Subgroups — C₂³×C₂²⋊Q₈×C₆₇ #30741908

Order 17152.

14078 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	C₂⁴×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂⁵×C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₂²⋊Q₈×C₆₇	C₂	1	56	yes	maximal, direct product factor
4	C₂³×C₄⋊C₄×C₆₇	C₂	1	2	yes	maximal, complemented
5	C₂³×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×C₂²⋊C₄×C₆₇	C₂	1	2	yes	maximal
7	C₂⁶×C₆₇	C₂²	1	1	yes	characteristic
8	C₂³×Q₈×C₆₇	—	2	16	no	—
9	C₂³×Q₈×C₆₇	C₂²	1	28	yes	complemented
10	C₂⁴×C₄×C₆₇	—	2	16	no	—
11	C₂⁴×C₄×C₆₇	C₂²	1	28	yes	—
12	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
13	C₂⁴×C₄×C₆₇	C₂²	1	2	yes	—
14	C₂⁴×C₄×C₆₇	C₂²	1	2	yes	—
15	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	448	yes	direct product factor
17	C₂²×C₄⋊C₄×C₆₇	C₂²	1	28	yes	complemented
18	C₂²×C₄⋊C₄×C₆₇	C₂²	1	56	yes	complemented
19	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	56	yes	—
20	C₂⁵×C₆₇	—	2	8	no	—
21	C₂⁵×C₆₇	—	2	8	no	—
22	C₂⁵×C₆₇	Q₈	1	16	yes	complemented
23	C₂⁵×C₆₇	C₂³	1	14	yes	—
24	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
25	C₂²×Q₈×C₆₇	—	2	224	no	—
26	C₂²×Q₈×C₆₇	C₂³	1	112	yes	complemented
27	C₂³×C₄×C₆₇	—	2	8	no	—
28	C₂³×C₄×C₆₇	—	2	16	no	—
29	C₂³×C₄×C₆₇	—	2	16	no	—
30	C₂³×C₄×C₆₇	—	2	224	no	—
31	C₂³×C₄×C₆₇	D₄	1	16	yes	—
32	C₂³×C₄×C₆₇	C₂³	1	28	yes	—
33	C₂³×C₄×C₆₇	C₂³	1	28	yes	—
34	C₂³×C₄×C₆₇	C₂³	1	14	yes	—
35	C₂³×C₄×C₆₇	C₂³	1	112	yes	—
36	C₂³×C₄×C₆₇	C₂³	1	14	yes	—
37	C₂²⋊Q₈×C₆₇	C₂³	1	512	yes	direct product factor
38	C₂×C₄⋊C₄×C₆₇	C₂³	1	112	yes	complemented
39	C₂×C₄⋊C₄×C₆₇	C₂³	1	224	yes	complemented
40	C₂×C₂²⋊C₄×C₆₇	C₂³	1	224	yes	—
41	C₂⁴×C₆₇	—	2	128	no	—
42	C₂⁴×C₆₇	—	2	56	no	—
43	C₂⁴×C₆₇	—	2	56	no	—
44	C₂⁴×C₆₇	C₂×D₄	1	8	yes	—
45	C₂⁴×C₆₇	C₂×Q₈	1	8	yes	—
46	C₂⁴×C₆₇	C₂×Q₈	1	112	yes	complemented
47	C₂⁴×C₆₇	#41	1	8	yes	—
48	C₂⁴×C₆₇	C₂⁴	1	28	yes	—
49	C₂⁴×C₆₇	C₂⁴	1	7	yes	—
50	C₂×Q₈×C₆₇	—	2	448	no	—
51	C₂×Q₈×C₆₇	C₂⁴	1	64	yes	complemented
52	C₂²×C₄×C₆₇	—	2	56	no	—
53	C₂²×C₄×C₆₇	—	2	112	no	—
54	C₂²×C₄×C₆₇	—	2	112	no	—
55	C₂²×C₄×C₆₇	—	2	448	no	—
56	C₂²×C₄×C₆₇	C₂×D₄	1	112	yes	—
57	C₂²×C₄×C₆₇	C₂⁴	1	56	yes	—
58	C₂²×C₄×C₆₇	C₂⁴	1	56	yes	—
59	C₂²×C₄×C₆₇	C₂⁴	1	28	yes	—
60	C₂²×C₄×C₆₇	C₂⁴	1	64	yes	—
61	C₂²×C₄×C₆₇	C₂⁴	1	28	yes	—
62	C₄⋊C₄×C₆₇	C₂⁴	1	64	yes	complemented
63	C₄⋊C₄×C₆₇	C₂⁴	1	128	yes	complemented
64	C₂²⋊C₄×C₆₇	C₂⁴	1	128	yes	—
65	C₂³×C₆₇	—	2	448	no	—
66	C₂³×C₆₇	—	2	56	no	—
67	C₂³×C₆₇	—	2	56	no	—
68	C₂³×C₆₇	C₂²⋊Q₈	1	64	yes	direct product factor
69	C₂³×C₆₇	C₂²×D₄	1	28	yes	—
70	C₂³×C₆₇	C₂²×Q₈	1	28	yes	—
71	C₂³×C₆₇	C₂²×Q₈	1	112	yes	complemented
72	C₂³×C₆₇	C₂×#41	1	28	yes	—
73	C₂³×C₆₇	C₂⁵	1	8	yes	—
74	C₂³×C₆₇	C₂⁵	1	7	yes	—
75	Q₈×C₆₇	—	2	128	no	—
76	C₂×C₄×C₆₇	—	2	56	no	—
77	C₂×C₄×C₆₇	—	2	112	no	—
78	C₂×C₄×C₆₇	—	2	112	no	—
79	C₂×C₄×C₆₇	—	2	128	no	—
80	C₂×C₄×C₆₇	C₂²×D₄	1	112	yes	—
81	C₂×C₄×C₆₇	C₂⁵	1	16	yes	—
82	C₂×C₄×C₆₇	C₂⁵	1	16	yes	—
83	C₂×C₄×C₆₇	C₂⁵	1	8	yes	—
84	C₂×C₄×C₆₇	C₂⁵	1	8	yes	—
85	C₂²×C₆₇	—	2	8	no	—
86	C₂²×C₆₇	—	2	8	no	—
87	C₂²×C₆₇	—	2	224	no	—
88	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	112	yes	direct product factor
89	C₂²×C₆₇	C₂³×D₄	1	14	yes	—
90	C₂²×C₆₇	C₂³×Q₈	1	14	yes	—
91	C₂²×C₆₇	C₂³×Q₈	1	16	yes	complemented
92	C₂²×C₆₇	C₂²×#41	1	14	yes	—
93	C₂²×C₆₇	C₂⁶	1	1	yes	characteristic
94	C₄×C₆₇	—	2	8	no	—
95	C₄×C₆₇	—	2	16	no	—
96	C₄×C₆₇	—	2	16	no	—
97	C₄×C₆₇	C₂³×D₄	1	16	yes	—
98	C₂³×C₂²⋊Q₈	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂×C₆₇	—	2	16	no	—
100	C₂×C₆₇	C₂²×C₂²⋊Q₈	1	28	yes	direct product factor
101	C₂×C₆₇	C₂⁴×D₄	1	1	yes	characteristic
102	C₂×C₆₇	C₂⁴×Q₈	1	1	yes	characteristic
103	C₂×C₆₇	C₂³×#41	1	1	yes	characteristic
104	C₂⁴×Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
105	C₂⁵×C₄	C₂×C₆₇	1	1	yes	characteristic
106	C₂²×C₂²⋊Q₈	C₂×C₆₇	1	56	yes	direct product factor
107	C₂³×C₄⋊C₄	C₂×C₆₇	1	2	yes	complemented
108	C₂³×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic, complemented
109	C₂³×C₂²⋊C₄	C₂×C₆₇	1	2	yes	—
110	C₆₇	C₂³×C₂²⋊Q₈	1	1	yes	characteristic, direct product factor, Sylow
111	C₂⁶	C₂²×C₆₇	1	1	yes	characteristic
112	C₂³×Q₈	—	2	16	no	—
113	C₂³×Q₈	C₂²×C₆₇	1	28	yes	complemented
114	C₂⁴×C₄	—	2	16	no	—
115	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
116	C₂⁴×C₄	C₂²×C₆₇	1	28	yes	—
117	C₂⁴×C₄	C₂²×C₆₇	1	2	yes	—
118	C₂⁴×C₄	C₂²×C₆₇	1	2	yes	—
119	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
120	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	448	yes	direct product factor
121	C₂²×C₄⋊C₄	C₂²×C₆₇	1	28	yes	complemented
122	C₂²×C₄⋊C₄	C₂²×C₆₇	1	56	yes	complemented
123	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	56	yes	—
124	C₂⁵	—	2	8	no	—
125	C₂⁵	—	2	8	no	—
126	C₂⁵	Q₈×C₆₇	1	16	yes	complemented
127	C₂⁵	C₂³×C₆₇	1	14	yes	—
128	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
129	C₂²×Q₈	—	2	224	no	—
130	C₂²×Q₈	C₂³×C₆₇	1	112	yes	complemented
131	C₂³×C₄	—	2	8	no	—
132	C₂³×C₄	—	2	16	no	—
133	C₂³×C₄	—	2	16	no	—
134	C₂³×C₄	—	2	224	no	—
135	C₂³×C₄	D₄×C₆₇	1	16	yes	—
136	C₂³×C₄	C₂³×C₆₇	1	28	yes	—
137	C₂³×C₄	C₂³×C₆₇	1	28	yes	—
138	C₂³×C₄	C₂³×C₆₇	1	14	yes	—
139	C₂³×C₄	C₂³×C₆₇	1	112	yes	—
140	C₂³×C₄	C₂³×C₆₇	1	14	yes	—
141	C₂²⋊Q₈	C₂³×C₆₇	1	512	yes	direct product factor
142	C₂×C₄⋊C₄	C₂³×C₆₇	1	112	yes	complemented
143	C₂×C₄⋊C₄	C₂³×C₆₇	1	224	yes	complemented
144	C₂×C₂²⋊C₄	C₂³×C₆₇	1	224	yes	—
145	C₂⁴	—	2	128	no	—
146	C₂⁴	—	2	56	no	—
147	C₂⁴	—	2	56	no	—
148	C₂⁴	C₂×D₄×C₆₇	1	8	yes	—
149	C₂⁴	C₂×Q₈×C₆₇	1	8	yes	—
150	C₂⁴	C₂×Q₈×C₆₇	1	112	yes	complemented
151	C₂⁴	#41×C₆₇	1	8	yes	—
152	C₂⁴	C₂⁴×C₆₇	1	28	yes	—
153	C₂⁴	C₂⁴×C₆₇	1	7	yes	—
154	C₂×Q₈	—	2	448	no	—
155	C₂×Q₈	C₂⁴×C₆₇	1	64	yes	complemented
156	C₂²×C₄	—	2	56	no	—
157	C₂²×C₄	—	2	112	no	—
158	C₂²×C₄	—	2	112	no	—
159	C₂²×C₄	—	2	448	no	—
160	C₂²×C₄	C₂×D₄×C₆₇	1	112	yes	—
161	C₂²×C₄	C₂⁴×C₆₇	1	56	yes	—
162	C₂²×C₄	C₂⁴×C₆₇	1	56	yes	—
163	C₂²×C₄	C₂⁴×C₆₇	1	28	yes	—
164	C₂²×C₄	C₂⁴×C₆₇	1	64	yes	—
165	C₂²×C₄	C₂⁴×C₆₇	1	28	yes	—
166	C₄⋊C₄	C₂⁴×C₆₇	1	64	yes	complemented
167	C₄⋊C₄	C₂⁴×C₆₇	1	128	yes	complemented
168	C₂²⋊C₄	C₂⁴×C₆₇	1	128	yes	—
169	C₂³	—	2	448	no	—
170	C₂³	—	2	56	no	—
171	C₂³	—	2	56	no	—
172	C₂³	C₂²⋊Q₈×C₆₇	1	64	yes	direct product factor
173	C₂³	C₂²×D₄×C₆₇	1	28	yes	—
174	C₂³	C₂²×Q₈×C₆₇	1	28	yes	—
175	C₂³	C₂²×Q₈×C₆₇	1	112	yes	complemented
176	C₂³	C₂×#41×C₆₇	1	28	yes	—
177	C₂³	C₂⁵×C₆₇	1	8	yes	—
178	C₂³	C₂⁵×C₆₇	1	7	yes	—
179	Q₈	—	2	128	no	—
180	C₂×C₄	—	2	56	no	—
181	C₂×C₄	—	2	112	no	—
182	C₂×C₄	—	2	112	no	—
183	C₂×C₄	—	2	128	no	—
184	C₂×C₄	C₂²×D₄×C₆₇	1	112	yes	—
185	C₂×C₄	C₂⁵×C₆₇	1	16	yes	—
186	C₂×C₄	C₂⁵×C₆₇	1	16	yes	—
187	C₂×C₄	C₂⁵×C₆₇	1	8	yes	—
188	C₂×C₄	C₂⁵×C₆₇	1	8	yes	—
189	C₂²	—	2	8	no	—
190	C₂²	—	2	8	no	—
191	C₂²	—	2	224	no	—
192	C₂²	C₂×C₂²⋊Q₈×C₆₇	1	112	yes	direct product factor
193	C₂²	C₂³×D₄×C₆₇	1	14	yes	—
194	C₂²	C₂³×Q₈×C₆₇	1	14	yes	—
195	C₂²	C₂³×Q₈×C₆₇	1	16	yes	complemented
196	C₂²	C₂²×#41×C₆₇	1	14	yes	—
197	C₂²	C₂⁶×C₆₇	1	1	yes	characteristic
198	C₄	—	2	8	no	—
199	C₄	—	2	16	no	—
200	C₄	—	2	16	no	—
201	C₄	C₂³×D₄×C₆₇	1	16	yes	—
202	C₂	—	2	16	no	—
203	C₂	C₂²×C₂²⋊Q₈×C₆₇	1	28	yes	direct product factor
204	C₂	C₂⁴×D₄×C₆₇	1	1	yes	characteristic
205	C₂	C₂⁴×Q₈×C₆₇	1	1	yes	characteristic
206	C₂	C₂³×#41×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁵×C₆₇	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁶×C₆₇
prime radical	C₂⁶×C₆₇	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂⁴×Q₈×C₆₇	C₂	semidirect
3. C₂²×C₂²⋊Q₈×C₆₇	C₂	direct (N × H)
4. C₂³×C₄⋊C₄×C₆₇	C₂	semidirect
5. C₂³×C₄⋊C₄×C₆₇	C₂	semidirect
9. C₂³×Q₈×C₆₇	C₂²	semidirect
16. C₂×C₂²⋊Q₈×C₆₇	C₂²	direct (N × H)
17. C₂²×C₄⋊C₄×C₆₇	C₂²	semidirect
18. C₂²×C₄⋊C₄×C₆₇	C₂²	semidirect
22. C₂⁵×C₆₇	Q₈	semidirect
26. C₂²×Q₈×C₆₇	C₂³	semidirect
37. C₂²⋊Q₈×C₆₇	C₂³	direct (N × H)
38. C₂×C₄⋊C₄×C₆₇	C₂³	semidirect
39. C₂×C₄⋊C₄×C₆₇	C₂³	semidirect
46. C₂⁴×C₆₇	C₂×Q₈	semidirect
51. C₂×Q₈×C₆₇	C₂⁴	semidirect
62. C₄⋊C₄×C₆₇	C₂⁴	semidirect
63. C₄⋊C₄×C₆₇	C₂⁴	semidirect
68. C₂³×C₆₇	C₂²⋊Q₈	direct (N × H)
71. C₂³×C₆₇	C₂²×Q₈	semidirect
88. C₂²×C₆₇	C₂×C₂²⋊Q₈	direct (N × H)
91. C₂²×C₆₇	C₂³×Q₈	semidirect
98. C₂³×C₂²⋊Q₈	C₆₇	direct (N × H)
100. C₂×C₆₇	C₂²×C₂²⋊Q₈	direct (N × H)
104. C₂⁴×Q₈	C₂×C₆₇	semidirect
106. C₂²×C₂²⋊Q₈	C₂×C₆₇	direct (N × H)
107. C₂³×C₄⋊C₄	C₂×C₆₇	semidirect
108. C₂³×C₄⋊C₄	C₂×C₆₇	semidirect
110. C₆₇	C₂³×C₂²⋊Q₈	direct (N × H)
113. C₂³×Q₈	C₂²×C₆₇	semidirect
120. C₂×C₂²⋊Q₈	C₂²×C₆₇	direct (N × H)
121. C₂²×C₄⋊C₄	C₂²×C₆₇	semidirect
122. C₂²×C₄⋊C₄	C₂²×C₆₇	semidirect
126. C₂⁵	Q₈×C₆₇	semidirect
130. C₂²×Q₈	C₂³×C₆₇	semidirect
141. C₂²⋊Q₈	C₂³×C₆₇	direct (N × H)
142. C₂×C₄⋊C₄	C₂³×C₆₇	semidirect
143. C₂×C₄⋊C₄	C₂³×C₆₇	semidirect
150. C₂⁴	C₂×Q₈×C₆₇	semidirect
155. C₂×Q₈	C₂⁴×C₆₇	semidirect
166. C₄⋊C₄	C₂⁴×C₆₇	semidirect
167. C₄⋊C₄	C₂⁴×C₆₇	semidirect
172. C₂³	C₂²⋊Q₈×C₆₇	direct (N × H)
175. C₂³	C₂²×Q₈×C₆₇	semidirect
192. C₂²	C₂×C₂²⋊Q₈×C₆₇	direct (N × H)
195. C₂²	C₂³×Q₈×C₆₇	semidirect
203. C₂	C₂²×C₂²⋊Q₈×C₆₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×Q₈×C₆₇	2
C₂⁵×C₄×C₆₇	2
C₂²×C₂²⋊Q₈×C₆₇	2
C₂³×C₄⋊C₄×C₆₇	2
C₂³×C₂²⋊C₄×C₆₇	2
C₂³×C₂²⋊Q₈	67

Subgroups — C23×C22⋊Q8×C67 #30741908

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₂²⋊Q₈×C₆₇ #30741908