Subgroups — C₂²×C₄×C₄⋊C₄×C₆₇ #30713240

Order 17152.

5758 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

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

	subgroup	G/H
center	C₂⁴×C₄×C₆₇	C₂²
commutator	C₂	C₂³×C₄²×C₆₇
Frattini, cosocle	C₂³	C₂⁵×C₆₇
socle, prime radical	C₂⁵×C₆₇	C₂³
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂×C₄×C₄⋊C₄×C₆₇	C₂	direct (N × H)
10. C₂²×C₄⋊C₄×C₆₇	C₄	direct (N × H)
13. C₂²×C₄²×C₆₇	C₄	semidirect
16. C₄×C₄⋊C₄×C₆₇	C₂²	direct (N × H)
33. C₂×C₄⋊C₄×C₆₇	C₂×C₄	direct (N × H)
36. C₂×C₄²×C₆₇	C₂×C₄	semidirect
51. C₂²×C₄×C₆₇	C₄²	semidirect
52. C₂²×C₄×C₆₇	C₄⋊C₄	direct (N × H)
64. C₄⋊C₄×C₆₇	C₂²×C₄	direct (N × H)
66. C₄²×C₆₇	C₂²×C₄	semidirect
83. C₂×C₄×C₆₇	C₂×C₄²	semidirect
84. C₂×C₄×C₆₇	C₂×C₄⋊C₄	direct (N × H)
92. C₂²×C₆₇	C₄×C₄⋊C₄	direct (N × H)
105. C₄×C₆₇	C₂²×C₄²	semidirect
106. C₄×C₆₇	C₂²×C₄⋊C₄	direct (N × H)
107. C₂²×C₄×C₄⋊C₄	C₆₇	direct (N × H)
108. C₂×C₆₇	C₂×C₄×C₄⋊C₄	direct (N × H)
117. C₂×C₄×C₄⋊C₄	C₂×C₆₇	direct (N × H)
119. C₆₇	C₂²×C₄×C₄⋊C₄	direct (N × H)
124. C₂²×C₄⋊C₄	C₄×C₆₇	direct (N × H)
127. C₂²×C₄²	C₄×C₆₇	semidirect
130. C₄×C₄⋊C₄	C₂²×C₆₇	direct (N × H)
147. C₂×C₄⋊C₄	C₂×C₄×C₆₇	direct (N × H)
150. C₂×C₄²	C₂×C₄×C₆₇	semidirect
165. C₂²×C₄	C₄²×C₆₇	semidirect
166. C₂²×C₄	C₄⋊C₄×C₆₇	direct (N × H)
178. C₄⋊C₄	C₂²×C₄×C₆₇	direct (N × H)
180. C₄²	C₂²×C₄×C₆₇	semidirect
197. C₂×C₄	C₂×C₄²×C₆₇	semidirect
198. C₂×C₄	C₂×C₄⋊C₄×C₆₇	direct (N × H)
206. C₂²	C₄×C₄⋊C₄×C₆₇	direct (N × H)
219. C₄	C₂²×C₄²×C₆₇	semidirect
220. C₄	C₂²×C₄⋊C₄×C₆₇	direct (N × H)
221. C₂	C₂×C₄×C₄⋊C₄×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×C₄⋊C₄×C₆₇	2
C₂³×C₄²×C₆₇	2
C₂×C₄×C₄⋊C₄×C₆₇	2
C₂²×#95×C₆₇	2
C₂²×C₄×C₄⋊C₄	67

Subgroups — C22×C4×C4⋊C4×C67 #30713240

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×C₄⋊C₄×C₆₇ #30713240