Subgroups — C₂²×#16863360 #31719262

Order 17152.

7214 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11761612	C₂	1	2	yes	maximal, complemented
2	C₂×#16863360	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₂²⋊(C₆₇⋊C₈)	C₂	1	4	yes	maximal, complemented
4	C₂⁵×C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11761612	—	2	16	no	—
6	C₂²×#11761612	C₂²	1	24	yes	complemented
7	C₂³×C₆₇⋊C₈	C₂²	1	4	yes	complemented
8	#16863360	C₂²	1	64	yes	direct product factor
9	C₂×C₂²⋊(C₆₇⋊C₈)	C₂²	1	48	yes	complemented
10	C₂⁶×C₆₇	C₄	1	1	yes	characteristic
11	C₂⁴×C₄×C₆₇	—	2	16	no	—
12	C₂⁴×C₄×C₆₇	C₄	1	1	yes	characteristic
13	C₂⁴×C₄×C₆₇	C₄	1	12	yes	—
14	C₂⁴×C₄×C₆₇	C₄	1	2	yes	—
15	C₂⁴×C₄×C₆₇	C₂²	1	2	yes	—
16	C₂⁴×C₄×C₆₇	C₂²	1	12	yes	—
17	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×#11761612	—	2	96	no	—
19	C₂×#11761612	C₂³	1	32	yes	complemented
20	C₂²×C₆₇⋊C₈	—	2	16	no	—
21	C₂²×C₆₇⋊C₈	C₂³	1	24	yes	complemented
22	C₂²⋊(C₆₇⋊C₈)	C₂³	1	64	yes	complemented
23	C₂⁵×C₆₇	—	2	8	no	—
24	C₂⁵×C₆₇	—	2	16	no	—
25	C₂⁵×C₆₇	C₂×C₄	1	12	yes	—
26	C₂⁵×C₆₇	C₂×C₄	1	1	yes	characteristic
27	C₂⁵×C₆₇	C₂×C₄	1	2	yes	—
28	C₂³×C₄×C₆₇	—	2	8	no	—
29	C₂³×C₄×C₆₇	—	2	96	no	—
30	C₂³×C₄×C₆₇	—	2	16	no	—
31	C₂³×C₄×C₆₇	—	2	16	no	—
32	C₂³×C₄×C₆₇	—	2	96	no	—
33	C₂³×C₄×C₆₇	C₂×C₄	1	12	yes	—
34	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
35	C₂³×C₄×C₆₇	C₂×C₄	1	48	yes	—
36	C₂³×C₄×C₆₇	C₂×C₄	1	12	yes	—
37	C₂³×C₄×C₆₇	C₂×C₄	1	24	yes	—
38	C₂³×C₄×C₆₇	C₂×C₄	1	6	yes	—
39	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₂³×C₄×C₆₇	D₄	1	16	yes	—
41	C₂³×C₄×C₆₇	C₂³	1	12	yes	—
42	C₂³×C₄×C₆₇	C₂³	1	16	yes	—
43	C₂³×C₄×C₆₇	C₂³	1	6	yes	—
44	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
45	#11761612	—	2	64	no	—
46	C₂×C₆₇⋊C₈	—	2	48	no	—
47	C₂×C₆₇⋊C₈	C₂⁴	1	16	yes	complemented
48	C₂⁴×C₆₇	—	2	128	no	—
49	C₂⁴×C₆₇	—	2	48	no	—
50	C₂⁴×C₆₇	—	2	8	no	—
51	C₂⁴×C₆₇	—	2	8	no	—
52	C₂⁴×C₆₇	—	2	8	no	—
53	C₂⁴×C₆₇	—	2	96	no	—
54	C₂⁴×C₆₇	C₂²⋊C₄	1	8	yes	—
55	C₂⁴×C₆₇	M₁₆	1	16	yes	complemented
56	C₂⁴×C₆₇	C₂²×C₄	1	16	yes	—
57	C₂⁴×C₆₇	C₂²×C₄	1	6	yes	—
58	C₂⁴×C₆₇	C₂²×C₄	1	12	yes	—
59	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₆₇	—	2	48	no	—
61	C₂²×C₄×C₆₇	—	2	8	no	—
62	C₂²×C₄×C₆₇	—	2	192	no	—
63	C₂²×C₄×C₆₇	—	2	48	no	—
64	C₂²×C₄×C₆₇	—	2	96	no	—
65	C₂²×C₄×C₆₇	—	2	48	no	—
66	C₂²×C₄×C₆₇	—	2	64	no	—
67	C₂²×C₄×C₆₇	C₂²⋊C₄	1	48	yes	—
68	C₂²×C₄×C₆₇	C₂²⋊C₄	1	8	yes	—
69	C₂²×C₄×C₆₇	C₂²×C₄	1	16	yes	—
70	C₂²×C₄×C₆₇	C₂²×C₄	1	12	yes	—
71	C₂²×C₄×C₆₇	C₂²×C₄	1	24	yes	—
72	C₂²×C₄×C₆₇	C₂²×C₄	1	32	yes	—
73	C₂²×C₄×C₆₇	C₂²×C₄	1	12	yes	—
74	C₂²×C₄×C₆₇	C₂²×C₄	1	6	yes	—
75	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
76	C₂²×C₄×C₆₇	C₂×D₄	1	8	yes	—
77	C₂²×C₄×C₆₇	C₂×D₄	1	48	yes	—
78	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
79	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
80	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
81	C₂³×C₆₇	—	2	384	no	—
82	C₂³×C₆₇	—	2	32	no	—
83	C₂³×C₆₇	—	2	24	no	—
84	C₂³×C₆₇	—	2	24	no	—
85	C₂³×C₆₇	—	2	24	no	—
86	C₂³×C₆₇	—	2	64	no	—
87	C₂³×C₆₇	—	2	32	no	—
88	C₂³×C₆₇	—	2	64	no	—
89	C₂³×C₆₇	C₂×C₂²⋊C₄	1	24	yes	—
90	C₂³×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
91	C₂³×C₆₇	C₂×M₁₆	1	8	yes	—
92	C₂³×C₆₇	C₂×M₁₆	1	48	yes	complemented
93	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
94	C₂³×C₆₇	C₂³×C₄	1	8	yes	—
95	C₂³×C₆₇	C₂³×C₄	1	3	yes	—
96	C₂×C₄×C₆₇	—	2	32	no	—
97	C₂×C₄×C₆₇	—	2	24	no	—
98	C₂×C₄×C₆₇	—	2	48	no	—
99	C₂×C₄×C₆₇	—	2	64	no	—
100	C₂×C₄×C₆₇	—	2	16	no	—
101	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	48	yes	—
102	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	12	yes	—
103	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	24	yes	—
104	C₂×C₄×C₆₇	C₂³×C₄	1	8	yes	—
105	C₂×C₄×C₆₇	C₂³×C₄	1	3	yes	—
106	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
107	C₂×C₄×C₆₇	C₂²×D₄	1	12	yes	—
108	C₂×C₄×C₆₇	C₂²×D₄	1	16	yes	—
109	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
110	C₆₇⋊C₈	—	2	16	no	—
111	C₂²×C₆₇	—	2	128	no	—
112	C₂²×C₆₇	—	2	48	no	—
113	C₂²×C₆₇	—	2	96	no	—
114	C₂²×C₆₇	—	2	8	no	—
115	C₂²×C₆₇	—	2	8	no	—
116	C₂²×C₆₇	—	2	8	no	—
117	C₂²×C₆₇	#407	1	16	yes	complemented
118	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	8	yes	—
119	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	6	yes	—
120	C₂²×C₆₇	C₂²×M₁₆	1	12	yes	—
121	C₂²×C₆₇	C₂²×M₁₆	1	16	yes	complemented
122	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
123	C₄×C₆₇	—	2	8	no	—
124	C₄×C₆₇	C₂²×C₂²⋊C₄	1	6	yes	—
125	C₄×C₆₇	C₂²×C₂²⋊C₄	1	8	yes	—
126	C₄×C₆₇	C₂³×D₄	1	2	yes	—
127	C₂²×#407	—	67	1	no	maximal, Sylow
128	C₂×C₆₇	—	2	8	no	—
129	C₂×C₆₇	—	2	16	no	—
130	C₂×C₆₇	C₂×#407	1	12	yes	complemented
131	C₂×C₆₇	C₂³×C₂²⋊C₄	1	1	yes	characteristic
132	C₂×C₆₇	C₂³×M₁₆	1	2	yes	—
133	C₂⁵×C₄	D₆₇	1	1	yes	characteristic
134	C₂³×M₁₆	—	67	2	no	—
135	C₂×#407	—	67	24	no	—
136	C₂²×C₂²⋊C₈	—	67	4	no	—
137	C₆₇	C₂²×#407	1	1	yes	characteristic, complemented, Sylow
138	C₂⁶	Dic₆₇	1	1	yes	characteristic
139	C₂⁴×C₄	—	2	16	no	—
140	C₂⁴×C₄	Dic₆₇	1	1	yes	characteristic
141	C₂⁴×C₄	Dic₆₇	1	12	yes	—
142	C₂⁴×C₄	Dic₆₇	1	2	yes	—
143	C₂⁴×C₄	C₂×D₆₇	1	12	yes	—
144	C₂⁴×C₄	C₂×D₆₇	1	2	yes	—
145	C₂⁴×C₄	C₂×D₆₇	1	1	yes	characteristic
146	C₂²×M₁₆	—	67	24	no	—
147	C₂²×M₁₆	—	134	16	no	—
148	C₂³×C₈	—	67	4	no	—
149	#407	—	67	64	no	—
150	C₂×C₂²⋊C₈	—	67	48	no	—
151	C₂⁵	—	2	8	no	—
152	C₂⁵	—	2	16	no	—
153	C₂⁵	C₂×Dic₆₇	1	12	yes	—
154	C₂⁵	C₂×Dic₆₇	1	1	yes	characteristic
155	C₂⁵	C₂×Dic₆₇	1	2	yes	—
156	C₂³×C₄	—	2	8	no	—
157	C₂³×C₄	—	2	96	no	—
158	C₂³×C₄	—	2	16	no	—
159	C₂³×C₄	—	2	16	no	—
160	C₂³×C₄	—	2	96	no	—
161	C₂³×C₄	C₂×Dic₆₇	1	12	yes	—
162	C₂³×C₄	C₂×Dic₆₇	1	2	yes	—
163	C₂³×C₄	C₂×Dic₆₇	1	48	yes	—
164	C₂³×C₄	C₂×Dic₆₇	1	24	yes	—
165	C₂³×C₄	C₂×Dic₆₇	1	12	yes	—
166	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
167	C₂³×C₄	C₂×Dic₆₇	1	6	yes	—
168	C₂³×C₄	C₂²⋊D₆₇	1	16	yes	—
169	C₂³×C₄	C₂²×D₆₇	1	12	yes	—
170	C₂³×C₄	C₂²×D₆₇	1	16	yes	—
171	C₂³×C₄	C₂²×D₆₇	1	6	yes	—
172	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
173	C₂×M₁₆	—	67	32	no	—
174	C₂×M₁₆	—	134	96	no	—
175	C₂²×C₈	—	67	24	no	—
176	C₂²×C₈	—	134	16	no	—
177	C₂²⋊C₈	—	67	64	no	—
178	C₂⁴	—	2	128	no	—
179	C₂⁴	—	2	48	no	—
180	C₂⁴	—	2	8	no	—
181	C₂⁴	—	2	8	no	—
182	C₂⁴	—	2	8	no	—
183	C₂⁴	—	2	96	no	—
184	C₂⁴	C₂²⋊Dic₆₇	1	8	yes	—
185	C₂⁴	#11761612	1	16	yes	complemented
186	C₂⁴	C₂²×Dic₆₇	1	16	yes	—
187	C₂⁴	C₂²×Dic₆₇	1	6	yes	—
188	C₂⁴	C₂²×Dic₆₇	1	12	yes	—
189	C₂⁴	C₂²×Dic₆₇	1	1	yes	characteristic
190	C₂²×C₄	—	2	48	no	—
191	C₂²×C₄	—	2	8	no	—
192	C₂²×C₄	—	2	192	no	—
193	C₂²×C₄	—	2	96	no	—
194	C₂²×C₄	—	2	48	no	—
195	C₂²×C₄	—	2	48	no	—
196	C₂²×C₄	—	2	64	no	—
197	C₂²×C₄	C₂²⋊Dic₆₇	1	48	yes	—
198	C₂²×C₄	C₂²⋊Dic₆₇	1	8	yes	—
199	C₂²×C₄	C₂²×Dic₆₇	1	16	yes	—
200	C₂²×C₄	C₂²×Dic₆₇	1	12	yes	—
201	C₂²×C₄	C₂²×Dic₆₇	1	24	yes	—
202	C₂²×C₄	C₂²×Dic₆₇	1	32	yes	—
203	C₂²×C₄	C₂²×Dic₆₇	1	12	yes	—
204	C₂²×C₄	C₂²×Dic₆₇	1	6	yes	—
205	C₂²×C₄	C₂²×Dic₆₇	1	3	yes	—
206	C₂²×C₄	C₂×C₂²⋊D₆₇	1	8	yes	—
207	C₂²×C₄	C₂×C₂²⋊D₆₇	1	48	yes	—
208	C₂²×C₄	C₂³×D₆₇	1	8	yes	—
209	C₂²×C₄	C₂³×D₆₇	1	3	yes	—
210	C₂²×C₄	C₂³×D₆₇	1	4	yes	—
211	M₁₆	—	134	64	no	—
212	C₂×C₈	—	67	16	no	—
213	C₂×C₈	—	134	48	no	—
214	C₂³	—	2	384	no	—
215	C₂³	—	2	32	no	—
216	C₂³	—	2	24	no	—
217	C₂³	—	2	32	no	—
218	C₂³	—	2	64	no	—
219	C₂³	—	2	24	no	—
220	C₂³	—	2	24	no	—
221	C₂³	—	2	64	no	—
222	C₂³	C₂×C₂²⋊Dic₆₇	1	24	yes	—
223	C₂³	C₂×C₂²⋊Dic₆₇	1	4	yes	—
224	C₂³	C₂×#11761612	1	8	yes	—
225	C₂³	C₂×#11761612	1	48	yes	complemented
226	C₂³	C₂³×Dic₆₇	1	4	yes	—
227	C₂³	C₂³×Dic₆₇	1	8	yes	—
228	C₂³	C₂³×Dic₆₇	1	3	yes	—
229	C₂×C₄	—	2	32	no	—
230	C₂×C₄	—	2	24	no	—
231	C₂×C₄	—	2	48	no	—
232	C₂×C₄	—	2	64	no	—
233	C₂×C₄	—	2	16	no	—
234	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	48	yes	—
235	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	24	yes	—
236	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	12	yes	—
237	C₂×C₄	C₂³×Dic₆₇	1	8	yes	—
238	C₂×C₄	C₂³×Dic₆₇	1	4	yes	—
239	C₂×C₄	C₂³×Dic₆₇	1	3	yes	—
240	C₂×C₄	C₂²×C₂²⋊D₆₇	1	12	yes	—
241	C₂×C₄	C₂²×C₂²⋊D₆₇	1	16	yes	—
242	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
243	C₈	—	134	16	no	—
244	C₂²	—	2	128	no	—
245	C₂²	—	2	8	no	—
246	C₂²	—	2	8	no	—
247	C₂²	—	2	8	no	—
248	C₂²	—	2	48	no	—
249	C₂²	—	2	96	no	—
250	C₂²	#16863360	1	16	yes	direct product factor
251	C₂²	C₂²×C₂²⋊Dic₆₇	1	8	yes	—
252	C₂²	C₂²×C₂²⋊Dic₆₇	1	6	yes	—
253	C₂²	C₂²×#11761612	1	12	yes	—
254	C₂²	C₂²×#11761612	1	16	yes	complemented
255	C₂²	C₂⁴×Dic₆₇	1	1	yes	characteristic
256	C₄	—	2	8	no	—
257	C₄	C₂²×C₂²⋊Dic₆₇	1	6	yes	—
258	C₄	C₂²×C₂²⋊Dic₆₇	1	8	yes	—
259	C₄	C₂³×C₂²⋊D₆₇	1	2	yes	—
260	C₂	—	2	8	no	—
261	C₂	—	2	16	no	—
262	C₂	C₂×#16863360	1	12	yes	direct product factor
263	C₂	C₂³×C₂²⋊Dic₆₇	1	1	yes	characteristic
264	C₂	C₂³×#11761612	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₆₇
commutator	C₂²×C₆₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₆₇
Fitting	C₂⁵×C₄×C₆₇	C₂
socle	C₂⁴×C₆₇	C₂²×C₄
prime radical	C₂⁶×C₆₇	C₄
cosocle	C₂×C₄×C₆₇	C₂⁵
hypercenter	C₂⁵×C₄	D₆₇
hypocenter	C₆₇	C₂²×#407
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁵×C₄

Complements

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

N	H ≅ G/N	type
1. C₂³×#11761612	C₂	semidirect
2. C₂×#16863360	C₂	direct (N × H)
3. C₂²×C₂²⋊(C₆₇⋊C₈)	C₂	semidirect
6. C₂²×#11761612	C₂²	semidirect
7. C₂³×C₆₇⋊C₈	C₂²	semidirect
8. #16863360	C₂²	direct (N × H)
9. C₂×C₂²⋊(C₆₇⋊C₈)	C₂²	semidirect
19. C₂×#11761612	C₂³	semidirect
21. C₂²×C₆₇⋊C₈	C₂³	semidirect
22. C₂²⋊(C₆₇⋊C₈)	C₂³	semidirect
47. C₂×C₆₇⋊C₈	C₂⁴	semidirect
55. C₂⁴×C₆₇	M₁₆	semidirect
92. C₂³×C₆₇	C₂×M₁₆	semidirect
117. C₂²×C₆₇	#407	semidirect
121. C₂²×C₆₇	C₂²×M₁₆	semidirect
130. C₂×C₆₇	C₂×#407	semidirect
137. C₆₇	C₂²×#407	semidirect
185. C₂⁴	#11761612	semidirect
225. C₂³	C₂×#11761612	semidirect
250. C₂²	#16863360	direct (N × H)
254. C₂²	C₂²×#11761612	semidirect
262. C₂	C₂×#16863360	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×#11761612	2
C₂×#16863360	2
C₂²×C₂²⋊(C₆₇⋊C₈)	2
C₂⁵×C₄×C₆₇	2
C₂²×#407	67

Subgroups — C22×#16863360 #31719262

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16863360 #31719262