Subgroups — #30773861

Order 17152.

326 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863222	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863222	C₂	1	1	yes	maximal, characteristic
3	#22560942	C₂	1	2	yes	maximal
4	#22558301	C₂	1	2	yes	maximal
5	C₂×C₄×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523977	C₂²	1	2	yes	—
7	C₂×#14523977	C₂²	1	2	yes	—
8	#16863294	—	2	2	no	—
9	#16863222	—	2	2	no	—
10	#16863222	C₄	1	4	yes	complemented
11	#16863159	—	2	2	no	—
12	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₄×C₁₆×C₆₇	—	2	2	no	—
16	C₂×C₄⋊Dic₆₇	—	2	2	no	—
17	C₂×C₄⋊Dic₆₇	—	2	2	no	—
18	C₂×Dic₆₇⋊C₄	—	4	2	no	—
19	#14523977	—	2	2	no	—
20	#14523977	—	2	2	no	—
21	#14523977	—	2	2	no	—
22	#14523977	C₂×C₄	1	4	yes	—
23	C₆₇⋊#95	—	4	2	no	—
24	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
27	C₂×C₁₆×C₆₇	—	2	1	no	—
28	C₂×C₁₆×C₆₇	—	2	1	no	—
29	C₂×C₁₆×C₆₇	C₂×C₄	1	2	yes	—
30	C₂×C₁₆×C₆₇	D₄	1	2	yes	—
31	C₂×C₁₆×C₆₇	Q₈	1	2	yes	—
32	C₂×C₁₆×C₆₇	Q₈	1	2	yes	—
33	C₄×C₈×C₆₇	—	2	1	no	—
34	C₄×C₈×C₆₇	—	2	1	no	—
35	C₂²×Dic₆₇	—	4	2	no	—
36	C₂²×Dic₆₇	—	4	2	no	—
37	C₄⋊Dic₆₇	—	2	4	no	—
38	C₄⋊Dic₆₇	—	4	2	no	—
39	C₄⋊Dic₆₇	—	4	2	no	—
40	C₄⋊Dic₆₇	—	4	2	no	—
41	Dic₆₇⋊C₄	—	8	4	no	—
42	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
43	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
44	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈×C₆₇	—	2	1	no	—
46	C₂×C₈×C₆₇	—	2	1	no	—
47	C₂×C₈×C₆₇	Q₁₆	1	2	yes	—
48	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
53	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
54	C₄²×C₆₇	—	2	1	no	—
55	C₄²×C₆₇	—	2	1	no	—
56	C₁₆×C₆₇	—	2	2	no	—
57	C₁₆×C₆₇	C₄⋊C₄	1	4	yes	complemented
58	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
59	C₂×C₄×C₆₇	—	2	1	no	—
60	C₂×C₄×C₆₇	—	2	1	no	—
61	C₂×C₄×C₆₇	—	2	1	no	—
62	C₂×C₄×C₆₇	—	2	1	no	—
63	C₂×C₄×C₆₇	D₁₆	1	2	yes	—
64	C₂×C₄×C₆₇	Q₃₂	1	2	yes	—
65	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
66	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
67	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
69	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
70	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
71	C₂×Dic₆₇	—	4	4	no	—
72	C₂×Dic₆₇	—	8	2	no	—
73	C₂×Dic₆₇	—	8	2	no	—
74	C₂×Dic₆₇	—	8	2	no	—
75	C₂×Dic₆₇	—	8	2	no	—
76	C₂×Dic₆₇	—	8	2	no	—
77	C₈×C₆₇	—	2	2	no	—
78	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₈×C₆₇	C₄×Q₈	1	1	yes	characteristic
80	C₈×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
81	C₈×C₆₇	#125	1	1	yes	characteristic
82	C₂²×C₆₇	C₄×D₈	1	1	yes	characteristic
83	C₂²×C₆₇	#466	1	1	yes	characteristic
84	C₂²×C₆₇	#500	1	1	yes	characteristic
85	C₂²×C₆₇	C₂×D₁₆	1	1	yes	characteristic
86	C₂²×C₆₇	C₂×Q₃₂	1	1	yes	characteristic
87	C₂²×C₆₇	#508	1	1	yes	characteristic
88	C₂²×C₆₇	#508	1	1	yes	characteristic
89	C₄×C₆₇	—	2	2	no	—
90	C₄×C₆₇	—	2	2	no	—
91	C₄×C₆₇	#389	1	1	yes	characteristic
92	C₄×C₆₇	C₄×Q₁₆	1	1	yes	characteristic
93	C₄×C₆₇	#467	1	1	yes	characteristic
94	C₄×C₆₇	#499	1	1	yes	characteristic
95	Dic₆₇	—	8	4	no	—
96	Dic₆₇	—	16	2	no	—
97	#13165	—	67	1	no	maximal, Sylow
98	C₂×C₆₇	#1942	1	1	yes	characteristic
99	C₂×C₆₇	C₄×D₁₆	1	1	yes	characteristic
100	C₂×C₆₇	C₄×Q₃₂	1	1	yes	characteristic
101	C₂×C₆₇	#2215	1	1	yes	characteristic
102	C₂×C₆₇	#2216	1	1	yes	characteristic
103	C₂×C₆₇	#2252	1	1	yes	characteristic
104	C₂×C₆₇	#2253	1	1	yes	characteristic
105	C₂×#366	—	67	1	no	—
106	C₂×#366	—	67	1	no	—
107	C₂×C₄×C₁₆	D₆₇	1	1	yes	characteristic
108	#1942	—	67	2	no	—
109	#1380	—	67	2	no	—
110	C₆₇	#13165	1	1	yes	characteristic, complemented, Sylow
111	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
112	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
113	C₂×#107	—	67	2	no	—
114	C₂×#107	—	67	2	no	—
115	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
116	#389	—	134	2	no	—
117	#366	—	67	4	no	—
118	#366	—	134	2	no	—
119	C₄×C₁₆	—	2	2	no	—
120	#340	—	134	2	no	—
121	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
122	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
123	C₂×C₄⋊C₄	—	268	2	no	—
124	C₂×C₄⋊C₄	—	134	2	no	—
125	C₂×C₄⋊C₄	—	134	2	no	—
126	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
127	C₂×C₁₆	—	2	1	no	—
128	C₂×C₁₆	—	2	1	no	—
129	C₂×C₁₆	Dic₁₃₄	1	2	yes	—
130	C₂×C₁₆	Dic₁₃₄	1	2	yes	—
131	C₂×C₁₆	C₄×D₆₇	1	2	yes	—
132	C₂×C₁₆	C₂²⋊D₆₇	1	2	yes	—
133	#107	—	67	4	no	—
134	#107	—	134	2	no	—
135	#107	—	134	2	no	—
136	#107	—	134	2	no	—
137	C₄×C₈	—	2	1	no	—
138	C₄×C₈	—	2	1	no	—
139	#95	—	268	2	no	—
140	C₂²×C₄	—	268	2	no	—
141	C₂²×C₄	—	268	2	no	—
142	C₂²×C₄	D₅₃₆	1	1	yes	characteristic
143	C₂²×C₄	D₅₃₆	1	1	yes	characteristic
144	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	2	1	no	—
147	C₂×C₈	Dic₂₆₈	1	2	yes	—
148	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
149	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
150	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
151	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
152	C₂×C₈	#11761628	1	1	yes	characteristic
153	C₂×C₈	#11761628	1	1	yes	characteristic
154	C₄⋊C₄	—	268	2	no	—
155	C₄⋊C₄	—	268	2	no	—
156	C₄⋊C₄	—	268	2	no	—
157	C₄⋊C₄	—	536	4	no	—
158	C₄⋊C₄	—	134	4	no	—
159	C₄²	—	2	1	no	—
160	C₄²	—	2	1	no	—
161	C₁₆	—	2	2	no	—
162	C₁₆	Dic₆₇⋊C₄	1	4	yes	complemented
163	C₂³	C₂×D₅₃₆	1	1	yes	characteristic
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	268	4	no	—
169	C₂×C₄	—	536	2	no	—
170	C₂×C₄	—	536	2	no	—
171	C₂×C₄	—	536	2	no	—
172	C₂×C₄	—	536	2	no	—
173	C₂×C₄	—	536	2	no	—
174	C₂×C₄	D₁₀₇₂	1	2	yes	—
175	C₂×C₄	Dic₅₃₆	1	2	yes	—
176	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
177	C₂×C₄	#14524017	1	1	yes	characteristic
178	C₂×C₄	#14524043	1	1	yes	characteristic
179	C₂×C₄	C₂×Dic₂₆₈	1	1	yes	characteristic
180	C₂×C₄	#14524065	1	1	yes	characteristic
181	C₂×C₄	#14524065	1	1	yes	characteristic
182	C₈	—	2	2	no	—
183	C₈	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
184	C₈	C₄×Dic₁₃₄	1	1	yes	characteristic
185	C₈	#14524022	1	1	yes	characteristic
186	C₈	#14524035	1	1	yes	characteristic
187	C₂²	C₄×D₅₃₆	1	1	yes	characteristic
188	C₂²	#16863632	1	1	yes	characteristic
189	C₂²	#16863801	1	1	yes	characteristic
190	C₂²	C₂×D₁₀₇₂	1	1	yes	characteristic
191	C₂²	C₂×Dic₅₃₆	1	1	yes	characteristic
192	C₂²	#16863828	1	1	yes	characteristic
193	C₂²	#16863828	1	1	yes	characteristic
194	C₄	—	2	2	no	—
195	C₄	—	2	2	no	—
196	C₄	—	536	4	no	—
197	C₄	—	1072	2	no	—
198	C₄	#16863294	1	1	yes	characteristic
199	C₄	C₄×Dic₂₆₈	1	1	yes	characteristic
200	C₄	#16863637	1	1	yes	characteristic
201	C₄	#16863798	1	1	yes	characteristic
202	C₂	#22560942	1	1	yes	characteristic
203	C₂	C₄×D₁₀₇₂	1	1	yes	characteristic
204	C₂	C₄×Dic₅₃₆	1	1	yes	characteristic
205	C₂	#22562295	1	1	yes	characteristic
206	C₂	#22562300	1	1	yes	characteristic
207	C₂	#22562484	1	1	yes	characteristic
208	C₂	#22562488	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₅₃₆
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂×C₄×C₁₆×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂×D₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×C₄×C₁₆	D₆₇
hypocenter	C₆₇	#13165
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
10. #16863222	C₄	semidirect
57. C₁₆×C₆₇	C₄⋊C₄	semidirect
110. C₆₇	#13165	semidirect
162. C₁₆	Dic₆₇⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863222	2
#22560942	2
#22558301	2
C₂×C₄×C₁₆×C₆₇	2
#13165	67