Subgroups — C₆₇×#20351 #30699602

Order 17152.

1978 conjugacy classes of subgroups.

The subgroup lattice has 188 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	C₂²×#131×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#1738	C₂	1	6	yes	maximal, complemented
4	C₂×#341×C₆₇	C₂	1	6	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	C₂×#131×C₆₇	—	2	8	no	—
9	C₂×#131×C₆₇	—	2	8	no	—
10	C₂×#131×C₆₇	C₄	1	8	yes	—
11	C₂²×M₁₆×C₆₇	C₂²	1	12	yes	—
12	C₂²×M₁₆×C₆₇	C₂²	1	3	yes	—
13	C₂³×C₈×C₆₇	C₂²	1	3	yes	—
14	#341×C₆₇	C₂²	1	4	yes	—
15	#341×C₆₇	C₂²	1	12	yes	—
16	C₂×#41×C₆₇	C₂×C₄	1	4	yes	—
17	C₂×#41×C₆₇	C₂×C₄	1	8	yes	—
18	C₂×#41×C₆₇	D₄	1	4	yes	—
19	C₂×#41×C₆₇	D₄	1	8	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	32	no	—
24	#131×C₆₇	—	2	64	no	—
25	C₂×M₁₆×C₆₇	—	2	12	no	—
26	C₂×M₁₆×C₆₇	—	2	12	no	—
27	C₂×M₁₆×C₆₇	—	2	24	no	—
28	C₂×M₁₆×C₆₇	C₂×C₄	1	48	yes	—
29	C₂×M₁₆×C₆₇	C₂×C₄	1	12	yes	—
30	C₂×M₁₆×C₆₇	C₂³	1	8	yes	—
31	C₂²×C₈×C₆₇	—	2	12	no	—
32	C₂²×C₈×C₆₇	—	2	12	no	—
33	C₂²×C₈×C₆₇	C₂×C₄	1	12	yes	—
34	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₆₇	C₂³	1	3	yes	—
36	C₂⁴×C₆₇	C₂×Q₈	1	3	yes	—
37	#41×C₆₇	C₄²	1	16	yes	—
38	#41×C₆₇	C₂²⋊C₄	1	16	yes	—
39	#41×C₆₇	C₂²⋊C₄	1	32	yes	—
40	C₂×Q₈×C₆₇	C₄⋊C₄	1	4	yes	—
41	C₂×Q₈×C₆₇	C₄⋊C₄	1	8	yes	—
42	C₂×D₄×C₆₇	C₄⋊C₄	1	12	yes	—
43	C₂×D₄×C₆₇	C₄⋊C₄	1	24	yes	—
44	C₂²×C₄×C₆₇	C₂²×C₄	1	6	yes	—
45	C₂²×C₄×C₆₇	C₂²×C₄	1	12	yes	—
46	C₂²×C₄×C₆₇	C₂×D₄	1	12	yes	—
47	C₂²×C₄×C₆₇	C₂×D₄	1	6	yes	—
48	C₂²×C₄×C₆₇	C₂×Q₈	1	3	yes	—
49	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
50	M₁₆×C₆₇	—	2	48	no	—
51	M₁₆×C₆₇	—	2	24	no	—
52	M₁₆×C₆₇	—	2	8	no	—
53	M₁₆×C₆₇	C₂²×C₄	1	16	yes	—
54	C₂×C₈×C₆₇	—	2	24	no	—
55	C₂×C₈×C₆₇	—	2	48	no	—
56	C₂×C₈×C₆₇	—	2	2	no	—
57	C₂×C₈×C₆₇	—	2	6	no	—
58	C₂×C₈×C₆₇	—	2	8	no	—
59	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
60	C₂×C₈×C₆₇	C₂²×C₄	1	6	yes	—
61	C₂³×C₆₇	—	4	6	no	—
62	C₂³×C₆₇	C₂×C₄⋊C₄	1	6	yes	—
63	C₂³×C₆₇	C₂×C₄⋊C₄	1	12	yes	—
64	C₂³×C₆₇	C₂²×Q₈	1	1	yes	characteristic
65	Q₈×C₆₇	#95	1	16	yes	—
66	D₄×C₆₇	#95	1	48	yes	—
67	C₂×C₄×C₆₇	C₂×C₄²	1	12	yes	—
68	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	12	yes	—
69	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	24	yes	—
70	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	12	yes	—
71	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	6	yes	—
72	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
73	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
74	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
75	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
76	C₈×C₆₇	—	2	2	no	—
77	C₈×C₆₇	—	2	16	no	—
78	C₈×C₆₇	—	2	6	no	—
79	C₂²×C₆₇	—	2	2	no	—
80	C₂²×C₆₇	—	2	12	no	—
81	C₂²×C₆₇	—	4	12	no	—
82	C₂²×C₆₇	C₂×#95	1	12	yes	—
83	C₂²×C₆₇	C₂²×C₄⋊C₄	1	2	yes	—
84	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
85	C₄×C₆₇	C₂×#95	1	12	yes	—
86	C₄×C₆₇	C₂²×C₄²	1	1	yes	characteristic
87	C₄×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
88	C₄×C₆₇	C₂²×C₂²⋊C₄	1	2	yes	—
89	#20351	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂×C₆₇	—	2	2	no	—
91	C₂×C₆₇	—	2	12	no	—
92	C₂×C₆₇	C₂²×#95	1	1	yes	characteristic
93	C₂×C₆₇	#2916	1	2	yes	—
94	C₂×#568	C₂×C₆₇	1	2	yes	—
95	C₂²×#131	C₂×C₆₇	1	1	yes	characteristic
96	#1738	C₂×C₆₇	1	6	yes	complemented
97	C₂×#341	C₂×C₆₇	1	6	yes	complemented
98	C₆₇	#20351	1	1	yes	characteristic, direct product factor, Sylow
99	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
100	#568	—	2	8	no	—
101	#568	C₄×C₆₇	1	16	yes	—
102	C₂×#131	—	2	8	no	—
103	C₂×#131	—	2	8	no	—
104	C₂×#131	C₄×C₆₇	1	8	yes	—
105	C₂²×M₁₆	C₂²×C₆₇	1	12	yes	—
106	C₂²×M₁₆	C₂²×C₆₇	1	3	yes	—
107	C₂³×C₈	C₂²×C₆₇	1	3	yes	—
108	#341	C₂²×C₆₇	1	4	yes	—
109	#341	C₂²×C₆₇	1	12	yes	—
110	C₂×#41	C₂×C₄×C₆₇	1	4	yes	—
111	C₂×#41	C₂×C₄×C₆₇	1	8	yes	—
112	C₂×#41	D₄×C₆₇	1	4	yes	—
113	C₂×#41	D₄×C₆₇	1	8	yes	—
114	C₂²×Q₈	Q₈×C₆₇	1	1	yes	characteristic
115	C₂²×D₄	Q₈×C₆₇	1	3	yes	—
116	C₂³×C₄	C₂³×C₆₇	1	3	yes	—
117	#131	—	2	32	no	—
118	#131	—	2	64	no	—
119	C₂×M₁₆	—	2	12	no	—
120	C₂×M₁₆	—	2	12	no	—
121	C₂×M₁₆	—	2	24	no	—
122	C₂×M₁₆	C₂×C₄×C₆₇	1	48	yes	—
123	C₂×M₁₆	C₂×C₄×C₆₇	1	12	yes	—
124	C₂×M₁₆	C₂³×C₆₇	1	8	yes	—
125	C₂²×C₈	—	2	12	no	—
126	C₂²×C₈	—	2	12	no	—
127	C₂²×C₈	C₂×C₄×C₆₇	1	12	yes	—
128	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
129	C₂²×C₈	C₂³×C₆₇	1	3	yes	—
130	C₂⁴	C₂×Q₈×C₆₇	1	3	yes	—
131	#41	C₄²×C₆₇	1	16	yes	—
132	#41	C₂²⋊C₄×C₆₇	1	16	yes	—
133	#41	C₂²⋊C₄×C₆₇	1	32	yes	—
134	C₂×Q₈	C₄⋊C₄×C₆₇	1	4	yes	—
135	C₂×Q₈	C₄⋊C₄×C₆₇	1	8	yes	—
136	C₂×D₄	C₄⋊C₄×C₆₇	1	12	yes	—
137	C₂×D₄	C₄⋊C₄×C₆₇	1	24	yes	—
138	C₂²×C₄	C₂²×C₄×C₆₇	1	6	yes	—
139	C₂²×C₄	C₂²×C₄×C₆₇	1	12	yes	—
140	C₂²×C₄	C₂×D₄×C₆₇	1	6	yes	—
141	C₂²×C₄	C₂×D₄×C₆₇	1	12	yes	—
142	C₂²×C₄	C₂×Q₈×C₆₇	1	3	yes	—
143	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
144	M₁₆	—	2	48	no	—
145	M₁₆	—	2	24	no	—
146	M₁₆	—	2	8	no	—
147	M₁₆	C₂²×C₄×C₆₇	1	16	yes	—
148	C₂×C₈	—	2	24	no	—
149	C₂×C₈	—	2	48	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	2	8	no	—
152	C₂×C₈	—	2	6	no	—
153	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
154	C₂×C₈	C₂²×C₄×C₆₇	1	6	yes	—
155	C₂³	—	4	6	no	—
156	C₂³	C₂×C₄⋊C₄×C₆₇	1	6	yes	—
157	C₂³	C₂×C₄⋊C₄×C₆₇	1	12	yes	—
158	C₂³	C₂²×Q₈×C₆₇	1	1	yes	characteristic
159	Q₈	#95×C₆₇	1	16	yes	—
160	D₄	#95×C₆₇	1	48	yes	—
161	C₂×C₄	C₂×C₄²×C₆₇	1	12	yes	—
162	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	12	yes	—
163	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	24	yes	—
164	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	12	yes	—
165	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	6	yes	—
166	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
167	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
168	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
169	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
170	C₈	—	2	2	no	—
171	C₈	—	2	16	no	—
172	C₈	—	2	6	no	—
173	C₂²	—	2	2	no	—
174	C₂²	—	2	12	no	—
175	C₂²	—	4	12	no	—
176	C₂²	C₂×#95×C₆₇	1	12	yes	—
177	C₂²	C₂²×C₄⋊C₄×C₆₇	1	2	yes	—
178	C₂²	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
179	C₄	C₂×#95×C₆₇	1	12	yes	—
180	C₄	C₂²×C₄²×C₆₇	1	1	yes	characteristic
181	C₄	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
182	C₄	C₂²×C₂²⋊C₄×C₆₇	1	2	yes	—
183	C₂	—	2	2	no	—
184	C₂	—	2	12	no	—
185	C₂	C₂²×#95×C₆₇	1	1	yes	characteristic
186	C₂	C₆₇×#2916	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂²×D₄
commutator	C₄	C₂²×C₄²×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂²×C₄⋊C₄
prime radical	C₂²×#41×C₆₇	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂²×#41×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₇×#1738	C₂	semidirect
4. C₂×#341×C₆₇	C₂	semidirect
89. #20351	C₆₇	direct (N × H)
96. #1738	C₂×C₆₇	semidirect
97. C₂×#341	C₂×C₆₇	semidirect
98. C₆₇	#20351	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×#568×C₆₇	2
C₂²×#131×C₆₇	2
C₆₇×#1738	2
C₂×#341×C₆₇	2
#20351	67

Subgroups — C67×#20351 #30699602

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#20351 #30699602