Subgroups — C₂×C₆₇×#2183 #30712724

Order 17152.

542 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2183	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊C₁₆×C₆₇	C₂	1	2	yes	maximal
4	C₂×C₄⋊C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#346×C₆₇	C₂	1	2	yes	maximal
6	C₂×C₄×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄×Q₈×C₆₇	C₄	1	1	yes	characteristic
8	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
9	C₂²×C₁₆×C₆₇	C₂²	1	2	yes	—
10	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
11	C₈×Q₈×C₆₇	C₄	1	4	yes	—
12	C₈×Q₈×C₆₇	C₂²	1	4	yes	—
13	C₂×C₄⋊C₈×C₆₇	C₄	1	2	yes	—
14	C₂×C₄⋊C₈×C₆₇	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
17	C₄⋊C₁₆×C₆₇	C₂²	1	8	yes	—
18	C₄⋊C₁₆×C₆₇	C₂²	1	4	yes	—
19	#346×C₆₇	C₂²	1	8	yes	—
20	C₄×C₁₆×C₆₇	C₂²	1	4	yes	—
21	C₂²×Q₈×C₆₇	C₈	1	1	yes	characteristic
22	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
25	C₄×Q₈×C₆₇	C₈	1	4	yes	—
26	C₄×Q₈×C₆₇	C₂×C₄	1	4	yes	—
27	C₂×C₄⋊C₄×C₆₇	C₈	1	2	yes	—
28	C₂×C₄⋊C₄×C₆₇	C₈	1	1	yes	characteristic
29	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₆×C₆₇	—	2	2	no	—
32	C₂×C₁₆×C₆₇	—	2	4	no	—
33	C₂×C₁₆×C₆₇	Q₈	1	4	yes	complemented
34	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
35	C₂×C₁₆×C₆₇	C₂³	1	4	yes	—
36	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
37	C₄⋊C₈×C₆₇	C₂×C₄	1	8	yes	—
38	C₄⋊C₈×C₆₇	C₂×C₄	1	4	yes	—
39	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
41	C₄×C₈×C₆₇	C₂³	1	2	yes	—
42	C₄×C₈×C₆₇	C₂³	1	4	yes	—
43	C₂×Q₈×C₆₇	—	2	4	no	—
44	C₂×Q₈×C₆₇	C₂×C₈	1	4	yes	—
45	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
46	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
47	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
48	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
49	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₆₇	—	2	2	no	—
51	C₂×C₈×C₆₇	—	2	4	no	—
52	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
53	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
54	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
56	C₂×C₈×C₆₇	#41	1	2	yes	—
57	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
58	C₄⋊C₄×C₆₇	C₂×C₈	1	8	yes	—
59	C₄⋊C₄×C₆₇	C₂×C₈	1	4	yes	—
60	C₄²×C₆₇	C₂×C₈	1	4	yes	—
61	C₄²×C₆₇	C₂×C₈	1	2	yes	—
62	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
63	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
64	C₁₆×C₆₇	—	2	2	no	—
65	C₁₆×C₆₇	—	2	4	no	—
66	C₁₆×C₆₇	C₂×Q₈	1	4	yes	complemented
67	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
68	Q₈×C₆₇	—	2	8	no	—
69	C₂×C₄×C₆₇	—	2	4	no	—
70	C₂×C₄×C₆₇	—	2	4	no	—
71	C₂×C₄×C₆₇	—	2	2	no	—
72	C₂×C₄×C₆₇	M₃₂	1	4	yes	—
73	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
74	C₂×C₄×C₆₇	C₂²×C₈	1	4	yes	—
75	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
76	C₂×C₄×C₆₇	C₂²×C₈	1	4	yes	—
77	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
78	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
79	C₂×C₄×C₆₇	#131	1	2	yes	—
80	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
81	C₈×C₆₇	—	2	2	no	—
82	C₈×C₆₇	—	2	4	no	—
83	C₈×C₆₇	C₄×Q₈	1	2	yes	—
84	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
85	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
86	C₂²×C₆₇	C₈×Q₈	1	2	yes	—
87	C₂²×C₆₇	C₂×M₃₂	1	2	yes	—
88	C₂²×C₆₇	#504	1	2	yes	—
89	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
90	C₄×C₆₇	—	2	4	no	—
91	C₄×C₆₇	—	2	4	no	—
92	C₄×C₆₇	—	2	2	no	—
93	C₄×C₆₇	C₈×Q₈	1	2	yes	—
94	C₄×C₆₇	C₂×M₃₂	1	4	yes	—
95	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
96	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
97	C₂×#2183	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₂×C₆₇	#2183	1	4	yes	direct product factor
99	C₂×C₆₇	C₂×C₈×Q₈	1	1	yes	characteristic
100	C₂×C₆₇	C₂²×M₃₂	1	1	yes	characteristic
101	C₂×C₆₇	C₂×#504	1	1	yes	characteristic
102	C₂×C₈×Q₈	C₂×C₆₇	1	1	yes	characteristic
103	#2183	C₂×C₆₇	1	8	yes	direct product factor
104	C₂×C₄⋊C₁₆	C₂×C₆₇	1	1	yes	characteristic
105	C₂×C₄⋊C₁₆	C₂×C₆₇	1	2	yes	—
106	C₂×#346	C₂×C₆₇	1	2	yes	—
107	C₂×C₄×C₁₆	C₂×C₆₇	1	1	yes	characteristic
108	C₆₇	C₂×#2183	1	1	yes	characteristic, direct product factor, Sylow
109	C₂×C₄×Q₈	C₄×C₆₇	1	1	yes	characteristic
110	C₂²×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
111	C₂²×C₁₆	C₂²×C₆₇	1	2	yes	—
112	C₂²×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
113	C₈×Q₈	C₄×C₆₇	1	4	yes	—
114	C₈×Q₈	C₂²×C₆₇	1	4	yes	—
115	C₂×C₄⋊C₈	C₄×C₆₇	1	2	yes	—
116	C₂×C₄⋊C₈	C₄×C₆₇	1	1	yes	characteristic
117	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
118	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
119	C₄⋊C₁₆	C₂²×C₆₇	1	4	yes	—
120	C₄⋊C₁₆	C₂²×C₆₇	1	8	yes	—
121	#346	C₂²×C₆₇	1	8	yes	—
122	C₄×C₁₆	C₂²×C₆₇	1	4	yes	—
123	C₂²×Q₈	C₈×C₆₇	1	1	yes	characteristic
124	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
125	C₂²×C₈	C₂×C₄×C₆₇	1	2	yes	—
126	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
127	C₄×Q₈	C₈×C₆₇	1	4	yes	—
128	C₄×Q₈	C₂×C₄×C₆₇	1	4	yes	—
129	C₂×C₄⋊C₄	C₈×C₆₇	1	2	yes	—
130	C₂×C₄⋊C₄	C₈×C₆₇	1	1	yes	characteristic
131	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
132	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
133	C₂×C₁₆	—	2	2	no	—
134	C₂×C₁₆	—	2	4	no	—
135	C₂×C₁₆	Q₈×C₆₇	1	4	yes	complemented
136	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
137	C₂×C₁₆	C₂³×C₆₇	1	4	yes	—
138	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
139	C₄⋊C₈	C₂×C₄×C₆₇	1	8	yes	—
140	C₄⋊C₈	C₂×C₄×C₆₇	1	4	yes	—
141	C₄×C₈	C₂×C₄×C₆₇	1	2	yes	—
142	C₄×C₈	C₂×C₄×C₆₇	1	4	yes	—
143	C₄×C₈	C₂³×C₆₇	1	2	yes	—
144	C₄×C₈	C₂³×C₆₇	1	4	yes	—
145	C₂×Q₈	—	2	4	no	—
146	C₂×Q₈	C₂×C₈×C₆₇	1	4	yes	—
147	C₂²×C₄	C₂×C₈×C₆₇	1	2	yes	—
148	C₂²×C₄	C₂×C₈×C₆₇	1	1	yes	characteristic
149	C₂²×C₄	C₂×C₈×C₆₇	1	2	yes	—
150	C₂²×C₄	C₂×C₈×C₆₇	1	1	yes	characteristic
151	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	2	4	no	—
154	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
155	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
156	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
157	C₂×C₈	C₂×Q₈×C₆₇	1	2	yes	—
158	C₂×C₈	#41×C₆₇	1	2	yes	—
159	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
160	C₄⋊C₄	C₂×C₈×C₆₇	1	8	yes	—
161	C₄⋊C₄	C₂×C₈×C₆₇	1	4	yes	—
162	C₄²	C₂×C₈×C₆₇	1	4	yes	—
163	C₄²	C₂×C₈×C₆₇	1	2	yes	—
164	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
165	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
166	C₁₆	—	2	2	no	—
167	C₁₆	—	2	4	no	—
168	C₁₆	C₂×Q₈×C₆₇	1	4	yes	complemented
169	C₂³	C₂²×C₈×C₆₇	1	1	yes	characteristic
170	Q₈	—	2	8	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	M₃₂×C₆₇	1	4	yes	—
175	C₂×C₄	C₄×Q₈×C₆₇	1	2	yes	—
176	C₂×C₄	C₂²×C₈×C₆₇	1	4	yes	—
177	C₂×C₄	C₂²×C₈×C₆₇	1	2	yes	—
178	C₂×C₄	C₂²×C₈×C₆₇	1	4	yes	—
179	C₂×C₄	C₂²×C₈×C₆₇	1	2	yes	—
180	C₂×C₄	C₂²×C₈×C₆₇	1	1	yes	characteristic
181	C₂×C₄	#131×C₆₇	1	2	yes	—
182	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
183	C₈	—	2	2	no	—
184	C₈	—	2	4	no	—
185	C₈	C₄×Q₈×C₆₇	1	2	yes	—
186	C₈	C₂²×Q₈×C₆₇	1	1	yes	characteristic
187	C₈	C₂×#41×C₆₇	1	1	yes	characteristic
188	C₂²	C₈×Q₈×C₆₇	1	2	yes	—
189	C₂²	C₂×M₃₂×C₆₇	1	2	yes	—
190	C₂²	#504×C₆₇	1	2	yes	—
191	C₂²	C₂³×C₈×C₆₇	1	1	yes	characteristic
192	C₄	—	2	4	no	—
193	C₄	—	2	4	no	—
194	C₄	—	2	2	no	—
195	C₄	C₈×Q₈×C₆₇	1	2	yes	—
196	C₄	C₂×M₃₂×C₆₇	1	4	yes	—
197	C₄	C₂×C₄×Q₈×C₆₇	1	1	yes	characteristic
198	C₄	C₂×#131×C₆₇	1	1	yes	characteristic
199	C₂	C₆₇×#2183	1	4	yes	direct product factor
200	C₂	C₂×C₈×Q₈×C₆₇	1	1	yes	characteristic
201	C₂	C₂²×M₃₂×C₆₇	1	1	yes	characteristic
202	C₂	C₂×#504×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈×C₆₇	C₂³
commutator	C₂²	C₂³×C₈×C₆₇
Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
socle, prime radical	C₂³×C₆₇	C₂²×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
2. C₆₇×#2183	C₂	direct (N × H)
33. C₂×C₁₆×C₆₇	Q₈	semidirect
66. C₁₆×C₆₇	C₂×Q₈	semidirect
97. C₂×#2183	C₆₇	direct (N × H)
98. C₂×C₆₇	#2183	direct (N × H)
103. #2183	C₂×C₆₇	direct (N × H)
108. C₆₇	C₂×#2183	direct (N × H)
135. C₂×C₁₆	Q₈×C₆₇	semidirect
168. C₁₆	C₂×Q₈×C₆₇	semidirect
199. C₂	C₆₇×#2183	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×C₈×Q₈×C₆₇	2
C₆₇×#2183	2
C₂×C₄⋊C₁₆×C₆₇	2
C₂×#346×C₆₇	2
C₂×C₄×C₁₆×C₆₇	2
C₂×#2183	67

Subgroups — C2×C67×#2183 #30712724

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2183 #30712724