Subgroups — C₆₇×#34642 #30713893

Order 17152.

2110 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	C₂×#533×C₆₇	C₂	1	2	yes	maximal
2	C₆₇×#2660	C₂	1	8	yes	maximal, complemented
3	C₆₇×#2659	C₂	1	8	yes	maximal
4	C₆₇×#2399	C₂	1	8	yes	maximal, complemented
5	C₆₇×#2285	C₂	1	4	yes	maximal
6	C₂×#376×C₆₇	C₂	1	1	yes	maximal, characteristic
7	#533×C₆₇	—	2	16	no	—
8	C₂×C₄⋊Q₈×C₆₇	C₂²	1	4	yes	—
9	C₂×#125×C₆₇	C₂²	1	4	yes	—
10	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	8	yes	—
11	C₂×C₄×Q₈×C₆₇	C₂²	1	8	yes	—
12	C₂×#117×C₆₇	C₂²	1	4	yes	—
13	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
15	#398×C₆₇	C₂²	1	32	yes	—
16	#395×C₆₇	C₂²	1	32	yes	—
17	#384×C₆₇	C₂²	1	8	yes	—
18	#382×C₆₇	C₂²	1	8	yes	—
19	#380×C₆₇	C₂²	1	16	yes	—
20	C₄×C₄⋊C₄×C₆₇	C₂²	1	16	yes	—
21	#376×C₆₇	C₂²	1	4	yes	—
22	#376×C₆₇	C₂²	1	4	yes	—
23	C₂×#95×C₆₇	C₂²	1	4	yes	—
24	C₂²×Q₈×C₆₇	C₂³	1	8	yes	—
25	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
26	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
27	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
28	C₄⋊Q₈×C₆₇	—	2	16	no	—
29	#125×C₆₇	—	2	16	no	—
30	C₂²⋊Q₈×C₆₇	—	2	32	no	—
31	C₄×Q₈×C₆₇	—	2	32	no	—
32	#117×C₆₇	—	2	16	no	—
33	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
34	C₂×C₄⋊C₄×C₆₇	C₂³	1	16	yes	—
35	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
36	C₂×C₄⋊C₄×C₆₇	C₂³	1	32	yes	—
37	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
38	C₂×C₄²×C₆₇	D₄	1	8	yes	—
39	C₂×C₄²×C₆₇	Q₈	1	16	yes	complemented
40	C₂×C₄²×C₆₇	C₂³	1	8	yes	—
41	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
42	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
43	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
44	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
45	#95×C₆₇	C₂³	1	16	yes	—
46	#95×C₆₇	C₂³	1	32	yes	—
47	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
48	C₂×Q₈×C₆₇	—	2	16	no	—
49	C₂×Q₈×C₆₇	—	4	16	no	—
50	C₂²×C₄×C₆₇	—	4	8	no	—
51	C₂²×C₄×C₆₇	—	4	4	no	—
52	C₂²×C₄×C₆₇	C₂×D₄	1	8	yes	—
53	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
54	C₂²×C₄×C₆₇	C₂×Q₈	1	16	yes	—
55	C₂²×C₄×C₆₇	C₂×Q₈	1	8	yes	—
56	C₂²×C₄×C₆₇	C₂⁴	1	16	yes	—
57	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
58	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
59	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
60	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
61	C₄⋊C₄×C₆₇	—	2	16	no	—
62	C₄⋊C₄×C₆₇	—	2	32	no	—
63	C₄⋊C₄×C₆₇	—	2	16	no	—
64	C₄⋊C₄×C₆₇	—	4	32	no	—
65	C₂²⋊C₄×C₆₇	—	2	16	no	—
66	C₄²×C₆₇	—	2	16	no	—
67	C₄²×C₆₇	C₂×D₄	1	8	yes	—
68	C₄²×C₆₇	C₂×D₄	1	8	yes	—
69	C₄²×C₆₇	C₂×Q₈	1	16	yes	—
70	C₄²×C₆₇	C₂×Q₈	1	16	yes	—
71	C₂³×C₆₇	—	2	4	no	—
72	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
73	C₂³×C₆₇	C₂²×Q₈	1	4	yes	—
74	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
75	Q₈×C₆₇	—	4	32	no	—
76	C₂×C₄×C₆₇	—	2	16	no	—
77	C₂×C₄×C₆₇	—	2	8	no	—
78	C₂×C₄×C₆₇	—	2	16	no	—
79	C₂×C₄×C₆₇	—	2	32	no	—
80	C₂×C₄×C₆₇	—	4	8	no	—
81	C₂×C₄×C₆₇	—	4	16	no	—
82	C₂×C₄×C₆₇	C₄⋊Q₈	1	16	yes	—
83	C₂×C₄×C₆₇	C₂²×D₄	1	8	yes	—
84	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
85	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
86	C₂×C₄×C₆₇	C₂²×Q₈	1	16	yes	—
87	C₂×C₄×C₆₇	C₂²×Q₈	1	4	yes	—
88	C₂×C₄×C₆₇	C₂²×Q₈	1	4	yes	—
89	C₂²×C₆₇	—	2	4	no	—
90	C₂²×C₆₇	—	2	8	no	—
91	C₂²×C₆₇	C₂×C₄⋊Q₈	1	4	yes	—
92	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
93	C₂²×C₆₇	C₂³×Q₈	1	2	yes	—
94	C₂²×C₆₇	#585	1	4	yes	—
95	C₄×C₆₇	—	2	8	no	—
96	C₄×C₆₇	—	2	16	no	—
97	C₄×C₆₇	—	4	16	no	—
98	C₄×C₆₇	C₂×C₄⋊Q₈	1	4	yes	—
99	C₄×C₆₇	C₂×C₄⋊Q₈	1	4	yes	—
100	#34642	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
101	C₂×C₆₇	—	2	4	no	—
102	C₂×C₆₇	C₂²×C₄⋊Q₈	1	1	yes	characteristic
103	C₂×C₆₇	#3449	1	2	yes	—
104	C₂×C₆₇	#3458	1	4	yes	—
105	C₂×#533	C₂×C₆₇	1	2	yes	—
106	#2660	C₂×C₆₇	1	8	yes	complemented
107	#2659	C₂×C₆₇	1	8	yes	—
108	#2399	C₂×C₆₇	1	8	yes	complemented
109	#2285	C₂×C₆₇	1	4	yes	—
110	C₂×#376	C₂×C₆₇	1	1	yes	characteristic
111	C₆₇	#34642	1	1	yes	characteristic, direct product factor, Sylow
112	#533	—	2	16	no	—
113	C₂×C₄⋊Q₈	C₂²×C₆₇	1	4	yes	—
114	C₂×#125	C₂²×C₆₇	1	4	yes	—
115	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	8	yes	—
116	C₂×C₄×Q₈	C₂²×C₆₇	1	8	yes	—
117	C₂×#117	C₂²×C₆₇	1	4	yes	—
118	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
119	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
120	#398	C₂²×C₆₇	1	32	yes	—
121	#395	C₂²×C₆₇	1	32	yes	—
122	#384	C₂²×C₆₇	1	8	yes	—
123	#382	C₂²×C₆₇	1	8	yes	—
124	#380	C₂²×C₆₇	1	16	yes	—
125	C₄×C₄⋊C₄	C₂²×C₆₇	1	16	yes	—
126	#376	C₂²×C₆₇	1	4	yes	—
127	#376	C₂²×C₆₇	1	4	yes	—
128	C₂×#95	C₂²×C₆₇	1	4	yes	—
129	C₂²×Q₈	C₂³×C₆₇	1	8	yes	—
130	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
131	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
132	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
133	C₄⋊Q₈	—	2	16	no	—
134	#125	—	2	16	no	—
135	C₂²⋊Q₈	—	2	32	no	—
136	C₄×Q₈	—	2	32	no	—
137	#117	—	2	16	no	—
138	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
139	C₂×C₄⋊C₄	C₂³×C₆₇	1	16	yes	—
140	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
141	C₂×C₄⋊C₄	C₂³×C₆₇	1	32	yes	—
142	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
143	C₂×C₄²	D₄×C₆₇	1	8	yes	—
144	C₂×C₄²	Q₈×C₆₇	1	16	yes	complemented
145	C₂×C₄²	C₂³×C₆₇	1	8	yes	—
146	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
147	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
148	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
149	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
150	#95	C₂³×C₆₇	1	16	yes	—
151	#95	C₂³×C₆₇	1	32	yes	—
152	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
153	C₂×Q₈	—	2	16	no	—
154	C₂×Q₈	—	4	16	no	—
155	C₂²×C₄	—	4	8	no	—
156	C₂²×C₄	—	4	4	no	—
157	C₂²×C₄	C₂×D₄×C₆₇	1	8	yes	—
158	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
159	C₂²×C₄	C₂×Q₈×C₆₇	1	16	yes	—
160	C₂²×C₄	C₂×Q₈×C₆₇	1	8	yes	—
161	C₂²×C₄	C₂⁴×C₆₇	1	16	yes	—
162	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
163	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
164	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
165	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
166	C₄⋊C₄	—	2	16	no	—
167	C₄⋊C₄	—	2	32	no	—
168	C₄⋊C₄	—	2	16	no	—
169	C₄⋊C₄	—	4	32	no	—
170	C₂²⋊C₄	—	2	16	no	—
171	C₄²	—	2	16	no	—
172	C₄²	C₂×D₄×C₆₇	1	8	yes	—
173	C₄²	C₂×D₄×C₆₇	1	8	yes	—
174	C₄²	C₂×Q₈×C₆₇	1	16	yes	—
175	C₄²	C₂×Q₈×C₆₇	1	16	yes	—
176	C₂³	—	2	4	no	—
177	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
178	C₂³	C₂²×Q₈×C₆₇	1	4	yes	—
179	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
180	Q₈	—	4	32	no	—
181	C₂×C₄	—	2	16	no	—
182	C₂×C₄	—	2	8	no	—
183	C₂×C₄	—	2	16	no	—
184	C₂×C₄	—	2	32	no	—
185	C₂×C₄	—	4	8	no	—
186	C₂×C₄	—	4	16	no	—
187	C₂×C₄	C₄⋊Q₈×C₆₇	1	16	yes	—
188	C₂×C₄	C₂²×D₄×C₆₇	1	8	yes	—
189	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
190	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
191	C₂×C₄	C₂²×Q₈×C₆₇	1	16	yes	—
192	C₂×C₄	C₂²×Q₈×C₆₇	1	4	yes	—
193	C₂×C₄	C₂²×Q₈×C₆₇	1	4	yes	—
194	C₂²	—	2	4	no	—
195	C₂²	—	2	8	no	—
196	C₂²	C₂×C₄⋊Q₈×C₆₇	1	4	yes	—
197	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
198	C₂²	C₂³×Q₈×C₆₇	1	2	yes	—
199	C₂²	#585×C₆₇	1	4	yes	—
200	C₄	—	2	8	no	—
201	C₄	—	2	16	no	—
202	C₄	—	4	16	no	—
203	C₄	C₂×C₄⋊Q₈×C₆₇	1	4	yes	—
204	C₄	C₂×C₄⋊Q₈×C₆₇	1	4	yes	—
205	C₂	—	2	4	no	—
206	C₂	C₂²×C₄⋊Q₈×C₆₇	1	1	yes	characteristic
207	C₂	C₆₇×#3449	1	2	yes	—
208	C₂	C₆₇×#3458	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
socle	C₂³×C₆₇	C₂⁵
prime radical	C₂⁴×C₆₇	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2660	C₂	semidirect
4. C₆₇×#2399	C₂	semidirect
39. C₂×C₄²×C₆₇	Q₈	semidirect
100. #34642	C₆₇	direct (N × H)
106. #2660	C₂×C₆₇	semidirect
108. #2399	C₂×C₆₇	semidirect
111. C₆₇	#34642	direct (N × H)
144. C₂×C₄²	Q₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#533×C₆₇	2
C₆₇×#2660	2
C₆₇×#2659	2
C₆₇×#2399	2
C₆₇×#2285	2
C₂×#376×C₆₇	2
#34642	67

Subgroups — C67×#34642 #30713893

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#34642 #30713893