Subgroups — #31008931

Order 17152.

582 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	#22562016	C₂	1	4	yes	maximal, complemented
2	#22562012	C₂	1	4	yes	maximal, complemented
3	#22561968	C₂	1	2	yes	maximal, complemented
4	#22561823	C₂	1	2	yes	maximal
5	C₄×#14523983	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16863172	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×#131×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#14523983	C₂²	1	2	yes	—
9	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
10	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	8	yes	—
11	C₂²⋊(C₆₇⋊C₁₆)	C₂²	1	8	yes	—
12	#16863172	C₂²	1	2	yes	—
13	#16863172	C₂²	1	2	yes	—
14	#16863172	C₂²	1	2	yes	—
15	C₄×C₆₇⋊C₁₆	C₂²	1	2	yes	—
16	C₂×#131×C₆₇	C₄	1	1	yes	characteristic
17	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
18	C₄×#41×C₆₇	C₄	1	1	yes	characteristic
19	#405×C₆₇	C₄	1	2	yes	—
20	#405×C₆₇	C₄	1	1	yes	characteristic
21	#405×C₆₇	C₂²	1	1	yes	characteristic
22	#405×C₆₇	C₂²	1	2	yes	—
23	C₄×M₁₆×C₆₇	C₄	1	2	yes	—
24	C₄×M₁₆×C₆₇	C₄	1	1	yes	characteristic
25	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
27	#14523983	—	2	4	no	—
28	C₂×C₆₇⋊C₁₆	—	2	4	no	—
29	C₂×C₆₇⋊C₁₆	C₂³	1	4	yes	—
30	C₂×C₆₇⋊C₁₆	C₂³	1	4	yes	—
31	C₂×#41×C₆₇	C₂×C₄	1	1	yes	characteristic
32	#131×C₆₇	—	2	4	no	—
33	#131×C₆₇	—	2	4	no	—
34	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
35	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
36	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
37	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
38	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
39	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
41	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
42	C₄×Q₈×C₆₇	C₈	1	2	yes	—
43	C₄×D₄×C₆₇	C₈	1	4	yes	—
44	C₄×D₄×C₆₇	C₈	1	2	yes	—
45	#117×C₆₇	C₂×C₄	1	2	yes	—
46	#117×C₆₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
49	#97×C₆₇	C₂×C₄	1	2	yes	—
50	#97×C₆₇	C₂×C₄	1	2	yes	—
51	#97×C₆₇	C₂×C₄	1	2	yes	—
52	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
53	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
55	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
57	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
58	C₄×C₈×C₆₇	C₂³	1	2	yes	—
59	C₆₇⋊C₁₆	—	2	4	no	—
60	C₆₇⋊C₁₆	—	2	4	no	—
61	#41×C₆₇	—	2	2	no	—
62	#41×C₆₇	—	2	2	no	—
63	C₂×Q₈×C₆₇	C₂×C₈	1	1	yes	characteristic
64	C₂×D₄×C₆₇	C₂×C₈	1	2	yes	—
65	C₂×D₄×C₆₇	C₂×C₈	1	1	yes	characteristic
66	C₂²×C₄×C₆₇	C₂×C₈	1	4	yes	—
67	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
68	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
70	M₁₆×C₆₇	—	2	2	no	—
71	M₁₆×C₆₇	—	2	2	no	—
72	M₁₆×C₆₇	—	2	4	no	—
73	M₁₆×C₆₇	—	2	4	no	—
74	C₂×C₈×C₆₇	—	2	4	no	—
75	C₂×C₈×C₆₇	—	2	4	no	—
76	C₂×C₈×C₆₇	M₁₆	1	4	yes	—
77	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₆₇	#41	1	4	yes	—
83	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
84	C₄⋊C₄×C₆₇	C₂×C₈	1	4	yes	—
85	C₄⋊C₄×C₆₇	C₂×C₈	1	2	yes	—
86	C₂²⋊C₄×C₆₇	C₂×C₈	1	4	yes	—
87	C₂²⋊C₄×C₆₇	C₂×C₈	1	2	yes	—
88	C₄²×C₆₇	C₂×C₈	1	4	yes	—
89	C₄²×C₆₇	C₂×C₈	1	2	yes	—
90	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
91	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
93	C₂³×C₆₇	C₂²×C₈	1	2	yes	—
94	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
95	Q₈×C₆₇	—	2	2	no	—
96	D₄×C₆₇	—	2	4	no	—
97	D₄×C₆₇	—	2	2	no	—
98	C₂×C₄×C₆₇	—	2	2	no	—
99	C₂×C₄×C₆₇	—	2	2	no	—
100	C₂×C₄×C₆₇	—	4	2	no	—
101	C₂×C₄×C₆₇	—	4	4	no	—
102	C₂×C₄×C₆₇	#117	1	2	yes	—
103	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
104	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
105	C₂×C₄×C₆₇	C₂²×C₈	1	4	yes	—
106	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
107	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
109	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
110	C₈×C₆₇	—	2	2	no	—
111	C₈×C₆₇	—	2	2	no	—
112	C₈×C₆₇	#117	1	2	yes	—
113	C₈×C₆₇	C₂×M₁₆	1	2	yes	—
114	C₈×C₆₇	C₂×M₁₆	1	2	yes	—
115	C₈×C₆₇	C₂×#41	1	2	yes	—
116	C₂²×C₆₇	—	2	2	no	—
117	C₂²×C₆₇	—	2	4	no	—
118	C₂²×C₆₇	—	2	2	no	—
119	C₂²×C₆₇	#431	1	2	yes	—
120	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
121	C₄×C₆₇	—	2	2	no	—
122	C₄×C₆₇	—	2	2	no	—
123	C₄×C₆₇	—	2	4	no	—
124	C₄×C₆₇	—	2	2	no	—
125	C₄×C₆₇	#431	1	2	yes	—
126	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
127	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
128	#33421	—	67	1	no	maximal, Sylow
129	C₂×C₆₇	—	2	2	no	—
130	C₂×C₆₇	—	4	2	no	—
131	C₂×C₆₇	C₂×#431	1	1	yes	characteristic
132	C₂×C₆₇	#3407	1	2	yes	—
133	C₄×#131	D₆₇	1	1	yes	characteristic
134	#2164	—	67	4	no	—
135	#2163	—	67	4	no	—
136	#2151	—	67	2	no	—
137	#2113	—	67	2	no	—
138	C₄×M₃₂	—	67	1	no	—
139	C₂×#346	—	67	1	no	—
140	C₆₇	#33421	1	1	yes	characteristic, complemented, Sylow
141	C₂×#131	Dic₆₇	1	1	yes	characteristic
142	C₂×#131	C₂×D₆₇	1	1	yes	characteristic
143	C₄×#41	Dic₆₇	1	1	yes	characteristic
144	C₂×M₃₂	—	67	2	no	—
145	C₂²×C₁₆	—	67	2	no	—
146	#405	Dic₆₇	1	2	yes	—
147	#405	Dic₆₇	1	1	yes	characteristic
148	#405	C₂×D₆₇	1	1	yes	characteristic
149	#405	C₂×D₆₇	1	2	yes	—
150	C₄×M₁₆	Dic₆₇	1	1	yes	characteristic
151	C₄×M₁₆	Dic₆₇	1	2	yes	—
152	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
153	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
154	C₄⋊C₁₆	—	67	8	no	—
155	C₂²⋊C₁₆	—	67	8	no	—
156	#346	—	67	2	no	—
157	#346	—	67	2	no	—
158	#346	—	67	2	no	—
159	C₄×C₁₆	—	67	2	no	—
160	C₂×#41	C₂×Dic₆₇	1	1	yes	characteristic
161	#131	—	2	4	no	—
162	#131	—	2	4	no	—
163	C₂×M₁₆	C₂×Dic₆₇	1	1	yes	characteristic
164	C₂×M₁₆	C₂×Dic₆₇	1	2	yes	—
165	C₂×M₁₆	C₂×Dic₆₇	1	1	yes	characteristic
166	C₂×M₁₆	C₂×Dic₆₇	1	2	yes	—
167	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
168	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
169	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
170	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
171	C₄×Q₈	C₆₇⋊C₈	1	2	yes	—
172	C₄×D₄	C₆₇⋊C₈	1	4	yes	—
173	C₄×D₄	C₆₇⋊C₈	1	2	yes	—
174	#117	C₂×Dic₆₇	1	2	yes	—
175	#117	C₂×Dic₆₇	1	1	yes	characteristic
176	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
177	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
178	M₃₂	—	134	4	no	—
179	C₂×C₁₆	—	67	4	no	—
180	C₂×C₁₆	—	67	4	no	—
181	C₂×C₁₆	—	134	4	no	—
182	#97	C₂×Dic₆₇	1	2	yes	—
183	#97	C₂×Dic₆₇	1	2	yes	—
184	#97	C₂×Dic₆₇	1	2	yes	—
185	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
186	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
187	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
188	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
189	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
190	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
191	C₄×C₈	C₂²×D₆₇	1	2	yes	—
192	#41	—	2	2	no	—
193	#41	—	2	2	no	—
194	C₂×Q₈	C₂×C₆₇⋊C₈	1	1	yes	characteristic
195	C₂×D₄	C₂×C₆₇⋊C₈	1	2	yes	—
196	C₂×D₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
197	C₂²×C₄	C₂×C₆₇⋊C₈	1	4	yes	—
198	C₂²×C₄	C₂×C₆₇⋊C₈	1	2	yes	—
199	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
200	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
201	M₁₆	—	2	2	no	—
202	M₁₆	—	2	2	no	—
203	M₁₆	—	2	4	no	—
204	M₁₆	—	2	4	no	—
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	#11761612	1	4	yes	—
208	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
209	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
210	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
211	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
212	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
213	C₂×C₈	#11761628	1	4	yes	—
214	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
215	C₄⋊C₄	C₂×C₆₇⋊C₈	1	4	yes	—
216	C₄⋊C₄	C₂×C₆₇⋊C₈	1	2	yes	—
217	C₂²⋊C₄	C₂×C₆₇⋊C₈	1	4	yes	—
218	C₂²⋊C₄	C₂×C₆₇⋊C₈	1	2	yes	—
219	C₄²	C₂×C₆₇⋊C₈	1	4	yes	—
220	C₄²	C₂×C₆₇⋊C₈	1	2	yes	—
221	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
222	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
223	C₄²	C₂²×Dic₆₇	1	2	yes	—
224	C₁₆	—	134	4	no	—
225	C₁₆	—	134	4	no	—
226	C₂³	C₂²×C₆₇⋊C₈	1	2	yes	—
227	C₂³	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
228	Q₈	—	2	2	no	—
229	D₄	—	2	4	no	—
230	D₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	#14524002	1	2	yes	—
236	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
237	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
238	C₂×C₄	C₂²×C₆₇⋊C₈	1	4	yes	—
239	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
240	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
241	C₂×C₄	C₂×#11761612	1	2	yes	—
242	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
243	C₈	—	2	2	no	—
244	C₈	—	2	2	no	—
245	C₈	#14524002	1	2	yes	—
246	C₈	C₂×#11761612	1	2	yes	—
247	C₈	C₂×#11761612	1	2	yes	—
248	C₈	C₂×#11761628	1	2	yes	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	4	no	—
251	C₂²	—	2	2	no	—
252	C₂²	#16863449	1	2	yes	—
253	C₂²	C₂³×C₆₇⋊C₈	1	1	yes	characteristic
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	2	4	no	—
257	C₄	—	2	2	no	—
258	C₄	#16863449	1	2	yes	—
259	C₄	C₂×#14524002	1	1	yes	characteristic
260	C₄	C₂²×#11761612	1	1	yes	characteristic
261	C₂	—	2	2	no	—
262	C₂	—	4	2	no	—
263	C₂	C₂×#16863449	1	1	yes	characteristic
264	C₂	#22573312	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₆₇
commutator, socle	C₂²×C₆₇	C₂³×C₈
Fitting	C₄×#131×C₆₇	C₂
prime radical	C₂×#41×C₆₇	C₂×C₄
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	C₄×#131	D₆₇
hypocenter	C₆₇	#33421
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
1. #22562016	C₂	semidirect
2. #22562012	C₂	semidirect
3. #22561968	C₂	semidirect
5. C₄×#14523983	C₂	semidirect
6. C₂×#16863172	C₂	semidirect
140. C₆₇	#33421	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22562016	2
#22562012	2
#22561968	2
#22561823	2
C₄×#14523983	2
C₂×#16863172	2
C₄×#131×C₆₇	2
#33421	67