Subgroups — C₂²×#357×C₆₇ #30712625

Order 17152.

1470 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#107×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#357×C₆₇	C₂	1	12	yes	maximal, direct product factor
5	C₂³×D₄×C₆₇	—	2	1	no	—
6	C₂²×D₈×C₆₇	—	2	3	no	—
7	C₂²×D₈×C₆₇	—	2	4	no	—
8	C₂²×D₈×C₆₇	C₄	1	8	yes	complemented
9	C₂²×D₈×C₆₇	C₂²	1	6	yes	—
10	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
11	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
12	C₂²×C₁₆×C₆₇	—	2	4	no	—
13	C₂²×C₁₆×C₆₇	C₂²	1	6	yes	complemented
14	C₂×#107×C₆₇	—	2	3	no	—
15	C₂×#107×C₆₇	C₂²	1	6	yes	complemented
16	#357×C₆₇	C₂²	1	16	yes	direct product factor
17	C₂⁵×C₆₇	—	4	1	no	—
18	C₂²×D₄×C₆₇	—	2	8	no	—
19	C₂²×D₄×C₆₇	—	2	6	no	—
20	C₂²×D₄×C₆₇	—	4	3	no	—
21	C₂²×D₄×C₆₇	—	4	4	no	—
22	C₂³×C₄×C₆₇	—	4	1	no	—
23	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
24	C₂×D₈×C₆₇	—	2	24	no	—
25	C₂×D₈×C₆₇	—	2	6	no	—
26	C₂×D₈×C₆₇	—	2	12	no	—
27	C₂×D₈×C₆₇	C₂×C₄	1	24	yes	complemented
28	C₂×D₈×C₆₇	C₂³	1	4	yes	—
29	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
30	C₂²×C₈×C₆₇	D₄	1	3	yes	—
31	C₂²×C₈×C₆₇	D₄	1	4	yes	—
32	C₂²×C₈×C₆₇	C₂³	1	3	yes	—
33	C₂×C₄⋊C₄×C₆₇	—	2	6	no	—
34	C₂×C₄⋊C₄×C₆₇	—	4	3	no	—
35	C₂×C₁₆×C₆₇	—	2	12	no	—
36	C₂×C₁₆×C₆₇	C₂³	1	4	yes	complemented
37	#107×C₆₇	—	2	6	no	—
38	#107×C₆₇	C₂³	1	4	yes	complemented
39	C₂⁴×C₆₇	—	4	8	no	—
40	C₂⁴×C₆₇	—	4	6	no	—
41	C₂⁴×C₆₇	—	8	4	no	—
42	C₂⁴×C₆₇	—	8	4	no	—
43	C₂⁴×C₆₇	D₈	1	1	yes	characteristic
44	C₂×D₄×C₆₇	—	2	24	no	—
45	C₂×D₄×C₆₇	—	2	4	no	—
46	C₂×D₄×C₆₇	—	4	24	no	—
47	C₂×D₄×C₆₇	—	4	6	no	—
48	C₂×D₄×C₆₇	—	4	12	no	—
49	C₂²×C₄×C₆₇	—	4	6	no	—
50	C₂²×C₄×C₆₇	—	8	4	no	—
51	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
52	C₂²×C₄×C₆₇	D₈	1	3	yes	—
53	C₂²×C₄×C₆₇	SD₁₆	1	4	yes	—
54	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
55	D₈×C₆₇	—	2	24	no	—
56	D₈×C₆₇	—	2	4	no	—
57	D₈×C₆₇	C₂²×C₄	1	8	yes	complemented
58	C₂×C₈×C₆₇	C₂²⋊C₄	1	12	yes	—
59	C₂×C₈×C₆₇	C₂²×C₄	1	6	yes	—
60	C₂×C₈×C₆₇	C₂×D₄	1	3	yes	—
61	C₂×C₈×C₆₇	C₂×D₄	1	6	yes	—
62	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
63	C₄⋊C₄×C₆₇	—	2	4	no	—
64	C₄⋊C₄×C₆₇	—	4	6	no	—
65	C₁₆×C₆₇	—	2	4	no	—
66	C₂³×C₆₇	—	4	4	no	—
67	C₂³×C₆₇	—	4	24	no	—
68	C₂³×C₆₇	—	8	32	no	—
69	C₂³×C₆₇	—	8	12	no	—
70	C₂³×C₆₇	—	8	12	no	—
71	C₂³×C₆₇	#102	1	4	yes	—
72	C₂³×C₆₇	D₁₆	1	4	yes	—
73	C₂³×C₆₇	SD₃₂	1	4	yes	—
74	C₂³×C₆₇	C₂×D₈	1	3	yes	—
75	D₄×C₆₇	—	2	8	no	—
76	D₄×C₆₇	—	4	4	no	—
77	D₄×C₆₇	—	4	24	no	—
78	C₂×C₄×C₆₇	—	4	4	no	—
79	C₂×C₄×C₆₇	—	8	12	no	—
80	C₂×C₄×C₆₇	#102	1	12	yes	—
81	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	6	yes	—
82	C₂×C₄×C₆₇	C₂×D₈	1	3	yes	—
83	C₂×C₄×C₆₇	C₂×SD₁₆	1	6	yes	—
84	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
85	C₈×C₆₇	C₂×C₂²⋊C₄	1	6	yes	—
86	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
87	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
88	C₂²×C₆₇	—	4	8	no	—
89	C₂²×C₆₇	—	8	4	no	—
90	C₂²×C₆₇	—	8	4	no	—
91	C₂²×C₆₇	—	8	48	no	—
92	C₂²×C₆₇	#357	1	16	yes	direct product factor
93	C₂²×C₆₇	C₂×#102	1	6	yes	—
94	C₂²×C₆₇	C₂×D₁₆	1	6	yes	—
95	C₂²×C₆₇	C₂×SD₃₂	1	6	yes	—
96	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
97	C₄×C₆₇	—	8	4	no	—
98	C₄×C₆₇	C₂×#102	1	6	yes	—
99	C₄×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
100	C₄×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
101	C₂²×#357	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
102	C₂×C₆₇	—	8	8	no	—
103	C₂×C₆₇	C₂×#357	1	12	yes	direct product factor
104	C₂×C₆₇	C₂²×#102	1	1	yes	characteristic
105	C₂×C₆₇	C₂²×D₁₆	1	1	yes	characteristic
106	C₂×C₆₇	C₂²×SD₃₂	1	1	yes	characteristic
107	C₂³×D₈	C₂×C₆₇	1	1	yes	characteristic
108	C₂³×C₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
109	C₂²×#107	C₂×C₆₇	1	1	yes	characteristic, complemented
110	C₂×#357	C₂×C₆₇	1	12	yes	direct product factor
111	C₆₇	C₂²×#357	1	1	yes	characteristic, direct product factor, Sylow
112	C₂³×D₄	—	2	1	no	—
113	C₂²×D₈	—	2	3	no	—
114	C₂²×D₈	—	2	4	no	—
115	C₂²×D₈	C₄×C₆₇	1	8	yes	complemented
116	C₂²×D₈	C₂²×C₆₇	1	6	yes	—
117	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
118	C₂²×C₄⋊C₄	—	2	1	no	—
119	C₂²×C₁₆	—	2	4	no	—
120	C₂²×C₁₆	C₂²×C₆₇	1	6	yes	complemented
121	C₂×#107	—	2	3	no	—
122	C₂×#107	C₂²×C₆₇	1	6	yes	complemented
123	#357	C₂²×C₆₇	1	16	yes	direct product factor
124	C₂⁵	—	4	1	no	—
125	C₂²×D₄	—	2	8	no	—
126	C₂²×D₄	—	2	6	no	—
127	C₂²×D₄	—	4	3	no	—
128	C₂²×D₄	—	4	4	no	—
129	C₂³×C₄	—	4	1	no	—
130	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
131	C₂×D₈	—	2	24	no	—
132	C₂×D₈	—	2	6	no	—
133	C₂×D₈	—	2	12	no	—
134	C₂×D₈	C₂×C₄×C₆₇	1	24	yes	complemented
135	C₂×D₈	C₂³×C₆₇	1	4	yes	—
136	C₂²×C₈	C₂×C₄×C₆₇	1	4	yes	—
137	C₂²×C₈	D₄×C₆₇	1	3	yes	—
138	C₂²×C₈	D₄×C₆₇	1	4	yes	—
139	C₂²×C₈	C₂³×C₆₇	1	3	yes	—
140	C₂×C₄⋊C₄	—	2	6	no	—
141	C₂×C₄⋊C₄	—	4	3	no	—
142	C₂×C₁₆	—	2	12	no	—
143	C₂×C₁₆	C₂³×C₆₇	1	4	yes	complemented
144	#107	—	2	6	no	—
145	#107	C₂³×C₆₇	1	4	yes	complemented
146	C₂⁴	—	4	8	no	—
147	C₂⁴	—	4	6	no	—
148	C₂⁴	—	8	4	no	—
149	C₂⁴	—	8	4	no	—
150	C₂⁴	D₈×C₆₇	1	1	yes	characteristic
151	C₂×D₄	—	2	24	no	—
152	C₂×D₄	—	2	4	no	—
153	C₂×D₄	—	4	24	no	—
154	C₂×D₄	—	4	6	no	—
155	C₂×D₄	—	4	12	no	—
156	C₂²×C₄	—	4	6	no	—
157	C₂²×C₄	—	8	4	no	—
158	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
159	C₂²×C₄	D₈×C₆₇	1	3	yes	—
160	C₂²×C₄	SD₁₆×C₆₇	1	4	yes	—
161	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
162	D₈	—	2	24	no	—
163	D₈	—	2	4	no	—
164	D₈	C₂²×C₄×C₆₇	1	8	yes	complemented
165	C₂×C₈	C₂²⋊C₄×C₆₇	1	12	yes	—
166	C₂×C₈	C₂²×C₄×C₆₇	1	6	yes	—
167	C₂×C₈	C₂×D₄×C₆₇	1	3	yes	—
168	C₂×C₈	C₂×D₄×C₆₇	1	6	yes	—
169	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
170	C₄⋊C₄	—	2	4	no	—
171	C₄⋊C₄	—	4	6	no	—
172	C₁₆	—	2	4	no	—
173	C₂³	—	4	4	no	—
174	C₂³	—	4	24	no	—
175	C₂³	—	8	12	no	—
176	C₂³	—	8	12	no	—
177	C₂³	—	8	32	no	—
178	C₂³	#102×C₆₇	1	4	yes	—
179	C₂³	D₁₆×C₆₇	1	4	yes	—
180	C₂³	SD₃₂×C₆₇	1	4	yes	—
181	C₂³	C₂×D₈×C₆₇	1	3	yes	—
182	D₄	—	2	8	no	—
183	D₄	—	4	4	no	—
184	D₄	—	4	24	no	—
185	C₂×C₄	—	4	4	no	—
186	C₂×C₄	—	8	12	no	—
187	C₂×C₄	#102×C₆₇	1	12	yes	—
188	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	6	yes	—
189	C₂×C₄	C₂×D₈×C₆₇	1	3	yes	—
190	C₂×C₄	C₂×SD₁₆×C₆₇	1	6	yes	—
191	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
192	C₈	C₂×C₂²⋊C₄×C₆₇	1	6	yes	—
193	C₈	C₂³×C₄×C₆₇	1	1	yes	characteristic
194	C₈	C₂²×D₄×C₆₇	1	1	yes	characteristic
195	C₂²	—	4	8	no	—
196	C₂²	—	8	4	no	—
197	C₂²	—	8	4	no	—
198	C₂²	—	8	48	no	—
199	C₂²	#357×C₆₇	1	16	yes	direct product factor
200	C₂²	C₂×#102×C₆₇	1	6	yes	—
201	C₂²	C₂×D₁₆×C₆₇	1	6	yes	—
202	C₂²	C₂×SD₃₂×C₆₇	1	6	yes	—
203	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
204	C₄	—	8	4	no	—
205	C₄	C₂×#102×C₆₇	1	6	yes	—
206	C₄	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
207	C₄	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
208	C₂	—	8	8	no	—
209	C₂	C₂×#357×C₆₇	1	12	yes	direct product factor
210	C₂	C₂²×#102×C₆₇	1	1	yes	characteristic
211	C₂	C₂²×D₁₆×C₆₇	1	1	yes	characteristic
212	C₂	C₂²×SD₃₂×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₆₇	D₈
commutator	C₈	C₂³×C₄×C₆₇
Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
prime radical	C₂³×D₈×C₆₇	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₆₇ ◁ C₂³×C₄×C₆₇ ◁ C₂³×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂³×C₁₆×C₆₇	C₂	semidirect
3. C₂²×#107×C₆₇	C₂	semidirect
4. C₂×#357×C₆₇	C₂	direct (N × H)
8. C₂²×D₈×C₆₇	C₄	semidirect
13. C₂²×C₁₆×C₆₇	C₂²	semidirect
15. C₂×#107×C₆₇	C₂²	semidirect
16. #357×C₆₇	C₂²	direct (N × H)
27. C₂×D₈×C₆₇	C₂×C₄	semidirect
36. C₂×C₁₆×C₆₇	C₂³	semidirect
38. #107×C₆₇	C₂³	semidirect
57. D₈×C₆₇	C₂²×C₄	semidirect
92. C₂²×C₆₇	#357	direct (N × H)
101. C₂²×#357	C₆₇	direct (N × H)
103. C₂×C₆₇	C₂×#357	direct (N × H)
108. C₂³×C₁₆	C₂×C₆₇	semidirect
109. C₂²×#107	C₂×C₆₇	semidirect
110. C₂×#357	C₂×C₆₇	direct (N × H)
111. C₆₇	C₂²×#357	direct (N × H)
115. C₂²×D₈	C₄×C₆₇	semidirect
120. C₂²×C₁₆	C₂²×C₆₇	semidirect
122. C₂×#107	C₂²×C₆₇	semidirect
123. #357	C₂²×C₆₇	direct (N × H)
134. C₂×D₈	C₂×C₄×C₆₇	semidirect
143. C₂×C₁₆	C₂³×C₆₇	semidirect
145. #107	C₂³×C₆₇	semidirect
164. D₈	C₂²×C₄×C₆₇	semidirect
199. C₂²	#357×C₆₇	direct (N × H)
209. C₂	C₂×#357×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×D₈×C₆₇	2
C₂³×C₁₆×C₆₇	2
C₂²×#107×C₆₇	2
C₂×#357×C₆₇	2
C₂²×#357	67

Subgroups — C22×#357×C67 #30712625

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#357×C₆₇ #30712625