Subgroups — #30864625

Order 17152.

854 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863794	C₂	1	1	yes	maximal, characteristic
2	C₄×#14524043	C₂	1	2	yes	maximal
3	#22561170	C₂	1	2	yes	maximal
4	#22560952	C₂	1	4	yes	maximal
5	#22560369	C₂	1	1	yes	maximal, characteristic
6	#22560106	C₂	1	4	yes	maximal
7	C₄×#97×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#14524043	C₂²	1	2	yes	—
9	C₂×C₄×Dic₁₃₄	—	2	2	no	—
10	#16863794	C₄	1	8	yes	complemented
11	C₂×#14523977	C₂²	1	4	yes	—
12	C₂×#14523975	C₂²	1	4	yes	—
13	#16863302	—	2	2	no	—
14	#16863294	—	2	2	no	—
15	#16863294	C₂²	1	4	yes	—
16	#16863266	C₂²	1	2	yes	—
17	C₄×C₄⋊Dic₆₇	—	2	2	no	—
18	C₄×C₄⋊Dic₆₇	C₂²	1	4	yes	—
19	#16863159	C₂²	1	8	yes	—
20	C₂×#97×C₆₇	C₂²	1	2	yes	—
21	C₂×#97×C₆₇	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₄³×C₆₇	C₂²	1	1	yes	characteristic
25	C₂²×Dic₁₃₄	—	2	2	no	—
26	#14524043	—	2	4	no	—
27	#14524043	C₂×C₄	1	8	yes	—
28	C₄×Dic₁₃₄	—	4	8	no	—
29	C₂×C₄⋊Dic₆₇	—	2	2	no	—
30	C₂×C₄⋊Dic₆₇	—	2	2	no	—
31	C₂×C₄⋊Dic₆₇	—	2	2	no	—
32	C₂×C₄⋊Dic₆₇	C₂³	1	4	yes	—
33	C₂×C₄⋊Dic₆₇	C₂³	1	4	yes	—
34	C₂×Dic₆₇⋊C₄	—	2	4	no	—
35	C₂×C₄×Dic₆₇	—	2	4	no	—
36	#14523977	—	2	8	no	—
37	#14523975	C₂×C₄	1	16	yes	—
38	C₆₇⋊#95	—	2	4	no	—
39	C₆₇⋊#95	—	2	4	no	—
40	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
41	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
42	C₂×C₄²×C₆₇	D₄	1	2	yes	—
43	C₂×C₄²×C₆₇	D₄	1	2	yes	—
44	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
47	#97×C₆₇	—	2	4	no	—
48	#97×C₆₇	Q₈	1	8	yes	complemented
49	C₄×C₈×C₆₇	—	2	2	no	—
50	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
51	C₂×Dic₁₃₄	—	2	8	no	—
52	C₂×Dic₁₃₄	—	4	4	no	—
53	C₂×Dic₁₃₄	—	4	4	no	—
54	C₂²×Dic₆₇	—	2	4	no	—
55	C₂²×Dic₆₇	—	2	4	no	—
56	C₄⋊Dic₆₇	—	2	8	no	—
57	C₄⋊Dic₆₇	—	2	4	no	—
58	C₄⋊Dic₆₇	—	2	8	no	—
59	C₄⋊Dic₆₇	—	4	4	no	—
60	C₄⋊Dic₆₇	—	4	4	no	—
61	C₄⋊Dic₆₇	C₂²×C₄	1	8	yes	—
62	Dic₆₇⋊C₄	—	4	8	no	—
63	C₄×Dic₆₇	—	4	8	no	—
64	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
69	C₂×C₈×C₆₇	—	2	2	no	—
70	C₂×C₈×C₆₇	—	2	2	no	—
71	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
72	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
73	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
74	C₂×C₈×C₆₇	#41	1	4	yes	—
75	C₄²×C₆₇	—	2	4	no	—
76	C₄²×C₆₇	—	2	4	no	—
77	C₄²×C₆₇	—	2	2	no	—
78	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
79	C₄²×C₆₇	C₂×Q₈	1	4	yes	—
80	C₄²×C₆₇	#41	1	2	yes	—
81	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
82	C₂×C₄×C₆₇	—	2	2	no	—
83	C₂×C₄×C₆₇	—	2	2	no	—
84	C₂×C₄×C₆₇	—	2	1	no	—
85	C₂×C₄×C₆₇	—	2	1	no	—
86	C₂×C₄×C₆₇	—	2	2	no	—
87	C₂×C₄×C₆₇	—	2	2	no	—
88	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
89	C₂×C₄×C₆₇	C₄⋊Q₈	1	4	yes	—
90	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
91	C₂×C₄×C₆₇	#137	1	4	yes	—
92	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
93	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
94	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
95	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
96	C₂×Dic₆₇	—	2	8	no	—
97	C₂×Dic₆₇	—	4	4	no	—
98	C₂×Dic₆₇	—	4	4	no	—
99	C₂×Dic₆₇	—	4	4	no	—
100	C₂×Dic₆₇	—	4	4	no	—
101	C₂×Dic₆₇	—	4	4	no	—
102	Dic₁₃₄	—	4	16	no	—
103	C₈×C₆₇	—	2	4	no	—
104	C₈×C₆₇	C₄×Q₈	1	8	yes	complemented
105	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
106	C₂²×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
107	C₂²×C₆₇	#532	1	1	yes	characteristic
108	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
109	C₂²×C₆₇	C₂×#137	1	2	yes	—
110	C₄×C₆₇	—	2	4	no	—
111	C₄×C₆₇	—	4	1	no	—
112	C₄×C₆₇	—	4	1	no	—
113	C₄×C₆₇	#440	1	8	yes	—
114	C₄×C₆₇	C₂×C₄×Q₈	1	2	yes	—
115	C₄×C₆₇	#533	1	2	yes	—
116	Dic₆₇	—	4	8	no	—
117	Dic₆₇	—	8	4	no	—
118	#21727	—	67	1	no	maximal, Sylow
119	C₂×C₆₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
120	C₂×C₆₇	C₂×#440	1	2	yes	—
121	C₂×C₆₇	#3165	1	2	yes	—
122	C₂×C₆₇	#3184	1	1	yes	characteristic
123	C₂×C₆₇	#3185	1	1	yes	characteristic
124	C₂×#498	—	67	1	no	—
125	C₄×C₄⋊Q₈	—	67	2	no	—
126	#1990	—	67	2	no	—
127	#1944	—	67	4	no	—
128	#1845	—	67	1	no	—
129	#1776	—	67	4	no	—
130	C₄×#97	D₆₇	1	1	yes	characteristic
131	C₆₇	#21727	1	1	yes	characteristic, complemented, Sylow
132	C₂×C₄⋊Q₈	—	67	2	no	—
133	C₂×C₄×Q₈	—	134	2	no	—
134	#498	—	67	8	no	—
135	C₂×#107	—	67	4	no	—
136	C₂×#106	—	67	4	no	—
137	C₂×#97	C₂×D₆₇	1	2	yes	—
138	C₂×#97	C₂×D₆₇	1	2	yes	—
139	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
140	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
141	#391	—	134	2	no	—
142	#389	—	67	4	no	—
143	#389	—	134	2	no	—
144	#384	—	67	2	no	—
145	C₄×C₄⋊C₄	—	67	4	no	—
146	C₄×C₄⋊C₄	—	134	2	no	—
147	C₄³	C₂×D₆₇	1	1	yes	characteristic
148	#340	—	67	8	no	—
149	C₂²×Q₈	—	134	2	no	—
150	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
151	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
152	C₄⋊Q₈	—	67	8	no	—
153	C₄⋊Q₈	—	134	4	no	—
154	C₄×Q₈	—	268	8	no	—
155	C₂×C₄⋊C₄	—	67	4	no	—
156	C₂×C₄⋊C₄	—	67	4	no	—
157	C₂×C₄⋊C₄	—	134	4	no	—
158	C₂×C₄⋊C₄	—	134	2	no	—
159	C₂×C₄⋊C₄	—	134	2	no	—
160	C₂×C₄⋊C₄	—	134	2	no	—
161	C₂×C₄²	—	134	4	no	—
162	C₂×C₄²	D₂₆₈	1	2	yes	—
163	C₂×C₄²	D₂₆₈	1	2	yes	—
164	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
165	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
166	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
167	#107	—	134	8	no	—
168	#106	—	67	16	no	—
169	#97	—	2	4	no	—
170	#97	Dic₁₃₄	1	8	yes	complemented
171	C₄×C₈	—	2	2	no	—
172	C₄×C₈	C₄×D₆₇	1	4	yes	—
173	#95	—	134	4	no	—
174	#95	—	134	4	no	—
175	C₂×Q₈	—	268	4	no	—
176	C₂×Q₈	—	268	4	no	—
177	C₂×Q₈	—	134	8	no	—
178	C₂²×C₄	—	134	4	no	—
179	C₂²×C₄	—	134	4	no	—
180	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
181	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
182	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
183	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
184	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	C₂×C₄×D₆₇	1	4	yes	—
188	C₂×C₈	C₂×Dic₁₃₄	1	4	yes	—
189	C₂×C₈	C₂×Dic₁₃₄	1	4	yes	—
190	C₂×C₈	#11761628	1	4	yes	—
191	C₄⋊C₄	—	268	8	no	—
192	C₄⋊C₄	—	268	4	no	—
193	C₄⋊C₄	—	268	4	no	—
194	C₄⋊C₄	—	67	8	no	—
195	C₄⋊C₄	—	134	8	no	—
196	C₄⋊C₄	—	134	8	no	—
197	C₄⋊C₄	—	134	4	no	—
198	C₄²	—	2	4	no	—
199	C₄²	—	2	4	no	—
200	C₄²	—	2	2	no	—
201	C₄²	—	268	8	no	—
202	C₄²	C₂×C₄×D₆₇	1	2	yes	—
203	C₄²	C₂×Dic₁₃₄	1	4	yes	—
204	C₄²	#11761628	1	2	yes	—
205	C₂³	C₂²×D₂₆₈	1	1	yes	characteristic
206	Q₈	—	268	16	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	268	4	no	—
214	C₂×C₄	—	268	4	no	—
215	C₂×C₄	—	268	4	no	—
216	C₂×C₄	—	268	4	no	—
217	C₂×C₄	—	268	4	no	—
218	C₂×C₄	—	134	8	no	—
219	C₂×C₄	C₄×D₂₆₈	1	4	yes	—
220	C₂×C₄	#14524043	1	4	yes	—
221	C₂×C₄	#14524071	1	4	yes	—
222	C₂×C₄	#14524076	1	4	yes	—
223	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
224	C₂×C₄	C₂²×Dic₁₃₄	1	2	yes	—
225	C₂×C₄	C₂×#11761628	1	2	yes	—
226	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
227	C₈	—	2	4	no	—
228	C₈	C₄×Dic₁₃₄	1	8	yes	complemented
229	C₂²	C₂×C₄×D₂₆₈	1	1	yes	characteristic
230	C₂²	C₂×#14524043	1	1	yes	characteristic
231	C₂²	#16863946	1	1	yes	characteristic
232	C₂²	C₂×#14524071	1	2	yes	—
233	C₂²	C₂×#14524076	1	2	yes	—
234	C₄	—	2	4	no	—
235	C₄	—	4	1	no	—
236	C₄	—	4	1	no	—
237	C₄	—	268	8	no	—
238	C₄	—	536	4	no	—
239	C₄	#16863498	1	8	yes	—
240	C₄	C₂×C₄×Dic₁₃₄	1	2	yes	—
241	C₄	#16863953	1	2	yes	—
242	C₂	C₄×#14524043	1	1	yes	characteristic
243	C₂	C₂×#16863498	1	2	yes	—
244	C₂	#22570755	1	2	yes	—
245	C₂	#22570947	1	1	yes	characteristic
246	C₂	#22570954	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₄×#97×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂²×D₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₄×#97	D₆₇
hypocenter	C₆₇	#21727
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
10. #16863794	C₄	semidirect
48. #97×C₆₇	Q₈	semidirect
104. C₈×C₆₇	C₄×Q₈	semidirect
131. C₆₇	#21727	semidirect
170. #97	Dic₁₃₄	semidirect
228. C₈	C₄×Dic₁₃₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863794	2
C₄×#14524043	2
#22561170	2
#22560952	2
#22560369	2
#22560106	2
C₄×#97×C₆₇	2
#21727	67