Subgroups — #30760082

Order 17152.

422 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863801	C₂	1	1	yes	maximal, characteristic
2	#22561157	C₂	1	2	yes	maximal
3	C₂×#16863144	C₂	1	2	yes	maximal
4	#22558044	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1939	C₂	1	1	yes	maximal, characteristic
6	C₂×#14524043	C₂²	1	2	yes	—
7	#16863801	—	2	4	no	—
8	C₂×#14523977	—	2	1	no	—
9	C₂×#14523977	—	2	1	no	—
10	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
11	#16863158	—	2	2	no	—
12	#16863158	—	2	2	no	—
13	#16863144	—	2	4	no	—
14	C₂×C₄⋊C₈×C₆₇	—	2	1	no	—
15	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
18	#336×C₆₇	—	2	1	no	—
19	C₂²×Dic₁₃₄	—	2	2	no	—
20	C₂²×C₆₇⋊C₈	—	2	2	no	—
21	#14524043	—	2	2	no	—
22	#14524043	—	2	2	no	—
23	#14524043	—	2	2	no	—
24	#14524043	D₄	1	4	yes	—
25	C₂×C₄⋊Dic₆₇	—	2	2	no	—
26	C₂×C₄⋊Dic₆₇	—	2	2	no	—
27	#14523977	—	2	4	no	—
28	#14523977	—	2	4	no	—
29	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
30	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
31	C₂²×C₈×C₆₇	—	2	1	no	—
32	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
34	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₆₇	—	2	1	no	—
37	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
38	C₄⋊C₈×C₆₇	—	2	2	no	—
39	C₄⋊C₈×C₆₇	—	4	2	no	—
40	C₄×C₈×C₆₇	—	2	2	no	—
41	#95×C₆₇	—	2	1	no	—
42	C₂×Dic₁₃₄	—	2	4	no	—
43	C₂×Dic₁₃₄	—	4	2	no	—
44	C₂×Dic₁₃₄	—	8	2	no	—
45	C₂×Dic₁₃₄	—	8	2	no	—
46	C₂²×Dic₆₇	—	4	2	no	—
47	C₂×C₆₇⋊C₈	—	4	2	no	—
48	C₂×C₆₇⋊C₈	—	4	2	no	—
49	C₂×C₆₇⋊C₈	—	4	2	no	—
50	C₄⋊Dic₆₇	—	2	4	no	—
51	C₄⋊Dic₆₇	—	2	4	no	—
52	C₄⋊Dic₆₇	—	4	2	no	—
53	C₄⋊Dic₆₇	—	4	2	no	—
54	C₂²×C₄×C₆₇	—	2	1	no	—
55	C₂²×C₄×C₆₇	—	2	1	no	—
56	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
59	C₂×C₈×C₆₇	—	2	1	no	—
60	C₂×C₈×C₆₇	—	2	1	no	—
61	C₂×C₈×C₆₇	—	2	1	no	—
62	C₂×C₈×C₆₇	—	2	1	no	—
63	C₂×C₈×C₆₇	—	4	1	no	—
64	C₂×C₈×C₆₇	—	4	1	no	—
65	C₂×C₈×C₆₇	—	4	1	no	—
66	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
67	C₂×C₈×C₆₇	Q₁₆	1	2	yes	—
68	C₄⋊C₄×C₆₇	—	2	1	no	—
69	C₄⋊C₄×C₆₇	—	2	2	no	—
70	C₄⋊C₄×C₆₇	Q₁₆	1	2	yes	—
71	C₄²×C₆₇	—	4	1	no	—
72	C₄²×C₆₇	—	4	1	no	—
73	C₄²×C₆₇	C₂×D₄	1	2	yes	—
74	C₄²×C₆₇	#41	1	2	yes	—
75	C₂³×C₆₇	#121	1	1	yes	characteristic
76	C₂×C₄×C₆₇	—	2	1	no	—
77	C₂×C₄×C₆₇	—	2	1	no	—
78	C₂×C₄×C₆₇	—	2	2	no	—
79	C₂×C₄×C₆₇	—	4	1	no	—
80	C₂×C₄×C₆₇	—	4	1	no	—
81	C₂×C₄×C₆₇	—	4	1	no	—
82	C₂×C₄×C₆₇	—	4	1	no	—
83	C₂×C₄×C₆₇	—	4	1	no	—
84	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
85	C₂×C₄×C₆₇	#121	1	2	yes	—
86	C₂×C₄×C₆₇	#123	1	2	yes	—
87	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
88	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
89	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
90	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
91	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
92	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
93	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
94	C₂×C₄×C₆₇	#137	1	2	yes	—
95	C₂×Dic₆₇	—	4	4	no	—
96	C₂×Dic₆₇	—	8	2	no	—
97	C₂×Dic₆₇	—	8	2	no	—
98	Dic₁₃₄	—	8	4	no	—
99	Dic₁₃₄	—	8	4	no	—
100	C₈×C₆₇	—	2	2	no	—
101	C₈×C₆₇	—	2	2	no	—
102	C₈×C₆₇	—	8	1	no	—
103	C₆₇⋊C₈	—	8	2	no	—
104	C₂²×C₆₇	#394	1	1	yes	characteristic
105	C₂²×C₆₇	#461	1	1	yes	characteristic
106	C₂²×C₆₇	#462	1	1	yes	characteristic
107	C₂²×C₆₇	#467	1	1	yes	characteristic
108	C₂²×C₆₇	#470	1	1	yes	characteristic
109	C₂²×C₆₇	#472	1	1	yes	characteristic
110	C₂²×C₆₇	#472	1	1	yes	characteristic
111	C₄×C₆₇	—	2	2	no	—
112	C₄×C₆₇	—	4	2	no	—
113	C₄×C₆₇	—	8	1	no	—
114	C₄×C₆₇	#448	1	1	yes	characteristic
115	C₄×C₆₇	#451	1	1	yes	characteristic
116	C₄×C₆₇	#462	1	2	yes	—
117	C₄×C₆₇	#464	1	2	yes	—
118	C₄×C₆₇	#487	1	1	yes	characteristic
119	C₄×C₆₇	#491	1	1	yes	characteristic
120	Dic₆₇	—	8	4	no	—
121	#10543	—	67	1	no	maximal, Sylow
122	C₂×C₆₇	#1675	1	1	yes	characteristic
123	C₂×C₆₇	#1676	1	1	yes	characteristic
124	C₂×C₆₇	#1695	1	1	yes	characteristic
125	C₂×C₆₇	#1696	1	1	yes	characteristic
126	C₂×C₆₇	#2043	1	1	yes	characteristic
127	C₂×C₆₇	#2045	1	1	yes	characteristic
128	C₂×C₆₇	#2056	1	1	yes	characteristic
129	C₂×#500	—	67	1	no	—
130	#1987	—	67	2	no	—
131	#1939	D₆₇	1	1	yes	characteristic
132	C₂×#333	—	67	2	no	—
133	#1277	—	67	1	no	—
134	C₆₇	#10543	1	1	yes	characteristic, complemented, Sylow
135	C₂×C₄⋊Q₈	—	67	2	no	—
136	#500	—	134	4	no	—
137	C₂×#107	—	134	1	no	—
138	C₂×#107	—	134	1	no	—
139	C₂×C₄⋊C₈	—	2	1	no	—
140	C₂×C₄⋊C₈	—	67	2	no	—
141	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
142	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
143	C₄×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
144	#340	—	134	2	no	—
145	#340	—	134	2	no	—
146	#336	—	2	1	no	—
147	#333	—	134	4	no	—
148	C₂²×Q₈	—	134	2	no	—
149	C₂²×C₈	—	2	1	no	—
150	C₂²×C₈	—	134	2	no	—
151	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
152	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
153	C₄⋊Q₈	—	67	4	no	—
154	C₄⋊Q₈	—	134	2	no	—
155	C₄⋊Q₈	—	134	2	no	—
156	C₄⋊Q₈	—	134	2	no	—
157	C₂×C₄⋊C₄	—	134	2	no	—
158	C₂×C₄⋊C₄	—	134	2	no	—
159	C₂×C₄⋊C₄	D₂₆₈	1	1	yes	characteristic
160	C₂×C₄⋊C₄	D₂₆₈	1	1	yes	characteristic
161	C₂×C₄²	—	2	1	no	—
162	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
163	#107	—	134	4	no	—
164	#107	—	134	4	no	—
165	C₄⋊C₈	—	2	2	no	—
166	C₄⋊C₈	—	4	2	no	—
167	C₄⋊C₈	—	134	2	no	—
168	C₄⋊C₈	—	134	2	no	—
169	C₄×C₈	—	2	2	no	—
170	#95	—	2	1	no	—
171	C₂×Q₈	—	268	2	no	—
172	C₂×Q₈	—	536	2	no	—
173	C₂×Q₈	—	536	2	no	—
174	C₂×Q₈	—	134	4	no	—
175	C₂²×C₄	—	2	1	no	—
176	C₂²×C₄	—	2	1	no	—
177	C₂²×C₄	—	268	2	no	—
178	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
179	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
180	C₂²×C₄	#11761628	1	1	yes	characteristic
181	C₂×C₈	—	2	1	no	—
182	C₂×C₈	—	2	1	no	—
183	C₂×C₈	—	2	1	no	—
184	C₂×C₈	—	2	1	no	—
185	C₂×C₈	—	4	1	no	—
186	C₂×C₈	—	4	1	no	—
187	C₂×C₈	—	4	1	no	—
188	C₂×C₈	—	268	2	no	—
189	C₂×C₈	—	268	2	no	—
190	C₂×C₈	—	268	2	no	—
191	C₂×C₈	#11761616	1	2	yes	—
192	C₂×C₈	#11761618	1	2	yes	—
193	C₄⋊C₄	—	2	1	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	—	268	2	no	—
196	C₄⋊C₄	—	268	2	no	—
197	C₄⋊C₄	—	134	4	no	—
198	C₄⋊C₄	—	134	4	no	—
199	C₄⋊C₄	Dic₂₆₈	1	2	yes	—
200	C₄²	—	4	1	no	—
201	C₄²	—	4	1	no	—
202	C₄²	D₄×D₆₇	1	2	yes	—
203	C₄²	#11761629	1	2	yes	—
204	C₂³	#14524017	1	1	yes	characteristic
205	Q₈	—	536	4	no	—
206	Q₈	—	536	4	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	4	1	no	—
211	C₂×C₄	—	4	1	no	—
212	C₂×C₄	—	4	1	no	—
213	C₂×C₄	—	4	1	no	—
214	C₂×C₄	—	4	1	no	—
215	C₂×C₄	—	268	4	no	—
216	C₂×C₄	—	536	2	no	—
217	C₂×C₄	—	536	2	no	—
218	C₂×C₄	#14524011	1	1	yes	characteristic
219	C₂×C₄	#14524018	1	2	yes	—
220	C₂×C₄	#14524028	1	2	yes	—
221	C₂×C₄	#14524032	1	1	yes	characteristic
222	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
223	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
224	C₂×C₄	C₂×Dic₂₆₈	1	1	yes	characteristic
225	C₂×C₄	#14524065	1	1	yes	characteristic
226	C₂×C₄	#14524073	1	1	yes	characteristic
227	C₂×C₄	#14524073	1	1	yes	characteristic
228	C₂×C₄	#14524076	1	2	yes	—
229	C₈	—	2	2	no	—
230	C₈	—	2	2	no	—
231	C₈	—	8	1	no	—
232	C₈	—	536	2	no	—
233	C₂²	#16863313	1	1	yes	characteristic
234	C₂²	#16863601	1	1	yes	characteristic
235	C₂²	#16863606	1	1	yes	characteristic
236	C₂²	#16863637	1	1	yes	characteristic
237	C₂²	#16863654	1	1	yes	characteristic
238	C₂²	#16863666	1	1	yes	characteristic
239	C₂²	#16863666	1	1	yes	characteristic
240	C₄	—	2	2	no	—
241	C₄	—	4	2	no	—
242	C₄	—	8	1	no	—
243	C₄	—	536	4	no	—
244	C₄	#16863534	1	1	yes	characteristic
245	C₄	#16863551	1	1	yes	characteristic
246	C₄	#16863607	1	2	yes	—
247	C₄	#16863619	1	2	yes	—
248	C₄	#16863748	1	1	yes	characteristic
249	C₄	#16863766	1	1	yes	characteristic
250	C₂	#22559608	1	1	yes	characteristic
251	C₂	#22559613	1	1	yes	characteristic
252	C₂	#22559732	1	1	yes	characteristic
253	C₂	#22559737	1	1	yes	characteristic
254	C₂	#22561455	1	1	yes	characteristic
255	C₂	#22561465	1	1	yes	characteristic
256	C₂	#22561532	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14524017
commutator, cosocle	C₂×C₄²×C₆₇	C₂³
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1939	C₂
socle, prime radical	C₂³×C₆₇	#121
hypercenter	#1939	D₆₇
hypocenter	C₆₇	#10543
LCS	G ▷ C₂×C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1939

Complements

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

N	H ≅ G/N	type
134. C₆₇	#10543	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863801	2
#22561157	2
C₂×#16863144	2
#22558044	2
C₆₇×#1939	2
#10543	67