	#41	C₂×D₄×C₆₇	1	16	yes	—
129	C₂×Q₈	C₄⋊C₄×C₆₇	1	8	yes	—
130	C₂×Q₈	C₂×Q₈×C₆₇	1	4	yes	—
131	C₂×D₄	C₄⋊C₄×C₆₇	1	24	yes	—
132	C₂×D₄	C₂×Q₈×C₆₇	1	12	yes	—
133	C₂²×C₄	C₂²×C₄×C₆₇	1	12	yes	—
134	C₂²×C₄	C₂×D₄×C₆₇	1	12	yes	—
135	C₂²×C₄	C₂×Q₈×C₆₇	1	3	yes	—
136	C₂²×C₄	C₂×Q₈×C₆₇	1	6	yes	—
137	C₂²×C₄	C₂⁴×C₆₇	1	6	yes	—
138	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
139	M₁₆	—	2	8	no	—
140	M₁₆	—	2	48	no	—
141	M₁₆	C₂²×C₄×C₆₇	1	16	yes	—
142	M₁₆	C₂²×C₄×C₆₇	1	48	yes	—
143	M₁₆	C₂⁴×C₆₇	1	16	yes	—
144	C₂×C₈	—	2	8	no	—
145	C₂×C₈	—	2	48	no	—
146	C₂×C₈	C₂²×C₄×C₆₇	1	48	yes	—
147	C₂×C₈	C₂⁴×C₆₇	1	8	yes	—
148	C₂³	—	8	3	no	—
149	C₂³	C₂×C₄⋊C₄×C₆₇	1	12	yes	—
150	C₂³	C₂²×Q₈×C₆₇	1	1	yes	characteristic
151	C₂³	C₂²×Q₈×C₆₇	1	6	yes	—
152	Q₈	C₂×C₄⋊C₄×C₆₇	1	16	yes	—
153	D₄	C₂×C₄⋊C₄×C₆₇	1	48	yes	—
154	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	12	yes	—
155	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	24	yes	—
156	C₂×C₄	C₂³×C₄×C₆₇	1	12	yes	—
157	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
158	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
159	C₂×C₄	C₂²×D₄×C₆₇	1	12	yes	—
160	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
161	C₂×C₄	C₂²×Q₈×C₆₇	1	6	yes	—
162	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
163	C₈	—	2	16	no	—
164	C₈	C₂³×C₄×C₆₇	1	16	yes	—
165	C₂²	—	4	1	no	—
166	C₂²	—	4	12	no	—
167	C₂²	—	4	6	no	—
168	C₂²	C₂²×C₄⋊C₄×C₆₇	1	2	yes	—
169	C₂²	C₂²×C₄⋊C₄×C₆₇	1	12	yes	—
170	C₂²	C₂³×Q₈×C₆₇	1	1	yes	characteristic
171	C₄	C₂²×C₄⋊C₄×C₆₇	1	2	yes	—
172	C₄	C₂²×C₄⋊C₄×C₆₇	1	12	yes	—
173	C₄	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
174	C₄	C₂³×D₄×C₆₇	1	1	yes	characteristic
175	C₂	—	2	2	no	—
176	C₂	—	2	1	no	—
177	C₂	—	2	12	no	—
178	C₂	C₂³×C₄⋊C₄×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₆₇	C₂³×D₄
commutator	C₄	C₂⁴×C₄×C₆₇
Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
socle	C₂×C₆₇	C₂³×C₄⋊C₄
prime radical	C₂²×#41×C₆₇	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₆₇ ◁ C₂²×#41×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₇×#2916	C₂	semidirect
4. C₂×#430×C₆₇	C₂	semidirect
14. #430×C₆₇	C₂²	semidirect
15. C₂×#108×C₆₇	C₂²	semidirect
85. #60113	C₆₇	direct (N × H)
92. #2916	C₂×C₆₇	semidirect
93. C₂×#430	C₂×C₆₇	semidirect
94. C₆₇	#60113	direct (N × H)
104. #430	C₂²×C₆₇	semidirect
105. C₂×#108	C₂²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#568×C₆₇	2
C₆₇×#2916	2
C₂×#430×C₆₇	2
#60113	67

Subgroups — C67×#60113 #30739364

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#60113 #30739364