Subgroups — C₂×C₄×Q₃₂×C₆₇ #30712697

Order 17152.

678 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×Q₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₁₆×C₆₇	C₂	1	2	yes	maximal
3	C₄×Q₃₂×C₆₇	C₂	1	8	yes	maximal, direct product factor
4	C₂×#366×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#358×C₆₇	C₂	1	2	yes	maximal
6	C₂×C₄×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₁₆×C₆₇	C₂²	1	2	yes	—
8	C₂×C₄×Q₈×C₆₇	—	2	2	no	—
9	C₂×Q₃₂×C₆₇	C₄	1	8	yes	direct product factor
10	C₂×Q₃₂×C₆₇	C₂²	1	4	yes	—
11	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
12	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
13	C₄×Q₁₆×C₆₇	—	2	4	no	—
14	C₄×Q₁₆×C₆₇	C₂²	1	8	yes	—
15	C₂×#107×C₆₇	C₂²	1	2	yes	—
16	C₂×#103×C₆₇	—	2	2	no	—
17	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	#366×C₆₇	C₂²	1	4	yes	—
19	#358×C₆₇	C₂²	1	8	yes	—
20	C₄×C₁₆×C₆₇	C₂²	1	4	yes	—
21	C₂²×Q₈×C₆₇	—	2	2	no	—
22	C₂×Q₁₆×C₆₇	—	2	2	no	—
23	C₂×Q₁₆×C₆₇	—	2	4	no	—
24	C₂×Q₁₆×C₆₇	C₂×C₄	1	8	yes	—
25	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
26	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
28	C₄×Q₈×C₆₇	—	2	8	no	—
29	C₄×Q₈×C₆₇	—	4	4	no	—
30	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
31	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
32	C₂×C₄²×C₆₇	—	4	2	no	—
33	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
34	Q₃₂×C₆₇	C₂×C₄	1	16	yes	direct product factor
35	C₂×C₁₆×C₆₇	—	2	2	no	—
36	C₂×C₁₆×C₆₇	C₂×C₄	1	4	yes	—
37	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
38	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
39	#107×C₆₇	—	2	2	no	—
40	#107×C₆₇	C₂³	1	4	yes	—
41	#103×C₆₇	—	2	8	no	—
42	C₄×C₈×C₆₇	D₄	1	2	yes	—
43	C₄×C₈×C₆₇	C₂³	1	2	yes	—
44	C₂×Q₈×C₆₇	—	2	8	no	—
45	C₂×Q₈×C₆₇	—	2	4	no	—
46	C₂×Q₈×C₆₇	—	4	2	no	—
47	C₂×Q₈×C₆₇	—	4	4	no	—
48	C₂²×C₄×C₆₇	—	4	2	no	—
49	C₂²×C₄×C₆₇	—	4	2	no	—
50	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
51	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
52	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
53	Q₁₆×C₆₇	—	2	8	no	—
54	Q₁₆×C₆₇	—	2	4	no	—
55	Q₁₆×C₆₇	C₂²×C₄	1	8	yes	—
56	C₂×C₈×C₆₇	—	2	2	no	—
57	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
58	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
59	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₆₇	#41	1	2	yes	—
61	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
62	C₄⋊C₄×C₆₇	—	2	4	no	—
63	C₄⋊C₄×C₆₇	—	4	2	no	—
64	C₄⋊C₄×C₆₇	—	4	8	no	—
65	C₄²×C₆₇	—	4	8	no	—
66	C₄²×C₆₇	D₈	1	2	yes	—
67	C₄²×C₆₇	C₂×D₄	1	2	yes	—
68	C₁₆×C₆₇	—	2	2	no	—
69	C₁₆×C₆₇	C₂²×C₄	1	4	yes	—
70	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
71	Q₈×C₆₇	—	2	8	no	—
72	Q₈×C₆₇	—	4	4	no	—
73	Q₈×C₆₇	—	4	8	no	—
74	C₂×C₄×C₆₇	—	2	2	no	—
75	C₂×C₄×C₆₇	—	4	4	no	—
76	C₂×C₄×C₆₇	—	4	8	no	—
77	C₂×C₄×C₆₇	—	4	4	no	—
78	C₂×C₄×C₆₇	—	8	4	no	—
79	C₂×C₄×C₆₇	Q₃₂	1	4	yes	direct product factor
80	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
81	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
82	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
83	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
84	C₂×C₄×C₆₇	#135	1	2	yes	—
85	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
86	C₈×C₆₇	—	2	2	no	—
87	C₈×C₆₇	C₄×D₄	1	2	yes	—
88	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
89	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
90	C₂²×C₆₇	C₄×D₈	1	2	yes	—
91	C₂²×C₆₇	C₂×Q₃₂	1	2	yes	—
92	C₂²×C₆₇	#508	1	2	yes	—
93	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
94	C₄×C₆₇	—	2	2	no	—
95	C₄×C₆₇	—	4	8	no	—
96	C₄×C₆₇	—	8	4	no	—
97	C₄×C₆₇	C₄×D₈	1	2	yes	—
98	C₄×C₆₇	C₂×Q₃₂	1	4	yes	direct product factor
99	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
100	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
101	C₂×C₄×Q₃₂	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
102	C₂×C₆₇	C₄×Q₃₂	1	4	yes	direct product factor
103	C₂×C₆₇	C₂×C₄×D₈	1	1	yes	characteristic
104	C₂×C₆₇	C₂²×Q₃₂	1	1	yes	characteristic
105	C₂×C₆₇	C₂×#508	1	1	yes	characteristic
106	C₂²×Q₃₂	C₂×C₆₇	1	1	yes	characteristic
107	C₂×C₄×Q₁₆	C₂×C₆₇	1	2	yes	—
108	C₄×Q₃₂	C₂×C₆₇	1	8	yes	direct product factor
109	C₂×#366	C₂×C₆₇	1	1	yes	characteristic
110	C₂×#358	C₂×C₆₇	1	2	yes	—
111	C₂×C₄×C₁₆	C₂×C₆₇	1	1	yes	characteristic
112	C₆₇	C₂×C₄×Q₃₂	1	1	yes	characteristic, direct product factor, Sylow
113	C₂²×Q₁₆	C₂²×C₆₇	1	2	yes	—
114	C₂×C₄×Q₈	—	2	2	no	—
115	C₂×Q₃₂	C₄×C₆₇	1	8	yes	direct product factor
116	C₂×Q₃₂	C₂²×C₆₇	1	4	yes	—
117	C₂²×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
118	C₂²×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
119	C₄×Q₁₆	—	2	4	no	—
120	C₄×Q₁₆	C₂²×C₆₇	1	8	yes	—
121	C₂×#107	C₂²×C₆₇	1	2	yes	—
122	C₂×#103	—	2	2	no	—
123	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
124	#366	C₂²×C₆₇	1	4	yes	—
125	#358	C₂²×C₆₇	1	8	yes	—
126	C₄×C₁₆	C₂²×C₆₇	1	4	yes	—
127	C₂²×Q₈	—	2	2	no	—
128	C₂×Q₁₆	—	2	2	no	—
129	C₂×Q₁₆	—	2	4	no	—
130	C₂×Q₁₆	C₂×C₄×C₆₇	1	8	yes	—
131	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
132	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
133	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
134	C₄×Q₈	—	2	8	no	—
135	C₄×Q₈	—	4	4	no	—
136	C₂×C₄⋊C₄	—	2	2	no	—
137	C₂×C₄⋊C₄	—	4	2	no	—
138	C₂×C₄²	—	4	2	no	—
139	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
140	Q₃₂	C₂×C₄×C₆₇	1	16	yes	direct product factor
141	C₂×C₁₆	—	2	2	no	—
142	C₂×C₁₆	C₂×C₄×C₆₇	1	4	yes	—
143	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
144	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
145	#107	—	2	2	no	—
146	#107	C₂³×C₆₇	1	4	yes	—
147	#103	—	2	8	no	—
148	C₄×C₈	D₄×C₆₇	1	2	yes	—
149	C₄×C₈	C₂³×C₆₇	1	2	yes	—
150	C₂×Q₈	—	2	8	no	—
151	C₂×Q₈	—	2	4	no	—
152	C₂×Q₈	—	4	2	no	—
153	C₂×Q₈	—	4	4	no	—
154	C₂²×C₄	—	4	2	no	—
155	C₂²×C₄	—	4	2	no	—
156	C₂²×C₄	D₈×C₆₇	1	1	yes	characteristic
157	C₂²×C₄	D₈×C₆₇	1	1	yes	characteristic
158	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
159	Q₁₆	—	2	8	no	—
160	Q₁₆	—	2	4	no	—
161	Q₁₆	C₂²×C₄×C₆₇	1	8	yes	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
164	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
165	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
166	C₂×C₈	#41×C₆₇	1	2	yes	—
167	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
168	C₄⋊C₄	—	2	4	no	—
169	C₄⋊C₄	—	4	2	no	—
170	C₄⋊C₄	—	4	8	no	—
171	C₄²	—	4	8	no	—
172	C₄²	D₈×C₆₇	1	2	yes	—
173	C₄²	C₂×D₄×C₆₇	1	2	yes	—
174	C₁₆	—	2	2	no	—
175	C₁₆	C₂²×C₄×C₆₇	1	4	yes	—
176	C₂³	C₂×D₈×C₆₇	1	1	yes	characteristic
177	Q₈	—	2	8	no	—
178	Q₈	—	4	4	no	—
179	Q₈	—	4	8	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	4	4	no	—
182	C₂×C₄	—	4	8	no	—
183	C₂×C₄	—	4	4	no	—
184	C₂×C₄	—	8	4	no	—
185	C₂×C₄	Q₃₂×C₆₇	1	4	yes	direct product factor
186	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
187	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
188	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
189	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
190	C₂×C₄	#135×C₆₇	1	2	yes	—
191	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
192	C₈	—	2	2	no	—
193	C₈	C₄×D₄×C₆₇	1	2	yes	—
194	C₈	C₂³×C₄×C₆₇	1	1	yes	characteristic
195	C₈	C₂×#41×C₆₇	1	1	yes	characteristic
196	C₂²	C₄×D₈×C₆₇	1	2	yes	—
197	C₂²	C₂×Q₃₂×C₆₇	1	2	yes	—
198	C₂²	#508×C₆₇	1	2	yes	—
199	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
200	C₄	—	2	2	no	—
201	C₄	—	4	8	no	—
202	C₄	—	8	4	no	—
203	C₄	C₄×D₈×C₆₇	1	2	yes	—
204	C₄	C₂×Q₃₂×C₆₇	1	4	yes	direct product factor
205	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
206	C₄	C₂×#135×C₆₇	1	1	yes	characteristic
207	C₂	C₄×Q₃₂×C₆₇	1	4	yes	direct product factor
208	C₂	C₂×C₄×D₈×C₆₇	1	1	yes	characteristic
209	C₂	C₂²×Q₃₂×C₆₇	1	1	yes	characteristic
210	C₂	C₂×#508×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₆₇	D₈
commutator	C₈	C₂³×C₄×C₆₇
Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
socle, prime radical	C₂³×C₆₇	C₂×D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ C₂×C₄×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄×Q₃₂×C₆₇	C₂	direct (N × H)
9. C₂×Q₃₂×C₆₇	C₄	direct (N × H)
34. Q₃₂×C₆₇	C₂×C₄	direct (N × H)
79. C₂×C₄×C₆₇	Q₃₂	direct (N × H)
98. C₄×C₆₇	C₂×Q₃₂	direct (N × H)
101. C₂×C₄×Q₃₂	C₆₇	direct (N × H)
102. C₂×C₆₇	C₄×Q₃₂	direct (N × H)
108. C₄×Q₃₂	C₂×C₆₇	direct (N × H)
112. C₆₇	C₂×C₄×Q₃₂	direct (N × H)
115. C₂×Q₃₂	C₄×C₆₇	direct (N × H)
140. Q₃₂	C₂×C₄×C₆₇	direct (N × H)
185. C₂×C₄	Q₃₂×C₆₇	direct (N × H)
204. C₄	C₂×Q₃₂×C₆₇	direct (N × H)
207. C₂	C₄×Q₃₂×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
C₂²×Q₃₂×C₆₇	2
C₂×C₄×Q₁₆×C₆₇	2
C₄×Q₃₂×C₆₇	2
C₂×#366×C₆₇	2
C₂×#358×C₆₇	2
C₂×C₄×C₁₆×C₆₇	2
C₂×C₄×Q₃₂	67

Subgroups — C2×C4×Q32×C67 #30712697

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×Q₃₂×C₆₇ #30712697