Subgroups — #31009142

Order 17152.

682 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#135×Dic₆₇	C₂	1	2	yes	maximal, complemented
2	#22562085	C₂	1	2	yes	maximal, complemented
3	#22562080	C₂	1	2	yes	maximal, complemented
4	#22562074	C₂	1	4	yes	maximal, complemented
5	#22561993	C₂	1	1	yes	maximal, characteristic, complemented
6	#22561916	C₂	1	2	yes	maximal, complemented
7	C₂×#16863171	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#508×C₆₇	C₂	1	1	yes	maximal, characteristic
9	#41×Dic₆₇	—	2	2	no	—
10	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
11	Q₁₆×Dic₆₇	C₂²	1	4	yes	—
12	SD₁₆×Dic₆₇	—	2	4	no	—
13	D₈×Dic₆₇	C₂²	1	4	yes	—
14	#16863434	C₂²	1	2	yes	—
15	#16863392	—	2	2	no	—
16	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
17	#16863224	C₂²	1	2	yes	—
18	#16863222	C₂²	1	2	yes	—
19	#16863200	C₂²	1	4	yes	—
20	#16863197	C₂²	1	4	yes	—
21	#16863171	C₂²	1	2	yes	—
22	#16863171	C₂²	1	2	yes	—
23	C₂×#135×C₆₇	C₂²	1	2	yes	—
24	#508×C₆₇	C₄	1	8	yes	complemented
25	C₂×Q₃₂×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×SD₃₂×C₆₇	C₂²	1	2	yes	—
27	C₂×D₁₆×C₆₇	C₂²	1	1	yes	characteristic
28	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
29	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
30	Q₈×Dic₆₇	—	2	4	no	—
31	D₄×Dic₆₇	—	2	4	no	—
32	D₄×Dic₆₇	—	4	4	no	—
33	#14524002	—	2	2	no	—
34	#14524002	—	4	2	no	—
35	C₂×C₄×Dic₆₇	—	4	2	no	—
36	C₂×C₄×Dic₆₇	D₄	1	1	yes	characteristic
37	C₂×C₆₇⋊C₁₆	—	2	2	no	—
38	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
39	#14523977	C₂³	1	4	yes	—
40	#14523975	—	2	2	no	—
41	#14523967	—	2	4	no	—
42	#14523964	—	2	4	no	—
43	C₈×Dic₆₇	D₄	1	2	yes	—
44	C₈×Dic₆₇	C₂³	1	2	yes	—
45	C₂×#41×C₆₇	—	2	2	no	—
46	#135×C₆₇	—	2	4	no	—
47	#135×C₆₇	C₂×C₄	1	8	yes	complemented
48	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
49	C₂×SD₁₆×C₆₇	—	2	2	no	—
50	C₂×D₈×C₆₇	C₂³	1	2	yes	—
51	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
52	Q₃₂×C₆₇	C₂×C₄	1	4	yes	complemented
53	SD₃₂×C₆₇	C₂×C₄	1	8	yes	complemented
54	D₁₆×C₆₇	C₂×C₄	1	4	yes	complemented
55	C₂×C₁₆×C₆₇	C₂×C₄	1	4	yes	complemented
56	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
57	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
58	C₂²×Dic₆₇	—	4	4	no	—
59	C₂²×Dic₆₇	D₈	1	2	yes	—
60	C₂×C₆₇⋊C₈	—	2	2	no	—
61	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
62	C₄⋊Dic₆₇	—	2	4	no	—
63	C₄⋊Dic₆₇	—	4	4	no	—
64	C₂²⋊Dic₆₇	—	4	2	no	—
65	C₂²⋊Dic₆₇	—	4	4	no	—
66	C₄×Dic₆₇	—	4	2	no	—
67	C₄×Dic₆₇	—	4	4	no	—
68	C₄×Dic₆₇	D₈	1	2	yes	—
69	C₄×Dic₆₇	C₂×D₄	1	2	yes	—
70	C₆₇⋊C₁₆	—	2	2	no	—
71	#41×C₆₇	—	2	8	no	—
72	#41×C₆₇	—	4	4	no	—
73	C₂×Q₈×C₆₇	—	2	2	no	—
74	C₂×D₄×C₆₇	—	2	2	no	—
75	C₂×D₄×C₆₇	—	4	2	no	—
76	C₂²×C₄×C₆₇	—	4	2	no	—
77	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
78	Q₁₆×C₆₇	—	2	2	no	—
79	Q₁₆×C₆₇	C₂²×C₄	1	4	yes	—
80	SD₁₆×C₆₇	—	2	8	no	—
81	D₈×C₆₇	—	2	2	no	—
82	D₈×C₆₇	C₂²×C₄	1	4	yes	—
83	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₆₇	#41	1	2	yes	—
86	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
87	C₁₆×C₆₇	C₂²×C₄	1	2	yes	—
88	C₁₆×C₆₇	C₂²×C₄	1	2	yes	—
89	C₂³×C₆₇	—	4	2	no	—
90	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
91	Q₈×C₆₇	—	2	4	no	—
92	Q₈×C₆₇	—	4	2	no	—
93	D₄×C₆₇	—	2	4	no	—
94	D₄×C₆₇	—	4	8	no	—
95	D₄×C₆₇	—	4	2	no	—
96	C₂×C₄×C₆₇	—	4	2	no	—
97	C₂×C₄×C₆₇	—	4	8	no	—
98	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
99	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
100	C₂×C₄×C₆₇	#135	1	2	yes	—
101	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
102	C₂×Dic₆₇	—	4	2	no	—
103	C₂×Dic₆₇	—	4	4	no	—
104	C₂×Dic₆₇	—	8	8	no	—
105	C₂×Dic₆₇	C₂×D₈	1	4	yes	—
106	C₈×C₆₇	C₄×D₄	1	2	yes	—
107	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
108	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
109	C₆₇⋊C₈	—	2	2	no	—
110	C₂²×C₆₇	—	2	2	no	—
111	C₂²×C₆₇	—	4	4	no	—
112	C₂²×C₆₇	—	8	4	no	—
113	C₂²×C₆₇	C₄×D₈	1	2	yes	—
114	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
115	C₄×C₆₇	—	4	4	no	—
116	C₄×C₆₇	C₄×D₈	1	2	yes	—
117	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
118	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
119	Dic₆₇	—	2	4	no	—
120	Dic₆₇	—	8	4	no	—
121	#33451	—	67	1	no	maximal, Sylow
122	C₂×C₆₇	—	2	2	no	—
123	C₂×C₆₇	—	8	4	no	—
124	C₂×C₆₇	C₂×C₄×D₈	1	1	yes	characteristic
125	C₂×C₆₇	#3414	1	2	yes	—
126	C₂×#508	D₆₇	1	1	yes	characteristic
127	C₄×#135	—	67	2	no	—
128	#2177	—	67	2	no	—
129	#2176	—	67	2	no	—
130	#2175	—	67	4	no	—
131	#2158	—	67	1	no	—
132	#2138	—	67	2	no	—
133	C₂×#346	—	67	1	no	—
134	C₆₇	#33451	1	1	yes	characteristic, complemented, Sylow
135	C₂×#135	C₂×D₆₇	1	2	yes	—
136	C₄×#41	—	134	2	no	—
137	#508	Dic₆₇	1	8	yes	complemented
138	C₂×Q₃₂	C₂×D₆₇	1	1	yes	characteristic
139	C₂×SD₃₂	C₂×D₆₇	1	2	yes	—
140	C₂×D₁₆	C₂×D₆₇	1	1	yes	characteristic
141	C₂²×C₁₆	—	67	1	no	—
142	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
143	C₄×Q₁₆	—	67	4	no	—
144	C₄×SD₁₆	—	134	4	no	—
145	C₄×D₈	—	67	4	no	—
146	#427	—	67	2	no	—
147	#416	—	134	2	no	—
148	C₂×C₄×C₈	—	67	1	no	—
149	#367	—	67	2	no	—
150	#366	—	67	2	no	—
151	#358	—	67	4	no	—
152	#357	—	67	4	no	—
153	#346	—	67	2	no	—
154	#346	—	67	2	no	—
155	C₂×#41	—	2	2	no	—
156	#135	—	2	4	no	—
157	#135	C₂×Dic₆₇	1	8	yes	complemented
158	C₂×Q₁₆	C₂²×D₆₇	1	2	yes	—
159	C₂×SD₁₆	—	2	2	no	—
160	C₂×D₈	C₂²×D₆₇	1	2	yes	—
161	C₂²×C₈	—	67	1	no	—
162	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
163	C₄×Q₈	—	134	4	no	—
164	C₄×D₄	—	268	4	no	—
165	C₄×D₄	—	134	4	no	—
166	#117	—	268	2	no	—
167	#117	—	134	2	no	—
168	C₂×C₄²	—	268	2	no	—
169	C₂×C₄²	—	67	1	no	—
170	Q₃₂	C₂×Dic₆₇	1	4	yes	complemented
171	SD₃₂	C₂×Dic₆₇	1	8	yes	complemented
172	D₁₆	C₂×Dic₆₇	1	4	yes	complemented
173	C₂×C₁₆	—	67	2	no	—
174	C₂×C₁₆	—	134	2	no	—
175	C₂×C₁₆	C₂×Dic₆₇	1	4	yes	complemented
176	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
177	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
178	#107	—	67	4	no	—
179	#106	—	134	2	no	—
180	#103	—	134	4	no	—
181	#102	—	134	4	no	—
182	C₄×C₈	—	67	2	no	—
183	C₄×C₈	—	67	2	no	—
184	#41	—	2	8	no	—
185	#41	—	4	4	no	—
186	C₂×Q₈	—	2	2	no	—
187	C₂×D₄	—	2	2	no	—
188	C₂×D₄	—	4	2	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	—	268	4	no	—
191	C₂²×C₄	—	67	2	no	—
192	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
193	Q₁₆	—	2	2	no	—
194	Q₁₆	C₂²×Dic₆₇	1	4	yes	—
195	SD₁₆	—	2	8	no	—
196	D₈	—	2	2	no	—
197	D₈	C₂²×Dic₆₇	1	4	yes	—
198	C₂×C₈	—	67	2	no	—
199	C₂×C₈	—	134	2	no	—
200	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
201	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
202	C₂×C₈	#11761627	1	2	yes	—
203	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
204	C₄⋊C₄	—	268	4	no	—
205	C₄⋊C₄	—	134	4	no	—
206	C₂²⋊C₄	—	268	2	no	—
207	C₂²⋊C₄	—	268	4	no	—
208	C₄²	—	268	4	no	—
209	C₄²	—	268	2	no	—
210	C₄²	—	67	2	no	—
211	C₄²	—	67	2	no	—
212	C₁₆	—	134	2	no	—
213	C₁₆	C₂²×Dic₆₇	1	2	yes	—
214	C₁₆	C₂²×Dic₆₇	1	2	yes	—
215	C₂³	—	4	2	no	—
216	C₂³	D₈×D₆₇	1	1	yes	characteristic
217	Q₈	—	2	4	no	—
218	Q₈	—	4	2	no	—
219	D₄	—	2	4	no	—
220	D₄	—	4	8	no	—
221	D₄	—	4	2	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	4	8	no	—
224	C₂×C₄	—	268	4	no	—
225	C₂×C₄	—	268	2	no	—
226	C₂×C₄	—	536	8	no	—
227	C₂×C₄	—	67	4	no	—
228	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
229	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
230	C₂×C₄	#14524062	1	2	yes	—
231	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
232	C₈	—	134	2	no	—
233	C₈	D₄×Dic₆₇	1	2	yes	—
234	C₈	C₂³×Dic₆₇	1	1	yes	characteristic
235	C₈	C₂×#11761627	1	1	yes	characteristic
236	C₂²	—	2	2	no	—
237	C₂²	—	4	4	no	—
238	C₂²	—	8	4	no	—
239	C₂²	D₈×Dic₆₇	1	2	yes	—
240	C₂²	C₂×D₈×D₆₇	1	1	yes	characteristic
241	C₄	—	4	4	no	—
242	C₄	—	536	4	no	—
243	C₄	—	134	4	no	—
244	C₄	D₈×Dic₆₇	1	2	yes	—
245	C₄	C₂×D₄×Dic₆₇	1	1	yes	characteristic
246	C₄	C₂×#14524062	1	1	yes	characteristic
247	C₂	—	2	2	no	—
248	C₂	—	8	4	no	—
249	C₂	C₂×D₈×Dic₆₇	1	1	yes	characteristic
250	C₂	#22573342	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₈×D₆₇
commutator	C₈×C₆₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₆₇
Fitting, prime radical	C₂×#508×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₈
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	C₂×#508	D₆₇
hypocenter	C₆₇	#33451
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#508

Complements

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

N	H ≅ G/N	type
1. #135×Dic₆₇	C₂	semidirect
2. #22562085	C₂	semidirect
3. #22562080	C₂	semidirect
4. #22562074	C₂	semidirect
5. #22561993	C₂	semidirect
6. #22561916	C₂	semidirect
7. C₂×#16863171	C₂	semidirect
24. #508×C₆₇	C₄	semidirect
47. #135×C₆₇	C₂×C₄	semidirect
52. Q₃₂×C₆₇	C₂×C₄	semidirect
53. SD₃₂×C₆₇	C₂×C₄	semidirect
54. D₁₆×C₆₇	C₂×C₄	semidirect
55. C₂×C₁₆×C₆₇	C₂×C₄	semidirect
134. C₆₇	#33451	semidirect
137. #508	Dic₆₇	semidirect
157. #135	C₂×Dic₆₇	semidirect
170. Q₃₂	C₂×Dic₆₇	semidirect
171. SD₃₂	C₂×Dic₆₇	semidirect
172. D₁₆	C₂×Dic₆₇	semidirect
175. C₂×C₁₆	C₂×Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
#135×Dic₆₇	2
#22562085	2
#22562080	2
#22562074	2
#22561993	2
#22561916	2
C₂×#16863171	2
C₂×#508×C₆₇	2
#33451	67