Subgroups — C₆₇×#33409 #30712660

Order 17152.

674 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#427×C₆₇	C₂	1	2	yes	maximal
3	C₆₇×#2159	C₂	1	8	yes	maximal, complemented
4	C₆₇×#2158	C₂	1	4	yes	maximal, complemented
5	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
6	C₂×#117×C₆₇	—	2	2	no	—
7	C₂×M₃₂×C₆₇	C₄	1	8	yes	complemented
8	C₂×M₃₂×C₆₇	C₂²	1	4	yes	—
9	C₂²×C₁₆×C₆₇	C₂²	1	2	yes	—
10	#427×C₆₇	—	2	2	no	—
11	#427×C₆₇	—	2	2	no	—
12	#427×C₆₇	C₂²	1	4	yes	—
13	#427×C₆₇	C₂²	1	4	yes	—
14	C₂×#107×C₆₇	C₂²	1	4	yes	—
15	C₂×#106×C₆₇	—	2	2	no	—
16	#367×C₆₇	C₂²	1	8	yes	complemented
17	#366×C₆₇	C₂²	1	8	yes	complemented
18	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
20	C₂²×C₈×C₆₇	D₄	1	2	yes	—
21	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₆₇	Q₈	1	4	yes	—
23	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
24	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
25	#117×C₆₇	—	2	4	no	—
26	#117×C₆₇	—	2	4	no	—
27	#117×C₆₇	—	4	2	no	—
28	#117×C₆₇	—	4	2	no	—
29	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
30	C₂×C₂²⋊C₄×C₆₇	—	4	2	no	—
31	C₂×C₄²×C₆₇	—	4	2	no	—
32	M₃₂×C₆₇	C₂×C₄	1	16	yes	complemented
33	C₂×C₁₆×C₆₇	C₂×C₄	1	8	yes	complemented
34	C₂×C₁₆×C₆₇	C₂³	1	4	yes	—
35	#107×C₆₇	—	2	4	no	—
36	#107×C₆₇	C₂³	1	8	yes	—
37	#106×C₆₇	—	2	4	no	—
38	#106×C₆₇	—	2	4	no	—
39	C₂⁴×C₆₇	D₈	1	1	yes	characteristic
40	C₂²×C₄×C₆₇	—	4	4	no	—
41	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
42	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
43	C₂²×C₄×C₆₇	D₈	1	2	yes	—
44	C₂²×C₄×C₆₇	Q₁₆	1	4	yes	—
45	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
46	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₆₇	C₄⋊C₄	1	8	yes	—
48	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
49	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
50	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
51	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
53	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
54	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
55	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
56	C₄⋊C₄×C₆₇	—	2	8	no	—
57	C₄⋊C₄×C₆₇	—	4	4	no	—
58	C₂²⋊C₄×C₆₇	—	4	4	no	—
59	C₂²⋊C₄×C₆₇	—	4	4	no	—
60	C₄²×C₆₇	—	4	4	no	—
61	C₄²×C₆₇	—	4	4	no	—
62	C₁₆×C₆₇	C₂²×C₄	1	8	yes	—
63	C₂³×C₆₇	—	4	2	no	—
64	C₂³×C₆₇	#107	1	4	yes	—
65	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
66	C₂³×C₆₇	C₂×D₈	1	2	yes	—
67	C₂×C₄×C₆₇	—	4	8	no	—
68	C₂×C₄×C₆₇	—	8	8	no	—
69	C₂×C₄×C₆₇	#107	1	4	yes	—
70	C₂×C₄×C₆₇	#107	1	8	yes	—
71	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
72	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
73	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
74	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
75	C₂×C₄×C₆₇	C₂×Q₁₆	1	4	yes	—
76	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
77	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
78	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
79	C₈×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
80	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
81	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
82	C₂²×C₆₇	—	2	4	no	—
83	C₂²×C₆₇	—	2	2	no	—
84	C₂²×C₆₇	—	4	4	no	—
85	C₂²×C₆₇	C₂×#107	1	2	yes	—
86	C₂²×C₆₇	C₂×#107	1	4	yes	—
87	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
88	C₄×C₆₇	—	8	8	no	—
89	C₄×C₆₇	C₂×#107	1	4	yes	—
90	C₄×C₆₇	C₂×#107	1	2	yes	—
91	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
92	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
93	#33409	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₂×C₆₇	—	2	4	no	—
95	C₂×C₆₇	—	2	2	no	—
96	C₂×C₆₇	C₂²×#107	1	1	yes	characteristic
97	C₂×C₆₇	#3414	1	2	yes	—
98	C₂²×M₃₂	C₂×C₆₇	1	1	yes	characteristic
99	C₂×#427	C₂×C₆₇	1	2	yes	—
100	#2159	C₂×C₆₇	1	8	yes	complemented
101	#2158	C₂×C₆₇	1	4	yes	complemented
102	C₆₇	#33409	1	1	yes	characteristic, direct product factor, Sylow
103	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
104	C₂×#117	—	2	2	no	—
105	C₂×M₃₂	C₄×C₆₇	1	8	yes	complemented
106	C₂×M₃₂	C₂²×C₆₇	1	4	yes	—
107	C₂²×C₁₆	C₂²×C₆₇	1	2	yes	—
108	#427	—	2	2	no	—
109	#427	—	2	2	no	—
110	#427	C₂²×C₆₇	1	4	yes	—
111	#427	C₂²×C₆₇	1	4	yes	—
112	C₂×#107	C₂²×C₆₇	1	4	yes	—
113	C₂×#106	—	2	2	no	—
114	#367	C₂²×C₆₇	1	8	yes	complemented
115	#366	C₂²×C₆₇	1	8	yes	complemented
116	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
117	C₂²×C₈	C₂×C₄×C₆₇	1	4	yes	—
118	C₂²×C₈	D₄×C₆₇	1	2	yes	—
119	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
120	C₂²×C₈	Q₈×C₆₇	1	4	yes	—
121	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
122	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
123	#117	—	2	4	no	—
124	#117	—	2	4	no	—
125	#117	—	4	2	no	—
126	#117	—	4	2	no	—
127	C₂×C₄⋊C₄	—	2	4	no	—
128	C₂×C₂²⋊C₄	—	4	2	no	—
129	C₂×C₄²	—	4	2	no	—
130	M₃₂	C₂×C₄×C₆₇	1	16	yes	complemented
131	C₂×C₁₆	C₂×C₄×C₆₇	1	8	yes	complemented
132	C₂×C₁₆	C₂³×C₆₇	1	4	yes	—
133	#107	—	2	4	no	—
134	#107	C₂³×C₆₇	1	8	yes	—
135	#106	—	2	4	no	—
136	#106	—	2	4	no	—
137	C₂⁴	D₈×C₆₇	1	1	yes	characteristic
138	C₂²×C₄	—	4	4	no	—
139	C₂²×C₄	C₄⋊C₄×C₆₇	1	4	yes	—
140	C₂²×C₄	D₈×C₆₇	1	2	yes	—
141	C₂²×C₄	D₈×C₆₇	1	1	yes	characteristic
142	C₂²×C₄	Q₁₆×C₆₇	1	4	yes	—
143	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
144	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
145	C₂×C₈	C₄⋊C₄×C₆₇	1	8	yes	—
146	C₂×C₈	C₄⋊C₄×C₆₇	1	4	yes	—
147	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
148	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
149	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
150	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
151	C₂×C₈	C₂×Q₈×C₆₇	1	4	yes	—
152	C₂×C₈	C₂×Q₈×C₆₇	1	2	yes	—
153	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
154	C₄⋊C₄	—	2	8	no	—
155	C₄⋊C₄	—	4	4	no	—
156	C₂²⋊C₄	—	4	4	no	—
157	C₂²⋊C₄	—	4	4	no	—
158	C₄²	—	4	4	no	—
159	C₄²	—	4	4	no	—
160	C₁₆	C₂²×C₄×C₆₇	1	8	yes	—
161	C₂³	—	4	2	no	—
162	C₂³	#107×C₆₇	1	4	yes	—
163	C₂³	C₂×D₈×C₆₇	1	1	yes	characteristic
164	C₂³	C₂×D₈×C₆₇	1	2	yes	—
165	C₂×C₄	—	4	8	no	—
166	C₂×C₄	—	8	8	no	—
167	C₂×C₄	#107×C₆₇	1	8	yes	—
168	C₂×C₄	#107×C₆₇	1	4	yes	—
169	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
170	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
171	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
172	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
173	C₂×C₄	C₂×Q₁₆×C₆₇	1	4	yes	—
174	C₂×C₄	C₂×Q₁₆×C₆₇	1	2	yes	—
175	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
176	C₈	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
177	C₈	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
178	C₈	C₂³×C₄×C₆₇	1	1	yes	characteristic
179	C₈	C₂²×Q₈×C₆₇	1	1	yes	characteristic
180	C₂²	—	2	4	no	—
181	C₂²	—	2	2	no	—
182	C₂²	—	4	4	no	—
183	C₂²	C₂×#107×C₆₇	1	2	yes	—
184	C₂²	C₂×#107×C₆₇	1	4	yes	—
185	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
186	C₄	—	8	8	no	—
187	C₄	C₂×#107×C₆₇	1	2	yes	—
188	C₄	C₂×#107×C₆₇	1	4	yes	—
189	C₄	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
190	C₄	C₂²×Q₁₆×C₆₇	1	1	yes	characteristic
191	C₂	—	2	4	no	—
192	C₂	—	2	2	no	—
193	C₂	C₂²×#107×C₆₇	1	1	yes	characteristic
194	C₂	C₆₇×#3414	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂×D₈
commutator	C₈	C₂³×C₄×C₆₇
Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂²×D₈
prime radical	C₂⁴×C₆₇	D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂³×C₄×C₆₇ ◁ C₂³×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₇×#2159	C₂	semidirect
4. C₆₇×#2158	C₂	semidirect
7. C₂×M₃₂×C₆₇	C₄	semidirect
16. #367×C₆₇	C₂²	semidirect
17. #366×C₆₇	C₂²	semidirect
32. M₃₂×C₆₇	C₂×C₄	semidirect
33. C₂×C₁₆×C₆₇	C₂×C₄	semidirect
93. #33409	C₆₇	direct (N × H)
100. #2159	C₂×C₆₇	semidirect
101. #2158	C₂×C₆₇	semidirect
102. C₆₇	#33409	direct (N × H)
105. C₂×M₃₂	C₄×C₆₇	semidirect
114. #367	C₂²×C₆₇	semidirect
115. #366	C₂²×C₆₇	semidirect
130. M₃₂	C₂×C₄×C₆₇	semidirect
131. C₂×C₁₆	C₂×C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×M₃₂×C₆₇	2
C₂×#427×C₆₇	2
C₆₇×#2159	2
C₆₇×#2158	2
#33409	67

Subgroups — C67×#33409 #30712660

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33409 #30712660