Subgroups — C₂²×#492×C₆₇ #30739879

Order 17152.

3950 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	C₂³×SD₁₆×C₆₇	C₂	1	4	yes	maximal, complemented
2	C₂²×C₄⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#127×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#492×C₆₇	C₂	1	24	yes	maximal, direct product factor
5	C₂²×C₄×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×Q₈×C₆₇	C₂²	1	2	yes	—
7	C₂³×D₄×C₆₇	—	2	2	no	—
8	C₂³×D₄×C₆₇	C₂²	1	2	yes	—
9	C₂²×SD₁₆×C₆₇	—	2	32	no	—
10	C₂²×SD₁₆×C₆₇	C₂²	1	48	yes	complemented
11	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
12	C₂×C₄⋊Q₈×C₆₇	—	2	6	no	—
13	C₂×C₄⋊Q₈×C₆₇	C₂²	1	12	yes	complemented
14	C₂×#127×C₆₇	—	2	6	no	—
15	C₂×#127×C₆₇	C₂²	1	12	yes	—
16	C₂²×C₄⋊C₄×C₆₇	—	2	2	no	—
17	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
18	#492×C₆₇	C₂²	1	64	yes	direct product factor
19	C₂×C₄×C₈×C₆₇	C₂²	1	12	yes	complemented
20	C₂⁵×C₆₇	—	2	2	no	—
21	C₂²×Q₈×C₆₇	—	2	6	no	—
22	C₂²×Q₈×C₆₇	—	2	16	no	—
23	C₂²×Q₈×C₆₇	C₂³	1	12	yes	—
24	C₂²×D₄×C₆₇	—	2	16	no	—
25	C₂²×D₄×C₆₇	—	2	6	no	—
26	C₂²×D₄×C₆₇	—	2	24	no	—
27	C₂²×D₄×C₆₇	—	4	16	no	—
28	C₂²×D₄×C₆₇	C₂³	1	12	yes	—
29	C₂³×C₄×C₆₇	—	2	2	no	—
30	C₂³×C₄×C₆₇	D₄	1	2	yes	—
31	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
32	C₂×SD₁₆×C₆₇	—	2	192	no	—
33	C₂×SD₁₆×C₆₇	C₂³	1	64	yes	complemented
34	C₂²×C₈×C₆₇	D₄	1	16	yes	complemented
35	C₂²×C₈×C₆₇	C₂³	1	12	yes	—
36	C₄⋊Q₈×C₆₇	—	2	24	no	—
37	C₄⋊Q₈×C₆₇	C₂³	1	16	yes	complemented
38	#127×C₆₇	—	2	24	no	—
39	#127×C₆₇	C₂³	1	16	yes	—
40	C₂×C₄⋊C₄×C₆₇	—	2	24	no	—
41	C₂×C₄²×C₆₇	D₄	1	6	yes	—
42	C₂×C₄²×C₆₇	C₂³	1	6	yes	—
43	C₄×C₈×C₆₇	C₂³	1	16	yes	complemented
44	C₂⁴×C₆₇	—	2	12	no	—
45	C₂⁴×C₆₇	—	4	8	no	—
46	C₂⁴×C₆₇	—	4	16	no	—
47	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
48	C₂×Q₈×C₆₇	—	2	12	no	—
49	C₂×Q₈×C₆₇	—	2	48	no	—
50	C₂×Q₈×C₆₇	—	4	24	no	—
51	C₂×Q₈×C₆₇	C₂⁴	1	8	yes	—
52	C₂×D₄×C₆₇	—	2	48	no	—
53	C₂×D₄×C₆₇	—	2	12	no	—
54	C₂×D₄×C₆₇	—	2	32	no	—
55	C₂×D₄×C₆₇	—	4	24	no	—
56	C₂×D₄×C₆₇	—	4	96	no	—
57	C₂×D₄×C₆₇	C₂⁴	1	8	yes	—
58	C₂²×C₄×C₆₇	—	2	12	no	—
59	C₂²×C₄×C₆₇	—	4	8	no	—
60	C₂²×C₄×C₆₇	SD₁₆	1	16	yes	complemented
61	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
62	C₂²×C₄×C₆₇	C₂×D₄	1	8	yes	—
63	C₂²×C₄×C₆₇	C₂×D₄	1	12	yes	—
64	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
65	SD₁₆×C₆₇	—	2	128	no	—
66	C₂×C₈×C₆₇	C₂×D₄	1	48	yes	complemented
67	C₂×C₈×C₆₇	C₂⁴	1	8	yes	—
68	C₄⋊C₄×C₆₇	—	2	32	no	—
69	C₄²×C₆₇	C₂×D₄	1	12	yes	—
70	C₄²×C₆₇	C₂⁴	1	4	yes	—
71	C₂³×C₆₇	—	2	8	no	—
72	C₂³×C₆₇	—	4	24	no	—
73	C₂³×C₆₇	—	4	48	no	—
74	C₂³×C₆₇	—	8	32	no	—
75	C₂³×C₆₇	#127	1	4	yes	—
76	C₂³×C₆₇	C₂×SD₁₆	1	8	yes	—
77	C₂³×C₆₇	C₂²×D₄	1	3	yes	—
78	Q₈×C₆₇	—	2	16	no	—
79	Q₈×C₆₇	—	4	24	no	—
80	D₄×C₆₇	—	2	16	no	—
81	D₄×C₆₇	—	4	24	no	—
82	D₄×C₆₇	—	4	64	no	—
83	C₂×C₄×C₆₇	—	2	8	no	—
84	C₂×C₄×C₆₇	—	4	24	no	—
85	C₂×C₄×C₆₇	#127	1	12	yes	—
86	C₂×C₄×C₆₇	C₂×SD₁₆	1	48	yes	complemented
87	C₂×C₄×C₆₇	C₂²×D₄	1	3	yes	—
88	C₂×C₄×C₆₇	C₂²×D₄	1	12	yes	—
89	C₂×C₄×C₆₇	C₂²×D₄	1	8	yes	—
90	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
91	C₈×C₆₇	C₂²×D₄	1	16	yes	complemented
92	C₂²×C₆₇	—	4	8	no	—
93	C₂²×C₆₇	—	4	16	no	—
94	C₂²×C₆₇	—	8	48	no	—
95	C₂²×C₆₇	#492	1	16	yes	direct product factor
96	C₂²×C₆₇	C₂×#127	1	6	yes	—
97	C₂²×C₆₇	C₂²×SD₁₆	1	12	yes	—
98	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
99	C₄×C₆₇	—	4	8	no	—
100	C₄×C₆₇	C₂×#127	1	6	yes	—
101	C₄×C₆₇	C₂²×SD₁₆	1	16	yes	complemented
102	C₄×C₆₇	C₂³×D₄	1	2	yes	—
103	C₂²×#492	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₂×C₆₇	—	8	8	no	—
105	C₂×C₆₇	C₂×#492	1	12	yes	direct product factor
106	C₂×C₆₇	C₂²×#127	1	1	yes	characteristic
107	C₂×C₆₇	C₂³×SD₁₆	1	2	yes	—
108	C₂³×SD₁₆	C₂×C₆₇	1	4	yes	complemented
109	C₂²×C₄⋊Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
110	C₂²×#127	C₂×C₆₇	1	1	yes	characteristic
111	C₂×#492	C₂×C₆₇	1	24	yes	direct product factor
112	C₂²×C₄×C₈	C₂×C₆₇	1	1	yes	characteristic, complemented
113	C₆₇	C₂²×#492	1	1	yes	characteristic, direct product factor, Sylow
114	C₂³×Q₈	C₂²×C₆₇	1	2	yes	—
115	C₂³×D₄	—	2	2	no	—
116	C₂³×D₄	C₂²×C₆₇	1	2	yes	—
117	C₂²×SD₁₆	—	2	32	no	—
118	C₂²×SD₁₆	C₂²×C₆₇	1	48	yes	complemented
119	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
120	C₂×C₄⋊Q₈	—	2	6	no	—
121	C₂×C₄⋊Q₈	C₂²×C₆₇	1	12	yes	complemented
122	C₂×#127	—	2	6	no	—
123	C₂×#127	C₂²×C₆₇	1	12	yes	—
124	C₂²×C₄⋊C₄	—	2	2	no	—
125	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
126	#492	C₂²×C₆₇	1	64	yes	direct product factor
127	C₂×C₄×C₈	C₂²×C₆₇	1	12	yes	complemented
128	C₂⁵	—	2	2	no	—
129	C₂²×Q₈	—	2	6	no	—
130	C₂²×Q₈	—	2	16	no	—
131	C₂²×Q₈	C₂³×C₆₇	1	12	yes	—
132	C₂²×D₄	—	2	16	no	—
133	C₂²×D₄	—	2	6	no	—
134	C₂²×D₄	—	2	24	no	—
135	C₂²×D₄	—	4	16	no	—
136	C₂²×D₄	C₂³×C₆₇	1	12	yes	—
137	C₂³×C₄	—	2	2	no	—
138	C₂³×C₄	D₄×C₆₇	1	2	yes	—
139	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
140	C₂×SD₁₆	—	2	192	no	—
141	C₂×SD₁₆	C₂³×C₆₇	1	64	yes	complemented
142	C₂²×C₈	D₄×C₆₇	1	16	yes	complemented
143	C₂²×C₈	C₂³×C₆₇	1	12	yes	—
144	C₄⋊Q₈	—	2	24	no	—
145	C₄⋊Q₈	C₂³×C₆₇	1	16	yes	complemented
146	#127	—	2	24	no	—
147	#127	C₂³×C₆₇	1	16	yes	—
148	C₂×C₄⋊C₄	—	2	24	no	—
149	C₂×C₄²	D₄×C₆₇	1	6	yes	—
150	C₂×C₄²	C₂³×C₆₇	1	6	yes	—
151	C₄×C₈	C₂³×C₆₇	1	16	yes	complemented
152	C₂⁴	—	2	12	no	—
153	C₂⁴	—	4	8	no	—
154	C₂⁴	—	4	16	no	—
155	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
156	C₂×Q₈	—	2	12	no	—
157	C₂×Q₈	—	2	48	no	—
158	C₂×Q₈	—	4	24	no	—
159	C₂×Q₈	C₂⁴×C₆₇	1	8	yes	—
160	C₂×D₄	—	2	48	no	—
161	C₂×D₄	—	2	12	no	—
162	C₂×D₄	—	2	32	no	—
163	C₂×D₄	—	4	24	no	—
164	C₂×D₄	—	4	96	no	—
165	C₂×D₄	C₂⁴×C₆₇	1	8	yes	—
166	C₂²×C₄	—	2	12	no	—
167	C₂²×C₄	—	4	8	no	—
168	C₂²×C₄	SD₁₆×C₆₇	1	16	yes	complemented
169	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
170	C₂²×C₄	C₂×D₄×C₆₇	1	8	yes	—
171	C₂²×C₄	C₂×D₄×C₆₇	1	12	yes	—
172	C₂²×C₄	C₂⁴×C₆₇	1	3	yes	—
173	SD₁₆	—	2	128	no	—
174	C₂×C₈	C₂×D₄×C₆₇	1	48	yes	complemented
175	C₂×C₈	C₂⁴×C₆₇	1	8	yes	—
176	C₄⋊C₄	—	2	32	no	—
177	C₄²	C₂×D₄×C₆₇	1	12	yes	—
178	C₄²	C₂⁴×C₆₇	1	4	yes	—
179	C₂³	—	2	8	no	—
180	C₂³	—	4	24	no	—
181	C₂³	—	4	48	no	—
182	C₂³	—	8	32	no	—
183	C₂³	#127×C₆₇	1	4	yes	—
184	C₂³	C₂×SD₁₆×C₆₇	1	8	yes	—
185	C₂³	C₂²×D₄×C₆₇	1	3	yes	—
186	Q₈	—	2	16	no	—
187	Q₈	—	4	24	no	—
188	D₄	—	2	16	no	—
189	D₄	—	4	24	no	—
190	D₄	—	4	64	no	—
191	C₂×C₄	—	2	8	no	—
192	C₂×C₄	—	4	24	no	—
193	C₂×C₄	#127×C₆₇	1	12	yes	—
194	C₂×C₄	C₂×SD₁₆×C₆₇	1	48	yes	complemented
195	C₂×C₄	C₂²×D₄×C₆₇	1	3	yes	—
196	C₂×C₄	C₂²×D₄×C₆₇	1	12	yes	—
197	C₂×C₄	C₂²×D₄×C₆₇	1	8	yes	—
198	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
199	C₈	C₂²×D₄×C₆₇	1	16	yes	complemented
200	C₂²	—	4	16	no	—
201	C₂²	—	4	8	no	—
202	C₂²	—	8	48	no	—
203	C₂²	#492×C₆₇	1	16	yes	direct product factor
204	C₂²	C₂×#127×C₆₇	1	6	yes	—
205	C₂²	C₂²×SD₁₆×C₆₇	1	12	yes	—
206	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
207	C₄	—	4	8	no	—
208	C₄	C₂×#127×C₆₇	1	6	yes	—
209	C₄	C₂²×SD₁₆×C₆₇	1	16	yes	complemented
210	C₄	C₂³×D₄×C₆₇	1	2	yes	—
211	C₂	—	8	8	no	—
212	C₂	C₂×#492×C₆₇	1	12	yes	direct product factor
213	C₂	C₂²×#127×C₆₇	1	1	yes	characteristic
214	C₂	C₂³×SD₁₆×C₆₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₆₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
prime radical	C₂²×#127×C₆₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×SD₁₆×C₆₇	C₂	semidirect
2. C₂²×C₄⋊Q₈×C₆₇	C₂	semidirect
4. C₂×#492×C₆₇	C₂	direct (N × H)
5. C₂²×C₄×C₈×C₆₇	C₂	semidirect
10. C₂²×SD₁₆×C₆₇	C₂²	semidirect
13. C₂×C₄⋊Q₈×C₆₇	C₂²	semidirect
18. #492×C₆₇	C₂²	direct (N × H)
19. C₂×C₄×C₈×C₆₇	C₂²	semidirect
33. C₂×SD₁₆×C₆₇	C₂³	semidirect
34. C₂²×C₈×C₆₇	D₄	semidirect
37. C₄⋊Q₈×C₆₇	C₂³	semidirect
43. C₄×C₈×C₆₇	C₂³	semidirect
60. C₂²×C₄×C₆₇	SD₁₆	semidirect
66. C₂×C₈×C₆₇	C₂×D₄	semidirect
86. C₂×C₄×C₆₇	C₂×SD₁₆	semidirect
91. C₈×C₆₇	C₂²×D₄	semidirect
95. C₂²×C₆₇	#492	direct (N × H)
101. C₄×C₆₇	C₂²×SD₁₆	semidirect
103. C₂²×#492	C₆₇	direct (N × H)
105. C₂×C₆₇	C₂×#492	direct (N × H)
108. C₂³×SD₁₆	C₂×C₆₇	semidirect
109. C₂²×C₄⋊Q₈	C₂×C₆₇	semidirect
111. C₂×#492	C₂×C₆₇	direct (N × H)
112. C₂²×C₄×C₈	C₂×C₆₇	semidirect
113. C₆₇	C₂²×#492	direct (N × H)
118. C₂²×SD₁₆	C₂²×C₆₇	semidirect
121. C₂×C₄⋊Q₈	C₂²×C₆₇	semidirect
126. #492	C₂²×C₆₇	direct (N × H)
127. C₂×C₄×C₈	C₂²×C₆₇	semidirect
141. C₂×SD₁₆	C₂³×C₆₇	semidirect
142. C₂²×C₈	D₄×C₆₇	semidirect
145. C₄⋊Q₈	C₂³×C₆₇	semidirect
151. C₄×C₈	C₂³×C₆₇	semidirect
168. C₂²×C₄	SD₁₆×C₆₇	semidirect
174. C₂×C₈	C₂×D₄×C₆₇	semidirect
194. C₂×C₄	C₂×SD₁₆×C₆₇	semidirect
199. C₈	C₂²×D₄×C₆₇	semidirect
203. C₂²	#492×C₆₇	direct (N × H)
209. C₄	C₂²×SD₁₆×C₆₇	semidirect
212. C₂	C₂×#492×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
C₂³×SD₁₆×C₆₇	2
C₂²×C₄⋊Q₈×C₆₇	2
C₂²×#127×C₆₇	2
C₂×#492×C₆₇	2
C₂²×C₄×C₈×C₆₇	2
C₂²×#492	67

Subgroups — C22×#492×C67 #30739879

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#492×C₆₇ #30739879