Subgroups — #29573119

Order 15616.

418 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16839892	C₂	1	1	yes	maximal, characteristic
2	#21445821	C₂	1	2	yes	maximal
3	C₂×#16839515	C₂	1	1	yes	maximal, characteristic
4	#21443702	C₂	1	2	yes	maximal
5	C₆₁×#1834	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₂₂	C₂²	1	1	yes	characteristic
7	#16839892	—	2	2	no	—
8	#16839892	C₄	1	4	yes	complemented
9	C₂×C₄⋊#92418	C₂²	1	2	yes	—
10	C₂×C₄⋊#92418	C₂²	1	1	yes	characteristic
11	C₂×#14519706	—	2	1	no	—
12	C₂×#14519700	—	2	2	no	—
13	C₂×#14519687	—	2	1	no	—
14	#16839524	—	2	2	no	—
15	#16839515	—	2	2	no	—
16	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₆₁×#384	C₂²	1	2	yes	—
18	C₆₁×#340	—	2	1	no	—
19	C₆₁×#340	—	2	1	no	—
20	C₂²×Dic₁₂₂	D₄	1	1	yes	characteristic
21	C₂²×#92418	—	2	1	no	—
22	C₂²×#92418	—	2	2	no	—
23	C₄×Dic₁₂₂	—	2	1	no	—
24	C₄×Dic₁₂₂	—	2	1	no	—
25	C₄×Dic₁₂₂	C₂×C₄	1	2	yes	—
26	C₄×Dic₁₂₂	Q₈	1	2	yes	—
27	C₂×C₄⋊Dic₆₁	D₄	1	1	yes	characteristic
28	C₂×#11758196	—	2	1	no	—
29	C₂×C₄×Dic₆₁	—	2	1	no	—
30	C₄⋊#92418	—	2	1	no	—
31	C₄⋊#92418	—	2	2	no	—
32	C₄⋊#92418	—	2	2	no	—
33	C₄⋊#92418	C₂×C₄	1	2	yes	—
34	#14519706	—	2	4	no	—
35	#14519700	—	4	4	no	—
36	#14519687	—	2	4	no	—
37	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
40	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
41	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
42	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
43	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₆₁	D₄	1	2	yes	—
45	C₂²×C₄×C₆₁	—	4	2	no	—
46	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
49	C₂²×Dic₆₁	—	2	1	no	—
50	C₂²×Dic₆₁	—	2	1	no	—
51	C₂×Dic₁₂₂	—	2	2	no	—
52	C₂×Dic₁₂₂	—	2	2	no	—
53	C₂×Dic₁₂₂	—	2	1	no	—
54	C₂×Dic₁₂₂	C₄⋊C₄	1	2	yes	—
55	C₂×C₈×C₆₁	—	2	2	no	—
56	C₂×C₈×C₆₁	—	2	2	no	—
57	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
58	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
59	C₄⋊C₄×C₆₁	—	4	2	no	—
60	C₄⋊C₄×C₆₁	—	4	2	no	—
61	C₄⋊C₄×C₆₁	—	4	2	no	—
62	C₄⋊C₄×C₆₁	—	4	2	no	—
63	C₄⋊C₄×C₆₁	—	8	2	no	—
64	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
65	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
67	C₄²×C₆₁	#41	1	1	yes	characteristic
68	C₄⋊Dic₆₁	—	2	1	no	—
69	C₄⋊Dic₆₁	C₄⋊C₄	1	2	yes	—
70	#11758196	—	2	2	no	—
71	#11758196	—	2	2	no	—
72	C₄×Dic₆₁	—	2	2	no	—
73	C₄×Dic₆₁	—	2	2	no	—
74	C₂×#92418	—	2	2	no	—
75	C₂×#92418	—	4	2	no	—
76	C₂×#92418	—	4	2	no	—
77	C₂×#92418	—	4	2	no	—
78	C₂×#92418	—	4	2	no	—
79	C₂³×C₆₁	#121	1	1	yes	characteristic
80	C₂×C₄×C₆₁	—	2	1	no	—
81	C₂×C₄×C₆₁	—	2	1	no	—
82	C₂×C₄×C₆₁	—	8	2	no	—
83	C₂×C₄×C₆₁	—	8	2	no	—
84	C₂×C₄×C₆₁	—	8	2	no	—
85	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
88	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
89	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
90	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
91	C₂×C₄×C₆₁	#135	1	2	yes	—
92	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
93	C₂×C₄×C₆₁	#137	1	2	yes	—
94	C₂×C₄×C₆₁	#137	1	2	yes	—
95	C₂×Dic₆₁	—	2	2	no	—
96	C₂×Dic₆₁	—	2	2	no	—
97	C₂×Dic₆₁	—	4	2	no	—
98	C₂×Dic₆₁	—	4	1	no	—
99	C₂×Dic₆₁	—	4	1	no	—
100	Dic₁₂₂	—	2	4	no	—
101	Dic₁₂₂	—	4	1	no	—
102	Dic₁₂₂	—	4	1	no	—
103	C₈×C₆₁	—	2	2	no	—
104	C₈×C₆₁	—	2	2	no	—
105	#92418	—	4	2	no	—
106	#92418	—	8	2	no	—
107	#8655	—	61	1	no	maximal, Sylow
108	C₂²×C₆₁	#380	1	1	yes	characteristic
109	C₂²×C₆₁	#464	1	2	yes	—
110	C₂²×C₆₁	#468	1	2	yes	—
111	C₂²×C₆₁	#472	1	1	yes	characteristic
112	C₂²×C₆₁	#473	1	1	yes	characteristic
113	C₄×C₆₁	—	2	1	no	—
114	C₄×C₆₁	—	2	1	no	—
115	C₄×C₆₁	—	16	2	no	—
116	C₄×C₆₁	#416	1	1	yes	characteristic
117	C₄×C₆₁	#419	1	1	yes	characteristic
118	C₄×C₆₁	#428	1	2	yes	—
119	C₄×C₆₁	#469	1	1	yes	characteristic
120	C₄×C₆₁	#470	1	1	yes	characteristic
121	C₄×C₆₁	#479	1	2	yes	—
122	Dic₆₁	—	4	2	no	—
123	Dic₆₁	—	4	2	no	—
124	C₂×#446	—	61	1	no	—
125	#1834	D₆₁	1	1	yes	characteristic
126	#1803	—	61	2	no	—
127	C₂×#335	—	61	1	no	—
128	#1296	—	61	2	no	—
129	C₂×C₆₁	#1698	1	2	yes	—
130	C₂×C₆₁	#1700	1	2	yes	—
131	C₂×C₆₁	#1805	1	1	yes	characteristic
132	C₂×C₆₁	#1809	1	1	yes	characteristic
133	C₂×C₆₁	#1816	1	1	yes	characteristic
134	C₂×C₄×Q₈	—	61	1	no	—
135	#446	—	61	4	no	—
136	#446	—	122	2	no	—
137	C₂×C₄⋊C₈	—	61	1	no	—
138	C₂×C₄⋊C₈	—	61	2	no	—
139	C₂×C₄⋊C₈	—	122	1	no	—
140	C₂×#103	—	122	2	no	—
141	C₂×#97	—	122	1	no	—
142	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
143	#384	C₂×D₆₁	1	2	yes	—
144	#340	—	2	1	no	—
145	#340	—	2	1	no	—
146	#340	—	122	2	no	—
147	#335	—	122	2	no	—
148	C₆₁	#8655	1	1	yes	characteristic, complemented, Sylow
149	C₂²×Q₈	—	61	1	no	—
150	C₂²×C₈	—	122	1	no	—
151	C₂²×C₈	—	122	2	no	—
152	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
153	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
154	C₄×Q₈	—	61	2	no	—
155	C₄×Q₈	—	61	2	no	—
156	C₄×Q₈	—	122	1	no	—
157	C₄×Q₈	—	122	1	no	—
158	C₂×C₄⋊C₄	—	2	2	no	—
159	C₂×C₄⋊C₄	—	2	2	no	—
160	C₂×C₄⋊C₄	—	2	2	no	—
161	C₂×C₄⋊C₄	—	61	1	no	—
162	C₂×C₄⋊C₄	—	122	1	no	—
163	C₂×C₄²	—	122	1	no	—
164	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
165	C₄⋊C₈	—	61	2	no	—
166	C₄⋊C₈	—	122	4	no	—
167	C₄⋊C₈	—	122	1	no	—
168	C₄⋊C₈	—	122	2	no	—
169	C₄⋊C₈	—	122	2	no	—
170	#103	—	244	4	no	—
171	#97	—	122	4	no	—
172	C₄×C₈	C₄×D₆₁	1	2	yes	—
173	C₄×C₈	D₂₄₄	1	2	yes	—
174	C₂×Q₈	—	61	2	no	—
175	C₂×Q₈	—	122	1	no	—
176	C₂×Q₈	—	122	2	no	—
177	C₂×Q₈	—	122	2	no	—
178	C₂²×C₄	—	4	2	no	—
179	C₂²×C₄	—	122	1	no	—
180	C₂²×C₄	—	122	1	no	—
181	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
182	C₂²×C₄	#11758224	1	1	yes	characteristic
183	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	122	2	no	—
187	C₂×C₈	—	244	2	no	—
188	C₂×C₈	—	244	2	no	—
189	C₂×C₈	—	244	2	no	—
190	C₂×C₈	—	244	2	no	—
191	C₂×C₈	#11758198	1	2	yes	—
192	C₂×C₈	#11758198	1	2	yes	—
193	C₄⋊C₄	—	4	2	no	—
194	C₄⋊C₄	—	4	2	no	—
195	C₄⋊C₄	—	4	2	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₄⋊C₄	—	8	2	no	—
198	C₄⋊C₄	—	61	2	no	—
199	C₄⋊C₄	—	122	2	no	—
200	C₄⋊C₄	—	122	1	no	—
201	C₄⋊C₄	—	122	2	no	—
202	C₄²	—	122	2	no	—
203	C₄²	—	122	2	no	—
204	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
205	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
206	C₄²	Q₈×D₆₁	1	1	yes	characteristic
207	C₄²	#11758226	1	1	yes	characteristic
208	C₂³	#14519748	1	1	yes	characteristic
209	Q₈	—	122	4	no	—
210	Q₈	—	244	1	no	—
211	Q₈	—	244	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	8	2	no	—
215	C₂×C₄	—	8	2	no	—
216	C₂×C₄	—	8	2	no	—
217	C₂×C₄	—	122	2	no	—
218	C₂×C₄	—	122	2	no	—
219	C₂×C₄	—	244	2	no	—
220	C₂×C₄	—	244	1	no	—
221	C₂×C₄	—	244	1	no	—
222	C₂×C₄	C₂×#11758198	1	1	yes	characteristic
223	C₂×C₄	C₄⋊C₄×D₆₁	1	1	yes	characteristic
224	C₂×C₄	#14519735	1	1	yes	characteristic
225	C₂×C₄	#14519752	1	1	yes	characteristic
226	C₂×C₄	#14519753	1	1	yes	characteristic
227	C₂×C₄	#14519757	1	1	yes	characteristic
228	C₂×C₄	#14519797	1	2	yes	—
229	C₂×C₄	#14519804	1	2	yes	—
230	C₂×C₄	#14519806	1	2	yes	—
231	C₂×C₄	#14519807	1	2	yes	—
232	C₈	—	2	2	no	—
233	C₈	—	2	2	no	—
234	C₈	—	488	2	no	—
235	C₈	—	244	2	no	—
236	C₂²	#16839619	1	1	yes	characteristic
237	C₂²	#16839981	1	2	yes	—
238	C₂²	#16840005	1	2	yes	—
239	C₂²	#16840029	1	1	yes	characteristic
240	C₂²	#16840034	1	1	yes	characteristic
241	C₄	—	2	1	no	—
242	C₄	—	2	1	no	—
243	C₄	—	16	2	no	—
244	C₄	—	244	2	no	—
245	C₄	—	244	2	no	—
246	C₄	#16839756	1	1	yes	characteristic
247	C₄	#16839770	1	1	yes	characteristic
248	C₄	#16839801	1	2	yes	—
249	C₄	#16840016	1	1	yes	characteristic
250	C₄	#16840021	1	1	yes	characteristic
251	C₄	#16840066	1	2	yes	—
252	C₂	#21445356	1	2	yes	—
253	C₂	#21445370	1	2	yes	—
254	C₂	#21445831	1	1	yes	characteristic
255	C₂	#21445849	1	1	yes	characteristic
256	C₂	#21445884	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14519748
commutator	C₄²×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1834	C₂
socle, prime radical	C₂³×C₆₁	#121
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#1834	D₆₁
hypocenter	C₆₁	#8655
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

Complements

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

N	H ≅ G/N	type
8. #16839892	C₄	semidirect
148. C₆₁	#8655	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16839892	2
#21445821	2
C₂×#16839515	2
#21443702	2
C₆₁×#1834	2
#8655	61