Subgroups — C₆₇×#33486 #30712737

Order 17152.

662 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic
2	D₄×C₁₆×C₆₇	C₂	1	8	yes	maximal, complemented
3	C₆₇×#2162	C₂	1	2	yes	maximal, complemented
4	C₂×C₂²⋊C₁₆×C₆₇	C₂	1	4	yes	maximal, complemented
5	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
6	C₂×C₄×D₄×C₆₇	C₄	1	1	yes	characteristic
7	C₂²×C₁₆×C₆₇	C₂²	1	8	yes	complemented
8	C₈×D₄×C₆₇	C₄	1	4	yes	—
9	C₈×D₄×C₆₇	C₂²	1	4	yes	—
10	C₂×C₄⋊C₈×C₆₇	C₄	1	1	yes	characteristic
11	C₂×C₂²⋊C₈×C₆₇	C₄	1	2	yes	—
12	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₄⋊C₁₆×C₆₇	C₂²	1	4	yes	complemented
14	C₂²⋊C₁₆×C₆₇	C₂²	1	4	yes	—
15	C₂²⋊C₁₆×C₆₇	C₂²	1	8	yes	complemented
16	C₄×C₁₆×C₆₇	C₂²	1	4	yes	complemented
17	C₂²×D₄×C₆₇	C₈	1	1	yes	characteristic
18	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
19	C₂²×C₈×C₆₇	—	4	4	no	—
20	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
21	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
22	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
24	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
25	C₄×D₄×C₆₇	C₈	1	4	yes	—
26	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
27	C₂×C₄⋊C₄×C₆₇	C₈	1	1	yes	characteristic
28	C₂×C₂²⋊C₄×C₆₇	C₈	1	2	yes	—
29	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₁₆×C₆₇	—	4	8	no	—
31	C₂×C₁₆×C₆₇	C₂³	1	8	yes	—
32	C₄⋊C₈×C₆₇	C₂×C₄	1	4	yes	—
33	C₂²⋊C₈×C₆₇	C₂×C₄	1	8	yes	—
34	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₆₇	C₂³	1	2	yes	—
36	C₂⁴×C₆₇	C₂×C₈	1	2	yes	—
37	C₂×D₄×C₆₇	C₁₆	1	8	yes	complemented
38	C₂×D₄×C₆₇	C₂×C₈	1	4	yes	—
39	C₂²×C₄×C₆₇	—	4	2	no	—
40	C₂²×C₄×C₆₇	—	4	2	no	—
41	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
42	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
43	C₂²×C₄×C₆₇	C₂×C₈	1	4	yes	—
44	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
47	C₂×C₈×C₆₇	—	2	8	no	—
48	C₂×C₈×C₆₇	—	2	2	no	—
49	C₂×C₈×C₆₇	—	4	4	no	—
50	C₂×C₈×C₆₇	—	4	2	no	—
51	C₂×C₈×C₆₇	—	4	1	no	—
52	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
53	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
55	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
56	C₄⋊C₄×C₆₇	C₂×C₈	1	4	yes	—
57	C₂²⋊C₄×C₆₇	C₂×C₈	1	8	yes	—
58	C₄²×C₆₇	C₂×C₈	1	2	yes	—
59	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
60	C₁₆×C₆₇	—	2	8	no	—
61	C₂³×C₆₇	—	4	2	no	—
62	C₂³×C₆₇	—	4	2	no	—
63	C₂³×C₆₇	C₂×C₁₆	1	8	yes	complemented
64	C₂³×C₆₇	C₂²×C₈	1	4	yes	—
65	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
66	D₄×C₆₇	C₂×C₁₆	1	16	yes	complemented
67	C₂×C₄×C₆₇	—	2	1	no	—
68	C₂×C₄×C₆₇	—	2	4	no	—
69	C₂×C₄×C₆₇	—	2	1	no	—
70	C₂×C₄×C₆₇	—	2	4	no	—
71	C₂×C₄×C₆₇	—	4	4	no	—
72	C₂×C₄×C₆₇	—	4	2	no	—
73	C₂×C₄×C₆₇	—	4	1	no	—
74	C₂×C₄×C₆₇	C₂×C₁₆	1	4	yes	complemented
75	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
77	C₂×C₄×C₆₇	C₂²×C₈	1	4	yes	—
78	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
79	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
80	C₈×C₆₇	—	2	1	no	—
81	C₈×C₆₇	—	2	4	no	—
82	C₈×C₆₇	—	2	2	no	—
83	C₈×C₆₇	H(5,𝔽₂)	1	2	yes	—
84	C₂²×C₆₇	—	2	1	no	—
85	C₂²×C₆₇	—	2	4	no	—
86	C₂²×C₆₇	—	2	1	no	—
87	C₂²×C₆₇	—	2	4	no	—
88	C₂²×C₆₇	—	4	8	no	—
89	C₂²×C₆₇	C₂²×C₁₆	1	2	yes	—
90	C₂²×C₆₇	C₂²×C₁₆	1	8	yes	—
91	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
92	C₄×C₆₇	—	2	2	no	—
93	C₄×C₆₇	—	2	1	no	—
94	C₄×C₆₇	—	2	4	no	—
95	C₄×C₆₇	C₂²×C₁₆	1	4	yes	—
96	C₄×C₆₇	#518	1	1	yes	characteristic
97	C₄×C₆₇	#568	1	1	yes	characteristic
98	#33486	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂×C₆₇	—	2	2	no	—
100	C₂×C₆₇	—	2	8	no	—
101	C₂×C₆₇	#2972	1	1	yes	characteristic
102	C₂×C₆₇	C₂³×C₁₆	1	1	yes	characteristic
103	C₂×C₆₇	#3407	1	1	yes	characteristic
104	C₂×C₈×D₄	C₂×C₆₇	1	1	yes	characteristic
105	D₄×C₁₆	C₂×C₆₇	1	8	yes	complemented
106	#2162	C₂×C₆₇	1	2	yes	complemented
107	C₂×C₂²⋊C₁₆	C₂×C₆₇	1	4	yes	complemented
108	C₆₇	#33486	1	1	yes	characteristic, direct product factor, Sylow
109	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
110	C₂×C₄×D₄	C₄×C₆₇	1	1	yes	characteristic
111	C₂²×C₁₆	C₂²×C₆₇	1	8	yes	complemented
112	C₈×D₄	C₄×C₆₇	1	4	yes	—
113	C₈×D₄	C₂²×C₆₇	1	4	yes	—
114	C₂×C₄⋊C₈	C₄×C₆₇	1	1	yes	characteristic
115	C₂×C₂²⋊C₈	C₄×C₆₇	1	2	yes	—
116	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
117	C₄⋊C₁₆	C₂²×C₆₇	1	4	yes	complemented
118	C₂²⋊C₁₆	C₂²×C₆₇	1	4	yes	—
119	C₂²⋊C₁₆	C₂²×C₆₇	1	8	yes	complemented
120	C₄×C₁₆	C₂²×C₆₇	1	4	yes	complemented
121	C₂²×D₄	C₈×C₆₇	1	1	yes	characteristic
122	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
123	C₂²×C₈	—	4	4	no	—
124	C₂²×C₈	C₂×C₄×C₆₇	1	2	yes	—
125	C₂²×C₈	C₂×C₄×C₆₇	1	4	yes	—
126	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
127	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
128	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
129	C₄×D₄	C₈×C₆₇	1	4	yes	—
130	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
131	C₂×C₄⋊C₄	C₈×C₆₇	1	1	yes	characteristic
132	C₂×C₂²⋊C₄	C₈×C₆₇	1	2	yes	—
133	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
134	C₂×C₁₆	—	4	8	no	—
135	C₂×C₁₆	C₂³×C₆₇	1	8	yes	—
136	C₄⋊C₈	C₂×C₄×C₆₇	1	4	yes	—
137	C₂²⋊C₈	C₂×C₄×C₆₇	1	8	yes	—
138	C₄×C₈	C₂×C₄×C₆₇	1	2	yes	—
139	C₄×C₈	C₂³×C₆₇	1	2	yes	—
140	C₂⁴	C₂×C₈×C₆₇	1	2	yes	—
141	C₂×D₄	C₁₆×C₆₇	1	8	yes	complemented
142	C₂×D₄	C₂×C₈×C₆₇	1	4	yes	—
143	C₂²×C₄	—	4	2	no	—
144	C₂²×C₄	—	4	2	no	—
145	C₂²×C₄	C₂×C₈×C₆₇	1	1	yes	characteristic
146	C₂²×C₄	C₂×C₈×C₆₇	1	2	yes	—
147	C₂²×C₄	C₂×C₈×C₆₇	1	4	yes	—
148	C₂²×C₄	C₂×C₈×C₆₇	1	1	yes	characteristic
149	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
150	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
151	C₂×C₈	—	2	8	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	4	4	no	—
154	C₂×C₈	—	4	2	no	—
155	C₂×C₈	—	4	1	no	—
156	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
157	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
158	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
159	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
160	C₄⋊C₄	C₂×C₈×C₆₇	1	4	yes	—
161	C₂²⋊C₄	C₂×C₈×C₆₇	1	8	yes	—
162	C₄²	C₂×C₈×C₆₇	1	2	yes	—
163	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
164	C₁₆	—	2	8	no	—
165	C₂³	—	4	2	no	—
166	C₂³	—	4	2	no	—
167	C₂³	C₂×C₁₆×C₆₇	1	8	yes	complemented
168	C₂³	C₂²×C₈×C₆₇	1	4	yes	—
169	C₂³	C₂²×C₈×C₆₇	1	1	yes	characteristic
170	D₄	C₂×C₁₆×C₆₇	1	16	yes	complemented
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	4	4	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	4	1	no	—
178	C₂×C₄	C₂×C₁₆×C₆₇	1	4	yes	complemented
179	C₂×C₄	C₂²×C₈×C₆₇	1	1	yes	characteristic
180	C₂×C₄	C₂²×C₈×C₆₇	1	2	yes	—
181	C₂×C₄	C₂²×C₈×C₆₇	1	4	yes	—
182	C₂×C₄	C₂²×C₈×C₆₇	1	2	yes	—
183	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
184	C₈	—	2	1	no	—
185	C₈	—	2	4	no	—
186	C₈	—	2	2	no	—
187	C₈	H(5,𝔽₂)×C₆₇	1	2	yes	—
188	C₂²	—	2	1	no	—
189	C₂²	—	2	4	no	—
190	C₂²	—	2	1	no	—
191	C₂²	—	2	4	no	—
192	C₂²	—	4	8	no	—
193	C₂²	C₂²×C₁₆×C₆₇	1	2	yes	—
194	C₂²	C₂²×C₁₆×C₆₇	1	8	yes	—
195	C₂²	C₂³×C₈×C₆₇	1	1	yes	characteristic
196	C₄	—	2	2	no	—
197	C₄	—	2	1	no	—
198	C₄	—	2	4	no	—
199	C₄	C₂²×C₁₆×C₆₇	1	4	yes	—
200	C₄	#518×C₆₇	1	1	yes	characteristic
201	C₄	#568×C₆₇	1	1	yes	characteristic
202	C₂	—	2	2	no	—
203	C₂	—	2	8	no	—
204	C₂	C₆₇×#2972	1	1	yes	characteristic
205	C₂	C₂³×C₁₆×C₆₇	1	1	yes	characteristic
206	C₂	C₆₇×#3407	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₆₇	C₂⁴
commutator	C₂	C₂³×C₁₆×C₆₇
Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂³×C₈
prime radical	C₂²×D₄×C₆₇	C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. D₄×C₁₆×C₆₇	C₂	semidirect
3. C₆₇×#2162	C₂	semidirect
4. C₂×C₂²⋊C₁₆×C₆₇	C₂	semidirect
7. C₂²×C₁₆×C₆₇	C₂²	semidirect
13. C₄⋊C₁₆×C₆₇	C₂²	semidirect
15. C₂²⋊C₁₆×C₆₇	C₂²	semidirect
16. C₄×C₁₆×C₆₇	C₂²	semidirect
37. C₂×D₄×C₆₇	C₁₆	semidirect
63. C₂³×C₆₇	C₂×C₁₆	semidirect
66. D₄×C₆₇	C₂×C₁₆	semidirect
74. C₂×C₄×C₆₇	C₂×C₁₆	semidirect
98. #33486	C₆₇	direct (N × H)
105. D₄×C₁₆	C₂×C₆₇	semidirect
106. #2162	C₂×C₆₇	semidirect
107. C₂×C₂²⋊C₁₆	C₂×C₆₇	semidirect
108. C₆₇	#33486	direct (N × H)
111. C₂²×C₁₆	C₂²×C₆₇	semidirect
117. C₄⋊C₁₆	C₂²×C₆₇	semidirect
119. C₂²⋊C₁₆	C₂²×C₆₇	semidirect
120. C₄×C₁₆	C₂²×C₆₇	semidirect
141. C₂×D₄	C₁₆×C₆₇	semidirect
167. C₂³	C₂×C₁₆×C₆₇	semidirect
170. D₄	C₂×C₁₆×C₆₇	semidirect
178. C₂×C₄	C₂×C₁₆×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₄×C₆₇	2
D₄×C₁₆×C₆₇	2
C₆₇×#2162	2
C₂×C₂²⋊C₁₆×C₆₇	2
#33486	67

Subgroups — C67×#33486 #30712737

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33486 #30712737