Subgroups — #30772176

Order 17152.

366 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₆₇⋊C₁₆	C₂	1	2	yes	maximal
2	C₄×C₈×Dic₆₇	C₂	1	1	yes	maximal, characteristic
3	#22558287	C₂	1	1	yes	maximal, characteristic
4	#22558286	C₂	1	2	yes	maximal
5	C₆₇×#1374	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
7	C₂×C₄×C₆₇⋊C₈	C₄	1	1	yes	characteristic
8	C₂×C₈×Dic₆₇	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₆₇	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
12	C₄²×Dic₆₇	C₄	1	1	yes	characteristic
13	Dic₆₇⋊C₁₆	—	2	4	no	—
14	C₂²×C₁₆×C₆₇	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×Dic₆₇	C₈	1	1	yes	characteristic
19	C₂×C₄×Dic₆₇	C₈	1	1	yes	characteristic
20	C₂×C₄×Dic₆₇	C₈	1	1	yes	characteristic
21	C₂×C₄×Dic₆₇	C₈	1	1	yes	characteristic
22	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₆₇⋊C₁₆	—	2	2	no	—
25	C₂×C₆₇⋊C₁₆	—	2	2	no	—
26	C₂×C₆₇⋊C₁₆	—	2	2	no	—
27	C₄×C₆₇⋊C₈	—	2	2	no	—
28	C₈×Dic₆₇	—	2	2	no	—
29	C₈×Dic₆₇	—	2	2	no	—
30	C₈×Dic₆₇	—	2	2	no	—
31	C₈×Dic₆₇	D₄	1	2	yes	—
32	C₈×Dic₆₇	Q₈	1	2	yes	—
33	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₁₆×C₆₇	—	2	2	no	—
37	C₂×C₁₆×C₆₇	—	2	2	no	—
38	C₂×C₁₆×C₆₇	—	2	2	no	—
39	C₄×C₈×C₆₇	—	2	2	no	—
40	C₂²×Dic₆₇	C₂×C₈	1	1	yes	characteristic
41	C₂²×Dic₆₇	C₂×C₈	1	1	yes	characteristic
42	C₂²×Dic₆₇	C₂×C₈	1	1	yes	characteristic
43	C₂²×Dic₆₇	C₂×C₈	1	1	yes	characteristic
44	C₂×C₆₇⋊C₈	—	2	1	no	—
45	C₂×C₆₇⋊C₈	—	2	1	no	—
46	C₂×C₆₇⋊C₈	—	2	1	no	—
47	C₂×C₆₇⋊C₈	—	2	1	no	—
48	C₂×C₆₇⋊C₈	C₄⋊C₄	1	2	yes	—
49	C₂×C₆₇⋊C₈	M₁₆	1	2	yes	—
50	C₄×Dic₆₇	—	2	2	no	—
51	C₄×Dic₆₇	—	2	1	no	—
52	C₄×Dic₆₇	—	2	1	no	—
53	C₄×Dic₆₇	—	2	2	no	—
54	C₄×Dic₆₇	—	2	2	no	—
55	C₄×Dic₆₇	C₁₆	1	4	yes	complemented
56	C₄×Dic₆₇	C₄⋊C₄	1	2	yes	—
57	C₄×Dic₆₇	M₁₆	1	2	yes	—
58	C₆₇⋊C₁₆	—	4	2	no	—
59	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
60	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₈×C₆₇	—	2	1	no	—
63	C₂×C₈×C₆₇	—	2	1	no	—
64	C₂×C₈×C₆₇	M₁₆	1	2	yes	—
65	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
67	C₂×C₈×C₆₇	#41	1	2	yes	—
68	C₂×C₈×C₆₇	#41	1	2	yes	—
69	C₄²×C₆₇	—	2	1	no	—
70	C₄²×C₆₇	—	2	1	no	—
71	C₁₆×C₆₇	—	4	2	no	—
72	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
73	C₂×C₄×C₆₇	—	2	1	no	—
74	C₂×C₄×C₆₇	—	2	1	no	—
75	C₂×C₄×C₆₇	—	2	1	no	—
76	C₂×C₄×C₆₇	—	2	1	no	—
77	C₂×C₄×C₆₇	C₂×C₁₆	1	2	yes	—
78	C₂×C₄×C₆₇	M₃₂	1	2	yes	—
79	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
81	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
82	C₂×C₄×C₆₇	C₂×M₁₆	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×Dic₆₇	—	2	1	no	—
86	C₂×Dic₆₇	—	2	1	no	—
87	C₂×Dic₆₇	—	2	1	no	—
88	C₂×Dic₆₇	—	2	1	no	—
89	C₂×Dic₆₇	—	2	1	no	—
90	C₂×Dic₆₇	—	2	1	no	—
91	C₂×Dic₆₇	C₄⋊C₈	1	2	yes	—
92	C₂×Dic₆₇	C₄⋊C₈	1	2	yes	—
93	C₂×Dic₆₇	C₂×C₁₆	1	2	yes	—
94	C₂×Dic₆₇	C₂×C₁₆	1	2	yes	—
95	C₂×Dic₆₇	M₃₂	1	2	yes	—
96	C₂×Dic₆₇	M₃₂	1	2	yes	—
97	C₈×C₆₇	—	2	2	no	—
98	C₈×C₆₇	#124	1	2	yes	—
99	C₈×C₆₇	#125	1	2	yes	—
100	C₆₇⋊C₈	—	2	2	no	—
101	C₆₇⋊C₈	—	2	2	no	—
102	C₂²×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
103	C₂²×C₆₇	#431	1	1	yes	characteristic
104	C₂²×C₆₇	#431	1	1	yes	characteristic
105	C₂²×C₆₇	C₂²×C₁₆	1	1	yes	characteristic
106	C₂²×C₆₇	C₂×M₃₂	1	1	yes	characteristic
107	C₂²×C₆₇	C₂×M₃₂	1	1	yes	characteristic
108	C₂²×C₆₇	C₂×M₃₂	1	1	yes	characteristic
109	C₄×C₆₇	—	2	2	no	—
110	C₄×C₆₇	—	2	2	no	—
111	C₄×C₆₇	#382	1	1	yes	characteristic
112	C₄×C₆₇	#423	1	1	yes	characteristic
113	C₄×C₆₇	#432	1	1	yes	characteristic
114	C₄×C₆₇	#432	1	1	yes	characteristic
115	Dic₆₇	—	2	2	no	—
116	Dic₆₇	—	2	2	no	—
117	Dic₆₇	—	2	2	no	—
118	Dic₆₇	C₄⋊C₁₆	1	4	yes	—
119	#12879	—	67	1	no	maximal, Sylow
120	C₂×C₆₇	#1831	1	1	yes	characteristic
121	C₂×C₆₇	C₂×C₄⋊C₁₆	1	1	yes	characteristic
122	C₂×C₆₇	#2150	1	1	yes	characteristic
123	C₂×C₆₇	#2162	1	1	yes	characteristic
124	C₂×C₆₇	#2162	1	1	yes	characteristic
125	C₂×C₆₇	#2163	1	1	yes	characteristic
126	C₂×C₆₇	#2163	1	1	yes	characteristic
127	C₂×C₄⋊C₁₆	—	67	2	no	—
128	C₄²×C₈	—	67	1	no	—
129	#1374	—	67	2	no	—
130	#1374	—	67	1	no	—
131	#1374	D₆₇	1	1	yes	characteristic
132	C₆₇	#12879	1	1	yes	characteristic, complemented, Sylow
133	C₂²×C₁₆	—	67	2	no	—
134	C₂²×C₁₆	C₂×D₆₇	1	2	yes	—
135	C₂×C₄×C₈	—	67	1	no	—
136	C₂×C₄×C₈	—	67	1	no	—
137	C₂×C₄×C₈	—	67	1	no	—
138	C₂×C₄×C₈	—	67	1	no	—
139	C₂×C₄×C₈	—	67	1	no	—
140	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
141	C₄³	—	67	1	no	—
142	C₄⋊C₁₆	—	134	4	no	—
143	C₂²×C₈	—	67	1	no	—
144	C₂²×C₈	—	67	1	no	—
145	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
146	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
147	C₂×C₄²	—	67	1	no	—
148	C₂×C₄²	—	67	1	no	—
149	C₂×C₄²	—	67	1	no	—
150	C₂×C₄²	—	67	1	no	—
151	C₂×C₄²	—	67	1	no	—
152	C₂×C₄²	—	67	1	no	—
153	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
154	C₂×C₁₆	—	2	2	no	—
155	C₂×C₁₆	—	2	2	no	—
156	C₂×C₁₆	—	2	2	no	—
157	C₂×C₁₆	—	134	2	no	—
158	C₂×C₁₆	—	134	2	no	—
159	C₂×C₁₆	—	134	2	no	—
160	C₄×C₈	—	2	2	no	—
161	C₄×C₈	—	67	2	no	—
162	C₄×C₈	—	67	2	no	—
163	C₄×C₈	—	134	2	no	—
164	C₄×C₈	—	134	2	no	—
165	C₄×C₈	—	134	2	no	—
166	C₄×C₈	—	134	2	no	—
167	C₂²×C₄	—	67	1	no	—
168	C₂²×C₄	—	67	1	no	—
169	C₂²×C₄	—	67	1	no	—
170	C₂²×C₄	—	67	1	no	—
171	C₂²×C₄	C₈×D₆₇	1	1	yes	characteristic
172	C₂²×C₄	C₈×D₆₇	1	1	yes	characteristic
173	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	67	2	no	—
177	C₂×C₈	—	67	2	no	—
178	C₂×C₈	—	134	1	no	—
179	C₂×C₈	—	134	1	no	—
180	C₂×C₈	—	134	1	no	—
181	C₂×C₈	—	134	1	no	—
182	C₂×C₈	#11761611	1	2	yes	—
183	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
184	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
185	C₂×C₈	#11761627	1	2	yes	—
186	C₂×C₈	#11761628	1	2	yes	—
187	C₄²	—	2	1	no	—
188	C₄²	—	2	1	no	—
189	C₄²	—	67	4	no	—
190	C₄²	—	67	2	no	—
191	C₄²	—	67	2	no	—
192	C₄²	—	134	2	no	—
193	C₄²	—	134	2	no	—
194	C₄²	—	134	2	no	—
195	C₄²	—	134	1	no	—
196	C₄²	—	134	1	no	—
197	C₁₆	—	4	2	no	—
198	C₁₆	—	268	2	no	—
199	C₂³	C₂×C₈×D₆₇	1	1	yes	characteristic
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	67	2	no	—
205	C₂×C₄	—	67	2	no	—
206	C₂×C₄	—	67	2	no	—
207	C₂×C₄	—	67	2	no	—
208	C₂×C₄	—	67	2	no	—
209	C₂×C₄	—	67	2	no	—
210	C₂×C₄	—	134	1	no	—
211	C₂×C₄	—	134	1	no	—
212	C₂×C₄	—	134	1	no	—
213	C₂×C₄	—	134	1	no	—
214	C₂×C₄	—	134	1	no	—
215	C₂×C₄	—	134	1	no	—
216	C₂×C₄	C₁₆×D₆₇	1	2	yes	—
217	C₂×C₄	#14523982	1	2	yes	—
218	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
219	C₂×C₄	#14523999	1	1	yes	characteristic
220	C₂×C₄	#14524000	1	1	yes	characteristic
221	C₂×C₄	M₁₆×D₆₇	1	1	yes	characteristic
222	C₂×C₄	M₁₆×D₆₇	1	1	yes	characteristic
223	C₂×C₄	C₂×#11761611	1	1	yes	characteristic
224	C₈	—	2	2	no	—
225	C₈	—	134	2	no	—
226	C₈	—	134	2	no	—
227	C₈	#14524029	1	2	yes	—
228	C₈	#14524033	1	2	yes	—
229	C₂²	C₄⋊C₈×D₆₇	1	1	yes	characteristic
230	C₂²	#16863446	1	1	yes	characteristic
231	C₂²	#16863447	1	1	yes	characteristic
232	C₂²	C₂×C₁₆×D₆₇	1	1	yes	characteristic
233	C₂²	M₃₂×D₆₇	1	1	yes	characteristic
234	C₂²	M₃₂×D₆₇	1	1	yes	characteristic
235	C₂²	C₂×#14523982	1	1	yes	characteristic
236	C₄	—	2	2	no	—
237	C₄	—	2	2	no	—
238	C₄	—	67	4	no	—
239	C₄	—	134	2	no	—
240	C₄	—	134	2	no	—
241	C₄	—	134	2	no	—
242	C₄	#16863260	1	1	yes	characteristic
243	C₄	#16863418	1	1	yes	characteristic
244	C₄	#16863450	1	1	yes	characteristic
245	C₄	#16863451	1	1	yes	characteristic
246	C₂	#22560324	1	1	yes	characteristic
247	C₂	C₄⋊C₁₆×D₆₇	1	1	yes	characteristic
248	C₂	#22561963	1	1	yes	characteristic
249	C₂	#22562005	1	1	yes	characteristic
250	C₂	#22562006	1	1	yes	characteristic
251	C₂	#22562009	1	1	yes	characteristic
252	C₂	#22562010	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₆₇
commutator	C₂²×C₆₇	C₂²×C₁₆
Fitting	C₆₇×#1374	C₂
socle, prime radical	C₂³×C₆₇	C₂²×C₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#1374	D₆₇
hypocenter	C₆₇	#12879
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₈ ◁ #1374

Complements

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

N	H ≅ G/N	type
55. C₄×Dic₆₇	C₁₆	semidirect
132. C₆₇	#12879	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₆₇⋊C₁₆	2
C₄×C₈×Dic₆₇	2
#22558287	2
#22558286	2
C₆₇×#1374	2
#12879	67