Subgroups — C₆₇×#33950 #30713201

Order 17152.

448 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3383	C₂	1	2	yes	maximal
2	C₆₇×#2253	C₂	1	2	yes	maximal
3	C₆₇×#2252	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2229	C₂	1	4	yes	maximal
5	C₆₇×#2225	C₂	1	2	yes	maximal
6	Q₈×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₄×Q₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₄×Q₃₂×C₆₇	C₂	1	2	yes	maximal
9	#557×C₆₇	—	2	2	no	—
10	C₂×Q₃₂×C₆₇	C₂²	1	1	yes	characteristic
11	#500×C₆₇	C₂²	1	2	yes	—
12	#499×C₆₇	C₂²	1	4	yes	—
13	#479×C₆₇	—	2	4	no	—
14	#475×C₆₇	—	2	2	no	—
15	C₈×Q₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₄×Q₁₆×C₆₇	C₂²	1	2	yes	—
17	C₄×Q₁₆×C₆₇	C₂²	1	4	yes	—
18	#367×C₆₇	C₂²	1	4	yes	—
19	#366×C₆₇	C₂²	1	2	yes	—
20	#366×C₆₇	C₂²	1	2	yes	—
21	#366×C₆₇	C₂²	1	1	yes	characteristic
22	C₄⋊C₁₆×C₆₇	C₂²	1	1	yes	characteristic
23	C₄⋊C₁₆×C₆₇	C₂²	1	2	yes	—
24	#358×C₆₇	C₂²	1	4	yes	—
25	#358×C₆₇	C₂²	1	2	yes	—
26	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
27	C₄×C₁₆×C₆₇	C₂²	1	2	yes	—
28	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
29	C₄⋊Q₈×C₆₇	—	2	2	no	—
30	C₄⋊Q₈×C₆₇	—	4	2	no	—
31	#125×C₆₇	—	2	4	no	—
32	#125×C₆₇	—	4	4	no	—
33	C₄×Q₈×C₆₇	—	2	2	no	—
34	C₄×Q₈×C₆₇	—	2	4	no	—
35	C₄×Q₈×C₆₇	—	4	2	no	—
36	C₄×Q₈×C₆₇	D₄	1	1	yes	characteristic
37	Q₃₂×C₆₇	Q₈	1	4	yes	complemented
38	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
39	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
40	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
41	#107×C₆₇	—	2	2	no	—
42	#107×C₆₇	C₂³	1	2	yes	—
43	#107×C₆₇	C₂³	1	4	yes	—
44	#106×C₆₇	—	2	4	no	—
45	C₄⋊C₈×C₆₇	D₄	1	1	yes	characteristic
46	C₄⋊C₈×C₆₇	D₄	1	2	yes	—
47	#103×C₆₇	—	2	4	no	—
48	#103×C₆₇	—	2	2	no	—
49	C₄×C₈×C₆₇	C₂³	1	2	yes	—
50	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
51	C₂×Q₈×C₆₇	—	2	2	no	—
52	C₂×Q₈×C₆₇	—	4	2	no	—
53	C₂×Q₈×C₆₇	D₈	1	1	yes	characteristic
54	Q₁₆×C₆₇	—	2	2	no	—
55	Q₁₆×C₆₇	C₂×Q₈	1	4	yes	—
56	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
57	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
59	C₄⋊C₄×C₆₇	—	2	2	no	—
60	C₄⋊C₄×C₆₇	—	2	4	no	—
61	C₄⋊C₄×C₆₇	—	4	2	no	—
62	C₄⋊C₄×C₆₇	—	4	4	no	—
63	C₄⋊C₄×C₆₇	—	4	4	no	—
64	C₄⋊C₄×C₆₇	—	4	4	no	—
65	C₄⋊C₄×C₆₇	—	4	2	no	—
66	C₄⋊C₄×C₆₇	D₈	1	1	yes	characteristic
67	C₄⋊C₄×C₆₇	D₈	1	2	yes	—
68	C₄²×C₆₇	—	4	4	no	—
69	C₄²×C₆₇	—	4	2	no	—
70	C₄²×C₆₇	C₂×D₄	1	2	yes	—
71	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
72	C₁₆×C₆₇	—	2	1	no	—
73	C₁₆×C₆₇	—	2	2	no	—
74	C₁₆×C₆₇	C₂×Q₈	1	2	yes	—
75	Q₈×C₆₇	—	2	2	no	—
76	Q₈×C₆₇	—	2	4	no	—
77	Q₈×C₆₇	—	4	2	no	—
78	Q₈×C₆₇	—	8	4	no	—
79	C₂×C₄×C₆₇	—	4	2	no	—
80	C₂×C₄×C₆₇	—	4	2	no	—
81	C₂×C₄×C₆₇	—	4	4	no	—
82	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
83	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
84	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
85	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
86	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
87	C₈×C₆₇	—	2	2	no	—
88	C₈×C₆₇	—	2	1	no	—
89	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
90	C₈×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
91	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
92	C₄×C₆₇	—	2	2	no	—
93	C₄×C₆₇	—	2	2	no	—
94	C₄×C₆₇	—	2	1	no	—
95	C₄×C₆₇	—	4	4	no	—
96	C₄×C₆₇	—	8	4	no	—
97	C₄×C₆₇	—	8	2	no	—
98	C₄×C₆₇	#508	1	2	yes	—
99	C₄×C₆₇	D₄×Q₈	1	1	yes	characteristic
100	C₄×C₆₇	#578	1	1	yes	characteristic
101	#33950	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
102	C₂×C₆₇	Q₈×D₈	1	1	yes	characteristic
103	C₂×C₆₇	C₂×#508	1	1	yes	characteristic
104	C₂×C₆₇	#3417	1	1	yes	characteristic
105	#3383	C₂×C₆₇	1	2	yes	—
106	#2253	C₂×C₆₇	1	2	yes	—
107	#2252	C₂×C₆₇	1	1	yes	characteristic
108	#2229	C₂×C₆₇	1	4	yes	—
109	#2225	C₂×C₆₇	1	2	yes	—
110	Q₈×C₁₆	C₂×C₆₇	1	1	yes	characteristic
111	C₄×Q₃₂	C₂×C₆₇	1	1	yes	characteristic
112	C₄×Q₃₂	C₂×C₆₇	1	2	yes	—
113	C₆₇	#33950	1	1	yes	characteristic, direct product factor, Sylow
114	#557	—	2	2	no	—
115	C₂×Q₃₂	C₂²×C₆₇	1	1	yes	characteristic
116	#500	C₂²×C₆₇	1	2	yes	—
117	#499	C₂²×C₆₇	1	4	yes	—
118	#479	—	2	4	no	—
119	#475	—	2	2	no	—
120	C₈×Q₈	C₂²×C₆₇	1	1	yes	characteristic
121	C₄×Q₁₆	C₂²×C₆₇	1	2	yes	—
122	C₄×Q₁₆	C₂²×C₆₇	1	4	yes	—
123	#367	C₂²×C₆₇	1	4	yes	—
124	#366	C₂²×C₆₇	1	2	yes	—
125	#366	C₂²×C₆₇	1	1	yes	characteristic
126	#366	C₂²×C₆₇	1	2	yes	—
127	C₄⋊C₁₆	C₂²×C₆₇	1	1	yes	characteristic
128	C₄⋊C₁₆	C₂²×C₆₇	1	2	yes	—
129	#358	C₂²×C₆₇	1	4	yes	—
130	#358	C₂²×C₆₇	1	2	yes	—
131	C₄×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
132	C₄×C₁₆	C₂²×C₆₇	1	2	yes	—
133	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
134	C₄⋊Q₈	—	2	2	no	—
135	C₄⋊Q₈	—	4	2	no	—
136	#125	—	2	4	no	—
137	#125	—	4	4	no	—
138	C₄×Q₈	—	2	2	no	—
139	C₄×Q₈	—	2	4	no	—
140	C₄×Q₈	—	4	2	no	—
141	C₄×Q₈	D₄×C₆₇	1	1	yes	characteristic
142	Q₃₂	Q₈×C₆₇	1	4	yes	complemented
143	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
144	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
145	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
146	#107	—	2	2	no	—
147	#107	C₂³×C₆₇	1	4	yes	—
148	#107	C₂³×C₆₇	1	2	yes	—
149	#106	—	2	4	no	—
150	C₄⋊C₈	D₄×C₆₇	1	1	yes	characteristic
151	C₄⋊C₈	D₄×C₆₇	1	2	yes	—
152	#103	—	2	4	no	—
153	#103	—	2	2	no	—
154	C₄×C₈	C₂³×C₆₇	1	2	yes	—
155	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
156	C₂×Q₈	—	2	2	no	—
157	C₂×Q₈	—	4	2	no	—
158	C₂×Q₈	D₈×C₆₇	1	1	yes	characteristic
159	Q₁₆	—	2	2	no	—
160	Q₁₆	C₂×Q₈×C₆₇	1	4	yes	—
161	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
162	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
163	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
164	C₄⋊C₄	—	2	2	no	—
165	C₄⋊C₄	—	2	4	no	—
166	C₄⋊C₄	—	4	2	no	—
167	C₄⋊C₄	—	4	4	no	—
168	C₄⋊C₄	—	4	4	no	—
169	C₄⋊C₄	—	4	4	no	—
170	C₄⋊C₄	—	4	2	no	—
171	C₄⋊C₄	D₈×C₆₇	1	1	yes	characteristic
172	C₄⋊C₄	D₈×C₆₇	1	2	yes	—
173	C₄²	—	4	4	no	—
174	C₄²	—	4	2	no	—
175	C₄²	C₂×D₄×C₆₇	1	2	yes	—
176	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
177	C₁₆	—	2	1	no	—
178	C₁₆	—	2	2	no	—
179	C₁₆	C₂×Q₈×C₆₇	1	2	yes	—
180	Q₈	—	2	2	no	—
181	Q₈	—	2	4	no	—
182	Q₈	—	4	2	no	—
183	Q₈	—	8	4	no	—
184	C₂×C₄	—	4	2	no	—
185	C₂×C₄	—	4	2	no	—
186	C₂×C₄	—	4	4	no	—
187	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
188	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
189	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
190	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
191	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
192	C₈	—	2	2	no	—
193	C₈	—	2	1	no	—
194	C₈	C₂²×Q₈×C₆₇	1	1	yes	characteristic
195	C₈	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
196	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
197	C₄	—	2	2	no	—
198	C₄	—	2	2	no	—
199	C₄	—	2	1	no	—
200	C₄	—	4	4	no	—
201	C₄	—	8	4	no	—
202	C₄	—	8	2	no	—
203	C₄	#508×C₆₇	1	2	yes	—
204	C₄	D₄×Q₈×C₆₇	1	1	yes	characteristic
205	C₄	#578×C₆₇	1	1	yes	characteristic
206	C₂	Q₈×D₈×C₆₇	1	1	yes	characteristic
207	C₂	C₂×#508×C₆₇	1	1	yes	characteristic
208	C₂	C₆₇×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₆₇	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₄×Q₈×C₆₇ ◁ C₈×Q₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
37. Q₃₂×C₆₇	Q₈	semidirect
101. #33950	C₆₇	direct (N × H)
113. C₆₇	#33950	direct (N × H)
142. Q₃₂	Q₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3383	2
C₆₇×#2253	2
C₆₇×#2252	2
C₆₇×#2229	2
C₆₇×#2225	2
Q₈×C₁₆×C₆₇	2
C₄×Q₃₂×C₆₇	2
#33950	67

Subgroups — C67×#33950 #30713201

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33950 #30713201