Subgroups — C₆₇×#33458 #30712709

Order 17152.

646 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	C₂²×Q₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2950	C₂	1	2	yes	maximal
3	C₆₇×#2176	C₂	1	4	yes	maximal, complemented
4	C₄×Q₃₂×C₆₇	C₂	1	4	yes	maximal, complemented
5	C₆₇×#2158	C₂	1	1	yes	maximal, characteristic
6	C₆₇×#2141	C₂	1	2	yes	maximal
7	C₆₇×#2108	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₁₆×C₆₇	C₂²	1	2	yes	—
9	#519×C₆₇	—	2	2	no	—
10	C₂×Q₃₂×C₆₇	C₄	1	8	yes	complemented
11	C₂×Q₃₂×C₆₇	C₂²	1	2	yes	—
12	C₂×Q₃₂×C₆₇	C₂²	1	2	yes	—
13	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
15	#441×C₆₇	—	2	4	no	—
16	C₄×Q₁₆×C₆₇	C₂²	1	8	yes	—
17	#427×C₆₇	C₂²	1	2	yes	—
18	#419×C₆₇	—	2	2	no	—
19	#405×C₆₇	C₂²	1	1	yes	characteristic
20	#367×C₆₇	C₂²	1	2	yes	—
21	#366×C₆₇	C₂²	1	2	yes	—
22	#358×C₆₇	C₂²	1	8	yes	—
23	#346×C₆₇	C₂²	1	2	yes	—
24	C₄×C₁₆×C₆₇	C₂²	1	2	yes	—
25	C₂²×Q₈×C₆₇	—	2	2	no	—
26	C₂×Q₁₆×C₆₇	—	2	2	no	—
27	C₂×Q₁₆×C₆₇	—	2	4	no	—
28	C₂×Q₁₆×C₆₇	C₂×C₄	1	8	yes	—
29	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
30	C₂×M₁₆×C₆₇	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
32	C₄×Q₈×C₆₇	—	2	8	no	—
33	C₄×Q₈×C₆₇	—	4	4	no	—
34	#117×C₆₇	—	2	2	no	—
35	#117×C₆₇	—	4	4	no	—
36	#117×C₆₇	D₄	1	1	yes	characteristic
37	Q₃₂×C₆₇	C₂×C₄	1	8	yes	—
38	Q₃₂×C₆₇	C₂×C₄	1	8	yes	—
39	M₃₂×C₆₇	—	2	2	no	—
40	C₂×C₁₆×C₆₇	C₂×C₄	1	4	yes	—
41	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
42	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
43	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
44	#107×C₆₇	C₂³	1	4	yes	—
45	#106×C₆₇	—	2	2	no	—
46	#103×C₆₇	—	2	8	no	—
47	#97×C₆₇	D₄	1	2	yes	—
48	C₄×C₈×C₆₇	C₂³	1	2	yes	—
49	C₂×Q₈×C₆₇	—	2	8	no	—
50	C₂×Q₈×C₆₇	—	2	4	no	—
51	C₂×Q₈×C₆₇	—	4	2	no	—
52	C₂×Q₈×C₆₇	—	4	4	no	—
53	C₂²×C₄×C₆₇	—	4	2	no	—
54	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
55	Q₁₆×C₆₇	—	2	8	no	—
56	Q₁₆×C₆₇	—	2	4	no	—
57	Q₁₆×C₆₇	C₂²×C₄	1	8	yes	—
58	M₁₆×C₆₇	—	2	2	no	—
59	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
60	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₆₇	#41	1	2	yes	—
63	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
64	C₄⋊C₄×C₆₇	—	2	4	no	—
65	C₄⋊C₄×C₆₇	—	4	8	no	—
66	C₄⋊C₄×C₆₇	D₈	1	2	yes	—
67	C₂²⋊C₄×C₆₇	—	4	2	no	—
68	C₂²⋊C₄×C₆₇	D₈	1	2	yes	—
69	C₄²×C₆₇	—	4	2	no	—
70	C₄²×C₆₇	—	4	8	no	—
71	C₄²×C₆₇	C₂×D₄	1	2	yes	—
72	C₁₆×C₆₇	—	2	2	no	—
73	C₁₆×C₆₇	C₂²×C₄	1	2	yes	—
74	C₁₆×C₆₇	C₂²×C₄	1	2	yes	—
75	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
76	Q₈×C₆₇	—	2	8	no	—
77	Q₈×C₆₇	—	4	4	no	—
78	Q₈×C₆₇	—	4	8	no	—
79	C₂×C₄×C₆₇	—	4	4	no	—
80	C₂×C₄×C₆₇	—	4	8	no	—
81	C₂×C₄×C₆₇	—	4	4	no	—
82	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
83	C₂×C₄×C₆₇	C₂×D₈	1	4	yes	—
84	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
85	C₂×C₄×C₆₇	#135	1	2	yes	—
86	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
87	C₈×C₆₇	—	2	2	no	—
88	C₈×C₆₇	C₄×D₄	1	2	yes	—
89	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
90	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
91	C₂²×C₆₇	—	2	2	no	—
92	C₂²×C₆₇	C₄×D₈	1	2	yes	—
93	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
94	C₄×C₆₇	—	2	4	no	—
95	C₄×C₆₇	—	4	8	no	—
96	C₄×C₆₇	—	8	4	no	—
97	C₄×C₆₇	C₄×D₈	1	2	yes	—
98	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
99	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
100	#33458	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
101	C₂×C₆₇	—	2	2	no	—
102	C₂×C₆₇	C₂×C₄×D₈	1	1	yes	characteristic
103	C₂×C₆₇	#3417	1	2	yes	—
104	C₂²×Q₃₂	C₂×C₆₇	1	1	yes	characteristic
105	#2950	C₂×C₆₇	1	2	yes	—
106	#2176	C₂×C₆₇	1	4	yes	complemented
107	C₄×Q₃₂	C₂×C₆₇	1	4	yes	complemented
108	#2158	C₂×C₆₇	1	1	yes	characteristic
109	#2141	C₂×C₆₇	1	2	yes	—
110	#2108	C₂×C₆₇	1	1	yes	characteristic
111	C₆₇	#33458	1	1	yes	characteristic, direct product factor, Sylow
112	C₂²×Q₁₆	C₂²×C₆₇	1	2	yes	—
113	#519	—	2	2	no	—
114	C₂×Q₃₂	C₄×C₆₇	1	8	yes	complemented
115	C₂×Q₃₂	C₂²×C₆₇	1	2	yes	—
116	C₂×Q₃₂	C₂²×C₆₇	1	2	yes	—
117	C₂×M₃₂	C₂²×C₆₇	1	1	yes	characteristic
118	C₂²×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
119	#441	—	2	4	no	—
120	C₄×Q₁₆	C₂²×C₆₇	1	8	yes	—
121	#427	C₂²×C₆₇	1	2	yes	—
122	#419	—	2	2	no	—
123	#405	C₂²×C₆₇	1	1	yes	characteristic
124	#367	C₂²×C₆₇	1	2	yes	—
125	#366	C₂²×C₆₇	1	2	yes	—
126	#358	C₂²×C₆₇	1	8	yes	—
127	#346	C₂²×C₆₇	1	2	yes	—
128	C₄×C₁₆	C₂²×C₆₇	1	2	yes	—
129	C₂²×Q₈	—	2	2	no	—
130	C₂×Q₁₆	—	2	2	no	—
131	C₂×Q₁₆	—	2	4	no	—
132	C₂×Q₁₆	C₂×C₄×C₆₇	1	8	yes	—
133	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
134	C₂×M₁₆	D₄×C₆₇	1	1	yes	characteristic
135	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
136	C₄×Q₈	—	2	8	no	—
137	C₄×Q₈	—	4	4	no	—
138	#117	—	2	2	no	—
139	#117	—	4	4	no	—
140	#117	D₄×C₆₇	1	1	yes	characteristic
141	Q₃₂	C₂×C₄×C₆₇	1	8	yes	—
142	Q₃₂	C₂×C₄×C₆₇	1	8	yes	—
143	M₃₂	—	2	2	no	—
144	C₂×C₁₆	C₂×C₄×C₆₇	1	4	yes	—
145	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
146	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
147	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
148	#107	C₂³×C₆₇	1	4	yes	—
149	#106	—	2	2	no	—
150	#103	—	2	8	no	—
151	#97	D₄×C₆₇	1	2	yes	—
152	C₄×C₈	C₂³×C₆₇	1	2	yes	—
153	C₂×Q₈	—	2	8	no	—
154	C₂×Q₈	—	2	4	no	—
155	C₂×Q₈	—	4	2	no	—
156	C₂×Q₈	—	4	4	no	—
157	C₂²×C₄	—	4	2	no	—
158	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
159	Q₁₆	—	2	8	no	—
160	Q₁₆	—	2	4	no	—
161	Q₁₆	C₂²×C₄×C₆₇	1	8	yes	—
162	M₁₆	—	2	2	no	—
163	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
164	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
165	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
166	C₂×C₈	#41×C₆₇	1	2	yes	—
167	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
168	C₄⋊C₄	—	2	4	no	—
169	C₄⋊C₄	—	4	8	no	—
170	C₄⋊C₄	D₈×C₆₇	1	2	yes	—
171	C₂²⋊C₄	—	4	2	no	—
172	C₂²⋊C₄	D₈×C₆₇	1	2	yes	—
173	C₄²	—	4	2	no	—
174	C₄²	—	4	8	no	—
175	C₄²	C₂×D₄×C₆₇	1	2	yes	—
176	C₁₆	—	2	2	no	—
177	C₁₆	C₂²×C₄×C₆₇	1	2	yes	—
178	C₁₆	C₂²×C₄×C₆₇	1	2	yes	—
179	C₂³	C₂×D₈×C₆₇	1	1	yes	characteristic
180	Q₈	—	2	8	no	—
181	Q₈	—	4	4	no	—
182	Q₈	—	4	8	no	—
183	C₂×C₄	—	4	4	no	—
184	C₂×C₄	—	4	8	no	—
185	C₂×C₄	—	4	4	no	—
186	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
187	C₂×C₄	C₂×D₈×C₆₇	1	4	yes	—
188	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
189	C₂×C₄	#135×C₆₇	1	2	yes	—
190	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
191	C₈	—	2	2	no	—
192	C₈	C₄×D₄×C₆₇	1	2	yes	—
193	C₈	C₂³×C₄×C₆₇	1	1	yes	characteristic
194	C₈	C₂×#41×C₆₇	1	1	yes	characteristic
195	C₂²	—	2	2	no	—
196	C₂²	C₄×D₈×C₆₇	1	2	yes	—
197	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
198	C₄	—	2	4	no	—
199	C₄	—	4	8	no	—
200	C₄	—	8	4	no	—
201	C₄	C₄×D₈×C₆₇	1	2	yes	—
202	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
203	C₄	C₂×#135×C₆₇	1	1	yes	characteristic
204	C₂	—	2	2	no	—
205	C₂	C₂×C₄×D₈×C₆₇	1	1	yes	characteristic
206	C₂	C₆₇×#3417	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂²×D₈
commutator	C₈	C₂³×C₄×C₆₇
Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
prime radical	C₂³×C₆₇	C₂×D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ #117×C₆₇ ◁ #405×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₇×#2176	C₂	semidirect
4. C₄×Q₃₂×C₆₇	C₂	semidirect
10. C₂×Q₃₂×C₆₇	C₄	semidirect
100. #33458	C₆₇	direct (N × H)
106. #2176	C₂×C₆₇	semidirect
107. C₄×Q₃₂	C₂×C₆₇	semidirect
111. C₆₇	#33458	direct (N × H)
114. C₂×Q₃₂	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×Q₃₂×C₆₇	2
C₆₇×#2950	2
C₆₇×#2176	2
C₄×Q₃₂×C₆₇	2
C₆₇×#2158	2
C₆₇×#2141	2
C₆₇×#2108	2
#33458	67

Subgroups — C67×#33458 #30712709

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33458 #30712709