Subgroups — C₆₇×#14353 #30693604

Order 17152.

2070 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#380×C₆₇	C₂	1	6	yes	maximal
2	C₂²×#95×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#1444	C₂	1	8	yes	maximal, complemented
4	C₂⁴×C₄×C₆₇	C₂²	1	3	yes	—
5	C₂²×C₄⋊C₄×C₆₇	C₂²	1	12	yes	—
6	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
7	#380×C₆₇	—	2	24	no	—
8	C₂×#95×C₆₇	C₄	1	8	yes	complemented
9	C₂×#95×C₆₇	C₂²	1	12	yes	—
10	C₂×#95×C₆₇	C₂²	1	4	yes	—
11	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
12	C₂³×C₄×C₆₇	—	4	12	no	—
13	C₂³×C₄×C₆₇	C₂×C₄	1	12	yes	—
14	C₂³×C₄×C₆₇	D₄	1	12	yes	—
15	C₂³×C₄×C₆₇	Q₈	1	12	yes	—
16	C₂³×C₄×C₆₇	C₂³	1	8	yes	—
17	C₂³×C₄×C₆₇	C₂³	1	6	yes	—
18	C₂×C₄⋊C₄×C₆₇	—	2	24	no	—
19	C₂×C₄⋊C₄×C₆₇	—	4	24	no	—
20	C₂×C₂²⋊C₄×C₆₇	—	2	24	no	—
21	#95×C₆₇	—	2	24	no	—
22	#95×C₆₇	C₂×C₄	1	16	yes	—
23	C₂⁴×C₆₇	—	2	6	no	—
24	C₂⁴×C₆₇	—	2	2	no	—
25	C₂⁴×C₆₇	C₂²×C₄	1	2	yes	—
26	C₂⁴×C₆₇	C₂×D₄	1	6	yes	—
27	C₂⁴×C₆₇	C₂×Q₈	1	6	yes	—
28	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
29	C₂²×C₄×C₆₇	—	2	24	no	—
30	C₂²×C₄×C₆₇	—	2	24	no	—
31	C₂²×C₄×C₆₇	—	2	24	no	—
32	C₂²×C₄×C₆₇	—	2	8	no	—
33	C₂²×C₄×C₆₇	—	4	24	no	—
34	C₂²×C₄×C₆₇	—	4	24	no	—
35	C₂²×C₄×C₆₇	—	4	24	no	—
36	C₂²×C₄×C₆₇	C₄⋊C₄	1	48	yes	complemented
37	C₂²×C₄×C₆₇	C₂²×C₄	1	12	yes	—
38	C₂²×C₄×C₆₇	C₂×D₄	1	12	yes	—
39	C₂²×C₄×C₆₇	C₂×Q₈	1	12	yes	—
40	C₄⋊C₄×C₆₇	—	4	48	no	—
41	C₂²⋊C₄×C₆₇	—	4	16	no	—
42	C₂³×C₆₇	—	2	6	no	—
43	C₂³×C₆₇	—	2	2	no	—
44	C₂³×C₆₇	—	2	12	no	—
45	C₂³×C₆₇	—	2	12	no	—
46	C₂³×C₆₇	—	2	12	no	—
47	C₂³×C₆₇	—	2	12	no	—
48	C₂³×C₆₇	C₂×C₄⋊C₄	1	12	yes	—
49	C₂³×C₆₇	#127	1	4	yes	—
50	C₂³×C₆₇	C₄⋊Q₈	1	12	yes	—
51	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
52	C₂³×C₆₇	C₂²×D₄	1	3	yes	—
53	C₂³×C₆₇	C₂²×Q₈	1	3	yes	—
54	C₂³×C₆₇	H(5,𝔽₂)	1	6	yes	—
55	C₂³×C₆₇	2₋¹⁺⁴	1	2	yes	—
56	C₂×C₄×C₆₇	—	2	24	no	—
57	C₂×C₄×C₆₇	—	2	24	no	—
58	C₂×C₄×C₆₇	—	2	24	no	—
59	C₂×C₄×C₆₇	—	4	48	no	—
60	C₂×C₄×C₆₇	—	4	8	no	—
61	C₂×C₄×C₆₇	—	4	24	no	—
62	C₂×C₄×C₆₇	—	4	24	no	—
63	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	24	yes	—
64	C₂²×C₆₇	—	2	12	no	—
65	C₂²×C₆₇	—	2	12	no	—
66	C₂²×C₆₇	—	2	8	no	—
67	C₂²×C₆₇	—	2	24	no	—
68	C₂²×C₆₇	#384	1	8	yes	—
69	C₂²×C₆₇	C₂²×C₄⋊C₄	1	3	yes	—
70	C₂²×C₆₇	#518	1	3	yes	—
71	C₂²×C₆₇	#519	1	1	yes	characteristic
72	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
73	C₂²×C₆₇	C₂×C₄⋊Q₈	1	3	yes	—
74	C₂²×C₆₇	#534	1	6	yes	—
75	C₂²×C₆₇	#537	1	6	yes	—
76	C₂²×C₆₇	#543	1	6	yes	—
77	C₂²×C₆₇	#544	1	6	yes	—
78	C₄×C₆₇	—	2	24	no	—
79	C₄×C₆₇	—	8	8	no	—
80	#14353	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₂×C₆₇	—	2	8	no	—
82	C₂×C₆₇	C₂×#384	1	1	yes	characteristic
83	C₂×C₆₇	#2315	1	3	yes	—
84	C₂×C₆₇	#2316	1	3	yes	—
85	C₂×C₆₇	#2656	1	2	yes	—
86	C₂×C₆₇	#2659	1	6	yes	—
87	C₂×#380	C₂×C₆₇	1	6	yes	—
88	C₂²×#95	C₂×C₆₇	1	1	yes	characteristic
89	#1444	C₂×C₆₇	1	8	yes	complemented
90	C₆₇	#14353	1	1	yes	characteristic, direct product factor, Sylow
91	C₂⁴×C₄	C₂²×C₆₇	1	3	yes	—
92	C₂²×C₄⋊C₄	C₂²×C₆₇	1	12	yes	—
93	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
94	#380	—	2	24	no	—
95	C₂×#95	C₄×C₆₇	1	8	yes	complemented
96	C₂×#95	C₂²×C₆₇	1	12	yes	—
97	C₂×#95	C₂²×C₆₇	1	4	yes	—
98	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
99	C₂³×C₄	—	4	12	no	—
100	C₂³×C₄	C₂×C₄×C₆₇	1	12	yes	—
101	C₂³×C₄	D₄×C₆₇	1	12	yes	—
102	C₂³×C₄	Q₈×C₆₇	1	12	yes	—
103	C₂³×C₄	C₂³×C₆₇	1	8	yes	—
104	C₂³×C₄	C₂³×C₆₇	1	6	yes	—
105	C₂×C₄⋊C₄	—	2	24	no	—
106	C₂×C₄⋊C₄	—	4	24	no	—
107	C₂×C₂²⋊C₄	—	2	24	no	—
108	#95	—	2	24	no	—
109	#95	C₂×C₄×C₆₇	1	16	yes	—
110	C₂⁴	—	2	6	no	—
111	C₂⁴	—	2	2	no	—
112	C₂⁴	C₂²×C₄×C₆₇	1	2	yes	—
113	C₂⁴	C₂×D₄×C₆₇	1	6	yes	—
114	C₂⁴	C₂×Q₈×C₆₇	1	6	yes	—
115	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
116	C₂²×C₄	—	2	24	no	—
117	C₂²×C₄	—	2	24	no	—
118	C₂²×C₄	—	2	24	no	—
119	C₂²×C₄	—	2	8	no	—
120	C₂²×C₄	—	4	24	no	—
121	C₂²×C₄	—	4	24	no	—
122	C₂²×C₄	—	4	24	no	—
123	C₂²×C₄	C₄⋊C₄×C₆₇	1	48	yes	complemented
124	C₂²×C₄	C₂²×C₄×C₆₇	1	12	yes	—
125	C₂²×C₄	C₂×D₄×C₆₇	1	12	yes	—
126	C₂²×C₄	C₂×Q₈×C₆₇	1	12	yes	—
127	C₄⋊C₄	—	4	48	no	—
128	C₂²⋊C₄	—	4	16	no	—
129	C₂³	—	2	6	no	—
130	C₂³	—	2	2	no	—
131	C₂³	—	2	12	no	—
132	C₂³	—	2	12	no	—
133	C₂³	—	2	12	no	—
134	C₂³	—	2	12	no	—
135	C₂³	C₂×C₄⋊C₄×C₆₇	1	12	yes	—
136	C₂³	#127×C₆₇	1	4	yes	—
137	C₂³	C₄⋊Q₈×C₆₇	1	12	yes	—
138	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
139	C₂³	C₂²×D₄×C₆₇	1	3	yes	—
140	C₂³	C₂²×Q₈×C₆₇	1	3	yes	—
141	C₂³	H(5,𝔽₂)×C₆₇	1	6	yes	—
142	C₂³	2₋¹⁺⁴×C₆₇	1	2	yes	—
143	C₂×C₄	—	2	24	no	—
144	C₂×C₄	—	2	24	no	—
145	C₂×C₄	—	2	24	no	—
146	C₂×C₄	—	4	48	no	—
147	C₂×C₄	—	4	8	no	—
148	C₂×C₄	—	4	24	no	—
149	C₂×C₄	—	4	24	no	—
150	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	24	yes	—
151	C₂²	—	2	12	no	—
152	C₂²	—	2	12	no	—
153	C₂²	—	2	8	no	—
154	C₂²	—	2	24	no	—
155	C₂²	#384×C₆₇	1	8	yes	—
156	C₂²	C₂²×C₄⋊C₄×C₆₇	1	3	yes	—
157	C₂²	#518×C₆₇	1	3	yes	—
158	C₂²	#519×C₆₇	1	1	yes	characteristic
159	C₂²	C₂×#127×C₆₇	1	1	yes	characteristic
160	C₂²	C₂×C₄⋊Q₈×C₆₇	1	3	yes	—
161	C₂²	#534×C₆₇	1	6	yes	—
162	C₂²	#537×C₆₇	1	6	yes	—
163	C₂²	#543×C₆₇	1	6	yes	—
164	C₂²	#544×C₆₇	1	6	yes	—
165	C₄	—	2	24	no	—
166	C₄	—	8	8	no	—
167	C₂	—	2	8	no	—
168	C₂	C₂×#384×C₆₇	1	1	yes	characteristic
169	C₂	C₆₇×#2315	1	3	yes	—
170	C₂	C₆₇×#2316	1	3	yes	—
171	C₂	C₆₇×#2656	1	2	yes	—
172	C₂	C₆₇×#2659	1	6	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₆₇	C₂⁴
commutator	C₂³	C₂³×C₄×C₆₇
Frattini, cosocle	C₂⁴	C₂⁴×C₆₇
prime radical	C₂⁵×C₆₇	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₇×#1444	C₂	semidirect
8. C₂×#95×C₆₇	C₄	semidirect
36. C₂²×C₄×C₆₇	C₄⋊C₄	semidirect
80. #14353	C₆₇	direct (N × H)
89. #1444	C₂×C₆₇	semidirect
90. C₆₇	#14353	direct (N × H)
95. C₂×#95	C₄×C₆₇	semidirect
123. C₂²×C₄	C₄⋊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×#380×C₆₇	2
C₂²×#95×C₆₇	2
C₆₇×#1444	2
#14353	67

Subgroups — C67×#14353 #30693604

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#14353 #30693604