Subgroups — C₂×C₆₇×#2150 #30712646

Order 17152.

798 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₃₂×C₆₇	C₂	1	2	yes	maximal
2	C₂²×C₄×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2150	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄⋊C₁₆×C₆₇	C₂	1	4	yes	maximal, complemented
5	C₂³×C₈×C₆₇	C₄	1	1	yes	characteristic
6	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
7	C₂²×C₄²×C₆₇	C₄	1	1	yes	characteristic
8	C₂×M₃₂×C₆₇	—	2	8	no	—
9	C₂×M₃₂×C₆₇	C₂²	1	8	yes	—
10	C₂²×C₁₆×C₆₇	C₂²	1	4	yes	—
11	C₂×C₄×C₈×C₆₇	C₄	1	4	yes	—
12	C₂×C₄×C₈×C₆₇	C₄	1	2	yes	—
13	C₂×C₄×C₈×C₆₇	C₂²	1	4	yes	—
14	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
15	C₄⋊C₁₆×C₆₇	C₂²	1	16	yes	complemented
16	C₂³×C₄×C₆₇	C₈	1	2	yes	—
17	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₆₇	—	2	4	no	—
19	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
20	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
21	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₆₇	D₄	1	4	yes	—
24	C₂²×C₈×C₆₇	Q₈	1	4	yes	—
25	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
26	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
27	C₂×C₄²×C₆₇	C₈	1	1	yes	characteristic
28	C₂×C₄²×C₆₇	C₈	1	4	yes	—
29	C₂×C₄²×C₆₇	C₈	1	1	yes	characteristic
30	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
32	M₃₂×C₆₇	—	2	16	no	—
33	C₂×C₁₆×C₆₇	—	2	8	no	—
34	C₂×C₁₆×C₆₇	C₂³	1	8	yes	—
35	C₄×C₈×C₆₇	C₂×C₄	1	8	yes	—
36	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
37	C₄×C₈×C₆₇	C₂³	1	4	yes	—
38	C₂⁴×C₆₇	C₂×C₈	1	1	yes	characteristic
39	C₂²×C₄×C₆₇	—	2	8	no	—
40	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
41	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
42	C₂²×C₄×C₆₇	C₂×C₈	1	8	yes	—
43	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
44	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
46	C₂²×C₄×C₆₇	M₁₆	1	4	yes	—
47	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
48	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₆₇	—	2	8	no	—
50	C₂×C₈×C₆₇	—	2	4	no	—
51	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
52	C₂×C₈×C₆₇	C₄⋊C₄	1	8	yes	—
53	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
54	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
55	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
57	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
58	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
59	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
60	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
61	C₄²×C₆₇	C₂×C₈	1	4	yes	—
62	C₄²×C₆₇	C₂×C₈	1	4	yes	—
63	C₄²×C₆₇	C₂×C₈	1	4	yes	—
64	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
65	C₁₆×C₆₇	—	2	8	no	—
66	C₂³×C₆₇	—	2	4	no	—
67	C₂³×C₆₇	C₄⋊C₈	1	4	yes	—
68	C₂³×C₆₇	C₂²×C₈	1	2	yes	—
69	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
70	C₂×C₄×C₆₇	—	2	4	no	—
71	C₂×C₄×C₆₇	—	2	16	no	—
72	C₂×C₄×C₆₇	C₄⋊C₈	1	8	yes	—
73	C₂×C₄×C₆₇	C₄⋊C₈	1	4	yes	—
74	C₂×C₄×C₆₇	M₃₂	1	8	yes	complemented
75	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
76	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
77	C₂×C₄×C₆₇	C₂²×C₈	1	4	yes	—
78	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
79	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄×C₆₇	C₂²×C₈	1	4	yes	—
81	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
82	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
83	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
84	C₈×C₆₇	—	2	4	no	—
85	C₈×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
86	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
87	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
88	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
89	C₂²×C₆₇	—	2	4	no	—
90	C₂²×C₆₇	—	2	8	no	—
91	C₂²×C₆₇	C₂×C₄⋊C₈	1	4	yes	—
92	C₂²×C₆₇	C₂×C₄⋊C₈	1	2	yes	—
93	C₂²×C₆₇	C₂×M₃₂	1	4	yes	—
94	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
95	C₄×C₆₇	—	2	4	no	—
96	C₄×C₆₇	C₂×C₄⋊C₈	1	2	yes	—
97	C₄×C₆₇	C₂×C₄⋊C₈	1	4	yes	—
98	C₄×C₆₇	C₂×M₃₂	1	8	yes	complemented
99	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
100	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
101	C₂×#2150	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
102	C₂×C₆₇	—	2	4	no	—
103	C₂×C₆₇	#2150	1	4	yes	direct product factor
104	C₂×C₆₇	C₂²×C₄⋊C₈	1	1	yes	characteristic
105	C₂×C₆₇	C₂²×M₃₂	1	2	yes	—
106	C₂²×M₃₂	C₂×C₆₇	1	2	yes	—
107	C₂²×C₄×C₈	C₂×C₆₇	1	1	yes	characteristic
108	#2150	C₂×C₆₇	1	8	yes	direct product factor
109	C₂×C₄⋊C₁₆	C₂×C₆₇	1	4	yes	complemented
110	C₆₇	C₂×#2150	1	1	yes	characteristic, direct product factor, Sylow
111	C₂³×C₈	C₄×C₆₇	1	1	yes	characteristic
112	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
113	C₂²×C₄²	C₄×C₆₇	1	1	yes	characteristic
114	C₂×M₃₂	—	2	8	no	—
115	C₂×M₃₂	C₂²×C₆₇	1	8	yes	—
116	C₂²×C₁₆	C₂²×C₆₇	1	4	yes	—
117	C₂×C₄×C₈	C₄×C₆₇	1	4	yes	—
118	C₂×C₄×C₈	C₄×C₆₇	1	2	yes	—
119	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
120	C₂×C₄×C₈	C₂²×C₆₇	1	4	yes	—
121	C₄⋊C₁₆	C₂²×C₆₇	1	16	yes	complemented
122	C₂³×C₄	C₈×C₆₇	1	2	yes	—
123	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
124	C₂²×C₈	—	2	4	no	—
125	C₂²×C₈	C₂×C₄×C₆₇	1	2	yes	—
126	C₂²×C₈	C₂×C₄×C₆₇	1	4	yes	—
127	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
128	C₂²×C₈	C₂×C₄×C₆₇	1	2	yes	—
129	C₂²×C₈	D₄×C₆₇	1	4	yes	—
130	C₂²×C₈	Q₈×C₆₇	1	4	yes	—
131	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
132	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
133	C₂×C₄²	C₈×C₆₇	1	1	yes	characteristic
134	C₂×C₄²	C₈×C₆₇	1	4	yes	—
135	C₂×C₄²	C₈×C₆₇	1	1	yes	characteristic
136	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
137	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
138	M₃₂	—	2	16	no	—
139	C₂×C₁₆	—	2	8	no	—
140	C₂×C₁₆	C₂³×C₆₇	1	8	yes	—
141	C₄×C₈	C₂×C₄×C₆₇	1	4	yes	—
142	C₄×C₈	C₂×C₄×C₆₇	1	8	yes	—
143	C₄×C₈	C₂³×C₆₇	1	4	yes	—
144	C₂⁴	C₂×C₈×C₆₇	1	1	yes	characteristic
145	C₂²×C₄	—	2	8	no	—
146	C₂²×C₄	C₄⋊C₄×C₆₇	1	4	yes	—
147	C₂²×C₄	C₂×C₈×C₆₇	1	2	yes	—
148	C₂²×C₄	C₂×C₈×C₆₇	1	8	yes	—
149	C₂²×C₄	C₂×C₈×C₆₇	1	2	yes	—
150	C₂²×C₄	C₂×C₈×C₆₇	1	1	yes	characteristic
151	C₂²×C₄	C₂×C₈×C₆₇	1	2	yes	—
152	C₂²×C₄	M₁₆×C₆₇	1	4	yes	—
153	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
154	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
155	C₂×C₈	—	2	8	no	—
156	C₂×C₈	—	2	4	no	—
157	C₂×C₈	C₄⋊C₄×C₆₇	1	8	yes	—
158	C₂×C₈	C₄⋊C₄×C₆₇	1	4	yes	—
159	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
160	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
161	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
162	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
163	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
164	C₂×C₈	C₂×Q₈×C₆₇	1	4	yes	—
165	C₂×C₈	C₂×Q₈×C₆₇	1	2	yes	—
166	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
167	C₄²	C₂×C₈×C₆₇	1	4	yes	—
168	C₄²	C₂×C₈×C₆₇	1	4	yes	—
169	C₄²	C₂×C₈×C₆₇	1	4	yes	—
170	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
171	C₁₆	—	2	8	no	—
172	C₂³	—	2	4	no	—
173	C₂³	C₄⋊C₈×C₆₇	1	4	yes	—
174	C₂³	C₂²×C₈×C₆₇	1	2	yes	—
175	C₂³	C₂²×C₈×C₆₇	1	1	yes	characteristic
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	16	no	—
178	C₂×C₄	C₄⋊C₈×C₆₇	1	8	yes	—
179	C₂×C₄	C₄⋊C₈×C₆₇	1	4	yes	—
180	C₂×C₄	M₃₂×C₆₇	1	8	yes	complemented
181	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
182	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
183	C₂×C₄	C₂²×C₈×C₆₇	1	4	yes	—
184	C₂×C₄	C₂²×C₈×C₆₇	1	4	yes	—
185	C₂×C₄	C₂²×C₈×C₆₇	1	2	yes	—
186	C₂×C₄	C₂²×C₈×C₆₇	1	1	yes	characteristic
187	C₂×C₄	C₂×M₁₆×C₆₇	1	4	yes	—
188	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
189	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
190	C₈	—	2	4	no	—
191	C₈	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
192	C₈	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
193	C₈	C₂²×D₄×C₆₇	1	1	yes	characteristic
194	C₈	C₂²×Q₈×C₆₇	1	1	yes	characteristic
195	C₂²	—	2	4	no	—
196	C₂²	—	2	8	no	—
197	C₂²	C₂×C₄⋊C₈×C₆₇	1	4	yes	—
198	C₂²	C₂×C₄⋊C₈×C₆₇	1	2	yes	—
199	C₂²	C₂×M₃₂×C₆₇	1	4	yes	—
200	C₂²	C₂³×C₈×C₆₇	1	1	yes	characteristic
201	C₄	—	2	4	no	—
202	C₄	C₂×C₄⋊C₈×C₆₇	1	2	yes	—
203	C₄	C₂×C₄⋊C₈×C₆₇	1	4	yes	—
204	C₄	C₂×M₃₂×C₆₇	1	8	yes	complemented
205	C₄	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
206	C₄	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
207	C₂	—	2	4	no	—
208	C₂	C₆₇×#2150	1	4	yes	direct product factor
209	C₂	C₂²×C₄⋊C₈×C₆₇	1	1	yes	characteristic
210	C₂	C₂²×M₃₂×C₆₇	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈×C₆₇	C₂³
commutator	C₂²	C₂³×C₈×C₆₇
Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
socle	C₂³×C₆₇	C₂²×C₈
prime radical	C₂⁴×C₆₇	C₂×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₇×#2150	C₂	direct (N × H)
4. C₂×C₄⋊C₁₆×C₆₇	C₂	semidirect
15. C₄⋊C₁₆×C₆₇	C₂²	semidirect
74. C₂×C₄×C₆₇	M₃₂	semidirect
98. C₄×C₆₇	C₂×M₃₂	semidirect
101. C₂×#2150	C₆₇	direct (N × H)
103. C₂×C₆₇	#2150	direct (N × H)
108. #2150	C₂×C₆₇	direct (N × H)
109. C₂×C₄⋊C₁₆	C₂×C₆₇	semidirect
110. C₆₇	C₂×#2150	direct (N × H)
121. C₄⋊C₁₆	C₂²×C₆₇	semidirect
180. C₂×C₄	M₃₂×C₆₇	semidirect
204. C₄	C₂×M₃₂×C₆₇	semidirect
208. C₂	C₆₇×#2150	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×M₃₂×C₆₇	2
C₂²×C₄×C₈×C₆₇	2
C₆₇×#2150	2
C₂×C₄⋊C₁₆×C₆₇	2
C₂×#2150	67

Subgroups — C2×C67×#2150 #30712646

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2150 #30712646