Subgroups — #30856314

Order 17152.

890 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	C₂×#16863449	C₂	1	2	yes	maximal
2	#22560363	C₂	1	2	yes	maximal, complemented
3	#22560329	C₂	1	2	yes	maximal, complemented
4	#22560193	C₂	1	4	yes	maximal
5	C₄×C₄⋊(C₆₇⋊C₈)	C₂	1	4	yes	maximal, complemented
6	C₄×#117×C₆₇	C₂	1	1	yes	maximal, characteristic
7	#16863449	—	2	8	no	—
8	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	8	yes	—
9	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	4	yes	—
10	C₂×C₂²⋊(C₆₇⋊C₈)	C₂²	1	4	yes	—
11	C₂×C₄×C₆₇⋊C₈	C₂²	1	4	yes	—
12	#16863151	C₂²	1	8	yes	—
13	C₂×#117×C₆₇	C₂²	1	2	yes	—
14	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₆₇	C₄	1	4	yes	—
16	C₄×C₂²⋊C₄×C₆₇	C₄	1	4	yes	—
17	#376×C₆₇	C₂²	1	1	yes	characteristic
18	#376×C₆₇	C₂²	1	1	yes	characteristic
19	C₄³×C₆₇	C₂²	1	2	yes	—
20	C₂²×C₆₇⋊C₈	C₂³	1	8	yes	—
21	C₄⋊(C₆₇⋊C₈)	—	2	8	no	—
22	C₄⋊(C₆₇⋊C₈)	—	2	16	no	—
23	C₂²⋊(C₆₇⋊C₈)	—	2	8	no	—
24	C₄×C₆₇⋊C₈	—	2	8	no	—
25	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
26	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
27	#117×C₆₇	—	2	8	no	—
28	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
29	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₆₇	C₂×C₄	1	8	yes	—
31	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₆₇	D₄	1	4	yes	—
33	C₂×C₄²×C₆₇	Q₈	1	4	yes	—
34	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
35	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
36	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
37	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
38	#95×C₆₇	C₂×C₄	1	4	yes	—
39	#95×C₆₇	C₂×C₄	1	4	yes	—
40	C₂×C₆₇⋊C₈	—	2	8	no	—
41	C₂×C₆₇⋊C₈	—	2	16	no	—
42	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₆₇	—	2	8	no	—
44	C₂²×C₄×C₆₇	—	2	4	no	—
45	C₂²×C₄×C₆₇	C₄⋊C₄	1	8	yes	—
46	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
47	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
49	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
50	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
51	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
52	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
53	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
54	C₄⋊C₄×C₆₇	—	2	8	no	—
55	C₂²⋊C₄×C₆₇	—	2	8	no	—
56	C₄²×C₆₇	—	2	16	no	—
57	C₄²×C₆₇	—	2	4	no	—
58	C₄²×C₆₇	C₄⋊C₄	1	8	yes	—
59	C₄²×C₆₇	C₂×D₄	1	2	yes	—
60	C₄²×C₆₇	C₂×D₄	1	2	yes	—
61	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
62	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
63	C₄²×C₆₇	#41	1	8	yes	—
64	C₂³×C₆₇	—	2	2	no	—
65	C₂³×C₆₇	—	2	4	no	—
66	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
67	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
68	C₂×C₄×C₆₇	—	2	2	no	—
69	C₂×C₄×C₆₇	—	2	8	no	—
70	C₂×C₄×C₆₇	—	2	4	no	—
71	C₂×C₄×C₆₇	—	2	4	no	—
72	C₂×C₄×C₆₇	—	2	4	no	—
73	C₂×C₄×C₆₇	—	2	16	no	—
74	C₂×C₄×C₆₇	—	2	4	no	—
75	C₂×C₄×C₆₇	—	2	8	no	—
76	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
77	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
78	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
79	C₂×C₄×C₆₇	#117	1	8	yes	—
80	C₂×C₄×C₆₇	#131	1	8	yes	—
81	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
82	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
83	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
84	C₆₇⋊C₈	—	4	8	no	—
85	C₂²×C₆₇	—	2	4	no	—
86	C₂²×C₆₇	—	2	2	no	—
87	C₂²×C₆₇	—	4	4	no	—
88	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
89	C₂²×C₆₇	C₂×#117	1	2	yes	—
90	C₂²×C₆₇	C₂×#131	1	4	yes	—
91	C₄×C₆₇	—	2	4	no	—
92	C₄×C₆₇	—	2	8	no	—
93	C₄×C₆₇	—	2	2	no	—
94	C₄×C₆₇	—	4	4	no	—
95	C₄×C₆₇	#424	1	8	yes	—
96	C₄×C₆₇	#532	1	2	yes	—
97	C₄×C₆₇	#533	1	2	yes	—
98	#20809	—	67	1	no	maximal, Sylow
99	C₂×C₆₇	—	4	2	no	—
100	C₂×C₆₇	#2285	1	1	yes	characteristic
101	C₂×C₆₇	C₂×#424	1	2	yes	—
102	C₂×C₆₇	#2922	1	4	yes	—
103	C₂×#431	—	67	2	no	—
104	C₄×#117	D₆₇	1	1	yes	characteristic
105	#1843	—	67	2	no	—
106	#1832	—	67	2	no	—
107	#1798	—	67	4	no	—
108	C₄×C₄⋊C₈	—	67	4	no	—
109	C₆₇	#20809	1	1	yes	characteristic, complemented, Sylow
110	C₂×#117	C₂×D₆₇	1	2	yes	—
111	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
112	#431	—	134	8	no	—
113	C₂×C₄⋊C₈	—	67	8	no	—
114	C₂×C₄⋊C₈	—	67	4	no	—
115	C₂×C₂²⋊C₈	—	67	4	no	—
116	C₂×C₄×C₈	—	67	4	no	—
117	C₄×C₄⋊C₄	Dic₆₇	1	4	yes	—
118	C₄×C₂²⋊C₄	Dic₆₇	1	4	yes	—
119	#376	C₂×D₆₇	1	1	yes	characteristic
120	#376	C₂×D₆₇	1	1	yes	characteristic
121	C₄³	C₂×D₆₇	1	2	yes	—
122	#336	—	67	8	no	—
123	C₂³×C₄	C₂×Dic₆₇	1	2	yes	—
124	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
125	C₂²×C₈	—	67	8	no	—
126	#117	—	2	8	no	—
127	C₂×C₄⋊C₄	C₂×Dic₆₇	1	4	yes	—
128	C₂×C₂²⋊C₄	C₂×Dic₆₇	1	4	yes	—
129	C₂×C₄²	Dic₁₃₄	1	4	yes	—
130	C₂×C₄²	D₂₆₈	1	4	yes	—
131	C₂×C₄²	C₂×Dic₆₇	1	8	yes	—
132	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
133	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
134	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
135	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
136	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
137	C₄⋊C₈	—	134	8	no	—
138	C₄⋊C₈	—	134	16	no	—
139	C₂²⋊C₈	—	134	8	no	—
140	C₄×C₈	—	134	8	no	—
141	#95	C₂×Dic₆₇	1	4	yes	—
142	#95	C₂×Dic₆₇	1	4	yes	—
143	C₂⁴	C₂²×Dic₆₇	1	1	yes	characteristic
144	C₂²×C₄	—	2	8	no	—
145	C₂²×C₄	—	2	4	no	—
146	C₂²×C₄	C₄⋊Dic₆₇	1	8	yes	—
147	C₂²×C₄	C₂²×Dic₆₇	1	4	yes	—
148	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
149	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
150	C₂²×C₄	C₂²×Dic₆₇	1	4	yes	—
151	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
152	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
153	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
154	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
155	C₂×C₈	—	134	8	no	—
156	C₂×C₈	—	134	16	no	—
157	C₄⋊C₄	—	2	8	no	—
158	C₂²⋊C₄	—	2	8	no	—
159	C₄²	—	2	16	no	—
160	C₄²	—	2	4	no	—
161	C₄²	C₄⋊Dic₆₇	1	8	yes	—
162	C₄²	C₂×D₂₆₈	1	2	yes	—
163	C₄²	C₂×D₂₆₈	1	2	yes	—
164	C₄²	C₂×Dic₁₃₄	1	2	yes	—
165	C₄²	C₂×Dic₁₃₄	1	2	yes	—
166	C₄²	#11761628	1	8	yes	—
167	C₂³	—	2	2	no	—
168	C₂³	—	2	4	no	—
169	C₂³	C₂×C₄⋊Dic₆₇	1	2	yes	—
170	C₂³	C₂³×Dic₆₇	1	1	yes	characteristic
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	8	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	2	16	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	2	8	no	—
179	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
180	C₂×C₄	C₂×C₄⋊Dic₆₇	1	4	yes	—
181	C₂×C₄	C₂×C₄⋊Dic₆₇	1	4	yes	—
182	C₂×C₄	#14524002	1	8	yes	—
183	C₂×C₄	#14524050	1	8	yes	—
184	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
185	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
186	C₂×C₄	C₂×#11761628	1	4	yes	—
187	C₈	—	268	8	no	—
188	C₂²	—	2	2	no	—
189	C₂²	—	2	4	no	—
190	C₂²	—	4	4	no	—
191	C₂²	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
192	C₂²	C₂×#14524002	1	2	yes	—
193	C₂²	C₂×#14524050	1	4	yes	—
194	C₄	—	2	2	no	—
195	C₄	—	2	4	no	—
196	C₄	—	2	8	no	—
197	C₄	—	4	4	no	—
198	C₄	#16863423	1	8	yes	—
199	C₄	#16863946	1	2	yes	—
200	C₄	#16863953	1	2	yes	—
201	C₂	—	4	2	no	—
202	C₂	#22562616	1	1	yes	characteristic
203	C₂	C₂×#16863423	1	2	yes	—
204	C₂	#22568853	1	4	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²×D₆₇
commutator, socle	C₂³×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₄×#117×C₆₇	C₂
prime radical	C₂⁴×C₆₇	C₂²×C₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₄×#117	D₆₇
hypocenter	C₆₇	#20809
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄² ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
2. #22560363	C₂	semidirect
3. #22560329	C₂	semidirect
5. C₄×C₄⋊(C₆₇⋊C₈)	C₂	semidirect
109. C₆₇	#20809	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863449	2
#22560363	2
#22560329	2
#22560193	2
C₄×C₄⋊(C₆₇⋊C₈)	2
C₄×#117×C₆₇	2
#20809	67