Subgroups — C₂²×#491×C₆₇ #30739859

Order 17152.

2142 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×#125×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#491×C₆₇	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#103×C₆₇	C₂	1	4	yes	maximal
5	C₂²×#97×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂³×Q₈×C₆₇	C₂²	1	2	yes	—
7	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
8	C₂×C₄⋊Q₈×C₆₇	—	2	6	no	—
9	C₂×C₄⋊Q₈×C₆₇	C₂²	1	12	yes	—
10	C₂×#125×C₆₇	—	2	6	no	—
11	C₂×#125×C₆₇	C₂²	1	12	yes	—
12	C₂²×C₄⋊C₄×C₆₇	—	2	2	no	—
13	C₂²×C₄⋊C₄×C₆₇	—	2	2	no	—
14	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
15	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
16	#491×C₆₇	C₂²	1	64	yes	direct product factor
17	C₂×#103×C₆₇	C₂²	1	48	yes	—
18	C₂×#97×C₆₇	C₂²	1	12	yes	—
19	C₂²×Q₈×C₆₇	—	2	6	no	—
20	C₂²×Q₈×C₆₇	—	2	16	no	—
21	C₂²×Q₈×C₆₇	C₂³	1	12	yes	—
22	C₂³×C₄×C₆₇	—	2	2	no	—
23	C₂³×C₄×C₆₇	—	2	2	no	—
24	C₂³×C₄×C₆₇	D₄	1	2	yes	—
25	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
26	C₂²×C₈×C₆₇	—	2	8	no	—
27	C₂²×C₈×C₆₇	C₂³	1	12	yes	—
28	C₄⋊Q₈×C₆₇	—	2	24	no	—
29	C₄⋊Q₈×C₆₇	C₂³	1	16	yes	—
30	#125×C₆₇	—	2	24	no	—
31	#125×C₆₇	C₂³	1	16	yes	—
32	C₂×C₄⋊C₄×C₆₇	—	2	24	no	—
33	C₂×C₄⋊C₄×C₆₇	—	2	24	no	—
34	C₂×C₄⋊C₄×C₆₇	—	2	6	no	—
35	C₂×C₄⋊C₄×C₆₇	C₂³	1	12	yes	—
36	C₂×C₄²×C₆₇	D₄	1	6	yes	—
37	C₂×C₄²×C₆₇	C₂³	1	6	yes	—
38	#103×C₆₇	C₂³	1	64	yes	—
39	#97×C₆₇	C₂³	1	16	yes	—
40	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
41	C₂×Q₈×C₆₇	—	2	12	no	—
42	C₂×Q₈×C₆₇	—	2	48	no	—
43	C₂×Q₈×C₆₇	—	4	24	no	—
44	C₂×Q₈×C₆₇	C₂⁴	1	8	yes	—
45	C₂²×C₄×C₆₇	—	2	8	no	—
46	C₂²×C₄×C₆₇	—	2	12	no	—
47	C₂²×C₄×C₆₇	—	2	12	no	—
48	C₂²×C₄×C₆₇	—	4	8	no	—
49	C₂²×C₄×C₆₇	—	4	8	no	—
50	C₂²×C₄×C₆₇	C₂×D₄	1	12	yes	—
51	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
52	C₂²×C₄×C₆₇	#41	1	8	yes	—
53	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
54	C₂×C₈×C₆₇	—	2	24	no	—
55	C₂×C₈×C₆₇	C₂⁴	1	8	yes	—
56	C₄⋊C₄×C₆₇	—	2	32	no	—
57	C₄⋊C₄×C₆₇	—	2	32	no	—
58	C₄⋊C₄×C₆₇	—	2	12	no	—
59	C₄⋊C₄×C₆₇	C₂⁴	1	8	yes	—
60	C₄²×C₆₇	C₂×D₄	1	12	yes	—
61	C₄²×C₆₇	C₂⁴	1	4	yes	—
62	C₂³×C₆₇	#124	1	4	yes	—
63	C₂³×C₆₇	#137	1	8	yes	—
64	C₂³×C₆₇	C₂²×D₄	1	3	yes	—
65	Q₈×C₆₇	—	2	16	no	—
66	Q₈×C₆₇	—	4	24	no	—
67	C₂×C₄×C₆₇	—	2	24	no	—
68	C₂×C₄×C₆₇	—	2	8	no	—
69	C₂×C₄×C₆₇	—	2	8	no	—
70	C₂×C₄×C₆₇	—	4	24	no	—
71	C₂×C₄×C₆₇	—	4	24	no	—
72	C₂×C₄×C₆₇	#124	1	12	yes	—
73	C₂×C₄×C₆₇	C₂²×D₄	1	8	yes	—
74	C₂×C₄×C₆₇	C₂²×D₄	1	3	yes	—
75	C₂×C₄×C₆₇	C₂×#41	1	12	yes	—
76	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
77	C₈×C₆₇	—	2	8	no	—
78	C₂²×C₆₇	#491	1	16	yes	direct product factor
79	C₂²×C₆₇	C₂×#124	1	6	yes	—
80	C₂²×C₆₇	C₂×#137	1	12	yes	—
81	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
82	C₄×C₆₇	—	2	8	no	—
83	C₄×C₆₇	—	4	8	no	—
84	C₄×C₆₇	—	4	8	no	—
85	C₄×C₆₇	C₂×#124	1	6	yes	—
86	C₄×C₆₇	C₂²×#41	1	2	yes	—
87	C₂²×#491	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂×C₆₇	C₂×#491	1	12	yes	direct product factor
89	C₂×C₆₇	C₂²×#124	1	1	yes	characteristic
90	C₂×C₆₇	C₂²×#137	1	2	yes	—
91	C₂²×C₄⋊Q₈	C₂×C₆₇	1	1	yes	characteristic
92	C₂²×#125	C₂×C₆₇	1	1	yes	characteristic
93	C₂×#491	C₂×C₆₇	1	24	yes	direct product factor
94	C₂²×#103	C₂×C₆₇	1	4	yes	—
95	C₂²×#97	C₂×C₆₇	1	1	yes	characteristic
96	C₆₇	C₂²×#491	1	1	yes	characteristic, direct product factor, Sylow
97	C₂³×Q₈	C₂²×C₆₇	1	2	yes	—
98	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
99	C₂×C₄⋊Q₈	—	2	6	no	—
100	C₂×C₄⋊Q₈	C₂²×C₆₇	1	12	yes	—
101	C₂×#125	—	2	6	no	—
102	C₂×#125	C₂²×C₆₇	1	12	yes	—
103	C₂²×C₄⋊C₄	—	2	2	no	—
104	C₂²×C₄⋊C₄	—	2	2	no	—
105	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
106	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
107	#491	C₂²×C₆₇	1	64	yes	direct product factor
108	C₂×#103	C₂²×C₆₇	1	48	yes	—
109	C₂×#97	C₂²×C₆₇	1	12	yes	—
110	C₂²×Q₈	—	2	6	no	—
111	C₂²×Q₈	—	2	16	no	—
112	C₂²×Q₈	C₂³×C₆₇	1	12	yes	—
113	C₂³×C₄	—	2	2	no	—
114	C₂³×C₄	—	2	2	no	—
115	C₂³×C₄	D₄×C₆₇	1	2	yes	—
116	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
117	C₂²×C₈	—	2	8	no	—
118	C₂²×C₈	C₂³×C₆₇	1	12	yes	—
119	C₄⋊Q₈	—	2	24	no	—
120	C₄⋊Q₈	C₂³×C₆₇	1	16	yes	—
121	#125	—	2	24	no	—
122	#125	C₂³×C₆₇	1	16	yes	—
123	C₂×C₄⋊C₄	—	2	24	no	—
124	C₂×C₄⋊C₄	—	2	24	no	—
125	C₂×C₄⋊C₄	—	2	6	no	—
126	C₂×C₄⋊C₄	C₂³×C₆₇	1	12	yes	—
127	C₂×C₄²	D₄×C₆₇	1	6	yes	—
128	C₂×C₄²	C₂³×C₆₇	1	6	yes	—
129	#103	C₂³×C₆₇	1	64	yes	—
130	#97	C₂³×C₆₇	1	16	yes	—
131	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
132	C₂×Q₈	—	2	12	no	—
133	C₂×Q₈	—	2	48	no	—
134	C₂×Q₈	—	4	24	no	—
135	C₂×Q₈	C₂⁴×C₆₇	1	8	yes	—
136	C₂²×C₄	—	2	8	no	—
137	C₂²×C₄	—	2	12	no	—
138	C₂²×C₄	—	2	12	no	—
139	C₂²×C₄	—	4	8	no	—
140	C₂²×C₄	—	4	8	no	—
141	C₂²×C₄	C₂×D₄×C₆₇	1	12	yes	—
142	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
143	C₂²×C₄	#41×C₆₇	1	8	yes	—
144	C₂²×C₄	C₂⁴×C₆₇	1	3	yes	—
145	C₂×C₈	—	2	24	no	—
146	C₂×C₈	C₂⁴×C₆₇	1	8	yes	—
147	C₄⋊C₄	—	2	32	no	—
148	C₄⋊C₄	—	2	32	no	—
149	C₄⋊C₄	—	2	12	no	—
150	C₄⋊C₄	C₂⁴×C₆₇	1	8	yes	—
151	C₄²	C₂×D₄×C₆₇	1	12	yes	—
152	C₄²	C₂⁴×C₆₇	1	4	yes	—
153	C₂³	#124×C₆₇	1	4	yes	—
154	C₂³	#137×C₆₇	1	8	yes	—
155	C₂³	C₂²×D₄×C₆₇	1	3	yes	—
156	Q₈	—	2	16	no	—
157	Q₈	—	4	24	no	—
158	C₂×C₄	—	2	24	no	—
159	C₂×C₄	—	2	8	no	—
160	C₂×C₄	—	2	8	no	—
161	C₂×C₄	—	4	24	no	—
162	C₂×C₄	—	4	24	no	—
163	C₂×C₄	#124×C₆₇	1	12	yes	—
164	C₂×C₄	C₂²×D₄×C₆₇	1	8	yes	—
165	C₂×C₄	C₂²×D₄×C₆₇	1	3	yes	—
166	C₂×C₄	C₂×#41×C₆₇	1	12	yes	—
167	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
168	C₈	—	2	8	no	—
169	C₂²	#491×C₆₇	1	16	yes	direct product factor
170	C₂²	C₂×#124×C₆₇	1	6	yes	—
171	C₂²	C₂×#137×C₆₇	1	12	yes	—
172	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
173	C₄	—	2	8	no	—
174	C₄	—	4	8	no	—
175	C₄	—	4	8	no	—
176	C₄	C₂×#124×C₆₇	1	6	yes	—
177	C₄	C₂²×#41×C₆₇	1	2	yes	—
178	C₂	C₂×#491×C₆₇	1	12	yes	direct product factor
179	C₂	C₂²×#124×C₆₇	1	1	yes	characteristic
180	C₂	C₂²×#137×C₆₇	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₆₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×#491×C₆₇	C₂	direct (N × H)
16. #491×C₆₇	C₂²	direct (N × H)
78. C₂²×C₆₇	#491	direct (N × H)
87. C₂²×#491	C₆₇	direct (N × H)
88. C₂×C₆₇	C₂×#491	direct (N × H)
93. C₂×#491	C₂×C₆₇	direct (N × H)
96. C₆₇	C₂²×#491	direct (N × H)
107. #491	C₂²×C₆₇	direct (N × H)
169. C₂²	#491×C₆₇	direct (N × H)
178. C₂	C₂×#491×C₆₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊Q₈×C₆₇	2
C₂²×#125×C₆₇	2
C₂×#491×C₆₇	2
C₂²×#103×C₆₇	2
C₂²×#97×C₆₇	2
C₂²×#491	67

Subgroups — C22×#491×C67 #30739859

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#491×C₆₇ #30739859