Subgroups — #30824820

Order 17152.

734 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	#22569320	C₂	1	2	yes	maximal
2	#22558961	C₂	1	6	yes	maximal, complemented
3	#22558957	C₂	1	3	yes	maximal, complemented
4	C₂×#16863147	C₂	1	3	yes	maximal, complemented
5	C₄×#131×C₆₇	C₂	1	1	yes	maximal, characteristic
6	Q₈×C₆₇⋊C₈	—	2	2	no	—
7	D₄×C₆₇⋊C₈	—	2	6	no	—
8	#16863449	C₂²	1	6	yes	—
9	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	6	yes	—
10	#16863149	C₂²	1	6	yes	—
11	#16863147	C₂²	1	3	yes	—
12	#16863147	C₂²	1	6	yes	—
13	#16863147	C₂²	1	1	yes	characteristic
14	C₂×#131×C₆₇	C₄	1	1	yes	characteristic
15	C₂×#131×C₆₇	C₄	1	1	yes	characteristic
16	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
17	#405×C₆₇	C₄	1	3	yes	—
18	#405×C₆₇	C₄	1	3	yes	—
19	C₄×M₁₆×C₆₇	C₂²	1	3	yes	—
20	C₂×C₄×C₈×C₆₇	C₂²	1	3	yes	—
21	C₂²×C₆₇⋊C₈	—	2	6	no	—
22	C₄⋊(C₆₇⋊C₈)	—	2	6	no	—
23	C₄⋊(C₆₇⋊C₈)	C₂³	1	6	yes	—
24	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
25	C₂²⋊(C₆₇⋊C₈)	—	2	6	no	—
26	C₄×C₆₇⋊C₈	—	2	6	no	—
27	C₂×#41×C₆₇	C₂×C₄	1	1	yes	characteristic
28	#131×C₆₇	—	2	4	no	—
29	#131×C₆₇	C₈	1	8	yes	complemented
30	C₂×M₁₆×C₆₇	C₂×C₄	1	3	yes	—
31	C₂×M₁₆×C₆₇	C₂×C₄	1	3	yes	—
32	C₂²×C₈×C₆₇	C₂×C₄	1	3	yes	—
33	C₂²×C₈×C₆₇	C₂×C₄	1	3	yes	—
34	C₄×Q₈×C₆₇	D₄	1	1	yes	characteristic
35	C₄×Q₈×C₆₇	Q₈	1	1	yes	characteristic
36	C₄×D₄×C₆₇	D₄	1	3	yes	—
37	C₄×D₄×C₆₇	Q₈	1	3	yes	—
38	#117×C₆₇	C₂×C₄	1	3	yes	—
39	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
40	#97×C₆₇	C₂×C₄	1	6	yes	—
41	C₄×C₈×C₆₇	C₂×C₄	1	3	yes	—
42	C₄×C₈×C₆₇	C₂×C₄	1	3	yes	—
43	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
44	C₄×C₈×C₆₇	C₂³	1	3	yes	—
45	C₂×C₆₇⋊C₈	—	2	2	no	—
46	C₂×C₆₇⋊C₈	—	2	6	no	—
47	C₂×C₆₇⋊C₈	—	8	6	no	—
48	#41×C₆₇	C₂×C₈	1	4	yes	—
49	#41×C₆₇	M₁₆	1	4	yes	—
50	C₂×Q₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
51	C₂×D₄×C₆₇	C₄⋊C₄	1	3	yes	—
52	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
53	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
54	C₂²×C₄×C₆₇	C₂×Q₈	1	3	yes	—
55	M₁₆×C₆₇	—	2	6	no	—
56	M₁₆×C₆₇	C₂×C₈	1	12	yes	complemented
57	C₂×C₈×C₆₇	—	2	6	no	—
58	C₂×C₈×C₆₇	C₂×C₈	1	12	yes	complemented
59	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₆₇	C₂²×C₄	1	3	yes	—
61	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₈×C₆₇	C₂²×C₄	1	3	yes	—
63	C₄⋊C₄×C₆₇	C₄⋊C₄	1	3	yes	—
64	C₄⋊C₄×C₆₇	C₄⋊C₄	1	3	yes	—
65	C₂²⋊C₄×C₆₇	C₄⋊C₄	1	6	yes	—
66	C₄²×C₆₇	C₂²×C₄	1	3	yes	—
67	C₄²×C₆₇	C₂×D₄	1	3	yes	—
68	C₄²×C₆₇	C₂×Q₈	1	3	yes	—
69	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
70	C₂³×C₆₇	C₂×C₄⋊C₄	1	3	yes	—
71	Q₈×C₆₇	C₄⋊C₈	1	4	yes	—
72	D₄×C₆₇	C₄⋊C₈	1	12	yes	—
73	C₂×C₄×C₆₇	—	2	6	no	—
74	C₂×C₄×C₆₇	—	4	3	no	—
75	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	3	yes	—
76	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	3	yes	—
77	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	3	yes	—
78	C₂×C₄×C₆₇	C₂²×C₈	1	6	yes	—
79	C₂×C₄×C₆₇	C₂×M₁₆	1	6	yes	—
80	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
81	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
82	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
83	C₈×C₆₇	—	2	1	no	—
84	C₈×C₆₇	—	2	3	no	—
85	C₈×C₆₇	C₂²×C₈	1	2	yes	—
86	C₈×C₆₇	C₂²×C₈	1	6	yes	—
87	C₆₇⋊C₈	—	4	2	no	—
88	C₆₇⋊C₈	—	4	6	no	—
89	C₂²×C₆₇	—	2	3	no	—
90	C₂²×C₆₇	—	2	3	no	—
91	C₂²×C₆₇	C₂×C₄⋊C₈	1	6	yes	—
92	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
93	C₄×C₆₇	—	2	3	no	—
94	C₄×C₆₇	—	2	1	no	—
95	C₄×C₆₇	—	2	3	no	—
96	C₄×C₆₇	C₂×C₄⋊C₈	1	6	yes	—
97	C₄×C₆₇	C₂³×C₈	1	1	yes	characteristic
98	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
99	C₄×C₆₇	#577	1	2	yes	—
100	#17788	—	67	1	no	maximal, Sylow
101	C₂×C₆₇	—	2	6	no	—
102	C₂×C₆₇	C₂²×C₄⋊C₈	1	1	yes	characteristic
103	C₂×C₆₇	#2912	1	1	yes	characteristic
104	C₂×C₆₇	#2916	1	1	yes	characteristic
105	#2988	—	67	2	no	—
106	C₄×#131	D₆₇	1	1	yes	characteristic
107	#1565	—	67	6	no	—
108	#1564	—	67	3	no	—
109	C₂×#334	—	67	3	no	—
110	C₆₇	#17788	1	1	yes	characteristic, complemented, Sylow
111	C₂×#131	Dic₆₇	1	1	yes	characteristic
112	C₂×#131	Dic₆₇	1	1	yes	characteristic
113	C₄×#41	C₂×D₆₇	1	1	yes	characteristic
114	C₈×Q₈	—	134	2	no	—
115	C₈×D₄	—	134	6	no	—
116	#431	—	67	6	no	—
117	C₂×C₄⋊C₈	—	67	6	no	—
118	#405	Dic₆₇	1	3	yes	—
119	#405	Dic₆₇	1	3	yes	—
120	C₄×M₁₆	C₂×D₆₇	1	3	yes	—
121	C₂×C₄×C₈	C₂×D₆₇	1	3	yes	—
122	#335	—	67	6	no	—
123	#334	—	67	3	no	—
124	#334	—	67	6	no	—
125	#334	—	67	1	no	—
126	C₂×#41	C₂×Dic₆₇	1	1	yes	characteristic
127	#131	—	2	4	no	—
128	#131	C₆₇⋊C₈	1	8	yes	complemented
129	C₂×M₁₆	C₂×Dic₆₇	1	3	yes	—
130	C₂×M₁₆	C₂×Dic₆₇	1	3	yes	—
131	C₂²×C₈	—	134	6	no	—
132	C₂²×C₈	C₂×Dic₆₇	1	3	yes	—
133	C₂²×C₈	C₂×Dic₆₇	1	3	yes	—
134	C₄×Q₈	Dic₁₃₄	1	1	yes	characteristic
135	C₄×Q₈	D₂₆₈	1	1	yes	characteristic
136	C₄×D₄	Dic₁₃₄	1	3	yes	—
137	C₄×D₄	D₂₆₈	1	3	yes	—
138	#117	C₂×Dic₆₇	1	3	yes	—
139	C₂×C₄²	C₂²×D₆₇	1	3	yes	—
140	C₄⋊C₈	—	67	6	no	—
141	C₄⋊C₈	—	67	2	no	—
142	C₄⋊C₈	—	134	6	no	—
143	C₂²⋊C₈	—	134	6	no	—
144	#97	C₂×Dic₆₇	1	6	yes	—
145	C₄×C₈	—	134	6	no	—
146	C₄×C₈	C₂×Dic₆₇	1	3	yes	—
147	C₄×C₈	C₂×Dic₆₇	1	3	yes	—
148	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
149	C₄×C₈	C₂²×D₆₇	1	3	yes	—
150	#41	C₂×C₆₇⋊C₈	1	4	yes	—
151	#41	#11761612	1	4	yes	—
152	C₂×Q₈	C₄⋊Dic₆₇	1	1	yes	characteristic
153	C₂×D₄	C₄⋊Dic₆₇	1	3	yes	—
154	C₂²×C₄	C₂²×Dic₆₇	1	3	yes	—
155	C₂²×C₄	C₂×D₂₆₈	1	3	yes	—
156	C₂²×C₄	C₂×Dic₁₃₄	1	3	yes	—
157	M₁₆	—	2	6	no	—
158	M₁₆	C₂×C₆₇⋊C₈	1	12	yes	complemented
159	C₂×C₈	—	2	6	no	—
160	C₂×C₈	—	536	6	no	—
161	C₂×C₈	—	134	2	no	—
162	C₂×C₈	—	134	6	no	—
163	C₂×C₈	C₂×C₆₇⋊C₈	1	12	yes	complemented
164	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
165	C₂×C₈	C₂²×Dic₆₇	1	3	yes	—
166	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
167	C₂×C₈	C₂²×Dic₆₇	1	3	yes	—
168	C₄⋊C₄	C₄⋊Dic₆₇	1	3	yes	—
169	C₄⋊C₄	C₄⋊Dic₆₇	1	3	yes	—
170	C₂²⋊C₄	C₄⋊Dic₆₇	1	6	yes	—
171	C₄²	C₂²×Dic₆₇	1	3	yes	—
172	C₄²	C₂×D₂₆₈	1	3	yes	—
173	C₄²	C₂×Dic₁₃₄	1	3	yes	—
174	C₄²	C₂³×D₆₇	1	1	yes	characteristic
175	C₂³	C₂×C₄⋊Dic₆₇	1	3	yes	—
176	Q₈	C₄⋊(C₆₇⋊C₈)	1	4	yes	—
177	D₄	C₄⋊(C₆₇⋊C₈)	1	12	yes	—
178	C₂×C₄	—	2	6	no	—
179	C₂×C₄	—	4	3	no	—
180	C₂×C₄	C₂×C₄⋊Dic₆₇	1	3	yes	—
181	C₂×C₄	C₂×C₄⋊Dic₆₇	1	3	yes	—
182	C₂×C₄	C₂×C₄⋊Dic₆₇	1	3	yes	—
183	C₂×C₄	C₂²×C₆₇⋊C₈	1	6	yes	—
184	C₂×C₄	C₂×#11761612	1	6	yes	—
185	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
186	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
187	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
188	C₈	—	2	1	no	—
189	C₈	—	2	3	no	—
190	C₈	—	268	6	no	—
191	C₈	—	268	2	no	—
192	C₈	C₂²×C₆₇⋊C₈	1	2	yes	—
193	C₈	C₂²×C₆₇⋊C₈	1	6	yes	—
194	C₂²	—	2	3	no	—
195	C₂²	—	2	3	no	—
196	C₂²	C₂×C₄⋊(C₆₇⋊C₈)	1	6	yes	—
197	C₂²	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
198	C₄	—	2	3	no	—
199	C₄	—	2	1	no	—
200	C₄	—	2	3	no	—
201	C₄	C₂×C₄⋊(C₆₇⋊C₈)	1	6	yes	—
202	C₄	C₂³×C₆₇⋊C₈	1	1	yes	characteristic
203	C₄	C₂²×#11761612	1	1	yes	characteristic
204	C₄	#16864208	1	2	yes	—
205	C₂	—	2	6	no	—
206	C₂	C₂²×C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
207	C₂	#22568804	1	1	yes	characteristic
208	C₂	#22568820	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₆₈
commutator	C₄×C₆₇	C₂³×C₈
Frattini	C₄²	C₂³×D₆₇
Fitting	C₄×#131×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×C₄⋊C₄
prime radical	C₂×#41×C₆₇	C₂×C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	C₄×#131	D₆₇
hypocenter	C₆₇	#17788
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
2. #22558961	C₂	semidirect
3. #22558957	C₂	semidirect
4. C₂×#16863147	C₂	semidirect
29. #131×C₆₇	C₈	semidirect
56. M₁₆×C₆₇	C₂×C₈	semidirect
58. C₂×C₈×C₆₇	C₂×C₈	semidirect
110. C₆₇	#17788	semidirect
128. #131	C₆₇⋊C₈	semidirect
158. M₁₆	C₂×C₆₇⋊C₈	semidirect
163. C₂×C₈	C₂×C₆₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#22569320	2
#22558961	2
#22558957	2
C₂×#16863147	2
C₄×#131×C₆₇	2
#17788	67