Subgroups — #29580108

Order 15616.

468 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21454182	C₂	1	1	yes	maximal, characteristic
2	#21445672	C₂	1	1	yes	maximal, characteristic
3	#21443701	C₂	1	4	yes	maximal
4	C₄×#97×C₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#14519769	C₄	1	2	yes	—
6	C₂×C₄⋊#92418	—	2	1	no	—
7	C₂×#14519688	C₂²	1	2	yes	—
8	#16839660	—	2	1	no	—
9	#16839660	—	2	1	no	—
10	#16839660	—	2	1	no	—
11	#16839660	—	2	1	no	—
12	#16839632	C₄	1	1	yes	characteristic
13	#16839632	C₄	1	1	yes	characteristic
14	#16839628	C₂²	1	2	yes	—
15	#16839517	—	2	1	no	—
16	C₂×#97×C₆₁	—	2	1	no	—
17	C₂×#97×C₆₁	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₆₁	—	2	1	no	—
19	C₄³×C₆₁	C₂²	1	1	yes	characteristic
20	C₂²×#92418	—	2	2	no	—
21	#14519769	—	4	4	no	—
22	C₂×C₄⋊Dic₆₁	—	2	1	no	—
23	C₂×C₄⋊Dic₆₁	—	2	1	no	—
24	C₂×C₄⋊Dic₆₁	—	2	1	no	—
25	C₂×C₄⋊Dic₆₁	—	2	1	no	—
26	C₂×C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
27	C₂×C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
28	C₂×#11758196	—	2	2	no	—
29	C₂×#11758196	—	2	2	no	—
30	C₄⋊#92418	—	2	4	no	—
31	#14519688	—	2	4	no	—
32	#14519684	—	2	2	no	—
33	#14519684	—	2	2	no	—
34	C₂²×C₈×C₆₁	—	2	2	no	—
35	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
42	#97×C₆₁	—	2	4	no	—
43	#97×C₆₁	—	2	2	no	—
44	#97×C₆₁	Q₈	1	4	yes	—
45	C₄×C₈×C₆₁	—	2	4	no	—
46	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
53	C₂²×Dic₆₁	—	2	2	no	—
54	C₂²×Dic₆₁	—	2	2	no	—
55	C₂×C₈×C₆₁	—	2	4	no	—
56	C₂×C₈×C₆₁	—	2	4	no	—
57	C₂×C₈×C₆₁	—	2	4	no	—
58	C₄²×C₆₁	—	2	2	no	—
59	C₄²×C₆₁	—	2	2	no	—
60	C₄²×C₆₁	—	2	2	no	—
61	C₄²×C₆₁	—	2	2	no	—
62	C₄²×C₆₁	—	2	2	no	—
63	C₄²×C₆₁	—	4	1	no	—
64	C₄²×C₆₁	C₂×Q₈	1	2	yes	—
65	C₄²×C₆₁	#41	1	2	yes	—
66	C₄⋊Dic₆₁	—	4	2	no	—
67	C₄⋊Dic₆₁	—	4	2	no	—
68	C₄⋊Dic₆₁	—	4	2	no	—
69	C₄⋊Dic₆₁	—	4	2	no	—
70	C₄⋊Dic₆₁	—	4	2	no	—
71	C₄⋊Dic₆₁	—	4	2	no	—
72	#11758196	—	4	4	no	—
73	#11758196	—	4	4	no	—
74	C₂×#92418	—	2	4	no	—
75	C₂×#92418	—	4	2	no	—
76	C₂×#92418	—	4	2	no	—
77	C₂³×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₆₁	—	2	1	no	—
79	C₂×C₄×C₆₁	—	2	1	no	—
80	C₂×C₄×C₆₁	—	2	1	no	—
81	C₂×C₄×C₆₁	—	2	1	no	—
82	C₂×C₄×C₆₁	—	2	1	no	—
83	C₂×C₄×C₆₁	—	2	1	no	—
84	C₂×C₄×C₆₁	—	2	1	no	—
85	C₂×C₄×C₆₁	—	2	1	no	—
86	C₂×C₄×C₆₁	—	2	1	no	—
87	C₂×C₄×C₆₁	—	2	1	no	—
88	C₂×C₄×C₆₁	—	2	1	no	—
89	C₂×C₄×C₆₁	—	2	1	no	—
90	C₂×C₄×C₆₁	—	2	1	no	—
91	C₂×C₄×C₆₁	#100	1	2	yes	—
92	C₂×C₄×C₆₁	#101	1	2	yes	—
93	C₂×C₄×C₆₁	#104	1	2	yes	—
94	C₂×C₄×C₆₁	#104	1	2	yes	—
95	C₂×C₄×C₆₁	#104	1	2	yes	—
96	C₂×C₄×C₆₁	C₄×Q₈	1	2	yes	—
97	C₂×C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
98	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
99	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
100	C₂×Dic₆₁	—	4	2	no	—
101	C₂×Dic₆₁	—	4	2	no	—
102	C₂×Dic₆₁	—	4	2	no	—
103	C₂×Dic₆₁	—	4	2	no	—
104	C₂×Dic₆₁	—	4	2	no	—
105	C₂×Dic₆₁	—	4	2	no	—
106	C₈×C₆₁	—	2	8	no	—
107	#92418	—	4	4	no	—
108	#9891	—	61	1	no	maximal, Sylow
109	C₂²×C₆₁	#391	1	1	yes	characteristic
110	C₂²×C₆₁	C₂×#100	1	1	yes	characteristic
111	C₂²×C₆₁	C₂×#101	1	1	yes	characteristic
112	C₂²×C₆₁	C₂×#104	1	1	yes	characteristic
113	C₂²×C₆₁	C₂×#104	1	1	yes	characteristic
114	C₂²×C₆₁	C₂×#104	1	1	yes	characteristic
115	C₂²×C₆₁	#421	1	1	yes	characteristic
116	C₄×C₆₁	—	2	1	no	—
117	C₄×C₆₁	—	2	1	no	—
118	C₄×C₆₁	—	2	2	no	—
119	C₄×C₆₁	—	2	2	no	—
120	C₄×C₆₁	—	2	2	no	—
121	C₄×C₆₁	—	2	2	no	—
122	C₄×C₆₁	—	4	1	no	—
123	C₄×C₆₁	#330	1	4	yes	—
124	Dic₆₁	—	8	2	no	—
125	Dic₆₁	—	8	2	no	—
126	#2859	—	61	1	no	—
127	#1767	—	61	1	no	—
128	C₄×#97	D₆₁	1	1	yes	characteristic
129	#1295	—	61	4	no	—
130	C₂×C₆₁	C₂×#330	1	1	yes	characteristic
131	C₂×C₆₁	#1525	1	1	yes	characteristic
132	C₂×C₆₁	#1528	1	1	yes	characteristic
133	C₂×C₆₁	#1530	1	1	yes	characteristic
134	C₂×C₆₁	#1986	1	1	yes	characteristic
135	C₂×C₆₁	#1994	1	1	yes	characteristic
136	C₂×C₆₁	#1997	1	1	yes	characteristic
137	C₂×#125	—	61	2	no	—
138	C₂×C₄⋊C₈	—	122	1	no	—
139	C₂×#97	—	2	1	no	—
140	C₂×#97	—	61	2	no	—
141	C₂×#97	C₂×D₆₁	1	2	yes	—
142	C₂×C₄×C₈	—	2	1	no	—
143	#389	—	122	1	no	—
144	#389	—	122	1	no	—
145	#389	—	122	1	no	—
146	#389	—	122	1	no	—
147	#384	—	61	1	no	—
148	#384	—	61	1	no	—
149	#382	—	61	2	no	—
150	C₄³	C₂×D₆₁	1	1	yes	characteristic
151	#336	—	122	1	no	—
152	C₆₁	#9891	1	1	yes	characteristic, complemented, Sylow
153	C₂²×C₈	—	2	2	no	—
154	C₂²×C₈	—	122	2	no	—
155	#125	—	244	4	no	—
156	C₂×C₄⋊C₄	—	61	2	no	—
157	C₂×C₄⋊C₄	—	61	2	no	—
158	C₂×C₄⋊C₄	—	122	2	no	—
159	C₂×C₄⋊C₄	—	122	2	no	—
160	C₂×C₄⋊C₄	—	122	1	no	—
161	C₂×C₄⋊C₄	—	122	1	no	—
162	C₂×C₄⋊C₄	—	122	1	no	—
163	C₂×C₄⋊C₄	—	122	1	no	—
164	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
165	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
166	C₂×C₄²	D₂₄₄	1	1	yes	characteristic
167	C₂×C₄²	D₂₄₄	1	1	yes	characteristic
168	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
169	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
170	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
171	C₄⋊C₈	—	122	4	no	—
172	#97	—	2	4	no	—
173	#97	—	2	2	no	—
174	#97	—	122	4	no	—
175	#97	Dic₁₂₂	1	4	yes	—
176	C₄×C₈	—	2	4	no	—
177	#95	—	122	2	no	—
178	#95	—	122	2	no	—
179	C₂²×C₄	—	122	2	no	—
180	C₂²×C₄	—	122	2	no	—
181	C₂²×C₄	#11758198	1	1	yes	characteristic
182	C₂²×C₄	#11758198	1	1	yes	characteristic
183	C₂²×C₄	#11758198	1	1	yes	characteristic
184	C₂²×C₄	#11758198	1	1	yes	characteristic
185	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
186	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
187	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	—	122	4	no	—
192	C₂×C₈	—	244	2	no	—
193	C₂×C₈	—	244	2	no	—
194	C₄⋊C₄	—	244	2	no	—
195	C₄⋊C₄	—	244	2	no	—
196	C₄⋊C₄	—	244	4	no	—
197	C₄⋊C₄	—	244	2	no	—
198	C₄⋊C₄	—	244	4	no	—
199	C₄⋊C₄	—	244	2	no	—
200	C₄⋊C₄	—	244	2	no	—
201	C₄⋊C₄	—	244	2	no	—
202	C₄²	—	2	2	no	—
203	C₄²	—	2	2	no	—
204	C₄²	—	2	2	no	—
205	C₄²	—	2	2	no	—
206	C₄²	—	2	2	no	—
207	C₄²	—	4	1	no	—
208	C₄²	C₂×Dic₁₂₂	1	2	yes	—
209	C₄²	#11758222	1	2	yes	—
210	C₂³	C₂×#11758198	1	1	yes	characteristic
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	244	2	no	—
225	C₂×C₄	—	244	2	no	—
226	C₂×C₄	—	244	2	no	—
227	C₂×C₄	—	244	2	no	—
228	C₂×C₄	—	244	2	no	—
229	C₂×C₄	—	244	2	no	—
230	C₂×C₄	#14519693	1	2	yes	—
231	C₂×C₄	#14519695	1	2	yes	—
232	C₂×C₄	#14519703	1	2	yes	—
233	C₂×C₄	#14519705	1	2	yes	—
234	C₂×C₄	#14519705	1	2	yes	—
235	C₂×C₄	C₄×Dic₁₂₂	1	2	yes	—
236	C₂×C₄	#14519756	1	2	yes	—
237	C₂×C₄	#14519765	1	1	yes	characteristic
238	C₂×C₄	#14519777	1	1	yes	characteristic
239	C₈	—	2	8	no	—
240	C₈	—	244	4	no	—
241	C₂²	#16839668	1	1	yes	characteristic
242	C₂²	C₂×#14519693	1	1	yes	characteristic
243	C₂²	C₂×#14519695	1	1	yes	characteristic
244	C₂²	C₂×#14519703	1	1	yes	characteristic
245	C₂²	C₂×#14519705	1	1	yes	characteristic
246	C₂²	C₂×#14519705	1	1	yes	characteristic
247	C₂²	#16839778	1	1	yes	characteristic
248	C₄	—	2	2	no	—
249	C₄	—	2	2	no	—
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	2	no	—
253	C₄	—	2	2	no	—
254	C₄	—	4	1	no	—
255	C₄	—	488	2	no	—
256	C₄	—	488	2	no	—
257	C₄	#16839503	1	4	yes	—
258	C₂	C₂×#16839503	1	1	yes	characteristic
259	C₂	#21444410	1	1	yes	characteristic
260	C₂	#21444420	1	1	yes	characteristic
261	C₂	#21444429	1	1	yes	characteristic
262	C₂	#21446763	1	1	yes	characteristic
263	C₂	#21446796	1	1	yes	characteristic
264	C₂	#21446808	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11758198
commutator	C₂²×C₄×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₄×#97×C₆₁	C₂
socle, prime radical	C₂³×C₆₁	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₄×#97	D₆₁
hypocenter	C₆₁	#9891
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
152. C₆₁	#9891	semidirect

Maximal subgroups

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

subgroup	index
#21454182	2
#21445672	2
#21443701	2
C₄×#97×C₆₁	2
#9891	61