Subgroups — #30853892

Order 17152.

1142 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	C₂×#16863434	C₂	1	2	yes	maximal
2	C₄×#14524002	C₂	1	2	yes	maximal
3	#22560106	C₂	1	8	yes	maximal, complemented
4	#22559926	C₂	1	2	yes	maximal
5	C₂²×#97×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14524002	C₂²	1	4	yes	—
7	#16863434	C₄	1	16	yes	complemented
8	C₂×#14523977	C₂²	1	4	yes	—
9	C₂×#14523975	C₂²	1	4	yes	—
10	C₄×C₄⋊Dic₆₇	C₂²	1	8	yes	—
11	C₄×C₂²⋊Dic₆₇	—	2	4	no	—
12	#16863242	—	2	2	no	—
13	C₄²×Dic₆₇	—	2	2	no	—
14	#16863159	C₂²	1	8	yes	—
15	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×#97×C₆₇	C₄	1	8	yes	complemented
19	C₂×#97×C₆₇	C₂²	1	4	yes	—
20	#14524002	—	2	8	no	—
21	#14524002	C₂×C₄	1	16	yes	—
22	C₂×C₄⋊Dic₆₇	C₂³	1	8	yes	—
23	C₂×C₂²⋊Dic₆₇	—	2	4	no	—
24	C₂×C₄×Dic₆₇	—	2	4	no	—
25	C₂×C₄×Dic₆₇	—	2	8	no	—
26	#14523977	C₂×C₄	1	16	yes	complemented
27	#14523975	C₂×C₄	1	16	yes	complemented
28	C₆₇⋊#95	—	2	8	no	—
29	C₂³×C₄×C₆₇	D₄	1	2	yes	—
30	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₆₇	—	2	4	no	—
32	C₂²×C₈×C₆₇	C₂×C₄	1	8	yes	—
33	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
34	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
35	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
36	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
37	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₆₇	D₄	1	2	yes	—
39	C₂×C₄²×C₆₇	Q₈	1	4	yes	—
40	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
41	#97×C₆₇	C₂×C₄	1	16	yes	—
42	C₂²×Dic₆₇	—	2	8	no	—
43	C₄⋊Dic₆₇	—	2	8	no	—
44	C₄⋊Dic₆₇	C₂²×C₄	1	16	yes	—
45	C₂²⋊Dic₆₇	—	2	16	no	—
46	C₄×Dic₆₇	—	2	16	no	—
47	C₄×Dic₆₇	—	4	16	no	—
48	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₆₇	—	4	4	no	—
50	C₂²×C₄×C₆₇	C₄⋊C₄	1	8	yes	—
51	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
52	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
53	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
54	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
56	C₂²×C₄×C₆₇	#41	1	4	yes	—
57	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
58	C₂×C₈×C₆₇	—	2	8	no	—
59	C₂×C₈×C₆₇	—	2	4	no	—
60	C₂×C₈×C₆₇	C₄²	1	16	yes	complemented
61	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
62	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
63	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
64	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
65	C₄²×C₆₇	C₄⋊C₄	1	8	yes	—
66	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
67	C₄²×C₆₇	C₂×Q₈	1	4	yes	—
68	C₂³×C₆₇	—	2	4	no	—
69	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
70	C₂³×C₆₇	C₄×D₄	1	4	yes	—
71	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
72	C₂×C₄×C₆₇	—	2	8	no	—
73	C₂×C₄×C₆₇	—	4	8	no	—
74	C₂×C₄×C₆₇	C₂×C₄²	1	4	yes	—
75	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	8	yes	—
76	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
77	C₂×C₄×C₆₇	#117	1	4	yes	—
78	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
79	C₂×C₄×C₆₇	C₄×Q₈	1	8	yes	—
80	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
81	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
82	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
83	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
84	C₂×Dic₆₇	—	2	16	no	—
85	C₂×Dic₆₇	—	4	16	no	—
86	C₈×C₆₇	—	2	4	no	—
87	C₈×C₆₇	C₂×C₄²	1	8	yes	—
88	C₂²×C₆₇	—	2	4	no	—
89	C₂²×C₆₇	—	2	8	no	—
90	C₂²×C₆₇	C₄×C₄⋊C₄	1	4	yes	—
91	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
92	C₂²×C₆₇	C₂×C₄×D₄	1	2	yes	—
93	C₂²×C₆₇	#575	1	4	yes	—
94	C₄×C₆₇	—	2	8	no	—
95	C₄×C₆₇	C₄×C₄⋊C₄	1	4	yes	—
96	C₄×C₆₇	C₂²×C₄²	1	1	yes	characteristic
97	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
98	C₄×C₆₇	C₂×C₄×Q₈	1	2	yes	—
99	Dic₆₇	—	4	16	no	—
100	#20451	—	67	1	no	maximal, Sylow
101	C₂×C₆₇	—	2	4	no	—
102	C₂×C₆₇	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₆₇	#2911	1	2	yes	—
104	C₂×C₆₇	#2943	1	4	yes	—
105	C₂×#427	—	67	2	no	—
106	C₂²×#97	D₆₇	1	1	yes	characteristic
107	C₄×#117	—	67	2	no	—
108	#1776	—	67	8	no	—
109	#1735	—	67	2	no	—
110	C₆₇	#20451	1	1	yes	characteristic, complemented, Sylow
111	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
112	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
113	C₂×#117	—	67	4	no	—
114	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
115	#427	—	67	16	no	—
116	C₂×#107	—	67	4	no	—
117	C₂×#106	—	67	4	no	—
118	C₂×#97	Dic₆₇	1	8	yes	complemented
119	C₂×#97	C₂×D₆₇	1	4	yes	—
120	C₄×C₄⋊C₄	—	67	8	no	—
121	C₄×C₂²⋊C₄	—	134	4	no	—
122	#376	—	134	2	no	—
123	C₄³	—	134	2	no	—
124	#340	—	67	8	no	—
125	C₂³×C₄	D₂₆₈	1	2	yes	—
126	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
127	C₂²×C₈	—	2	4	no	—
128	C₂²×C₈	C₄×D₆₇	1	8	yes	—
129	C₂²×C₈	C₂×Dic₆₇	1	4	yes	—
130	C₂²×C₈	C₂×Dic₆₇	1	4	yes	—
131	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
132	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
133	#117	—	67	16	no	—
134	#117	—	134	8	no	—
135	C₂×C₄⋊C₄	—	67	8	no	—
136	C₂×C₂²⋊C₄	—	134	4	no	—
137	C₂×C₄²	—	134	8	no	—
138	C₂×C₄²	—	134	4	no	—
139	C₂×C₄²	Dic₁₃₄	1	4	yes	—
140	C₂×C₄²	D₂₆₈	1	2	yes	—
141	C₂×C₄²	C₂×Dic₆₇	1	4	yes	—
142	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
143	#107	—	67	16	no	—
144	#106	—	67	16	no	—
145	#97	C₂×Dic₆₇	1	16	yes	—
146	#95	—	134	8	no	—
147	C₂⁴	C₂×D₂₆₈	1	1	yes	characteristic
148	C₂²×C₄	—	4	4	no	—
149	C₂²×C₄	—	134	8	no	—
150	C₂²×C₄	C₄⋊Dic₆₇	1	8	yes	—
151	C₂²×C₄	C₂×C₄×D₆₇	1	4	yes	—
152	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
153	C₂²×C₄	C₂×D₂₆₈	1	4	yes	—
154	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
155	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
156	C₂²×C₄	#11761628	1	4	yes	—
157	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
158	C₂×C₈	—	2	8	no	—
159	C₂×C₈	—	2	4	no	—
160	C₂×C₈	C₄×Dic₆₇	1	16	yes	complemented
161	C₂×C₈	C₂×C₄×D₆₇	1	4	yes	—
162	C₂×C₈	C₂×C₄×D₆₇	1	4	yes	—
163	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
164	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
165	C₄⋊C₄	—	67	16	no	—
166	C₄⋊C₄	—	134	8	no	—
167	C₂²⋊C₄	—	134	16	no	—
168	C₄²	—	268	16	no	—
169	C₄²	—	134	16	no	—
170	C₄²	C₄⋊Dic₆₇	1	8	yes	—
171	C₄²	C₂²×Dic₆₇	1	4	yes	—
172	C₄²	C₂×Dic₁₃₄	1	4	yes	—
173	C₂³	—	2	4	no	—
174	C₂³	C₂×C₄⋊Dic₆₇	1	2	yes	—
175	C₂³	C₄×D₂₆₈	1	4	yes	—
176	C₂³	C₂²×D₂₆₈	1	1	yes	characteristic
177	C₂×C₄	—	2	8	no	—
178	C₂×C₄	—	4	8	no	—
179	C₂×C₄	—	268	16	no	—
180	C₂×C₄	—	134	16	no	—
181	C₂×C₄	C₂×C₄×Dic₆₇	1	4	yes	—
182	C₂×C₄	C₂×C₄⋊Dic₆₇	1	8	yes	—
183	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
184	C₂×C₄	#14524002	1	4	yes	—
185	C₂×C₄	C₄×D₂₆₈	1	4	yes	—
186	C₂×C₄	C₄×Dic₁₃₄	1	8	yes	—
187	C₂×C₄	C₂²×C₄×D₆₇	1	2	yes	—
188	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
189	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
190	C₂×C₄	C₂×#11761628	1	2	yes	—
191	C₈	—	2	4	no	—
192	C₈	C₂×C₄×Dic₆₇	1	8	yes	—
193	C₂²	—	2	8	no	—
194	C₂²	—	2	4	no	—
195	C₂²	C₄×C₄⋊Dic₆₇	1	4	yes	—
196	C₂²	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
197	C₂²	C₂×C₄×D₂₆₈	1	2	yes	—
198	C₂²	#16864196	1	4	yes	—
199	C₄	—	2	8	no	—
200	C₄	—	268	16	no	—
201	C₄	C₄×C₄⋊Dic₆₇	1	4	yes	—
202	C₄	C₂²×C₄×Dic₆₇	1	1	yes	characteristic
203	C₄	C₂×#14524002	1	1	yes	characteristic
204	C₄	C₂×C₄×Dic₁₃₄	1	2	yes	—
205	C₂	—	2	4	no	—
206	C₂	C₂×C₄×C₄⋊Dic₆₇	1	1	yes	characteristic
207	C₂	#22568799	1	2	yes	—
208	C₂	#22569011	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₆₈
commutator	C₄×C₆₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂²×#97×C₆₇	C₂
socle	C₂³×C₆₇	C₂²×D₄
prime radical	C₂⁴×C₆₇	C₂×D₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂²×#97	D₆₇
hypocenter	C₆₇	#20451
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
3. #22560106	C₂	semidirect
7. #16863434	C₄	semidirect
18. C₂×#97×C₆₇	C₄	semidirect
26. #14523977	C₂×C₄	semidirect
27. #14523975	C₂×C₄	semidirect
60. C₂×C₈×C₆₇	C₄²	semidirect
110. C₆₇	#20451	semidirect
118. C₂×#97	Dic₆₇	semidirect
160. C₂×C₈	C₄×Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863434	2
C₄×#14524002	2
#22560106	2
#22559926	2
C₂²×#97×C₆₇	2
#20451	67