Subgroups — #30864142

Order 17152.

694 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	#22560928	C₂	1	2	yes	maximal
2	#22560768	C₂	1	4	yes	maximal
3	#22560326	C₂	1	1	yes	maximal, characteristic
4	#22560084	C₂	1	4	yes	maximal
5	#22560078	C₂	1	2	yes	maximal
6	C₄²×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
7	C₄×#125×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	4	yes	—
9	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
10	C₂×#14523954	C₂²	1	4	yes	—
11	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
12	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
13	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
15	#16863151	C₂²	1	4	yes	—
16	#16863151	C₂²	1	4	yes	—
17	#16863151	C₂²	1	4	yes	—
18	C₂×#125×C₆₇	C₄	1	1	yes	characteristic
19	#389×C₆₇	C₄	1	2	yes	—
20	#387×C₆₇	C₄	1	4	yes	—
21	#382×C₆₇	C₄	1	1	yes	characteristic
22	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
23	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
24	C₄³×C₆₇	C₂²	1	1	yes	characteristic
25	C₂²×C₆₇⋊C₈	C₂³	1	4	yes	—
26	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
27	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
28	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
29	C₄⋊(C₆₇⋊C₈)	—	2	8	no	—
30	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
31	#14523954	—	2	8	no	—
32	C₄×C₆₇⋊C₈	—	2	4	no	—
33	C₄×C₆₇⋊C₈	—	2	4	no	—
34	C₄×C₆₇⋊C₈	—	2	2	no	—
35	C₄×C₆₇⋊C₈	Q₈	1	4	yes	—
36	#125×C₆₇	—	2	4	no	—
37	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
38	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
39	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
40	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
41	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
45	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
46	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
47	#95×C₆₇	C₂×C₄	1	4	yes	—
48	#95×C₆₇	C₂×C₄	1	4	yes	—
49	#95×C₆₇	C₂×C₄	1	4	yes	—
50	C₂×C₆₇⋊C₈	—	2	2	no	—
51	C₂×C₆₇⋊C₈	—	2	2	no	—
52	C₂×C₆₇⋊C₈	—	2	2	no	—
53	C₂×C₆₇⋊C₈	—	2	4	no	—
54	C₂×C₆₇⋊C₈	—	2	4	no	—
55	C₂×C₆₇⋊C₈	—	2	4	no	—
56	C₂×C₆₇⋊C₈	C₂×Q₈	1	2	yes	—
57	C₂×C₆₇⋊C₈	C₂×Q₈	1	2	yes	—
58	C₂×C₆₇⋊C₈	#41	1	4	yes	—
59	C₂×C₆₇⋊C₈	#41	1	4	yes	—
60	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
61	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
62	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
67	C₄⋊C₄×C₆₇	—	2	8	no	—
68	C₄⋊C₄×C₆₇	—	2	4	no	—
69	C₄⋊C₄×C₆₇	—	2	8	no	—
70	C₄⋊C₄×C₆₇	—	2	4	no	—
71	C₄²×C₆₇	—	2	4	no	—
72	C₄²×C₆₇	—	2	2	no	—
73	C₄²×C₆₇	—	2	2	no	—
74	C₄²×C₆₇	—	2	2	no	—
75	C₄²×C₆₇	—	2	4	no	—
76	C₄²×C₆₇	—	2	4	no	—
77	C₄²×C₆₇	M₁₆	1	4	yes	—
78	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
79	C₄²×C₆₇	#41	1	2	yes	—
80	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
81	C₂×C₄×C₆₇	—	2	2	no	—
82	C₂×C₄×C₆₇	—	2	2	no	—
83	C₂×C₄×C₆₇	—	2	2	no	—
84	C₂×C₄×C₆₇	—	2	2	no	—
85	C₂×C₄×C₆₇	—	2	2	no	—
86	C₂×C₄×C₆₇	—	2	4	no	—
87	C₂×C₄×C₆₇	—	2	4	no	—
88	C₂×C₄×C₆₇	—	2	4	no	—
89	C₂×C₄×C₆₇	—	2	4	no	—
90	C₂×C₄×C₆₇	—	2	4	no	—
91	C₂×C₄×C₆₇	—	2	4	no	—
92	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
93	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
94	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
95	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
96	C₂×C₄×C₆₇	#131	1	4	yes	—
97	C₂×C₄×C₆₇	#131	1	2	yes	—
98	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
99	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
100	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
101	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
102	C₆₇⋊C₈	—	2	4	no	—
103	C₆₇⋊C₈	—	2	2	no	—
104	C₆₇⋊C₈	—	4	4	no	—
105	C₆₇⋊C₈	#125	1	4	yes	complemented
106	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
107	C₂²×C₆₇	C₄×#41	1	2	yes	—
108	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
109	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
110	C₂²×C₆₇	C₂×#131	1	2	yes	—
111	C₄×C₆₇	—	2	2	no	—
112	C₄×C₆₇	—	2	4	no	—
113	C₄×C₆₇	—	2	4	no	—
114	C₄×C₆₇	—	4	4	no	—
115	C₄×C₆₇	—	4	4	no	—
116	C₄×C₆₇	#446	1	4	yes	—
117	C₄×C₆₇	C₂×#125	1	1	yes	characteristic
118	C₄×C₆₇	#529	1	2	yes	—
119	C₄×C₆₇	#533	1	1	yes	characteristic
120	#21673	—	67	1	no	maximal, Sylow
121	C₂×C₆₇	C₄×#125	1	1	yes	characteristic
122	C₂×C₆₇	C₂×#446	1	1	yes	characteristic
123	C₂×C₆₇	#2960	1	1	yes	characteristic
124	C₂×C₆₇	#2965	1	2	yes	—
125	C₂×C₆₇	#2966	1	2	yes	—
126	C₄×#125	D₆₇	1	1	yes	characteristic
127	#1939	—	67	2	no	—
128	#1916	—	67	4	no	—
129	#1831	—	67	1	no	—
130	#1771	—	67	4	no	—
131	#1770	—	67	2	no	—
132	C₄²×C₈	—	67	1	no	—
133	C₆₇	#21673	1	1	yes	characteristic, complemented, Sylow
134	C₂×#125	Dic₆₇	1	1	yes	characteristic
135	C₂×C₄⋊C₈	—	67	2	no	—
136	C₂×C₄⋊C₈	—	67	4	no	—
137	C₂×#97	—	67	4	no	—
138	C₂×C₄×C₈	—	67	1	no	—
139	C₂×C₄×C₈	—	67	2	no	—
140	C₂×C₄×C₈	—	67	1	no	—
141	C₂×C₄×C₈	—	67	2	no	—
142	#389	Dic₆₇	1	2	yes	—
143	#387	Dic₆₇	1	4	yes	—
144	#382	Dic₆₇	1	1	yes	characteristic
145	C₄×C₄⋊C₄	C₂×D₆₇	1	4	yes	—
146	C₄×C₄⋊C₄	C₂×D₆₇	1	2	yes	—
147	C₄³	C₂×D₆₇	1	1	yes	characteristic
148	#336	—	67	4	no	—
149	#336	—	67	4	no	—
150	#336	—	67	4	no	—
151	C₂²×C₈	—	67	1	no	—
152	C₂²×C₈	—	67	1	no	—
153	C₂²×C₈	—	67	2	no	—
154	C₂²×C₈	—	67	4	no	—
155	#125	—	2	4	no	—
156	C₂×C₄⋊C₄	C₂×Dic₆₇	1	4	yes	—
157	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
158	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
159	C₂×C₄⋊C₄	C₂×Dic₆₇	1	4	yes	—
160	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
161	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
162	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
163	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
164	C₂×C₄²	C₂²×D₆₇	1	4	yes	—
165	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
166	C₄⋊C₈	—	134	8	no	—
167	C₄⋊C₈	—	134	4	no	—
168	#97	—	134	8	no	—
169	C₄×C₈	—	67	4	no	—
170	C₄×C₈	—	134	4	no	—
171	C₄×C₈	—	134	4	no	—
172	C₄×C₈	—	134	2	no	—
173	#95	C₂×Dic₆₇	1	4	yes	—
174	#95	C₂×Dic₆₇	1	4	yes	—
175	#95	C₂×Dic₆₇	1	4	yes	—
176	C₂²×C₄	C₂²×Dic₆₇	1	4	yes	—
177	C₂²×C₄	C₂²×Dic₆₇	1	4	yes	—
178	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
179	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
180	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
181	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
182	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
183	C₂×C₈	—	67	2	no	—
184	C₂×C₈	—	67	4	no	—
185	C₂×C₈	—	67	2	no	—
186	C₂×C₈	—	67	4	no	—
187	C₂×C₈	—	134	4	no	—
188	C₂×C₈	—	134	4	no	—
189	C₂×C₈	—	134	4	no	—
190	C₂×C₈	—	134	2	no	—
191	C₂×C₈	—	134	2	no	—
192	C₂×C₈	—	134	2	no	—
193	C₄⋊C₄	—	2	8	no	—
194	C₄⋊C₄	—	2	4	no	—
195	C₄⋊C₄	—	2	8	no	—
196	C₄⋊C₄	—	2	4	no	—
197	C₄²	—	2	4	no	—
198	C₄²	—	2	2	no	—
199	C₄²	—	2	2	no	—
200	C₄²	—	2	2	no	—
201	C₄²	—	2	4	no	—
202	C₄²	—	2	4	no	—
203	C₄²	#11761612	1	4	yes	—
204	C₄²	Q₈×D₆₇	1	2	yes	—
205	C₄²	#11761629	1	2	yes	—
206	C₂³	C₂³×Dic₆₇	1	1	yes	characteristic
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	2	4	no	—
218	C₂×C₄	Q₈×Dic₆₇	1	2	yes	—
219	C₂×C₄	Q₈×Dic₆₇	1	2	yes	—
220	C₂×C₄	C₂×#11761612	1	4	yes	—
221	C₂×C₄	C₂×#11761612	1	2	yes	—
222	C₂×C₄	#14524050	1	4	yes	—
223	C₂×C₄	#14524050	1	2	yes	—
224	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
225	C₂×C₄	#41×D₆₇	1	2	yes	—
226	C₂×C₄	#41×D₆₇	1	2	yes	—
227	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
228	C₈	—	268	4	no	—
229	C₈	—	67	4	no	—
230	C₈	—	134	4	no	—
231	C₈	—	134	2	no	—
232	C₂²	C₂×Q₈×Dic₆₇	1	1	yes	characteristic
233	C₂²	#41×Dic₆₇	1	2	yes	—
234	C₂²	C₂²×#11761612	1	1	yes	characteristic
235	C₂²	C₂×#14524050	1	1	yes	characteristic
236	C₂²	C₂×#14524050	1	2	yes	—
237	C₄	—	2	2	no	—
238	C₄	—	2	4	no	—
239	C₄	—	2	4	no	—
240	C₄	—	4	4	no	—
241	C₄	—	4	4	no	—
242	C₄	#16863527	1	4	yes	—
243	C₄	#125×D₆₇	1	1	yes	characteristic
244	C₄	#16863931	1	2	yes	—
245	C₄	#16863952	1	1	yes	characteristic
246	C₂	#125×Dic₆₇	1	1	yes	characteristic
247	C₂	C₂×#16863527	1	1	yes	characteristic
248	C₂	#22569144	1	1	yes	characteristic
249	C₂	#22569176	1	2	yes	—
250	C₂	#22569185	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₆₇
commutator, socle, prime radical	C₂³×C₆₇	C₂³×C₄
Fitting	C₄×#125×C₆₇	C₂
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₄×#125	D₆₇
hypocenter	C₆₇	#21673
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#125

Complements

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

N	H ≅ G/N	type
105. C₆₇⋊C₈	#125	semidirect
133. C₆₇	#21673	semidirect

Maximal subgroups

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

subgroup	index
#22560928	2
#22560768	2
#22560326	2
#22560084	2
#22560078	2
C₄²×C₆₇⋊C₈	2
C₄×#125×C₆₇	2
#21673	67