Subgroups — #30825055

Order 17152.

702 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22568777	C₂	1	2	yes	maximal
2	#22558980	C₂	1	6	yes	maximal, complemented
3	#22558965	C₂	1	3	yes	maximal, complemented
4	C₂×#16863147	C₂	1	3	yes	maximal, complemented
5	C₈×#41×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14524050	—	2	2	no	—
7	#16863423	—	2	6	no	—
8	#16863420	C₂²	1	6	yes	—
9	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	6	yes	—
10	#16863147	C₂²	1	6	yes	—
11	#16863147	C₂²	1	6	yes	—
12	#16863147	C₂²	1	1	yes	characteristic
13	#16863147	C₂²	1	3	yes	—
14	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
15	C₈×Q₈×C₆₇	C₄	1	1	yes	characteristic
16	C₈×Q₈×C₆₇	C₄	1	1	yes	characteristic
17	C₈×D₄×C₆₇	C₄	1	3	yes	—
18	C₈×D₄×C₆₇	C₄	1	3	yes	—
19	#431×C₆₇	C₂²	1	3	yes	—
20	C₂×C₄×C₈×C₆₇	C₂²	1	3	yes	—
21	#14524050	—	4	8	no	—
22	C₂×#11761612	—	2	6	no	—
23	C₂²×C₆₇⋊C₈	—	2	6	no	—
24	C₄⋊(C₆₇⋊C₈)	—	2	6	no	—
25	C₄⋊(C₆₇⋊C₈)	—	2	6	no	—
26	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
27	C₄⋊(C₆₇⋊C₈)	C₂³	1	6	yes	—
28	C₂×#41×C₆₇	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₆₇	C₂×C₄	1	3	yes	—
30	C₂²×C₈×C₆₇	C₂×C₄	1	3	yes	—
31	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
32	C₄×Q₈×C₆₇	Q₈	1	1	yes	characteristic
33	C₄×D₄×C₆₇	C₂×C₄	1	3	yes	—
34	C₄×D₄×C₆₇	Q₈	1	3	yes	—
35	#117×C₆₇	D₄	1	3	yes	—
36	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
37	C₄⋊C₈×C₆₇	C₂×C₄	1	3	yes	—
38	C₄⋊C₈×C₆₇	C₂×C₄	1	3	yes	—
39	C₂²⋊C₈×C₆₇	C₂×C₄	1	6	yes	—
40	C₄×C₈×C₆₇	C₂×C₄	1	3	yes	—
41	C₄×C₈×C₆₇	C₂×C₄	1	3	yes	—
42	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
43	C₄×C₈×C₆₇	C₂³	1	3	yes	—
44	#11761612	—	4	12	no	—
45	C₂×C₆₇⋊C₈	—	2	2	no	—
46	C₂×C₆₇⋊C₈	—	2	6	no	—
47	C₂×C₆₇⋊C₈	—	4	12	no	—
48	#41×C₆₇	SD₁₆	1	8	yes	—
49	C₂×Q₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
50	C₂×D₄×C₆₇	C₄⋊C₄	1	3	yes	—
51	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
52	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
53	C₂²×C₄×C₆₇	C₂×Q₈	1	3	yes	—
54	C₂×C₈×C₆₇	—	2	6	no	—
55	C₂×C₈×C₆₇	—	4	3	no	—
56	C₂×C₈×C₆₇	C₂²×C₄	1	3	yes	—
57	C₂×C₈×C₆₇	C₂²×C₄	1	3	yes	—
58	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
60	C₄⋊C₄×C₆₇	C₄⋊C₄	1	3	yes	—
61	C₄⋊C₄×C₆₇	C₄⋊C₄	1	3	yes	—
62	C₂²⋊C₄×C₆₇	C₄⋊C₄	1	6	yes	—
63	C₄²×C₆₇	C₂²×C₄	1	3	yes	—
64	C₄²×C₆₇	C₂×D₄	1	3	yes	—
65	C₄²×C₆₇	C₂×Q₈	1	3	yes	—
66	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
67	C₂³×C₆₇	C₂×C₄⋊C₄	1	3	yes	—
68	Q₈×C₆₇	#106	1	4	yes	—
69	D₄×C₆₇	#106	1	12	yes	—
70	C₂×C₄×C₆₇	—	2	3	no	—
71	C₂×C₄×C₆₇	—	2	3	no	—
72	C₂×C₄×C₆₇	—	4	3	no	—
73	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	3	yes	—
74	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	3	yes	—
75	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	3	yes	—
76	C₂×C₄×C₆₇	C₂×SD₁₆	1	12	yes	—
77	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
78	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
79	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
80	C₈×C₆₇	—	2	3	no	—
81	C₈×C₆₇	—	2	1	no	—
82	C₈×C₆₇	—	2	3	no	—
83	C₈×C₆₇	#131	1	2	yes	complemented
84	C₆₇⋊C₈	—	4	2	no	—
85	C₆₇⋊C₈	—	4	6	no	—
86	C₂²×C₆₇	—	2	3	no	—
87	C₂²×C₆₇	—	2	3	no	—
88	C₂²×C₆₇	C₂×#106	1	6	yes	—
89	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
90	C₄×C₆₇	—	2	3	no	—
91	C₄×C₆₇	—	2	1	no	—
92	C₄×C₆₇	—	2	3	no	—
93	C₄×C₆₇	C₂×#106	1	6	yes	—
94	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
95	C₄×C₆₇	#568	1	1	yes	characteristic
96	C₄×C₆₇	C₂²×SD₁₆	1	2	yes	—
97	#17826	—	67	1	no	maximal, Sylow
98	C₂×C₆₇	—	2	6	no	—
99	C₂×C₆₇	#2906	1	1	yes	characteristic
100	C₂×C₆₇	C₂²×#106	1	1	yes	characteristic
101	C₂×C₆₇	#2916	1	1	yes	characteristic
102	C₈×#41	D₆₇	1	1	yes	characteristic
103	#2906	—	67	2	no	—
104	#1570	—	67	6	no	—
105	#1566	—	67	3	no	—
106	C₂×#334	—	67	3	no	—
107	C₆₇	#17826	1	1	yes	characteristic, complemented, Sylow
108	C₂×#131	—	134	2	no	—
109	C₄×#41	C₂×D₆₇	1	1	yes	characteristic
110	C₈×Q₈	Dic₆₇	1	1	yes	characteristic
111	C₈×Q₈	Dic₆₇	1	1	yes	characteristic
112	C₈×D₄	Dic₆₇	1	3	yes	—
113	C₈×D₄	Dic₆₇	1	3	yes	—
114	#431	C₂×D₆₇	1	3	yes	—
115	#424	—	134	6	no	—
116	#423	—	67	6	no	—
117	C₂×C₄⋊C₈	—	67	6	no	—
118	C₂×C₄×C₈	C₂×D₆₇	1	3	yes	—
119	#334	—	67	1	no	—
120	#334	—	67	3	no	—
121	#334	—	67	6	no	—
122	#334	—	67	6	no	—
123	C₂×#41	D₂₆₈	1	1	yes	characteristic
124	#131	—	268	8	no	—
125	C₂×M₁₆	—	134	6	no	—
126	C₂²×C₈	—	134	6	no	—
127	C₂²×C₈	C₂×Dic₆₇	1	3	yes	—
128	C₂²×C₈	C₂×Dic₆₇	1	3	yes	—
129	C₄×Q₈	Dic₁₃₄	1	1	yes	characteristic
130	C₄×Q₈	C₂×Dic₆₇	1	1	yes	characteristic
131	C₄×D₄	Dic₁₃₄	1	3	yes	—
132	C₄×D₄	C₂×Dic₆₇	1	3	yes	—
133	#117	D₂₆₈	1	3	yes	—
134	C₂×C₄²	C₂²×D₆₇	1	3	yes	—
135	C₄⋊C₈	—	67	2	no	—
136	C₄⋊C₈	—	67	6	no	—
137	C₄⋊C₈	—	134	6	no	—
138	C₄⋊C₈	—	134	6	no	—
139	C₄⋊C₈	C₂×Dic₆₇	1	3	yes	—
140	C₄⋊C₈	C₂×Dic₆₇	1	3	yes	—
141	C₂²⋊C₈	C₂×Dic₆₇	1	6	yes	—
142	C₄×C₈	C₂×Dic₆₇	1	3	yes	—
143	C₄×C₈	C₂×Dic₆₇	1	3	yes	—
144	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
145	C₄×C₈	C₂²×D₆₇	1	3	yes	—
146	#41	#11761617	1	8	yes	—
147	C₂×Q₈	C₄⋊Dic₆₇	1	1	yes	characteristic
148	C₂×D₄	C₄⋊Dic₆₇	1	3	yes	—
149	C₂²×C₄	C₂²×Dic₆₇	1	3	yes	—
150	C₂²×C₄	C₂×D₂₆₈	1	3	yes	—
151	C₂²×C₄	C₂×Dic₁₃₄	1	3	yes	—
152	M₁₆	—	268	12	no	—
153	C₂×C₈	—	2	6	no	—
154	C₂×C₈	—	4	3	no	—
155	C₂×C₈	—	268	12	no	—
156	C₂×C₈	—	134	2	no	—
157	C₂×C₈	—	134	6	no	—
158	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
159	C₂×C₈	C₂²×Dic₆₇	1	3	yes	—
160	C₂×C₈	C₂²×Dic₆₇	1	3	yes	—
161	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
162	C₄⋊C₄	C₄⋊Dic₆₇	1	3	yes	—
163	C₄⋊C₄	C₄⋊Dic₆₇	1	3	yes	—
164	C₂²⋊C₄	C₄⋊Dic₆₇	1	6	yes	—
165	C₄²	C₂²×Dic₆₇	1	3	yes	—
166	C₄²	C₂×D₂₆₈	1	3	yes	—
167	C₄²	C₂×Dic₁₃₄	1	3	yes	—
168	C₄²	C₂³×D₆₇	1	1	yes	characteristic
169	C₂³	C₂×C₄⋊Dic₆₇	1	3	yes	—
170	Q₈	#14523975	1	4	yes	—
171	D₄	#14523975	1	12	yes	—
172	C₂×C₄	—	2	3	no	—
173	C₂×C₄	—	2	3	no	—
174	C₂×C₄	—	4	3	no	—
175	C₂×C₄	C₂×C₄⋊Dic₆₇	1	3	yes	—
176	C₂×C₄	C₂×C₄⋊Dic₆₇	1	3	yes	—
177	C₂×C₄	C₂×C₄⋊Dic₆₇	1	3	yes	—
178	C₂×C₄	C₂×#11761617	1	12	yes	—
179	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
180	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
181	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
182	C₈	—	2	3	no	—
183	C₈	—	2	1	no	—
184	C₈	—	2	3	no	—
185	C₈	—	268	2	no	—
186	C₈	—	268	6	no	—
187	C₈	#14524050	1	2	yes	complemented
188	C₂²	—	2	3	no	—
189	C₂²	—	2	3	no	—
190	C₂²	C₂×#14523975	1	6	yes	—
191	C₂²	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
192	C₄	—	2	3	no	—
193	C₄	—	2	1	no	—
194	C₄	—	2	3	no	—
195	C₄	C₂×#14523975	1	6	yes	—
196	C₄	C₂×#14524050	1	1	yes	characteristic
197	C₄	#16864163	1	1	yes	characteristic
198	C₄	C₂²×#11761617	1	2	yes	—
199	C₂	—	2	6	no	—
200	C₂	#22568777	1	1	yes	characteristic
201	C₂	C₂²×#14523975	1	1	yes	characteristic
202	C₂	#22568820	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₈×#41×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×C₄⋊C₄
prime radical	C₂×#41×C₆₇	D₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	C₈×#41	D₆₇
hypocenter	C₆₇	#17826
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ C₈×#41

Complements

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

N	H ≅ G/N	type
2. #22558980	C₂	semidirect
3. #22558965	C₂	semidirect
4. C₂×#16863147	C₂	semidirect
83. C₈×C₆₇	#131	semidirect
107. C₆₇	#17826	semidirect
187. C₈	#14524050	semidirect

Maximal subgroups

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

subgroup	index
#22568777	2
#22558980	2
#22558965	2
C₂×#16863147	2
C₈×#41×C₆₇	2
#17826	67