Subgroups — #30742736

Order 17152.

254 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	C₄×#14523983	C₂	1	1	yes	maximal, characteristic
2	#22560076	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2150	C₂	1	1	yes	maximal, characteristic
4	C₂×#14523983	—	2	1	no	—
5	C₂×C₄⋊(C₆₇⋊C₈)	C₄	1	1	yes	characteristic
6	C₂×#14523953	—	2	1	no	—
7	C₄×C₄⋊Dic₆₇	C₄	1	1	yes	characteristic
8	#16863172	—	2	1	no	—
9	C₄×C₆₇⋊C₁₆	C₄	1	2	yes	complemented
10	#16863150	—	2	1	no	—
11	C₂×M₃₂×C₆₇	—	2	1	no	—
12	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₄⋊C₁₆×C₆₇	—	2	1	no	—
14	C₄⋊C₁₆×C₆₇	C₄	1	2	yes	complemented
15	C₂²×C₆₇⋊C₈	—	2	1	no	—
16	C₂×C₄⋊Dic₆₇	—	2	1	no	—
17	C₂×C₄×Dic₆₇	—	2	1	no	—
18	#14523983	—	2	2	no	—
19	#14523983	—	2	2	no	—
20	C₂×C₆₇⋊C₁₆	—	2	2	no	—
21	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
22	C₄⋊(C₆₇⋊C₈)	C₈	1	2	yes	—
23	#14523953	—	4	1	no	—
24	#14523953	—	4	1	no	—
25	C₆₇⋊#95	C₈	1	2	yes	—
26	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
29	M₃₂×C₆₇	—	4	1	no	—
30	M₃₂×C₆₇	—	4	1	no	—
31	C₂×C₁₆×C₆₇	—	2	2	no	—
32	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
35	C₄×C₈×C₆₇	Q₈	1	1	yes	characteristic
36	C₂²×Dic₆₇	—	2	2	no	—
37	C₂×C₆₇⋊C₈	—	2	2	no	—
38	C₂×C₆₇⋊C₈	—	4	1	no	—
39	C₂×C₆₇⋊C₈	—	4	1	no	—
40	C₄⋊Dic₆₇	—	4	2	no	—
41	C₄×Dic₆₇	—	4	1	no	—
42	C₄×Dic₆₇	—	4	1	no	—
43	C₆₇⋊C₁₆	—	2	4	no	—
44	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
46	C₂²×C₄×C₆₇	M₁₆	1	1	yes	characteristic
47	C₂×C₈×C₆₇	—	2	1	no	—
48	C₂×C₈×C₆₇	—	2	1	no	—
49	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₆₇	D₈	1	1	yes	characteristic
54	C₂×C₈×C₆₇	SD₁₆	1	1	yes	characteristic
55	C₂×C₈×C₆₇	SD₁₆	1	1	yes	characteristic
56	C₂×C₈×C₆₇	Q₁₆	1	1	yes	characteristic
57	C₄²×C₆₇	C₄²	1	1	yes	characteristic
58	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
59	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
60	C₄²×C₆₇	M₁₆	1	1	yes	characteristic
61	C₁₆×C₆₇	—	4	2	no	—
62	C₂³×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
63	C₂×C₄×C₆₇	—	2	2	no	—
64	C₂×C₄×C₆₇	—	4	1	no	—
65	C₂×C₄×C₆₇	#95	1	1	yes	characteristic
66	C₂×C₄×C₆₇	C₄×C₈	1	1	yes	characteristic
67	C₂×C₄×C₆₇	#97	1	1	yes	characteristic
68	C₂×C₄×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
69	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
70	C₂×C₄×C₆₇	#103	1	1	yes	characteristic
71	C₂×C₄×C₆₇	#104	1	1	yes	characteristic
72	C₂×C₄×C₆₇	#104	1	1	yes	characteristic
73	C₂×C₄×C₆₇	C₄⋊C₈	1	1	yes	characteristic
74	C₂×C₄×C₆₇	C₄⋊C₈	1	1	yes	characteristic
75	C₂×Dic₆₇	—	4	2	no	—
76	C₂×Dic₆₇	—	4	2	no	—
77	C₂×Dic₆₇	—	4	2	no	—
78	C₈×C₆₇	—	2	1	no	—
79	C₈×C₆₇	—	2	1	no	—
80	C₈×C₆₇	#102	1	1	yes	characteristic
81	C₈×C₆₇	#103	1	1	yes	characteristic
82	C₈×C₆₇	#106	1	1	yes	characteristic
83	C₈×C₆₇	#107	1	1	yes	characteristic
84	C₆₇⋊C₈	—	4	2	no	—
85	C₂²×C₆₇	—	2	1	no	—
86	C₂²×C₆₇	—	2	1	no	—
87	C₂²×C₆₇	D₄⋊C₈	1	1	yes	characteristic
88	C₂²×C₆₇	Q₈⋊C₈	1	1	yes	characteristic
89	C₂²×C₆₇	#336	1	1	yes	characteristic
90	C₄×C₆₇	—	2	1	no	—
91	C₄×C₆₇	—	2	1	no	—
92	C₄×C₆₇	—	2	1	no	—
93	C₄×C₆₇	D₄⋊C₈	1	1	yes	characteristic
94	C₄×C₆₇	Q₈⋊C₈	1	1	yes	characteristic
95	C₄×C₆₇	#339	1	1	yes	characteristic
96	C₄×C₆₇	#340	1	1	yes	characteristic
97	C₄×C₆₇	#347	1	2	yes	—
98	Dic₆₇	—	8	2	no	—
99	#7168	—	67	1	no	maximal, Sylow
100	C₂×C₆₇	—	2	1	no	—
101	C₂×C₆₇	—	2	1	no	—
102	C₂×C₆₇	#1279	1	1	yes	characteristic
103	C₂×C₆₇	#1375	1	1	yes	characteristic
104	C₂×C₆₇	#1378	1	1	yes	characteristic
105	#2150	D₆₇	1	1	yes	characteristic
106	C₄×M₃₂	—	67	1	no	—
107	#1770	—	67	1	no	—
108	C₆₇	#7168	1	1	yes	characteristic, complemented, Sylow
109	C₂×M₃₂	—	2	1	no	—
110	C₂×M₃₂	—	134	1	no	—
111	C₂×C₄⋊C₈	—	67	1	no	—
112	C₂×#97	—	134	1	no	—
113	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
114	C₄×C₄⋊C₄	—	67	1	no	—
115	C₄⋊C₁₆	—	2	1	no	—
116	C₄⋊C₁₆	Dic₆₇	1	2	yes	complemented
117	#346	—	134	1	no	—
118	C₄×C₁₆	—	67	2	no	—
119	#336	—	134	1	no	—
120	C₂²×C₈	—	134	1	no	—
121	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
122	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
123	C₂×C₄⋊C₄	—	134	1	no	—
124	C₂×C₄²	—	134	1	no	—
125	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
126	M₃₂	—	4	1	no	—
127	M₃₂	—	4	1	no	—
128	M₃₂	—	134	2	no	—
129	M₃₂	—	134	2	no	—
130	C₂×C₁₆	—	2	2	no	—
131	C₂×C₁₆	—	134	2	no	—
132	C₄⋊C₈	—	67	2	no	—
133	C₄⋊C₈	—	134	1	no	—
134	#97	—	268	1	no	—
135	#97	—	268	1	no	—
136	C₄×C₈	Dic₁₃₄	1	1	yes	characteristic
137	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
138	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
139	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
140	#95	—	67	2	no	—
141	C₂²×C₄	—	134	2	no	—
142	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
143	C₂²×C₄	C₈×D₆₇	1	1	yes	characteristic
144	C₂²×C₄	#11761611	1	1	yes	characteristic
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	2	1	no	—
147	C₂×C₈	—	268	1	no	—
148	C₂×C₈	—	268	1	no	—
149	C₂×C₈	—	134	2	no	—
150	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
151	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
152	C₂×C₈	Dic₆₇⋊C₄	1	1	yes	characteristic
153	C₂×C₈	C₄⋊Dic₆₇	1	1	yes	characteristic
154	C₂×C₈	#11761613	1	1	yes	characteristic
155	C₂×C₈	#11761615	1	1	yes	characteristic
156	C₂×C₈	#11761616	1	1	yes	characteristic
157	C₂×C₈	#11761618	1	1	yes	characteristic
158	C₄⋊C₄	—	268	2	no	—
159	C₄²	—	268	1	no	—
160	C₄²	—	268	1	no	—
161	C₄²	C₄×Dic₆₇	1	1	yes	characteristic
162	C₄²	Dic₆₇⋊C₄	1	1	yes	characteristic
163	C₄²	C₈×D₆₇	1	1	yes	characteristic
164	C₄²	#11761611	1	1	yes	characteristic
165	C₁₆	—	4	2	no	—
166	C₁₆	—	134	4	no	—
167	C₂³	(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	4	1	no	—
170	C₂×C₄	—	268	2	no	—
171	C₂×C₄	—	268	2	no	—
172	C₂×C₄	—	268	2	no	—
173	C₂×C₄	C₆₇⋊#95	1	1	yes	characteristic
174	C₂×C₄	C₈×Dic₆₇	1	1	yes	characteristic
175	C₂×C₄	#14523953	1	1	yes	characteristic
176	C₂×C₄	(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
177	C₂×C₄	#14523963	1	1	yes	characteristic
178	C₂×C₄	#14523966	1	1	yes	characteristic
179	C₂×C₄	#14523969	1	1	yes	characteristic
180	C₂×C₄	#14523969	1	1	yes	characteristic
181	C₂×C₄	Dic₆₇⋊C₈	1	1	yes	characteristic
182	C₂×C₄	Dic₆₇⋊C₈	1	1	yes	characteristic
183	C₈	—	2	1	no	—
184	C₈	—	2	1	no	—
185	C₈	—	268	2	no	—
186	C₈	#14523964	1	1	yes	characteristic
187	C₈	#14523967	1	1	yes	characteristic
188	C₈	#14523974	1	1	yes	characteristic
189	C₈	#14523976	1	1	yes	characteristic
190	C₂²	—	2	1	no	—
191	C₂²	—	2	1	no	—
192	C₂²	#16863123	1	1	yes	characteristic
193	C₂²	#16863126	1	1	yes	characteristic
194	C₂²	#16863150	1	1	yes	characteristic
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	2	1	no	—
198	C₄	—	536	2	no	—
199	C₄	#16863123	1	1	yes	characteristic
200	C₄	#16863126	1	1	yes	characteristic
201	C₄	#16863157	1	1	yes	characteristic
202	C₄	#16863158	1	1	yes	characteristic
203	C₄	#16863173	1	2	yes	—
204	C₂	—	2	1	no	—
205	C₂	—	2	1	no	—
206	C₂	#22558048	1	1	yes	characteristic
207	C₂	#22558288	1	1	yes	characteristic
208	C₂	#22558296	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₆₇⋊#95
commutator	C₂×C₄×C₆₇	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₆₇
Fitting	C₆₇×#2150	C₂
socle	C₂²×C₆₇	#336
prime radical	C₂³×C₆₇	C₂²⋊C₈
cosocle	C₂×C₄×C₈×C₆₇	C₂²
hypercenter	#2150	D₆₇
hypocenter	C₆₇	#7168
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
9. C₄×C₆₇⋊C₁₆	C₄	semidirect
14. C₄⋊C₁₆×C₆₇	C₄	semidirect
108. C₆₇	#7168	semidirect
116. C₄⋊C₁₆	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14523983	2
#22560076	2
C₆₇×#2150	2
#7168	67