Subgroups — C₂²×C₄×C₂²⋊C₄×C₆₇ #30713239

Order 17152.

9534 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₂²⋊C₄×C₆₇	C₂	1	2	yes	maximal
3	C₂³×C₄²×C₆₇	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×C₂²⋊C₄×C₆₇	C₂	1	24	yes	maximal, direct product factor
5	C₂²×#95×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂⁶×C₆₇	C₂²	1	1	yes	characteristic
7	C₂⁴×C₄×C₆₇	—	2	16	no	—
8	C₂⁴×C₄×C₆₇	C₄	1	16	yes	complemented
9	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
10	C₂⁴×C₄×C₆₇	C₂²	1	4	yes	—
11	C₂⁴×C₄×C₆₇	C₂²	1	12	yes	—
12	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×C₂²⋊C₄×C₆₇	C₄	1	32	yes	direct product factor
14	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	24	yes	—
15	C₂²×C₄²×C₆₇	—	2	16	no	—
16	C₂²×C₄²×C₆₇	C₂²	1	24	yes	complemented
17	C₄×C₂²⋊C₄×C₆₇	C₂²	1	64	yes	direct product factor
18	C₂×#95×C₆₇	C₂²	1	24	yes	complemented
19	C₂⁵×C₆₇	—	2	8	no	—
20	C₂⁵×C₆₇	—	2	8	no	—
21	C₂⁵×C₆₇	C₂×C₄	1	16	yes	—
22	C₂⁵×C₆₇	C₂×C₄	1	8	yes	—
23	C₂⁵×C₆₇	C₂³	1	6	yes	—
24	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
25	C₂³×C₄×C₆₇	—	2	16	no	—
26	C₂³×C₄×C₆₇	—	2	8	no	—
27	C₂³×C₄×C₆₇	—	2	16	no	—
28	C₂³×C₄×C₆₇	—	2	128	no	—
29	C₂³×C₄×C₆₇	—	2	96	no	—
30	C₂³×C₄×C₆₇	C₂×C₄	1	8	yes	—
31	C₂³×C₄×C₆₇	C₂×C₄	1	8	yes	—
32	C₂³×C₄×C₆₇	C₂×C₄	1	96	yes	complemented
33	C₂³×C₄×C₆₇	C₂×C₄	1	32	yes	complemented
34	C₂³×C₄×C₆₇	D₄	1	16	yes	—
35	C₂³×C₄×C₆₇	C₂³	1	6	yes	—
36	C₂³×C₄×C₆₇	C₂³	1	24	yes	—
37	C₂³×C₄×C₆₇	C₂³	1	16	yes	—
38	C₂³×C₄×C₆₇	C₂³	1	6	yes	—
39	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	192	yes	direct product factor
40	C₂×C₂²⋊C₄×C₆₇	C₂³	1	32	yes	—
41	C₂×C₄²×C₆₇	—	2	96	no	—
42	C₂×C₄²×C₆₇	C₂³	1	32	yes	complemented
43	#95×C₆₇	C₂³	1	32	yes	complemented
44	C₂⁴×C₆₇	—	2	32	no	—
45	C₂⁴×C₆₇	—	2	32	no	—
46	C₂⁴×C₆₇	—	2	128	no	—
47	C₂⁴×C₆₇	—	2	24	no	—
48	C₂⁴×C₆₇	—	2	24	no	—
49	C₂⁴×C₆₇	C₄²	1	64	yes	complemented
50	C₂⁴×C₆₇	C₂²×C₄	1	48	yes	—
51	C₂⁴×C₆₇	C₂²×C₄	1	24	yes	—
52	C₂⁴×C₆₇	C₂²×C₄	1	4	yes	—
53	C₂⁴×C₆₇	C₂²×C₄	1	8	yes	—
54	C₂⁴×C₆₇	C₂×D₄	1	8	yes	—
55	C₂⁴×C₆₇	#41	1	8	yes	—
56	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
57	C₂⁴×C₆₇	C₂⁴	1	3	yes	—
58	C₂²×C₄×C₆₇	—	2	32	no	—
59	C₂²×C₄×C₆₇	—	2	128	no	—
60	C₂²×C₄×C₆₇	—	2	48	no	—
61	C₂²×C₄×C₆₇	—	2	24	no	—
62	C₂²×C₄×C₆₇	—	2	48	no	—
63	C₂²×C₄×C₆₇	—	2	384	no	—
64	C₂²×C₄×C₆₇	—	2	64	no	—
65	C₂²×C₄×C₆₇	C₂²⋊C₄	1	64	yes	direct product factor
66	C₂²×C₄×C₆₇	C₂²×C₄	1	24	yes	—
67	C₂²×C₄×C₆₇	C₂²×C₄	1	24	yes	—
68	C₂²×C₄×C₆₇	C₂²×C₄	1	64	yes	complemented
69	C₂²×C₄×C₆₇	C₂²×C₄	1	96	yes	complemented
70	C₂²×C₄×C₆₇	C₂×D₄	1	48	yes	—
71	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
72	C₂²×C₄×C₆₇	C₂⁴	1	16	yes	—
73	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
74	C₂²⋊C₄×C₆₇	C₂²×C₄	1	128	yes	direct product factor
75	C₄²×C₆₇	—	2	64	no	—
76	C₂³×C₆₇	—	2	48	no	—
77	C₂³×C₆₇	—	2	48	no	—
78	C₂³×C₆₇	—	2	192	no	—
79	C₂³×C₆₇	—	2	8	no	—
80	C₂³×C₆₇	—	2	8	no	—
81	C₂³×C₆₇	—	2	256	no	—
82	C₂³×C₆₇	C₂×C₄²	1	16	yes	—
83	C₂³×C₆₇	C₂×C₄²	1	96	yes	complemented
84	C₂³×C₆₇	C₂×C₂²⋊C₄	1	16	yes	—
85	C₂³×C₆₇	#117	1	16	yes	—
86	C₂³×C₆₇	C₄×D₄	1	64	yes	—
87	C₂³×C₆₇	C₂³×C₄	1	16	yes	—
88	C₂³×C₆₇	C₂³×C₄	1	8	yes	—
89	C₂³×C₆₇	C₂³×C₄	1	6	yes	—
90	C₂³×C₆₇	C₂³×C₄	1	12	yes	—
91	C₂³×C₆₇	C₂²×D₄	1	12	yes	—
92	C₂³×C₆₇	C₂×#41	1	12	yes	—
93	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
94	C₂×C₄×C₆₇	—	2	48	no	—
95	C₂×C₄×C₆₇	—	2	192	no	—
96	C₂×C₄×C₆₇	—	2	16	no	—
97	C₂×C₄×C₆₇	—	2	8	no	—
98	C₂×C₄×C₆₇	—	2	16	no	—
99	C₂×C₄×C₆₇	—	2	128	no	—
100	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	96	yes	direct product factor
101	C₂×C₄×C₆₇	C₂³×C₄	1	8	yes	—
102	C₂×C₄×C₆₇	C₂³×C₄	1	8	yes	—
103	C₂×C₄×C₆₇	C₂³×C₄	1	32	yes	complemented
104	C₂×C₄×C₆₇	C₂²×D₄	1	16	yes	—
105	C₂²×C₆₇	—	2	8	no	—
106	C₂²×C₆₇	—	2	8	no	—
107	C₂²×C₆₇	—	2	32	no	—
108	C₂²×C₆₇	—	2	192	no	—
109	C₂²×C₆₇	C₄×C₂²⋊C₄	1	64	yes	direct product factor
110	C₂²×C₆₇	C₂²×C₄²	1	12	yes	—
111	C₂²×C₆₇	C₂²×C₄²	1	16	yes	complemented
112	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	12	yes	—
113	C₂²×C₆₇	C₂×#117	1	12	yes	—
114	C₂²×C₆₇	C₂×C₄×D₄	1	48	yes	—
115	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
116	C₂²×C₆₇	C₂⁴×C₄	1	2	yes	—
117	C₂²×C₆₇	C₂³×D₄	1	2	yes	—
118	C₂²×C₆₇	C₂²×#41	1	2	yes	—
119	C₄×C₆₇	—	2	8	no	—
120	C₄×C₆₇	—	2	32	no	—
121	C₄×C₆₇	C₂²×C₂²⋊C₄	1	16	yes	direct product factor
122	C₂²×C₄×C₂²⋊C₄	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
123	C₂×C₆₇	—	2	16	no	—
124	C₂×C₆₇	C₂×C₄×C₂²⋊C₄	1	24	yes	direct product factor
125	C₂×C₆₇	C₂³×C₄²	1	1	yes	characteristic
126	C₂×C₆₇	C₂³×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₆₇	C₂²×#117	1	1	yes	characteristic
128	C₂×C₆₇	C₂²×C₄×D₄	1	4	yes	—
129	C₂⁵×C₄	C₂×C₆₇	1	1	yes	characteristic
130	C₂³×C₂²⋊C₄	C₂×C₆₇	1	2	yes	—
131	C₂³×C₄²	C₂×C₆₇	1	2	yes	complemented
132	C₂×C₄×C₂²⋊C₄	C₂×C₆₇	1	24	yes	direct product factor
133	C₂²×#95	C₂×C₆₇	1	2	yes	complemented
134	C₆₇	C₂²×C₄×C₂²⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
135	C₂⁶	C₂²×C₆₇	1	1	yes	characteristic
136	C₂⁴×C₄	—	2	16	no	—
137	C₂⁴×C₄	C₄×C₆₇	1	16	yes	complemented
138	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
139	C₂⁴×C₄	C₂²×C₆₇	1	12	yes	—
140	C₂⁴×C₄	C₂²×C₆₇	1	4	yes	—
141	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
142	C₂²×C₂²⋊C₄	C₄×C₆₇	1	32	yes	direct product factor
143	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	24	yes	—
144	C₂²×C₄²	—	2	16	no	—
145	C₂²×C₄²	C₂²×C₆₇	1	24	yes	complemented
146	C₄×C₂²⋊C₄	C₂²×C₆₇	1	64	yes	direct product factor
147	C₂×#95	C₂²×C₆₇	1	24	yes	complemented
148	C₂⁵	—	2	8	no	—
149	C₂⁵	—	2	8	no	—
150	C₂⁵	C₂×C₄×C₆₇	1	16	yes	—
151	C₂⁵	C₂×C₄×C₆₇	1	8	yes	—
152	C₂⁵	C₂³×C₆₇	1	6	yes	—
153	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
154	C₂³×C₄	—	2	8	no	—
155	C₂³×C₄	—	2	16	no	—
156	C₂³×C₄	—	2	16	no	—
157	C₂³×C₄	—	2	128	no	—
158	C₂³×C₄	—	2	96	no	—
159	C₂³×C₄	C₂×C₄×C₆₇	1	8	yes	—
160	C₂³×C₄	C₂×C₄×C₆₇	1	8	yes	—
161	C₂³×C₄	C₂×C₄×C₆₇	1	96	yes	complemented
162	C₂³×C₄	C₂×C₄×C₆₇	1	32	yes	complemented
163	C₂³×C₄	D₄×C₆₇	1	16	yes	—
164	C₂³×C₄	C₂³×C₆₇	1	6	yes	—
165	C₂³×C₄	C₂³×C₆₇	1	24	yes	—
166	C₂³×C₄	C₂³×C₆₇	1	16	yes	—
167	C₂³×C₄	C₂³×C₆₇	1	6	yes	—
168	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	192	yes	direct product factor
169	C₂×C₂²⋊C₄	C₂³×C₆₇	1	32	yes	—
170	C₂×C₄²	—	2	96	no	—
171	C₂×C₄²	C₂³×C₆₇	1	32	yes	complemented
172	#95	C₂³×C₆₇	1	32	yes	complemented
173	C₂⁴	—	2	32	no	—
174	C₂⁴	—	2	32	no	—
175	C₂⁴	—	2	128	no	—
176	C₂⁴	—	2	24	no	—
177	C₂⁴	—	2	24	no	—
178	C₂⁴	C₄²×C₆₇	1	64	yes	complemented
179	C₂⁴	C₂²×C₄×C₆₇	1	48	yes	—
180	C₂⁴	C₂²×C₄×C₆₇	1	24	yes	—
181	C₂⁴	C₂²×C₄×C₆₇	1	4	yes	—
182	C₂⁴	C₂²×C₄×C₆₇	1	8	yes	—
183	C₂⁴	C₂×D₄×C₆₇	1	8	yes	—
184	C₂⁴	#41×C₆₇	1	8	yes	—
185	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
186	C₂⁴	C₂⁴×C₆₇	1	3	yes	—
187	C₂²×C₄	—	2	32	no	—
188	C₂²×C₄	—	2	128	no	—
189	C₂²×C₄	—	2	48	no	—
190	C₂²×C₄	—	2	24	no	—
191	C₂²×C₄	—	2	48	no	—
192	C₂²×C₄	—	2	384	no	—
193	C₂²×C₄	—	2	64	no	—
194	C₂²×C₄	C₂²⋊C₄×C₆₇	1	64	yes	direct product factor
195	C₂²×C₄	C₂²×C₄×C₆₇	1	24	yes	—
196	C₂²×C₄	C₂²×C₄×C₆₇	1	24	yes	—
197	C₂²×C₄	C₂²×C₄×C₆₇	1	64	yes	complemented
198	C₂²×C₄	C₂²×C₄×C₆₇	1	96	yes	complemented
199	C₂²×C₄	C₂×D₄×C₆₇	1	48	yes	—
200	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
201	C₂²×C₄	C₂⁴×C₆₇	1	16	yes	—
202	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
203	C₂²⋊C₄	C₂²×C₄×C₆₇	1	128	yes	direct product factor
204	C₄²	—	2	64	no	—
205	C₂³	—	2	48	no	—
206	C₂³	—	2	48	no	—
207	C₂³	—	2	192	no	—
208	C₂³	—	2	8	no	—
209	C₂³	—	2	8	no	—
210	C₂³	—	2	256	no	—
211	C₂³	C₂×C₄²×C₆₇	1	16	yes	—
212	C₂³	C₂×C₄²×C₆₇	1	96	yes	complemented
213	C₂³	C₂×C₂²⋊C₄×C₆₇	1	16	yes	—
214	C₂³	#117×C₆₇	1	16	yes	—
215	C₂³	C₄×D₄×C₆₇	1	64	yes	—
216	C₂³	C₂³×C₄×C₆₇	1	16	yes	—
217	C₂³	C₂³×C₄×C₆₇	1	8	yes	—
218	C₂³	C₂³×C₄×C₆₇	1	6	yes	—
219	C₂³	C₂³×C₄×C₆₇	1	12	yes	—
220	C₂³	C₂²×D₄×C₆₇	1	12	yes	—
221	C₂³	C₂×#41×C₆₇	1	12	yes	—
222	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
223	C₂×C₄	—	2	48	no	—
224	C₂×C₄	—	2	192	no	—
225	C₂×C₄	—	2	16	no	—
226	C₂×C₄	—	2	8	no	—
227	C₂×C₄	—	2	16	no	—
228	C₂×C₄	—	2	128	no	—
229	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	96	yes	direct product factor
230	C₂×C₄	C₂³×C₄×C₆₇	1	8	yes	—
231	C₂×C₄	C₂³×C₄×C₆₇	1	8	yes	—
232	C₂×C₄	C₂³×C₄×C₆₇	1	32	yes	complemented
233	C₂×C₄	C₂²×D₄×C₆₇	1	16	yes	—
234	C₂²	—	2	192	no	—
235	C₂²	—	2	8	no	—
236	C₂²	—	2	8	no	—
237	C₂²	—	2	32	no	—
238	C₂²	C₄×C₂²⋊C₄×C₆₇	1	64	yes	direct product factor
239	C₂²	C₂²×C₄²×C₆₇	1	12	yes	—
240	C₂²	C₂²×C₄²×C₆₇	1	16	yes	complemented
241	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	12	yes	—
242	C₂²	C₂×#117×C₆₇	1	12	yes	—
243	C₂²	C₂×C₄×D₄×C₆₇	1	48	yes	—
244	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
245	C₂²	C₂⁴×C₄×C₆₇	1	2	yes	—
246	C₂²	C₂³×D₄×C₆₇	1	2	yes	—
247	C₂²	C₂²×#41×C₆₇	1	2	yes	—
248	C₄	—	2	8	no	—
249	C₄	—	2	32	no	—
250	C₄	C₂²×C₂²⋊C₄×C₆₇	1	16	yes	direct product factor
251	C₂	—	2	16	no	—
252	C₂	C₂×C₄×C₂²⋊C₄×C₆₇	1	24	yes	direct product factor
253	C₂	C₂³×C₄²×C₆₇	1	1	yes	characteristic
254	C₂	C₂³×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
255	C₂	C₂²×#117×C₆₇	1	1	yes	characteristic
256	C₂	C₂²×C₄×D₄×C₆₇	1	4	yes	—

	subgroup	G/H
center	C₂⁴×C₄×C₆₇	C₂²
commutator	C₂	C₂³×C₄²×C₆₇
Frattini, cosocle	C₂³	C₂⁵×C₆₇
socle	C₂⁵×C₆₇	C₂³
prime radical	C₂⁶×C₆₇	C₂²
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
3. C₂³×C₄²×C₆₇	C₂	semidirect
4. C₂×C₄×C₂²⋊C₄×C₆₇	C₂	direct (N × H)
5. C₂²×#95×C₆₇	C₂	semidirect
8. C₂⁴×C₄×C₆₇	C₄	semidirect
13. C₂²×C₂²⋊C₄×C₆₇	C₄	direct (N × H)
16. C₂²×C₄²×C₆₇	C₂²	semidirect
17. C₄×C₂²⋊C₄×C₆₇	C₂²	direct (N × H)
18. C₂×#95×C₆₇	C₂²	semidirect
32. C₂³×C₄×C₆₇	C₂×C₄	semidirect
33. C₂³×C₄×C₆₇	C₂×C₄	semidirect
39. C₂×C₂²⋊C₄×C₆₇	C₂×C₄	direct (N × H)
42. C₂×C₄²×C₆₇	C₂³	semidirect
43. #95×C₆₇	C₂³	semidirect
49. C₂⁴×C₆₇	C₄²	semidirect
65. C₂²×C₄×C₆₇	C₂²⋊C₄	direct (N × H)
68. C₂²×C₄×C₆₇	C₂²×C₄	semidirect
69. C₂²×C₄×C₆₇	C₂²×C₄	semidirect
74. C₂²⋊C₄×C₆₇	C₂²×C₄	direct (N × H)
83. C₂³×C₆₇	C₂×C₄²	semidirect
100. C₂×C₄×C₆₇	C₂×C₂²⋊C₄	direct (N × H)
103. C₂×C₄×C₆₇	C₂³×C₄	semidirect
109. C₂²×C₆₇	C₄×C₂²⋊C₄	direct (N × H)
111. C₂²×C₆₇	C₂²×C₄²	semidirect
121. C₄×C₆₇	C₂²×C₂²⋊C₄	direct (N × H)
122. C₂²×C₄×C₂²⋊C₄	C₆₇	direct (N × H)
124. C₂×C₆₇	C₂×C₄×C₂²⋊C₄	direct (N × H)
131. C₂³×C₄²	C₂×C₆₇	semidirect
132. C₂×C₄×C₂²⋊C₄	C₂×C₆₇	direct (N × H)
133. C₂²×#95	C₂×C₆₇	semidirect
134. C₆₇	C₂²×C₄×C₂²⋊C₄	direct (N × H)
137. C₂⁴×C₄	C₄×C₆₇	semidirect
142. C₂²×C₂²⋊C₄	C₄×C₆₇	direct (N × H)
145. C₂²×C₄²	C₂²×C₆₇	semidirect
146. C₄×C₂²⋊C₄	C₂²×C₆₇	direct (N × H)
147. C₂×#95	C₂²×C₆₇	semidirect
161. C₂³×C₄	C₂×C₄×C₆₇	semidirect
162. C₂³×C₄	C₂×C₄×C₆₇	semidirect
168. C₂×C₂²⋊C₄	C₂×C₄×C₆₇	direct (N × H)
171. C₂×C₄²	C₂³×C₆₇	semidirect
172. #95	C₂³×C₆₇	semidirect
178. C₂⁴	C₄²×C₆₇	semidirect
194. C₂²×C₄	C₂²⋊C₄×C₆₇	direct (N × H)
197. C₂²×C₄	C₂²×C₄×C₆₇	semidirect
198. C₂²×C₄	C₂²×C₄×C₆₇	semidirect
203. C₂²⋊C₄	C₂²×C₄×C₆₇	direct (N × H)
212. C₂³	C₂×C₄²×C₆₇	semidirect
229. C₂×C₄	C₂×C₂²⋊C₄×C₆₇	direct (N × H)
232. C₂×C₄	C₂³×C₄×C₆₇	semidirect
238. C₂²	C₄×C₂²⋊C₄×C₆₇	direct (N × H)
240. C₂²	C₂²×C₄²×C₆₇	semidirect
250. C₄	C₂²×C₂²⋊C₄×C₆₇	direct (N × H)
252. C₂	C₂×C₄×C₂²⋊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
C₂⁵×C₄×C₆₇	2
C₂³×C₂²⋊C₄×C₆₇	2
C₂³×C₄²×C₆₇	2
C₂×C₄×C₂²⋊C₄×C₆₇	2
C₂²×#95×C₆₇	2
C₂²×C₄×C₂²⋊C₄	67

Subgroups — C22×C4×C22⋊C4×C67 #30713239

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×C₂²⋊C₄×C₆₇ #30713239