Subgroups — #30754502

Order 17152.

370 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22560973	C₂	1	2	yes	maximal
2	#22560366	C₂	1	1	yes	maximal, characteristic
3	#22560076	C₂	1	1	yes	maximal, characteristic
4	#22558093	C₂	1	2	yes	maximal
5	C₂×#335×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523975	—	2	2	no	—
7	C₂×#14523975	—	2	1	no	—
8	C₂×#14523975	—	2	1	no	—
9	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
10	C₂×#14523953	—	2	1	no	—
11	#16863266	C₂²	1	2	yes	—
12	C₄×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
13	#16863159	—	2	2	no	—
14	#16863150	—	2	1	no	—
15	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	#335×C₆₇	C₄	1	4	yes	complemented
18	C₂²×C₆₇⋊C₈	—	2	1	no	—
19	C₂×C₄⋊Dic₆₇	—	2	2	no	—
20	C₂×C₄⋊Dic₆₇	—	2	2	no	—
21	C₂×C₄⋊Dic₆₇	—	2	2	no	—
22	C₂×C₄⋊Dic₆₇	D₄	1	2	yes	—
23	C₂×C₄×Dic₆₇	—	2	1	no	—
24	#14523975	—	4	2	no	—
25	#14523975	—	4	4	no	—
26	#14523975	—	4	2	no	—
27	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
28	C₄⋊(C₆₇⋊C₈)	Q₈	1	2	yes	—
29	#14523953	—	4	2	no	—
30	C₆₇⋊#95	—	2	1	no	—
31	C₂²×C₈×C₆₇	—	2	2	no	—
32	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
35	C₄⋊C₈×C₆₇	—	2	2	no	—
36	C₄⋊C₈×C₆₇	C₂×C₄	1	4	yes	—
37	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₆₇	Q₈	1	2	yes	—
39	C₂²×Dic₆₇	—	2	2	no	—
40	C₂²×Dic₆₇	—	4	2	no	—
41	C₂×C₆₇⋊C₈	—	2	2	no	—
42	C₂×C₆₇⋊C₈	—	4	1	no	—
43	C₂×C₆₇⋊C₈	—	4	1	no	—
44	C₄⋊Dic₆₇	—	2	2	no	—
45	C₄⋊Dic₆₇	—	2	2	no	—
46	C₄⋊Dic₆₇	—	4	2	no	—
47	C₄⋊Dic₆₇	—	4	2	no	—
48	C₄⋊Dic₆₇	—	4	2	no	—
49	C₄⋊Dic₆₇	—	4	2	no	—
50	C₄⋊Dic₆₇	—	8	2	no	—
51	C₄×Dic₆₇	—	4	1	no	—
52	C₄×Dic₆₇	—	4	1	no	—
53	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
56	C₂×C₈×C₆₇	—	2	1	no	—
57	C₂×C₈×C₆₇	—	2	1	no	—
58	C₂×C₈×C₆₇	—	2	4	no	—
59	C₂×C₈×C₆₇	—	4	2	no	—
60	C₂×C₈×C₆₇	—	4	2	no	—
61	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
62	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
63	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
64	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
65	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
66	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
67	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
68	C₄²×C₆₇	#41	1	1	yes	characteristic
69	C₂³×C₆₇	#121	1	1	yes	characteristic
70	C₂×C₄×C₆₇	—	2	2	no	—
71	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
73	C₂×C₄×C₆₇	C₄×Q₈	1	1	yes	characteristic
74	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₆₇	#125	1	1	yes	characteristic
76	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
77	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
78	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
80	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
81	C₂×C₄×C₆₇	#137	1	2	yes	—
82	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
83	C₂×Dic₆₇	—	4	2	no	—
84	C₂×Dic₆₇	—	4	2	no	—
85	C₂×Dic₆₇	—	4	2	no	—
86	C₂×Dic₆₇	—	8	2	no	—
87	C₂×Dic₆₇	—	8	2	no	—
88	C₂×Dic₆₇	—	8	2	no	—
89	C₈×C₆₇	—	2	2	no	—
90	C₈×C₆₇	—	4	4	no	—
91	C₈×C₆₇	#106	1	4	yes	complemented
92	C₆₇⋊C₈	—	4	2	no	—
93	C₂²×C₆₇	#389	1	1	yes	characteristic
94	C₂²×C₆₇	#460	1	1	yes	characteristic
95	C₂²×C₆₇	#461	1	1	yes	characteristic
96	C₂²×C₆₇	#468	1	2	yes	—
97	C₂²×C₆₇	#471	1	2	yes	—
98	C₄×C₆₇	—	2	1	no	—
99	C₄×C₆₇	—	2	1	no	—
100	C₄×C₆₇	C₂×#106	1	1	yes	characteristic
101	C₄×C₆₇	#428	1	1	yes	characteristic
102	C₄×C₆₇	#441	1	1	yes	characteristic
103	C₄×C₆₇	#442	1	1	yes	characteristic
104	C₄×C₆₇	#475	1	1	yes	characteristic
105	C₄×C₆₇	#476	1	1	yes	characteristic
106	C₄×C₆₇	#501	1	2	yes	—
107	Dic₆₇	—	8	2	no	—
108	Dic₆₇	—	16	2	no	—
109	#9618	—	67	1	no	maximal, Sylow
110	C₂×C₆₇	#1686	1	2	yes	—
111	C₂×C₆₇	#1691	1	2	yes	—
112	C₂×C₆₇	#1943	1	1	yes	characteristic
113	C₂×C₆₇	#1948	1	1	yes	characteristic
114	C₂×C₆₇	#1951	1	1	yes	characteristic
115	#1948	—	67	2	no	—
116	#1844	—	67	1	no	—
117	#1770	—	67	1	no	—
118	C₂×#335	D₆₇	1	1	yes	characteristic
119	#1296	—	67	2	no	—
120	C₆₇	#9618	1	1	yes	characteristic, complemented, Sylow
121	C₂×#106	—	134	2	no	—
122	C₂×#106	—	134	1	no	—
123	C₂×#106	—	134	1	no	—
124	C₂×C₄⋊C₈	—	67	1	no	—
125	C₂×C₄⋊C₈	C₂×D₆₇	1	2	yes	—
126	C₂×#97	—	134	1	no	—
127	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
128	#384	—	67	2	no	—
129	C₄×C₄⋊C₄	—	67	1	no	—
130	#340	—	134	2	no	—
131	#336	—	134	1	no	—
132	#335	Dic₆₇	1	4	yes	complemented
133	C₂²×C₈	—	2	2	no	—
134	C₂²×C₈	—	134	1	no	—
135	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
136	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
137	C₂×C₄⋊C₄	—	67	2	no	—
138	C₂×C₄⋊C₄	—	134	2	no	—
139	C₂×C₄⋊C₄	—	134	2	no	—
140	C₂×C₄⋊C₄	—	134	2	no	—
141	C₂×C₄²	—	134	1	no	—
142	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
143	#106	—	268	2	no	—
144	#106	—	268	2	no	—
145	#106	—	268	4	no	—
146	C₄⋊C₈	—	2	2	no	—
147	C₄⋊C₈	—	67	2	no	—
148	C₄⋊C₈	—	134	1	no	—
149	C₄⋊C₈	C₂×Dic₆₇	1	4	yes	—
150	#97	—	268	2	no	—
151	C₄×C₈	Dic₁₃₄	1	2	yes	—
152	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
153	#95	—	134	1	no	—
154	C₂²×C₄	—	268	2	no	—
155	C₂²×C₄	—	134	2	no	—
156	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
157	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
158	C₂²×C₄	#11761629	1	1	yes	characteristic
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	—	2	1	no	—
161	C₂×C₈	—	2	4	no	—
162	C₂×C₈	—	4	2	no	—
163	C₂×C₈	—	4	2	no	—
164	C₂×C₈	—	268	1	no	—
165	C₂×C₈	—	268	1	no	—
166	C₂×C₈	—	134	2	no	—
167	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
168	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
169	C₂×C₈	#11761617	1	2	yes	—
170	C₂×C₈	#11761617	1	2	yes	—
171	C₄⋊C₄	—	268	2	no	—
172	C₄⋊C₄	—	268	2	no	—
173	C₄⋊C₄	—	268	2	no	—
174	C₄⋊C₄	—	268	2	no	—
175	C₄⋊C₄	—	536	2	no	—
176	C₄⋊C₄	—	134	2	no	—
177	C₄⋊C₄	—	134	2	no	—
178	C₄²	—	268	1	no	—
179	C₄²	—	268	1	no	—
180	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
181	C₄²	Q₈×D₆₇	1	1	yes	characteristic
182	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
183	C₄²	#11761627	1	1	yes	characteristic
184	C₂³	#14524018	1	1	yes	characteristic
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	268	2	no	—
187	C₂×C₄	—	268	2	no	—
188	C₂×C₄	—	268	2	no	—
189	C₂×C₄	—	536	2	no	—
190	C₂×C₄	—	536	2	no	—
191	C₂×C₄	—	536	2	no	—
192	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
193	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
194	C₂×C₄	Q₈×Dic₆₇	1	1	yes	characteristic
195	C₂×C₄	#14524023	1	1	yes	characteristic
196	C₂×C₄	#14524034	1	1	yes	characteristic
197	C₂×C₄	#14524042	1	1	yes	characteristic
198	C₂×C₄	C₂×#11761617	1	1	yes	characteristic
199	C₂×C₄	C₂×#11761617	1	1	yes	characteristic
200	C₂×C₄	#14524067	1	2	yes	—
201	C₂×C₄	#14524071	1	1	yes	characteristic
202	C₂×C₄	#14524075	1	2	yes	—
203	C₂×C₄	#14524076	1	1	yes	characteristic
204	C₈	—	2	2	no	—
205	C₈	—	4	4	no	—
206	C₈	—	268	2	no	—
207	C₈	#14523975	1	4	yes	complemented
208	C₂²	#16863295	1	1	yes	characteristic
209	C₂²	#16863597	1	1	yes	characteristic
210	C₂²	#16863602	1	1	yes	characteristic
211	C₂²	#16863643	1	2	yes	—
212	C₂²	#16863660	1	2	yes	—
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	536	2	no	—
216	C₄	—	1072	2	no	—
217	C₄	C₂×#14523975	1	1	yes	characteristic
218	C₄	#16863438	1	1	yes	characteristic
219	C₄	#16863503	1	1	yes	characteristic
220	C₄	#16863508	1	1	yes	characteristic
221	C₄	#16863682	1	1	yes	characteristic
222	C₄	#16863687	1	1	yes	characteristic
223	C₄	#16863804	1	2	yes	—
224	C₂	#22559682	1	2	yes	—
225	C₂	#22559709	1	2	yes	—
226	C₂	#22560948	1	1	yes	characteristic
227	C₂	#22560973	1	1	yes	characteristic
228	C₂	#22560990	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14524018
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×#335×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	#121
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×#335	D₆₇
hypocenter	C₆₇	#9618
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
17. #335×C₆₇	C₄	semidirect
91. C₈×C₆₇	#106	semidirect
120. C₆₇	#9618	semidirect
132. #335	Dic₆₇	semidirect
207. C₈	#14523975	semidirect

Maximal subgroups

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

subgroup	index
#22560973	2
#22560366	2
#22560076	2
#22558093	2
C₂×#335×C₆₇	2
#9618	67