Subgroups — C₆₇×#33497 #30712748

Order 17152.

486 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2183	C₂	1	8	yes	maximal, complemented
3	C₆₇×#2163	C₂	1	4	yes	maximal
4	C₆₇×#2162	C₂	1	2	yes	maximal
5	C₂×C₄×Q₈×C₆₇	C₄	1	1	yes	characteristic
6	C₈×Q₈×C₆₇	C₄	1	4	yes	—
7	C₈×Q₈×C₆₇	C₂²	1	4	yes	—
8	C₂×C₄⋊C₈×C₆₇	C₄	1	2	yes	—
9	C₂×C₄⋊C₈×C₆₇	C₄	1	1	yes	characteristic
10	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
12	C₄⋊C₁₆×C₆₇	C₂²	1	4	yes	—
13	C₄⋊C₁₆×C₆₇	C₂²	1	8	yes	—
14	C₂²⋊C₁₆×C₆₇	C₂²	1	4	yes	—
15	#346×C₆₇	C₂²	1	8	yes	—
16	C₄×C₁₆×C₆₇	C₂²	1	4	yes	—
17	C₂²×Q₈×C₆₇	C₈	1	1	yes	characteristic
18	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
21	C₄×Q₈×C₆₇	C₈	1	4	yes	—
22	C₄×Q₈×C₆₇	C₂×C₄	1	4	yes	—
23	C₂×C₄⋊C₄×C₆₇	C₈	1	2	yes	—
24	C₂×C₄⋊C₄×C₆₇	C₈	1	1	yes	characteristic
25	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₆×C₆₇	C₂³	1	8	yes	—
28	C₄⋊C₈×C₆₇	C₂×C₄	1	8	yes	—
29	C₄⋊C₈×C₆₇	C₂×C₄	1	4	yes	—
30	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
32	C₄×C₈×C₆₇	C₂³	1	2	yes	—
33	C₄×C₈×C₆₇	C₂³	1	4	yes	—
34	C₂×Q₈×C₆₇	—	2	4	no	—
35	C₂×Q₈×C₆₇	C₂×C₈	1	4	yes	—
36	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
37	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
38	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
39	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
40	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₆₇	—	2	2	no	—
42	C₂×C₈×C₆₇	—	4	1	no	—
43	C₂×C₈×C₆₇	—	4	2	no	—
44	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
45	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
46	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
48	C₄⋊C₄×C₆₇	C₂×C₈	1	8	yes	—
49	C₄⋊C₄×C₆₇	C₂×C₈	1	4	yes	—
50	C₄²×C₆₇	C₂×C₈	1	4	yes	—
51	C₄²×C₆₇	C₂×C₈	1	2	yes	—
52	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
53	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
54	C₁₆×C₆₇	—	2	8	no	—
55	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
56	Q₈×C₆₇	—	2	8	no	—
57	C₂×C₄×C₆₇	—	2	4	no	—
58	C₂×C₄×C₆₇	—	2	1	no	—
59	C₂×C₄×C₆₇	—	2	1	no	—
60	C₂×C₄×C₆₇	—	4	2	no	—
61	C₂×C₄×C₆₇	—	4	1	no	—
62	C₂×C₄×C₆₇	M₃₂	1	4	yes	—
63	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
64	C₂×C₄×C₆₇	C₂²×C₈	1	4	yes	—
65	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
66	C₂×C₄×C₆₇	C₂²×C₈	1	4	yes	—
67	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
68	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
69	C₈×C₆₇	—	2	1	no	—
70	C₈×C₆₇	—	2	2	no	—
71	C₈×C₆₇	—	2	4	no	—
72	C₈×C₆₇	2₋¹⁺⁴	1	2	yes	—
73	C₂²×C₆₇	—	2	1	no	—
74	C₂²×C₆₇	—	2	1	no	—
75	C₂²×C₆₇	C₂×M₃₂	1	2	yes	—
76	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
77	C₄×C₆₇	—	2	4	no	—
78	C₄×C₆₇	—	2	4	no	—
79	C₄×C₆₇	—	2	2	no	—
80	C₄×C₆₇	—	2	1	no	—
81	C₄×C₆₇	C₂×M₃₂	1	4	yes	—
82	C₄×C₆₇	#519	1	1	yes	characteristic
83	C₄×C₆₇	#568	1	1	yes	characteristic
84	#33497	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₂×C₆₇	—	2	2	no	—
86	C₂×C₆₇	#2982	1	1	yes	characteristic
87	C₂×C₆₇	C₂²×M₃₂	1	1	yes	characteristic
88	C₂×C₆₇	#3407	1	1	yes	characteristic
89	C₂×C₈×Q₈	C₂×C₆₇	1	1	yes	characteristic
90	#2183	C₂×C₆₇	1	8	yes	complemented
91	#2163	C₂×C₆₇	1	4	yes	—
92	#2162	C₂×C₆₇	1	2	yes	—
93	C₆₇	#33497	1	1	yes	characteristic, direct product factor, Sylow
94	C₂×C₄×Q₈	C₄×C₆₇	1	1	yes	characteristic
95	C₈×Q₈	C₄×C₆₇	1	4	yes	—
96	C₈×Q₈	C₂²×C₆₇	1	4	yes	—
97	C₂×C₄⋊C₈	C₄×C₆₇	1	2	yes	—
98	C₂×C₄⋊C₈	C₄×C₆₇	1	1	yes	characteristic
99	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
100	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
101	C₄⋊C₁₆	C₂²×C₆₇	1	4	yes	—
102	C₄⋊C₁₆	C₂²×C₆₇	1	8	yes	—
103	C₂²⋊C₁₆	C₂²×C₆₇	1	4	yes	—
104	#346	C₂²×C₆₇	1	8	yes	—
105	C₄×C₁₆	C₂²×C₆₇	1	4	yes	—
106	C₂²×Q₈	C₈×C₆₇	1	1	yes	characteristic
107	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
108	C₂²×C₈	C₂×C₄×C₆₇	1	2	yes	—
109	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
110	C₄×Q₈	C₈×C₆₇	1	4	yes	—
111	C₄×Q₈	C₂×C₄×C₆₇	1	4	yes	—
112	C₂×C₄⋊C₄	C₈×C₆₇	1	2	yes	—
113	C₂×C₄⋊C₄	C₈×C₆₇	1	1	yes	characteristic
114	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
115	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
116	C₂×C₁₆	C₂³×C₆₇	1	8	yes	—
117	C₄⋊C₈	C₂×C₄×C₆₇	1	8	yes	—
118	C₄⋊C₈	C₂×C₄×C₆₇	1	4	yes	—
119	C₄×C₈	C₂×C₄×C₆₇	1	2	yes	—
120	C₄×C₈	C₂×C₄×C₆₇	1	4	yes	—
121	C₄×C₈	C₂³×C₆₇	1	2	yes	—
122	C₄×C₈	C₂³×C₆₇	1	4	yes	—
123	C₂×Q₈	—	2	4	no	—
124	C₂×Q₈	C₂×C₈×C₆₇	1	4	yes	—
125	C₂²×C₄	C₂×C₈×C₆₇	1	2	yes	—
126	C₂²×C₄	C₂×C₈×C₆₇	1	2	yes	—
127	C₂²×C₄	C₂×C₈×C₆₇	1	1	yes	characteristic
128	C₂²×C₄	C₂×C₈×C₆₇	1	1	yes	characteristic
129	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
130	C₂×C₈	—	2	2	no	—
131	C₂×C₈	—	4	1	no	—
132	C₂×C₈	—	4	2	no	—
133	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
134	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
135	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
136	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
137	C₄⋊C₄	C₂×C₈×C₆₇	1	8	yes	—
138	C₄⋊C₄	C₂×C₈×C₆₇	1	4	yes	—
139	C₄²	C₂×C₈×C₆₇	1	4	yes	—
140	C₄²	C₂×C₈×C₆₇	1	2	yes	—
141	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
142	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
143	C₁₆	—	2	8	no	—
144	C₂³	C₂²×C₈×C₆₇	1	1	yes	characteristic
145	Q₈	—	2	8	no	—
146	C₂×C₄	—	2	4	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	4	2	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	M₃₂×C₆₇	1	4	yes	—
152	C₂×C₄	C₂²×C₈×C₆₇	1	1	yes	characteristic
153	C₂×C₄	C₂²×C₈×C₆₇	1	4	yes	—
154	C₂×C₄	C₂²×C₈×C₆₇	1	2	yes	—
155	C₂×C₄	C₂²×C₈×C₆₇	1	4	yes	—
156	C₂×C₄	C₂²×C₈×C₆₇	1	2	yes	—
157	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
158	C₈	—	2	1	no	—
159	C₈	—	2	2	no	—
160	C₈	—	2	4	no	—
161	C₈	2₋¹⁺⁴×C₆₇	1	2	yes	—
162	C₂²	—	2	1	no	—
163	C₂²	—	2	1	no	—
164	C₂²	C₂×M₃₂×C₆₇	1	2	yes	—
165	C₂²	C₂³×C₈×C₆₇	1	1	yes	characteristic
166	C₄	—	2	4	no	—
167	C₄	—	2	4	no	—
168	C₄	—	2	2	no	—
169	C₄	—	2	1	no	—
170	C₄	C₂×M₃₂×C₆₇	1	4	yes	—
171	C₄	#519×C₆₇	1	1	yes	characteristic
172	C₄	#568×C₆₇	1	1	yes	characteristic
173	C₂	—	2	2	no	—
174	C₂	C₆₇×#2982	1	1	yes	characteristic
175	C₂	C₂²×M₃₂×C₆₇	1	1	yes	characteristic
176	C₂	C₆₇×#3407	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₆₇	C₂⁴
commutator	C₂²	C₂³×C₈×C₆₇
Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂³×C₈
prime radical	C₂³×C₆₇	C₂²×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2183	C₂	semidirect
84. #33497	C₆₇	direct (N × H)
90. #2183	C₂×C₆₇	semidirect
93. C₆₇	#33497	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×C₈×Q₈×C₆₇	2
C₆₇×#2183	2
C₆₇×#2163	2
C₆₇×#2162	2
#33497	67

Subgroups — C67×#33497 #30712748

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33497 #30712748