Subgroups — #30749480

Order 17152.

310 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22559895	C₂	1	1	yes	maximal, characteristic
2	#22558957	C₂	1	1	yes	maximal, characteristic
3	#22558053	C₂	1	2	yes	maximal
4	#22558037	C₂	1	2	yes	maximal
5	C₆₇×#1564	C₂	1	1	yes	maximal, characteristic
6	#16863449	C₂²	1	2	yes	—
7	C₂×#14523954	C₂²	1	1	yes	characteristic
8	C₂×#14523953	C₄	1	1	yes	characteristic
9	C₂×#14523953	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₆₇	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₆₇	C₄	1	1	yes	characteristic
12	#16863242	C₂²	1	1	yes	characteristic
13	#16863149	—	2	1	no	—
14	#16863147	—	2	1	no	—
15	#431×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	#335×C₆₇	—	2	1	no	—
18	#334×C₆₇	—	2	1	no	—
19	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
23	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
24	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
25	#14523954	—	2	1	no	—
26	#14523954	—	2	1	no	—
27	#14523953	—	2	2	no	—
28	#14523953	C₈	1	4	yes	complemented
29	C₄×C₆₇⋊C₈	—	2	2	no	—
30	C₈×Dic₆₇	—	2	2	no	—
31	C₈×Dic₆₇	—	2	2	no	—
32	C₆₇⋊#95	—	2	1	no	—
33	C₆₇⋊#95	—	2	1	no	—
34	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
37	C₄⋊C₈×C₆₇	—	2	2	no	—
38	C₂²⋊C₈×C₆₇	—	2	2	no	—
39	C₄×C₈×C₆₇	—	2	2	no	—
40	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
41	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
42	C₄×C₈×C₆₇	Q₈	1	1	yes	characteristic
43	C₄×C₈×C₆₇	Q₈	1	1	yes	characteristic
44	C₂²×Dic₆₇	—	2	1	no	—
45	C₂²×Dic₆₇	—	2	1	no	—
46	C₂×C₆₇⋊C₈	—	2	1	no	—
47	C₂×C₆₇⋊C₈	—	2	1	no	—
48	C₂×C₆₇⋊C₈	—	2	1	no	—
49	C₂×C₆₇⋊C₈	—	2	1	no	—
50	C₂×C₆₇⋊C₈	—	4	2	no	—
51	C₂×C₆₇⋊C₈	C₂×C₈	1	2	yes	—
52	C₂×C₆₇⋊C₈	M₁₆	1	2	yes	—
53	C₄×Dic₆₇	—	2	1	no	—
54	C₄×Dic₆₇	—	2	1	no	—
55	C₄×Dic₆₇	C₂×C₈	1	2	yes	—
56	C₄×Dic₆₇	M₁₆	1	2	yes	—
57	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
59	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
60	C₂×C₈×C₆₇	—	2	1	no	—
61	C₂×C₈×C₆₇	—	2	1	no	—
62	C₂×C₈×C₆₇	—	4	2	no	—
63	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₆₇	C₂×C₈	1	2	yes	—
68	C₂×C₈×C₆₇	M₁₆	1	2	yes	—
69	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
71	C₄²×C₆₇	#41	1	1	yes	characteristic
72	C₄²×C₆₇	#41	1	1	yes	characteristic
73	C₂³×C₆₇	#117	1	1	yes	characteristic
74	C₂×C₄×C₆₇	—	4	1	no	—
75	C₂×C₄×C₆₇	—	4	1	no	—
76	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
78	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
79	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
80	C₂×C₄×C₆₇	#125	1	1	yes	characteristic
81	C₂×C₄×C₆₇	#125	1	1	yes	characteristic
82	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
83	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
84	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
85	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
86	C₂×Dic₆₇	—	2	2	no	—
87	C₂×Dic₆₇	—	2	2	no	—
88	C₂×Dic₆₇	—	4	1	no	—
89	C₂×Dic₆₇	—	4	1	no	—
90	C₂×Dic₆₇	—	4	1	no	—
91	C₂×Dic₆₇	—	4	1	no	—
92	C₈×C₆₇	—	2	1	no	—
93	C₈×C₆₇	—	2	1	no	—
94	C₈×C₆₇	—	8	2	no	—
95	C₈×C₆₇	C₄⋊C₈	1	2	yes	—
96	C₈×C₆₇	C₄⋊C₈	1	2	yes	—
97	C₆₇⋊C₈	—	2	2	no	—
98	C₆₇⋊C₈	—	2	2	no	—
99	C₆₇⋊C₈	—	8	2	no	—
100	C₂²×C₆₇	—	2	1	no	—
101	C₂²×C₆₇	—	2	1	no	—
102	C₂²×C₆₇	#382	1	1	yes	characteristic
103	C₂²×C₆₇	#431	1	1	yes	characteristic
104	C₂²×C₆₇	#432	1	1	yes	characteristic
105	C₄×C₆₇	—	2	1	no	—
106	C₄×C₆₇	—	2	1	no	—
107	C₄×C₆₇	—	2	1	no	—
108	C₄×C₆₇	—	2	1	no	—
109	C₄×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
110	C₄×C₆₇	#423	1	1	yes	characteristic
111	C₄×C₆₇	#431	1	1	yes	characteristic
112	C₄×C₆₇	#432	1	1	yes	characteristic
113	Dic₆₇	—	4	2	no	—
114	Dic₆₇	—	4	2	no	—
115	#8781	—	67	1	no	maximal, Sylow
116	C₂×C₆₇	—	2	1	no	—
117	C₂×C₆₇	—	2	1	no	—
118	C₂×C₆₇	#1831	1	1	yes	characteristic
119	C₂×C₆₇	#1833	1	1	yes	characteristic
120	C₂×C₆₇	#1833	1	1	yes	characteristic
121	#1726	—	67	1	no	—
122	#1564	—	67	1	no	—
123	#1564	D₆₇	1	1	yes	characteristic
124	#1281	—	67	2	no	—
125	#1274	—	67	2	no	—
126	C₆₇	#8781	1	1	yes	characteristic, complemented, Sylow
127	#431	—	67	2	no	—
128	#431	C₂×D₆₇	1	2	yes	—
129	C₂×#97	—	67	1	no	—
130	C₂×#97	—	67	1	no	—
131	C₂×#97	—	67	1	no	—
132	C₂×C₄×C₈	—	67	1	no	—
133	C₂×C₄×C₈	—	67	1	no	—
134	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
135	#376	—	67	1	no	—
136	#335	—	2	1	no	—
137	#335	—	134	1	no	—
138	#334	—	2	1	no	—
139	#334	—	134	1	no	—
140	C₂²×C₈	—	67	1	no	—
141	C₂²×C₈	—	67	1	no	—
142	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
143	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
144	C₂×C₄²	—	67	1	no	—
145	C₂×C₄²	—	67	1	no	—
146	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
147	C₄⋊C₈	—	2	2	no	—
148	C₄⋊C₈	—	134	2	no	—
149	C₂²⋊C₈	—	2	2	no	—
150	C₂²⋊C₈	—	134	2	no	—
151	#97	—	67	4	no	—
152	#97	—	134	2	no	—
153	#97	—	134	1	no	—
154	#97	—	134	1	no	—
155	C₄×C₈	—	2	2	no	—
156	C₄×C₈	—	134	2	no	—
157	C₄×C₈	—	134	2	no	—
158	C₄×C₈	—	134	2	no	—
159	C₄×C₈	Dic₁₃₄	1	1	yes	characteristic
160	C₄×C₈	Dic₁₃₄	1	1	yes	characteristic
161	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
162	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
163	#95	—	134	1	no	—
164	#95	—	134	1	no	—
165	C₂²×C₄	—	134	1	no	—
166	C₂²×C₄	—	134	1	no	—
167	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
168	C₂²×C₄	#11761627	1	1	yes	characteristic
169	C₂²×C₄	#11761629	1	1	yes	characteristic
170	C₂×C₈	—	2	1	no	—
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	4	2	no	—
173	C₂×C₈	—	268	2	no	—
174	C₂×C₈	—	67	2	no	—
175	C₂×C₈	—	67	2	no	—
176	C₂×C₈	—	134	1	no	—
177	C₂×C₈	—	134	1	no	—
178	C₂×C₈	—	134	1	no	—
179	C₂×C₈	—	134	1	no	—
180	C₂×C₈	Dic₆₇⋊C₄	1	1	yes	characteristic
181	C₂×C₈	Dic₆₇⋊C₄	1	1	yes	characteristic
182	C₂×C₈	Dic₆₇⋊C₄	1	1	yes	characteristic
183	C₂×C₈	Dic₆₇⋊C₄	1	1	yes	characteristic
184	C₂×C₈	C₈×D₆₇	1	2	yes	—
185	C₂×C₈	#11761611	1	2	yes	—
186	C₄²	—	67	2	no	—
187	C₄²	—	67	2	no	—
188	C₄²	—	134	1	no	—
189	C₄²	—	134	1	no	—
190	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
191	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
192	C₄²	#11761627	1	1	yes	characteristic
193	C₄²	#11761629	1	1	yes	characteristic
194	C₂³	#14524001	1	1	yes	characteristic
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	4	1	no	—
197	C₂×C₄	—	268	1	no	—
198	C₂×C₄	—	268	1	no	—
199	C₂×C₄	—	268	1	no	—
200	C₂×C₄	—	268	1	no	—
201	C₂×C₄	—	134	2	no	—
202	C₂×C₄	—	134	2	no	—
203	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
204	C₂×C₄	#14524001	1	1	yes	characteristic
205	C₂×C₄	#14524030	1	1	yes	characteristic
206	C₂×C₄	#14524032	1	1	yes	characteristic
207	C₂×C₄	#14524034	1	1	yes	characteristic
208	C₂×C₄	#14524034	1	1	yes	characteristic
209	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
210	C₂×C₄	M₁₆×D₆₇	1	1	yes	characteristic
211	C₂×C₄	M₁₆×D₆₇	1	1	yes	characteristic
212	C₂×C₄	C₂×#11761611	1	1	yes	characteristic
213	C₈	—	2	1	no	—
214	C₈	—	2	1	no	—
215	C₈	—	8	2	no	—
216	C₈	—	536	2	no	—
217	C₈	—	134	2	no	—
218	C₈	—	134	2	no	—
219	C₈	Dic₆₇⋊C₈	1	2	yes	—
220	C₈	Dic₆₇⋊C₈	1	2	yes	—
221	C₂²	—	2	1	no	—
222	C₂²	—	2	1	no	—
223	C₂²	#16863261	1	1	yes	characteristic
224	C₂²	#16863448	1	1	yes	characteristic
225	C₂²	#16863452	1	1	yes	characteristic
226	C₄	—	2	1	no	—
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	268	2	no	—
231	C₄	—	268	2	no	—
232	C₄	C₂×Dic₆₇⋊C₈	1	1	yes	characteristic
233	C₄	#16863419	1	1	yes	characteristic
234	C₄	#16863448	1	1	yes	characteristic
235	C₄	#16863452	1	1	yes	characteristic
236	C₂	—	2	1	no	—
237	C₂	—	2	1	no	—
238	C₂	#22560325	1	1	yes	characteristic
239	C₂	#22560332	1	1	yes	characteristic
240	C₂	#22560334	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14524001
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1564	C₂
socle	C₂²×C₆₇	#382
prime radical	C₂³×C₆₇	#117
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1564	D₆₇
hypocenter	C₆₇	#8781
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
28. #14523953	C₈	semidirect
126. C₆₇	#8781	semidirect

Maximal subgroups

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

subgroup	index
#22559895	2
#22558957	2
#22558053	2
#22558037	2
C₆₇×#1564	2
#8781	67