Subgroups — C₂×Q₈²×D₆₇ #31762896

Order 17152.

7846 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×Dic₁₃₄	C₂	1	6	yes	maximal, complemented
2	C₂×#16864126	C₂	1	9	yes	maximal, complemented
3	Q₈²×D₆₇	C₂	1	32	yes	maximal, direct product factor
4	C₂×C₄⋊Q₈×D₆₇	C₂	1	9	yes	maximal
5	C₂×C₄×Q₈×D₆₇	C₂	1	6	yes	maximal
6	C₂×Q₈²×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×Q₈×D₆₇	C₂²	1	6	yes	—
8	C₂²×Q₈×D₆₇	C₂²	1	2	yes	—
9	Q₈×Dic₁₃₄	C₂²	1	96	yes	complemented
10	#16864126	C₂²	1	144	yes	complemented
11	C₂×#14524043	C₂²	1	9	yes	—
12	C₂×#14524042	C₂²	1	36	yes	—
13	C₂×#14524041	C₂²	1	18	yes	—
14	C₄⋊Q₈×D₆₇	C₂²	1	144	yes	—
15	C₂×Q₈×Dic₆₇	C₂²	1	6	yes	—
16	C₂×C₄×Dic₁₃₄	C₂²	1	18	yes	—
17	C₂×#14524008	C₂²	1	18	yes	—
18	C₄×Q₈×D₆₇	C₂²	1	96	yes	—
19	C₂×C₄⋊C₄×D₆₇	C₂²	1	18	yes	—
20	C₂×C₄²×D₆₇	C₂²	1	9	yes	—
21	Q₈²×C₆₇	C₂²	1	16	yes	complemented
22	C₂×C₄⋊Q₈×C₆₇	C₂²	1	9	yes	—
23	C₂×C₄×Q₈×C₆₇	C₂²	1	6	yes	—
24	C₂²×Dic₁₃₄	C₂³	1	18	yes	—
25	C₂²×Dic₁₃₄	C₂³	1	6	yes	—
26	C₂×Q₈×D₆₇	—	4	24	no	—
27	C₂×Q₈×D₆₇	Q₈	1	32	yes	direct product factor
28	C₂×Q₈×D₆₇	C₂³	1	48	yes	—
29	C₂×Q₈×D₆₇	C₂³	1	16	yes	—
30	C₂²×C₄×D₆₇	C₂³	1	9	yes	—
31	C₂²×C₄×D₆₇	C₂³	1	6	yes	—
32	#14524043	C₂³	1	72	yes	—
33	#14524042	C₂³	1	288	yes	—
34	#14524041	C₂³	1	144	yes	—
35	Q₈×Dic₆₇	C₂³	1	48	yes	—
36	C₄×Dic₁₃₄	C₂³	1	144	yes	—
37	#14524008	C₂³	1	144	yes	—
38	C₂×C₄⋊Dic₆₇	C₂³	1	18	yes	—
39	C₂×Dic₆₇⋊C₄	C₂³	1	36	yes	—
40	C₄⋊C₄×D₆₇	C₂³	1	144	yes	—
41	C₂×C₄×Dic₆₇	C₂³	1	9	yes	—
42	C₂×C₄×Dic₆₇	C₂³	1	18	yes	—
43	C₄²×D₆₇	C₂³	1	72	yes	—
44	C₂²×Q₈×C₆₇	C₂³	1	6	yes	—
45	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
46	C₄⋊Q₈×C₆₇	C₂³	1	72	yes	—
47	C₄×Q₈×C₆₇	C₂³	1	48	yes	—
48	C₂×C₄⋊C₄×C₆₇	C₂³	1	18	yes	—
49	C₂×C₄²×C₆₇	C₂³	1	9	yes	—
50	C₂³×D₆₇	C₂⁴	1	1	yes	characteristic
51	C₂×Dic₁₃₄	—	4	36	no	—
52	C₂×Dic₁₃₄	C₂×Q₈	1	48	yes	complemented
53	C₂×Dic₁₃₄	C₂⁴	1	72	yes	—
54	C₂×Dic₁₃₄	C₂⁴	1	24	yes	—
55	Q₈×D₆₇	—	4	96	no	—
56	Q₈×D₆₇	C₂×Q₈	1	128	yes	direct product factor
57	C₂²×Dic₆₇	C₂⁴	1	9	yes	—
58	C₂²×Dic₆₇	C₂⁴	1	6	yes	—
59	C₂×C₄×D₆₇	—	2	36	no	—
60	C₂×C₄×D₆₇	C₂×Q₈	1	48	yes	—
61	C₂×C₄×D₆₇	C₂⁴	1	36	yes	—
62	C₂×C₄×D₆₇	C₂⁴	1	24	yes	—
63	C₄⋊Dic₆₇	C₂⁴	1	72	yes	—
64	Dic₆₇⋊C₄	C₂⁴	1	144	yes	—
65	C₄×Dic₆₇	C₂⁴	1	36	yes	—
66	C₄×Dic₆₇	C₂⁴	1	72	yes	—
67	C₂×Q₈×C₆₇	—	4	12	no	—
68	C₂×Q₈×C₆₇	C₂×Q₈	1	16	yes	complemented
69	C₂×Q₈×C₆₇	C₂⁴	1	24	yes	—
70	C₂×Q₈×C₆₇	C₂⁴	1	8	yes	—
71	C₂²×C₄×C₆₇	C₂⁴	1	9	yes	—
72	C₂²×C₄×C₆₇	C₂⁴	1	6	yes	—
73	C₄⋊C₄×C₆₇	C₂⁴	1	72	yes	—
74	C₄²×C₆₇	C₂⁴	1	36	yes	—
75	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
76	C₂²×D₆₇	C₂²×Q₈	1	8	yes	—
77	C₂²×D₆₇	H(5,𝔽₂)	1	4	yes	—
78	C₂²×D₆₇	C₂⁵	1	2	yes	—
79	Q₈×C₆₇	—	4	24	no	—
80	Q₈×C₆₇	C₂²×Q₈	1	32	yes	complemented
81	C₂×C₄×C₆₇	—	2	18	no	—
82	C₂×C₄×C₆₇	C₂²×Q₈	1	24	yes	—
83	C₂×C₄×C₆₇	C₂⁵	1	18	yes	—
84	C₂×C₄×C₆₇	C₂⁵	1	12	yes	—
85	C₂×Dic₆₇	—	2	18	no	—
86	C₂×Dic₆₇	C₂²×Q₈	1	24	yes	—
87	C₂×Dic₆₇	C₂⁵	1	18	yes	—
88	C₂×Dic₆₇	C₂⁵	1	12	yes	—
89	C₄×D₆₇	—	2	72	no	—
90	C₄×D₆₇	C₂²×Q₈	1	96	yes	—
91	Dic₁₃₄	—	4	72	no	—
92	Dic₁₃₄	C₂²×Q₈	1	96	yes	complemented
93	C₂²×C₆₇	C₂³×Q₈	1	4	yes	—
94	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
95	C₂²×C₆₇	C₂⁶	1	1	yes	characteristic
96	C₂×D₆₇	Q₈²	1	16	yes	direct product factor
97	C₂×D₆₇	C₂³×Q₈	1	8	yes	—
98	C₂×D₆₇	C₂×H(5,𝔽₂)	1	4	yes	—
99	C₄×C₆₇	—	2	18	no	—
100	C₄×C₆₇	C₂³×Q₈	1	24	yes	—
101	Dic₆₇	—	2	18	no	—
102	Dic₆₇	C₂³×Q₈	1	24	yes	—
103	C₂²×Q₈²	—	67	1	no	maximal, Sylow
104	C₂×C₆₇	C₂×Q₈²	1	4	yes	complemented
105	C₂×C₆₇	C₂⁴×Q₈	1	2	yes	—
106	C₂×C₆₇	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
107	D₆₇	C₂×Q₈²	1	8	yes	direct product factor
108	C₂×Q₈²	—	67	32	no	—
109	C₂×Q₈²	—	67	9	no	—
110	C₂×Q₈²	—	67	6	no	—
111	C₂×Q₈²	D₆₇	1	1	yes	characteristic, direct product factor
112	C₂²×C₄⋊Q₈	—	67	9	no	—
113	C₂²×C₄×Q₈	—	67	6	no	—
114	C₆₇	C₂²×Q₈²	1	1	yes	characteristic, complemented, Sylow
115	C₂³×Q₈	—	67	6	no	—
116	C₂³×Q₈	—	67	2	no	—
117	Q₈²	—	67	144	no	—
118	Q₈²	—	67	96	no	—
119	Q₈²	C₂×D₆₇	1	16	yes	direct product factor
120	C₂×C₄⋊Q₈	—	67	18	no	—
121	C₂×C₄⋊Q₈	—	67	144	no	—
122	C₂×C₄⋊Q₈	—	67	36	no	—
123	C₂×C₄⋊Q₈	—	67	9	no	—
124	C₂×C₄⋊Q₈	C₂×D₆₇	1	9	yes	—
125	C₂×C₄×Q₈	—	67	6	no	—
126	C₂×C₄×Q₈	—	67	96	no	—
127	C₂×C₄×Q₈	—	67	18	no	—
128	C₂×C₄×Q₈	—	67	18	no	—
129	C₂×C₄×Q₈	C₂×D₆₇	1	6	yes	—
130	C₂²×C₄⋊C₄	—	67	18	no	—
131	C₂²×C₄²	—	67	9	no	—
132	C₂²×Q₈	—	268	24	no	—
133	C₂²×Q₈	—	67	48	no	—
134	C₂²×Q₈	—	67	32	no	—
135	C₂²×Q₈	—	67	16	no	—
136	C₂²×Q₈	—	67	18	no	—
137	C₂²×Q₈	—	67	6	no	—
138	C₂²×Q₈	C₂²×D₆₇	1	6	yes	—
139	C₂²×Q₈	C₂²×D₆₇	1	2	yes	—
140	C₂³×C₄	—	67	6	no	—
141	C₂³×C₄	—	67	9	no	—
142	C₄⋊Q₈	—	67	144	no	—
143	C₄⋊Q₈	—	67	288	no	—
144	C₄⋊Q₈	—	67	72	no	—
145	C₄⋊Q₈	C₂²×D₆₇	1	72	yes	—
146	C₄×Q₈	—	67	48	no	—
147	C₄×Q₈	—	67	144	no	—
148	C₄×Q₈	—	67	144	no	—
149	C₄×Q₈	C₂²×D₆₇	1	48	yes	—
150	C₂×C₄⋊C₄	—	67	36	no	—
151	C₂×C₄⋊C₄	—	67	18	no	—
152	C₂×C₄⋊C₄	—	67	144	no	—
153	C₂×C₄⋊C₄	C₂²×D₆₇	1	18	yes	—
154	C₂×C₄²	—	67	9	no	—
155	C₂×C₄²	—	67	18	no	—
156	C₂×C₄²	—	67	72	no	—
157	C₂×C₄²	C₂²×D₆₇	1	9	yes	—
158	C₂⁴	—	67	1	no	—
159	C₂×Q₈	—	4	12	no	—
160	C₂×Q₈	—	268	36	no	—
161	C₂×Q₈	—	268	96	no	—
162	C₂×Q₈	—	67	128	no	—
163	C₂×Q₈	—	67	48	no	—
164	C₂×Q₈	—	67	72	no	—
165	C₂×Q₈	—	67	24	no	—
166	C₂×Q₈	Q₈×D₆₇	1	16	yes	direct product factor
167	C₂×Q₈	C₂³×D₆₇	1	24	yes	—
168	C₂×Q₈	C₂³×D₆₇	1	8	yes	—
169	C₂²×C₄	—	67	6	no	—
170	C₂²×C₄	—	67	48	no	—
171	C₂²×C₄	—	67	24	no	—
172	C₂²×C₄	—	67	9	no	—
173	C₂²×C₄	—	67	36	no	—
174	C₂²×C₄	—	134	36	no	—
175	C₂²×C₄	C₂³×D₆₇	1	9	yes	—
176	C₂²×C₄	C₂³×D₆₇	1	6	yes	—
177	C₄⋊C₄	—	67	144	no	—
178	C₄⋊C₄	—	67	72	no	—
179	C₄⋊C₄	C₂³×D₆₇	1	72	yes	—
180	C₄²	—	67	36	no	—
181	C₄²	—	67	72	no	—
182	C₄²	C₂³×D₆₇	1	36	yes	—
183	C₂³	—	67	8	no	—
184	C₂³	—	67	4	no	—
185	C₂³	—	67	2	no	—
186	C₂³	C₂⁴×D₆₇	1	1	yes	characteristic
187	Q₈	—	4	24	no	—
188	Q₈	—	268	72	no	—
189	Q₈	—	67	96	no	—
190	Q₈	C₂×Q₈×D₆₇	1	32	yes	direct product factor
191	C₂×C₄	—	2	18	no	—
192	C₂×C₄	—	67	24	no	—
193	C₂×C₄	—	67	12	no	—
194	C₂×C₄	—	67	96	no	—
195	C₂×C₄	—	67	18	no	—
196	C₂×C₄	—	134	72	no	—
197	C₂×C₄	—	134	18	no	—
198	C₂×C₄	C₂×Q₈×D₆₇	1	24	yes	—
199	C₂×C₄	C₂⁴×D₆₇	1	18	yes	—
200	C₂×C₄	C₂⁴×D₆₇	1	12	yes	—
201	C₂²	—	67	8	no	—
202	C₂²	—	67	4	no	—
203	C₂²	—	67	16	no	—
204	C₂²	C₂²×Q₈×D₆₇	1	4	yes	—
205	C₂²	H(5,𝔽₂)×D₆₇	1	2	yes	—
206	C₂²	C₂⁵×D₆₇	1	1	yes	characteristic
207	C₄	—	2	18	no	—
208	C₄	—	67	24	no	—
209	C₄	—	134	18	no	—
210	C₄	C₂²×Q₈×D₆₇	1	24	yes	—
211	C₂	—	67	8	no	—
212	C₂	Q₈²×D₆₇	1	4	yes	direct product factor
213	C₂	C₂³×Q₈×D₆₇	1	2	yes	—
214	C₂	C₂×H(5,𝔽₂)×D₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂⁴×D₆₇
commutator, cosocle	C₂²×C₆₇	C₂⁶
Frattini	C₂²	C₂⁵×D₆₇
Fitting	C₂×Q₈²×C₆₇	C₂
socle	C₂³×C₆₇	C₂⁵
prime radical	C₂³×D₆₇	C₂⁴
hypercenter	C₂×Q₈²	D₆₇
hypocenter	C₆₇	C₂²×Q₈²
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×Q₈²

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×Dic₁₃₄	C₂	semidirect
2. C₂×#16864126	C₂	semidirect
3. Q₈²×D₆₇	C₂	direct (N × H)
6. C₂×Q₈²×C₆₇	C₂	semidirect
9. Q₈×Dic₁₃₄	C₂²	semidirect
10. #16864126	C₂²	semidirect
21. Q₈²×C₆₇	C₂²	semidirect
27. C₂×Q₈×D₆₇	Q₈	direct (N × H)
52. C₂×Dic₁₃₄	C₂×Q₈	semidirect
56. Q₈×D₆₇	C₂×Q₈	direct (N × H)
68. C₂×Q₈×C₆₇	C₂×Q₈	semidirect
80. Q₈×C₆₇	C₂²×Q₈	semidirect
92. Dic₁₃₄	C₂²×Q₈	semidirect
96. C₂×D₆₇	Q₈²	direct (N × H)
104. C₂×C₆₇	C₂×Q₈²	semidirect
107. D₆₇	C₂×Q₈²	direct (N × H)
111. C₂×Q₈²	D₆₇	direct (N × H)
114. C₆₇	C₂²×Q₈²	semidirect
119. Q₈²	C₂×D₆₇	direct (N × H)
166. C₂×Q₈	Q₈×D₆₇	direct (N × H)
190. Q₈	C₂×Q₈×D₆₇	direct (N × H)
212. C₂	Q₈²×D₆₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×Q₈×Dic₁₃₄	2
C₂×#16864126	2
Q₈²×D₆₇	2
C₂×C₄⋊Q₈×D₆₇	2
C₂×C₄×Q₈×D₆₇	2
C₂×Q₈²×C₆₇	2
C₂²×Q₈²	67

Subgroups — C2×Q82×D67 #31762896

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈²×D₆₇ #31762896