Subgroups — C₆₇×#22258 #30701509

Order 17152.

578 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	C₆₇×#3160	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2961	C₂	1	2	yes	maximal
3	C₆₇×#1995	C₂	1	2	yes	maximal
4	C₆₇×#1922	C₂	1	8	yes	maximal
5	C₆₇×#1837	C₂	1	1	yes	maximal, characteristic
6	C₆₇×#1746	C₂	1	1	yes	maximal, characteristic
7	#533×C₆₇	C₂²	1	2	yes	—
8	#500×C₆₇	C₄	1	2	yes	complemented
9	#499×C₆₇	C₄	1	4	yes	complemented
10	#498×C₆₇	C₄	1	2	yes	complemented
11	#446×C₆₇	—	2	8	no	—
12	#427×C₆₇	C₂²	1	4	yes	—
13	#424×C₆₇	C₂²	1	4	yes	—
14	#423×C₆₇	C₂²	1	2	yes	—
15	#405×C₆₇	C₂²	1	4	yes	—
16	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	#376×C₆₇	C₂²	1	1	yes	characteristic
19	#341×C₆₇	C₂²	1	4	yes	—
20	#339×C₆₇	C₂²	1	8	yes	—
21	#336×C₆₇	C₂²	1	4	yes	—
22	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
23	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
24	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
25	C₄⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
26	#125×C₆₇	C₂×C₄	1	4	yes	—
27	C₂²⋊Q₈×C₆₇	—	2	4	no	—
28	C₄×Q₈×C₆₇	—	2	4	no	—
29	#117×C₆₇	C₂³	1	4	yes	—
30	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
31	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
32	#107×C₆₇	C₂×C₄	1	8	yes	—
33	#106×C₆₇	C₂×C₄	1	8	yes	—
34	C₄⋊C₈×C₆₇	—	2	8	no	—
35	C₄⋊C₈×C₆₇	—	2	4	no	—
36	#97×C₆₇	—	2	4	no	—
37	#97×C₆₇	D₄	1	4	yes	—
38	C₄×C₈×C₆₇	—	2	4	no	—
39	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
42	#95×C₆₇	—	2	2	no	—
43	C₂×Q₈×C₆₇	—	2	4	no	—
44	C₂²×C₄×C₆₇	—	2	2	no	—
45	C₂²×C₄×C₆₇	#41	1	2	yes	—
46	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
47	M₁₆×C₆₇	—	2	8	no	—
48	C₂×C₈×C₆₇	—	2	4	no	—
49	C₂×C₈×C₆₇	—	2	4	no	—
50	C₂×C₈×C₆₇	—	2	4	no	—
51	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
52	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
53	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
54	C₄⋊C₄×C₆₇	—	2	8	no	—
55	C₄⋊C₄×C₆₇	—	2	4	no	—
56	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
57	C₂²⋊C₄×C₆₇	—	2	4	no	—
58	C₄²×C₆₇	—	2	4	no	—
59	C₄²×C₆₇	—	2	4	no	—
60	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
61	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
62	C₄²×C₆₇	C₂×D₄	1	2	yes	—
63	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
64	Q₈×C₆₇	—	8	4	no	—
65	C₂×C₄×C₆₇	—	2	8	no	—
66	C₂×C₄×C₆₇	—	4	2	no	—
67	C₂×C₄×C₆₇	—	4	4	no	—
68	C₂×C₄×C₆₇	—	4	2	no	—
69	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
70	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
71	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
72	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
73	C₂×C₄×C₆₇	2₋¹⁺⁴	1	4	yes	—
74	C₈×C₆₇	—	2	2	no	—
75	C₈×C₆₇	—	2	2	no	—
76	C₈×C₆₇	—	2	4	no	—
77	C₈×C₆₇	—	4	4	no	—
78	C₂²×C₆₇	—	2	2	no	—
79	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
80	C₂²×C₆₇	#541	1	2	yes	—
81	C₄×C₆₇	—	2	2	no	—
82	C₄×C₆₇	—	2	2	no	—
83	C₄×C₆₇	—	4	8	no	—
84	C₄×C₆₇	—	8	2	no	—
85	C₄×C₆₇	#519	1	2	yes	—
86	C₄×C₆₇	#536	1	2	yes	—
87	#22258	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂×C₆₇	—	2	2	no	—
89	C₂×C₆₇	#2327	1	1	yes	characteristic
90	C₂×C₆₇	#2957	1	2	yes	—
91	#3160	C₂×C₆₇	1	1	yes	characteristic
92	#2961	C₂×C₆₇	1	2	yes	—
93	#1995	C₂×C₆₇	1	2	yes	—
94	#1922	C₂×C₆₇	1	8	yes	—
95	#1837	C₂×C₆₇	1	1	yes	characteristic
96	#1746	C₂×C₆₇	1	1	yes	characteristic
97	C₆₇	#22258	1	1	yes	characteristic, direct product factor, Sylow
98	#533	C₂²×C₆₇	1	2	yes	—
99	#500	C₄×C₆₇	1	2	yes	complemented
100	#499	C₄×C₆₇	1	4	yes	complemented
101	#498	C₄×C₆₇	1	2	yes	complemented
102	#446	—	2	8	no	—
103	#427	C₂²×C₆₇	1	4	yes	—
104	#424	C₂²×C₆₇	1	4	yes	—
105	#423	C₂²×C₆₇	1	2	yes	—
106	#405	C₂²×C₆₇	1	4	yes	—
107	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
108	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
109	#376	C₂²×C₆₇	1	1	yes	characteristic
110	#341	C₂²×C₆₇	1	4	yes	—
111	#339	C₂²×C₆₇	1	8	yes	—
112	#336	C₂²×C₆₇	1	4	yes	—
113	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
114	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
115	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
116	C₄⋊Q₈	C₂×C₄×C₆₇	1	4	yes	—
117	#125	C₂×C₄×C₆₇	1	4	yes	—
118	C₂²⋊Q₈	—	2	4	no	—
119	C₄×Q₈	—	2	4	no	—
120	#117	C₂³×C₆₇	1	4	yes	—
121	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
122	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
123	#107	C₂×C₄×C₆₇	1	8	yes	—
124	#106	C₂×C₄×C₆₇	1	8	yes	—
125	C₄⋊C₈	—	2	8	no	—
126	C₄⋊C₈	—	2	4	no	—
127	#97	—	2	4	no	—
128	#97	D₄×C₆₇	1	4	yes	—
129	C₄×C₈	—	2	4	no	—
130	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
131	C₄×C₈	C₂×C₄×C₆₇	1	2	yes	—
132	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
133	#95	—	2	2	no	—
134	C₂×Q₈	—	2	4	no	—
135	C₂²×C₄	—	2	2	no	—
136	C₂²×C₄	#41×C₆₇	1	2	yes	—
137	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
138	M₁₆	—	2	8	no	—
139	C₂×C₈	—	2	4	no	—
140	C₂×C₈	—	2	4	no	—
141	C₂×C₈	—	2	4	no	—
142	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
143	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
144	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
145	C₄⋊C₄	—	2	8	no	—
146	C₄⋊C₄	—	2	4	no	—
147	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
148	C₂²⋊C₄	—	2	4	no	—
149	C₄²	—	2	4	no	—
150	C₄²	—	2	4	no	—
151	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
152	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
153	C₄²	C₂×D₄×C₆₇	1	2	yes	—
154	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
155	Q₈	—	8	4	no	—
156	C₂×C₄	—	2	8	no	—
157	C₂×C₄	—	4	2	no	—
158	C₂×C₄	—	4	4	no	—
159	C₂×C₄	—	4	2	no	—
160	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
161	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
162	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
163	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
164	C₂×C₄	2₋¹⁺⁴×C₆₇	1	4	yes	—
165	C₈	—	2	2	no	—
166	C₈	—	2	2	no	—
167	C₈	—	2	4	no	—
168	C₈	—	4	4	no	—
169	C₂²	—	2	2	no	—
170	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
171	C₂²	#541×C₆₇	1	2	yes	—
172	C₄	—	2	2	no	—
173	C₄	—	2	2	no	—
174	C₄	—	4	8	no	—
175	C₄	—	8	2	no	—
176	C₄	#519×C₆₇	1	2	yes	—
177	C₄	#536×C₆₇	1	2	yes	—
178	C₂	—	2	2	no	—
179	C₂	C₆₇×#2327	1	1	yes	characteristic
180	C₂	C₆₇×#2957	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂³×C₆₇	C₂×#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂×#97×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
8. #500×C₆₇	C₄	semidirect
9. #499×C₆₇	C₄	semidirect
10. #498×C₆₇	C₄	semidirect
87. #22258	C₆₇	direct (N × H)
97. C₆₇	#22258	direct (N × H)
99. #500	C₄×C₆₇	semidirect
100. #499	C₄×C₆₇	semidirect
101. #498	C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3160	2
C₆₇×#2961	2
C₆₇×#1995	2
C₆₇×#1922	2
C₆₇×#1837	2
C₆₇×#1746	2
#22258	67

Subgroups — C67×#22258 #30701509

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22258 #30701509