Subgroups — #30745200

Order 17152.

518 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22559954	C₂	1	1	yes	maximal, characteristic
2	#22559954	C₂	1	1	yes	maximal, characteristic
3	C₄×#14523953	C₂	1	1	yes	maximal, characteristic
4	C₂×#16863118	C₂	1	1	yes	maximal, characteristic
5	C₂×#16863118	C₂	1	1	yes	maximal, characteristic
6	C₂×C₈×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
7	C₂×#322×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#14523953	C₄	1	4	yes	—
9	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
12	C₄²×Dic₆₇	C₂²	1	1	yes	characteristic
13	#16863150	C₄	1	4	yes	—
14	#16863150	C₄	1	4	yes	—
15	#16863118	C₄	1	4	yes	complemented
16	#16863118	C₄	1	4	yes	complemented
17	C₈×C₆₇⋊C₈	C₄	1	4	yes	complemented
18	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
21	#322×C₆₇	C₄	1	4	yes	complemented
22	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
23	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
24	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
25	C₂×C₄×Dic₆₇	C₂×C₄	1	2	yes	—
26	C₂×C₄×Dic₆₇	C₂×C₄	1	2	yes	—
27	C₂×C₄×Dic₆₇	C₂×C₄	1	2	yes	—
28	#14523953	—	2	8	no	—
29	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
30	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
31	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
32	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
33	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
34	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
37	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
39	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
43	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
45	C₂²×Dic₆₇	C₄²	1	4	yes	—
46	C₂×C₆₇⋊C₈	—	2	4	no	—
47	C₂×C₆₇⋊C₈	—	2	4	no	—
48	C₂×C₆₇⋊C₈	C₄²	1	4	yes	—
49	C₂×C₆₇⋊C₈	C₄²	1	4	yes	—
50	C₂×C₆₇⋊C₈	C₄²	1	4	yes	—
51	C₂×C₆₇⋊C₈	M₁₆	1	8	yes	—
52	C₄×Dic₆₇	—	2	4	no	—
53	C₄×Dic₆₇	—	2	4	no	—
54	C₄×Dic₆₇	—	2	4	no	—
55	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₆₇	—	2	4	no	—
59	C₂×C₈×C₆₇	—	2	4	no	—
60	C₂×C₈×C₆₇	—	2	4	no	—
61	C₂×C₈×C₆₇	C₄²	1	4	yes	—
62	C₂×C₈×C₆₇	C₄²	1	4	yes	—
63	C₂×C₈×C₆₇	C₄²	1	4	yes	—
64	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
65	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
66	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
67	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
68	C₂³×C₆₇	C₂×C₄²	1	1	yes	characteristic
69	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
70	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
71	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
72	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
73	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
74	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
75	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
76	C₂×C₄×C₆₇	#131	1	4	yes	—
77	C₂×C₄×C₆₇	#131	1	4	yes	—
78	C₂×Dic₆₇	—	2	4	no	—
79	C₂×Dic₆₇	—	2	4	no	—
80	C₂×Dic₆₇	—	2	4	no	—
81	C₈×C₆₇	—	2	4	no	—
82	C₈×C₆₇	—	2	4	no	—
83	C₈×C₆₇	—	2	4	no	—
84	C₆₇⋊C₈	—	2	4	no	—
85	C₆₇⋊C₈	—	2	4	no	—
86	C₆₇⋊C₈	#97	1	8	yes	—
87	C₂²×C₆₇	C₄³	1	1	yes	characteristic
88	C₂²×C₆₇	C₄×M₁₆	1	2	yes	—
89	C₂²×C₆₇	#405	1	2	yes	—
90	C₂²×C₆₇	#405	1	2	yes	—
91	C₄×C₆₇	C₂×#97	1	2	yes	—
92	C₄×C₆₇	C₄×M₁₆	1	2	yes	—
93	C₄×C₆₇	#405	1	2	yes	—
94	C₄×C₆₇	#405	1	2	yes	—
95	C₄×C₆₇	#405	1	2	yes	—
96	C₄×C₆₇	#405	1	2	yes	—
97	Dic₆₇	—	4	4	no	—
98	#7844	—	67	1	no	maximal, Sylow
99	C₂×C₆₇	#1453	1	4	yes	—
100	C₂×C₆₇	C₄×#97	1	1	yes	characteristic
101	C₂×C₆₇	#1726	1	1	yes	characteristic
102	C₂×C₆₇	#1726	1	1	yes	characteristic
103	#1744	—	67	1	no	—
104	#1744	—	67	1	no	—
105	C₄×#97	—	67	1	no	—
106	C₂×#322	—	67	1	no	—
107	C₂×#322	—	67	1	no	—
108	C₂×#322	D₆₇	1	1	yes	characteristic
109	C₂×C₈²	—	67	1	no	—
110	C₆₇	#7844	1	1	yes	characteristic, complemented, Sylow
111	C₂×#97	—	67	4	no	—
112	C₂×C₄×C₈	—	67	1	no	—
113	C₂×C₄×C₈	—	67	1	no	—
114	C₂×C₄×C₈	—	67	1	no	—
115	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
116	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
117	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
118	C₄³	—	67	1	no	—
119	#336	—	67	4	no	—
120	#336	—	67	4	no	—
121	#322	—	67	4	no	—
122	#322	—	67	4	no	—
123	#322	Dic₆₇	1	4	yes	complemented
124	C₈²	—	67	4	no	—
125	C₂²×C₈	—	67	2	no	—
126	C₂²×C₈	—	67	2	no	—
127	C₂²×C₈	—	67	2	no	—
128	C₂²×C₈	C₄×D₆₇	1	2	yes	—
129	C₂²×C₈	C₄×D₆₇	1	2	yes	—
130	C₂²×C₈	C₄×D₆₇	1	2	yes	—
131	C₂×C₄²	—	67	2	no	—
132	C₂×C₄²	—	67	2	no	—
133	C₂×C₄²	—	67	2	no	—
134	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
135	#97	—	134	8	no	—
136	C₄×C₈	—	67	2	no	—
137	C₄×C₈	—	67	2	no	—
138	C₄×C₈	—	67	2	no	—
139	C₄×C₈	—	67	2	no	—
140	C₄×C₈	—	67	2	no	—
141	C₄×C₈	—	67	2	no	—
142	C₄×C₈	C₄×D₆₇	1	2	yes	—
143	C₄×C₈	C₄×D₆₇	1	2	yes	—
144	C₄×C₈	C₄×D₆₇	1	2	yes	—
145	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
146	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
147	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
148	C₂²×C₄	—	67	4	no	—
149	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
150	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
151	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	—	2	4	no	—
154	C₂×C₈	—	2	4	no	—
155	C₂×C₈	—	67	4	no	—
156	C₂×C₈	—	67	8	no	—
157	C₂×C₈	—	67	4	no	—
158	C₂×C₈	—	67	4	no	—
159	C₂×C₈	—	134	4	no	—
160	C₂×C₈	—	134	4	no	—
161	C₂×C₈	C₄×Dic₆₇	1	4	yes	—
162	C₂×C₈	C₄×Dic₆₇	1	4	yes	—
163	C₂×C₈	C₄×Dic₆₇	1	4	yes	—
164	C₄²	—	134	4	no	—
165	C₄²	—	134	4	no	—
166	C₄²	—	134	4	no	—
167	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
168	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
169	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
170	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
171	C₂³	C₄²×D₆₇	1	1	yes	characteristic
172	C₂×C₄	—	134	4	no	—
173	C₂×C₄	—	134	4	no	—
174	C₂×C₄	—	134	4	no	—
175	C₂×C₄	C₄²×D₆₇	1	1	yes	characteristic
176	C₂×C₄	C₄²×D₆₇	1	1	yes	characteristic
177	C₂×C₄	C₄²×D₆₇	1	1	yes	characteristic
178	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
179	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
180	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
181	C₂×C₄	M₁₆×D₆₇	1	4	yes	—
182	C₂×C₄	#14524049	1	4	yes	—
183	C₂×C₄	#14524051	1	4	yes	—
184	C₈	—	2	4	no	—
185	C₈	—	2	4	no	—
186	C₈	—	2	4	no	—
187	C₈	—	67	8	no	—
188	C₈	—	134	4	no	—
189	C₈	—	134	4	no	—
190	C₂²	C₄²×Dic₆₇	1	1	yes	characteristic
191	C₂²	#16863348	1	2	yes	—
192	C₂²	#16863350	1	2	yes	—
193	C₂²	#16863353	1	2	yes	—
194	C₄	—	268	4	no	—
195	C₄	#97×D₆₇	1	2	yes	—
196	C₄	M₁₆×Dic₆₇	1	2	yes	—
197	C₄	#16863350	1	2	yes	—
198	C₄	#16863351	1	2	yes	—
199	C₄	#16863353	1	2	yes	—
200	C₄	#16863354	1	2	yes	—
201	C₂	#22558505	1	4	yes	—
202	C₂	#97×Dic₆₇	1	1	yes	characteristic
203	C₂	#22559895	1	1	yes	characteristic
204	C₂	#22559897	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₆₇
commutator	C₂²×C₆₇	C₄³
Fitting	C₂×#322×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₄²
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×#322	D₆₇
hypocenter	C₆₇	#7844
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄² ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
15. #16863118	C₄	semidirect
16. #16863118	C₄	semidirect
17. C₈×C₆₇⋊C₈	C₄	semidirect
21. #322×C₆₇	C₄	semidirect
110. C₆₇	#7844	semidirect
123. #322	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22559954	2
C₄×#14523953	2
C₂×#16863118	2
C₂×C₈×C₆₇⋊C₈	2
C₂×#322×C₆₇	2
#7844	67