Subgroups — #30773871

Order 17152.

318 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863224	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×#346×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	#16863224	C₄	1	4	yes	complemented
10	#16863222	—	2	2	no	—
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	#346×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	2	no	—
62	C₂×C₄×C₆₇	—	2	2	no	—
63	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
64	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
65	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
67	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
68	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
69	C₂×Dic₆₇	—	4	4	no	—
70	C₂×Dic₆₇	—	8	2	no	—
71	C₂×Dic₆₇	—	8	2	no	—
72	C₂×Dic₆₇	—	8	2	no	—
73	C₂×Dic₆₇	—	8	2	no	—
74	C₂×Dic₆₇	—	8	2	no	—
75	C₈×C₆₇	—	2	2	no	—
76	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₈×C₆₇	C₄×Q₈	1	1	yes	characteristic
78	C₈×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
79	C₈×C₆₇	#125	1	1	yes	characteristic
80	C₂²×C₆₇	C₄×D₈	1	1	yes	characteristic
81	C₂²×C₆₇	#466	1	1	yes	characteristic
82	C₂²×C₆₇	#500	1	1	yes	characteristic
83	C₂²×C₆₇	#509	1	2	yes	—
84	C₂²×C₆₇	#510	1	2	yes	—
85	C₄×C₆₇	—	4	1	no	—
86	C₄×C₆₇	—	4	1	no	—
87	C₄×C₆₇	#389	1	1	yes	characteristic
88	C₄×C₆₇	C₄×Q₁₆	1	1	yes	characteristic
89	C₄×C₆₇	#467	1	1	yes	characteristic
90	C₄×C₆₇	#499	1	1	yes	characteristic
91	Dic₆₇	—	8	4	no	—
92	Dic₆₇	—	16	2	no	—
93	#13167	—	67	1	no	maximal, Sylow
94	C₂×C₆₇	#1942	1	1	yes	characteristic
95	C₂×C₆₇	#2175	1	2	yes	—
96	C₂×C₆₇	#2218	1	1	yes	characteristic
97	C₂×C₆₇	#2219	1	1	yes	characteristic
98	C₂×C₆₇	#2255	1	2	yes	—
99	C₂×#367	—	67	1	no	—
100	C₂×#366	—	67	1	no	—
101	C₂×#346	D₆₇	1	1	yes	characteristic
102	#1942	—	67	2	no	—
103	#1380	—	67	2	no	—
104	C₆₇	#13167	1	1	yes	characteristic, complemented, Sylow
105	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
106	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
107	C₂×#107	—	67	2	no	—
108	C₂×#107	—	67	2	no	—
109	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
110	#389	—	134	2	no	—
111	#367	—	67	4	no	—
112	#366	—	134	2	no	—
113	#346	—	2	2	no	—
114	#340	—	134	2	no	—
115	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
116	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
117	C₂×C₄⋊C₄	—	268	2	no	—
118	C₂×C₄⋊C₄	—	134	2	no	—
119	C₂×C₄⋊C₄	—	134	2	no	—
120	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
121	C₂×C₁₆	—	2	1	no	—
122	C₂×C₁₆	—	2	1	no	—
123	C₂×C₁₆	Dic₁₃₄	1	2	yes	—
124	C₂×C₁₆	Dic₁₃₄	1	2	yes	—
125	C₂×C₁₆	C₄×D₆₇	1	2	yes	—
126	C₂×C₁₆	C₂²⋊D₆₇	1	2	yes	—
127	#107	—	67	4	no	—
128	#107	—	134	2	no	—
129	#107	—	134	2	no	—
130	#107	—	134	2	no	—
131	C₄×C₈	—	2	1	no	—
132	C₄×C₈	—	2	1	no	—
133	#95	—	268	2	no	—
134	C₂²×C₄	—	268	2	no	—
135	C₂²×C₄	—	268	2	no	—
136	C₂²×C₄	D₅₃₆	1	1	yes	characteristic
137	C₂²×C₄	D₅₃₆	1	1	yes	characteristic
138	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	Dic₂₆₈	1	2	yes	—
142	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
143	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
144	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
145	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
146	C₂×C₈	#11761628	1	1	yes	characteristic
147	C₂×C₈	#11761628	1	1	yes	characteristic
148	C₄⋊C₄	—	268	2	no	—
149	C₄⋊C₄	—	268	2	no	—
150	C₄⋊C₄	—	268	2	no	—
151	C₄⋊C₄	—	536	4	no	—
152	C₄⋊C₄	—	134	4	no	—
153	C₄²	—	2	1	no	—
154	C₄²	—	2	1	no	—
155	C₁₆	—	2	2	no	—
156	C₁₆	Dic₆₇⋊C₄	1	4	yes	complemented
157	C₂³	C₂×D₅₃₆	1	1	yes	characteristic
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	268	4	no	—
163	C₂×C₄	—	536	2	no	—
164	C₂×C₄	—	536	2	no	—
165	C₂×C₄	—	536	2	no	—
166	C₂×C₄	—	536	2	no	—
167	C₂×C₄	—	536	2	no	—
168	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
169	C₂×C₄	#14524017	1	1	yes	characteristic
170	C₂×C₄	#14524043	1	1	yes	characteristic
171	C₂×C₄	C₂×Dic₂₆₈	1	1	yes	characteristic
172	C₂×C₄	#14524065	1	1	yes	characteristic
173	C₂×C₄	#14524065	1	1	yes	characteristic
174	C₈	—	2	2	no	—
175	C₈	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
176	C₈	C₄×Dic₁₃₄	1	1	yes	characteristic
177	C₈	#14524022	1	1	yes	characteristic
178	C₈	#14524035	1	1	yes	characteristic
179	C₂²	C₄×D₅₃₆	1	1	yes	characteristic
180	C₂²	#16863632	1	1	yes	characteristic
181	C₂²	#16863801	1	1	yes	characteristic
182	C₂²	#16863834	1	2	yes	—
183	C₂²	#16863839	1	2	yes	—
184	C₄	—	4	1	no	—
185	C₄	—	4	1	no	—
186	C₄	—	536	4	no	—
187	C₄	—	1072	2	no	—
188	C₄	#16863294	1	1	yes	characteristic
189	C₄	C₄×Dic₂₆₈	1	1	yes	characteristic
190	C₄	#16863637	1	1	yes	characteristic
191	C₄	#16863798	1	1	yes	characteristic
192	C₂	#22560942	1	1	yes	characteristic
193	C₂	#22562075	1	2	yes	—
194	C₂	#22562312	1	1	yes	characteristic
195	C₂	#22562317	1	1	yes	characteristic
196	C₂	#22562495	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₅₃₆
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂×#346×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂×D₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×#346	D₆₇
hypocenter	C₆₇	#13167
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
9. #16863224	C₄	semidirect
57. C₁₆×C₆₇	C₄⋊C₄	semidirect
104. C₆₇	#13167	semidirect
156. 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×#16863224	2
C₂×#16863222	2
#22560942	2
#22558301	2
C₂×#346×C₆₇	2
#13167	67