Subgroups — C₆₇×#22734 #30701985

Order 17152.

876 conjugacy classes of subgroups.

The subgroup lattice has 234 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	2	yes	maximal
2	C₄²×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#1939	C₂	1	6	yes	maximal
4	C₆₇×#1831	C₂	1	3	yes	maximal
5	C₄×C₄⋊C₈×C₆₇	C₂	1	3	yes	maximal
6	C₂×C₄×Q₈×C₆₇	C₄	1	1	yes	characteristic
7	C₂×C₄×Q₈×C₆₇	C₄	1	1	yes	characteristic
8	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
9	C₈×Q₈×C₆₇	—	2	8	no	—
10	C₂×C₄⋊C₈×C₆₇	C₂²	1	6	yes	—
11	C₂×C₄⋊C₈×C₆₇	C₂²	1	6	yes	—
12	C₂×C₄×C₈×C₆₇	C₂²	1	6	yes	—
13	C₄×C₄⋊C₄×C₆₇	C₄	1	3	yes	—
14	C₄×C₄⋊C₄×C₆₇	C₄	1	3	yes	—
15	C₄×C₄⋊C₄×C₆₇	C₂²	1	3	yes	—
16	C₄³×C₆₇	C₂²	1	3	yes	—
17	#336×C₆₇	C₂²	1	1	yes	characteristic
18	#336×C₆₇	C₂²	1	3	yes	—
19	#336×C₆₇	C₂²	1	6	yes	—
20	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₆₇	C₂³	1	6	yes	—
22	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
23	C₄×Q₈×C₆₇	—	2	4	no	—
24	C₄×Q₈×C₆₇	C₈	1	8	yes	complemented
25	C₄×Q₈×C₆₇	D₄	1	4	yes	—
26	C₄×Q₈×C₆₇	Q₈	1	4	yes	—
27	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	3	yes	—
28	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	3	yes	—
29	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	3	yes	—
30	C₂×C₄²×C₆₇	C₂×C₄	1	3	yes	—
31	C₂×C₄²×C₆₇	C₂×C₄	1	3	yes	—
32	C₂×C₄²×C₆₇	C₂×C₄	1	3	yes	—
33	C₂×C₄²×C₆₇	C₂×C₄	1	3	yes	—
34	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
36	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
37	C₄⋊C₈×C₆₇	—	2	12	no	—
38	C₄⋊C₈×C₆₇	—	2	12	no	—
39	C₄×C₈×C₆₇	—	2	12	no	—
40	#95×C₆₇	C₂×C₄	1	6	yes	—
41	C₂×Q₈×C₆₇	C₄⋊C₄	1	4	yes	—
42	C₂×Q₈×C₆₇	C₂×C₈	1	4	yes	—
43	C₂×Q₈×C₆₇	M₁₆	1	4	yes	—
44	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
45	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
46	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
48	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
50	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
51	C₂×C₈×C₆₇	—	2	2	no	—
52	C₂×C₈×C₆₇	—	2	2	no	—
53	C₂×C₈×C₆₇	—	2	6	no	—
54	C₂×C₈×C₆₇	—	2	6	no	—
55	C₂×C₈×C₆₇	—	2	6	no	—
56	C₂×C₈×C₆₇	#41	1	4	yes	—
57	C₄⋊C₄×C₆₇	—	2	6	no	—
58	C₄⋊C₄×C₆₇	C₄⋊C₄	1	12	yes	—
59	C₄⋊C₄×C₆₇	C₂×C₈	1	12	yes	—
60	C₄²×C₆₇	—	2	6	no	—
61	C₄²×C₆₇	—	2	6	no	—
62	C₄²×C₆₇	—	2	6	no	—
63	C₄²×C₆₇	—	2	2	no	—
64	C₄²×C₆₇	—	4	3	no	—
65	C₄²×C₆₇	C₂×C₈	1	12	yes	—
66	C₄²×C₆₇	C₂×D₄	1	6	yes	—
67	C₄²×C₆₇	C₂×Q₈	1	6	yes	—
68	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
69	Q₈×C₆₇	C₄⋊C₈	1	16	yes	complemented
70	C₂×C₄×C₆₇	—	2	3	no	—
71	C₂×C₄×C₆₇	—	2	1	no	—
72	C₂×C₄×C₆₇	—	2	1	no	—
73	C₂×C₄×C₆₇	—	2	1	no	—
74	C₂×C₄×C₆₇	—	2	3	no	—
75	C₂×C₄×C₆₇	—	2	3	no	—
76	C₂×C₄×C₆₇	—	2	1	no	—
77	C₂×C₄×C₆₇	—	2	3	no	—
78	C₂×C₄×C₆₇	—	2	3	no	—
79	C₂×C₄×C₆₇	—	2	3	no	—
80	C₂×C₄×C₆₇	—	2	3	no	—
81	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	6	yes	—
82	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	6	yes	—
83	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
84	C₂×C₄×C₆₇	C₂²×C₈	1	6	yes	—
85	C₂×C₄×C₆₇	C₂²×C₈	1	6	yes	—
86	C₂×C₄×C₆₇	C₂×M₁₆	1	6	yes	—
87	C₂×C₄×C₆₇	#131	1	2	yes	—
88	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
89	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
90	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
91	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
92	C₈×C₆₇	—	2	4	no	—
93	C₈×C₆₇	—	4	6	no	—
94	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
95	C₂²×C₆₇	C₄×#41	1	1	yes	characteristic
96	C₂²×C₆₇	#519	1	1	yes	characteristic
97	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
98	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
99	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
100	C₂²×C₆₇	#568	1	1	yes	characteristic
101	C₄×C₆₇	—	2	6	no	—
102	C₄×C₆₇	—	2	6	no	—
103	C₄×C₆₇	—	2	2	no	—
104	C₄×C₆₇	—	2	2	no	—
105	C₄×C₆₇	—	4	3	no	—
106	C₄×C₆₇	C₂×C₄⋊C₈	1	12	yes	—
107	C₄×C₆₇	#548	1	2	yes	—
108	C₄×C₆₇	#557	1	2	yes	—
109	#22734	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₂×C₆₇	#2351	1	1	yes	characteristic
111	C₂×C₆₇	C₂²×C₄⋊C₈	1	1	yes	characteristic
112	C₂×C₆₇	#2906	1	1	yes	characteristic
113	C₂×C₆₇	C₈×#41	1	1	yes	characteristic
114	C₂×C₆₇	#2971	1	1	yes	characteristic
115	C₂×C₆₇	#2982	1	1	yes	characteristic
116	C₂×C₆₇	#2984	1	1	yes	characteristic
117	C₂×C₈×Q₈	C₂×C₆₇	1	2	yes	—
118	C₄²×Q₈	C₂×C₆₇	1	1	yes	characteristic
119	#1939	C₂×C₆₇	1	6	yes	—
120	#1831	C₂×C₆₇	1	3	yes	—
121	C₄×C₄⋊C₈	C₂×C₆₇	1	3	yes	—
122	C₆₇	#22734	1	1	yes	characteristic, direct product factor, Sylow
123	C₂×C₄×Q₈	C₄×C₆₇	1	1	yes	characteristic
124	C₂×C₄×Q₈	C₄×C₆₇	1	1	yes	characteristic
125	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
126	C₈×Q₈	—	2	8	no	—
127	C₂×C₄⋊C₈	C₂²×C₆₇	1	6	yes	—
128	C₂×C₄⋊C₈	C₂²×C₆₇	1	6	yes	—
129	C₂×C₄×C₈	C₂²×C₆₇	1	6	yes	—
130	C₄×C₄⋊C₄	C₄×C₆₇	1	3	yes	—
131	C₄×C₄⋊C₄	C₄×C₆₇	1	3	yes	—
132	C₄×C₄⋊C₄	C₂²×C₆₇	1	3	yes	—
133	C₄³	C₂²×C₆₇	1	3	yes	—
134	#336	C₂²×C₆₇	1	3	yes	—
135	#336	C₂²×C₆₇	1	6	yes	—
136	#336	C₂²×C₆₇	1	1	yes	characteristic
137	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
138	C₂²×C₈	C₂³×C₆₇	1	6	yes	—
139	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
140	C₄×Q₈	—	2	4	no	—
141	C₄×Q₈	C₈×C₆₇	1	8	yes	complemented
142	C₄×Q₈	D₄×C₆₇	1	4	yes	—
143	C₄×Q₈	Q₈×C₆₇	1	4	yes	—
144	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	3	yes	—
145	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	3	yes	—
146	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	3	yes	—
147	C₂×C₄²	C₂×C₄×C₆₇	1	3	yes	—
148	C₂×C₄²	C₂×C₄×C₆₇	1	3	yes	—
149	C₂×C₄²	C₂×C₄×C₆₇	1	3	yes	—
150	C₂×C₄²	C₂×C₄×C₆₇	1	3	yes	—
151	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
152	C₂×C₄²	C₂³×C₆₇	1	3	yes	—
153	C₂×C₄²	C₂³×C₆₇	1	3	yes	—
154	C₄⋊C₈	—	2	12	no	—
155	C₄⋊C₈	—	2	12	no	—
156	C₄×C₈	—	2	12	no	—
157	#95	C₂×C₄×C₆₇	1	6	yes	—
158	C₂×Q₈	C₄⋊C₄×C₆₇	1	4	yes	—
159	C₂×Q₈	C₂×C₈×C₆₇	1	4	yes	—
160	C₂×Q₈	M₁₆×C₆₇	1	4	yes	—
161	C₂²×C₄	C₂²×C₄×C₆₇	1	3	yes	—
162	C₂²×C₄	C₂²×C₄×C₆₇	1	3	yes	—
163	C₂²×C₄	C₂²×C₄×C₆₇	1	3	yes	—
164	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
165	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
166	C₂²×C₄	C₂²×C₄×C₆₇	1	3	yes	—
167	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	6	no	—
171	C₂×C₈	—	2	6	no	—
172	C₂×C₈	—	2	6	no	—
173	C₂×C₈	#41×C₆₇	1	4	yes	—
174	C₄⋊C₄	—	2	6	no	—
175	C₄⋊C₄	C₄⋊C₄×C₆₇	1	12	yes	—
176	C₄⋊C₄	C₂×C₈×C₆₇	1	12	yes	—
177	C₄²	—	2	6	no	—
178	C₄²	—	2	6	no	—
179	C₄²	—	2	6	no	—
180	C₄²	—	2	2	no	—
181	C₄²	—	4	3	no	—
182	C₄²	C₂×C₈×C₆₇	1	12	yes	—
183	C₄²	C₂×D₄×C₆₇	1	6	yes	—
184	C₄²	C₂×Q₈×C₆₇	1	6	yes	—
185	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
186	Q₈	C₄⋊C₈×C₆₇	1	16	yes	complemented
187	C₂×C₄	—	2	3	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	3	no	—
192	C₂×C₄	—	2	3	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	3	no	—
195	C₂×C₄	—	2	3	no	—
196	C₂×C₄	—	2	3	no	—
197	C₂×C₄	—	2	3	no	—
198	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	6	yes	—
199	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	6	yes	—
200	C₂×C₄	C₂²×C₈×C₆₇	1	2	yes	—
201	C₂×C₄	C₂²×C₈×C₆₇	1	6	yes	—
202	C₂×C₄	C₂²×C₈×C₆₇	1	6	yes	—
203	C₂×C₄	C₂×M₁₆×C₆₇	1	6	yes	—
204	C₂×C₄	#131×C₆₇	1	2	yes	—
205	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
206	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
207	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
208	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
209	C₈	—	2	4	no	—
210	C₈	—	4	6	no	—
211	C₂²	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
212	C₂²	C₄×#41×C₆₇	1	1	yes	characteristic
213	C₂²	#519×C₆₇	1	1	yes	characteristic
214	C₂²	C₂³×C₈×C₆₇	1	1	yes	characteristic
215	C₂²	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
216	C₂²	C₂×#131×C₆₇	1	1	yes	characteristic
217	C₂²	#568×C₆₇	1	1	yes	characteristic
218	C₄	—	2	6	no	—
219	C₄	—	2	6	no	—
220	C₄	—	2	2	no	—
221	C₄	—	2	2	no	—
222	C₄	—	4	3	no	—
223	C₄	C₂×C₄⋊C₈×C₆₇	1	12	yes	—
224	C₄	#548×C₆₇	1	2	yes	—
225	C₄	#557×C₆₇	1	2	yes	—
226	C₂	C₆₇×#2351	1	1	yes	characteristic
227	C₂	C₂²×C₄⋊C₈×C₆₇	1	1	yes	characteristic
228	C₂	C₆₇×#2906	1	1	yes	characteristic
229	C₂	C₈×#41×C₆₇	1	1	yes	characteristic
230	C₂	C₆₇×#2971	1	1	yes	characteristic
231	C₂	C₆₇×#2982	1	1	yes	characteristic
232	C₂	C₆₇×#2984	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
24. C₄×Q₈×C₆₇	C₈	semidirect
69. Q₈×C₆₇	C₄⋊C₈	semidirect
109. #22734	C₆₇	direct (N × H)
122. C₆₇	#22734	direct (N × H)
141. C₄×Q₈	C₈×C₆₇	semidirect
186. Q₈	C₄⋊C₈×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×Q₈×C₆₇	2
C₄²×Q₈×C₆₇	2
C₆₇×#1939	2
C₆₇×#1831	2
C₄×C₄⋊C₈×C₆₇	2
#22734	67

Subgroups — C67×#22734 #30701985

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22734 #30701985