Subgroups — #30624464

Order 15616.

494 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21462236	C₂	1	4	yes	maximal, complemented
2	#21462235	C₂	1	2	yes	maximal, complemented
3	#21462233	C₂	1	2	yes	maximal, complemented
4	#21462181	C₂	1	2	yes	maximal
5	#21462180	C₂	1	1	yes	maximal, characteristic
6	#21462177	C₂	1	2	yes	maximal
7	#21462176	C₂	1	1	yes	maximal, characteristic
8	C₂×#16839886	C₂	1	1	yes	maximal, characteristic
9	#16840683	C₂²	1	4	yes	—
10	#16840683	C₂²	1	4	yes	—
11	C₄⋊#11758187	C₂²	1	2	yes	—
12	C₄⋊#11758187	C₂²	1	2	yes	—
13	C₂²⋊#11758187	C₂²	1	1	yes	characteristic
14	C₂²⋊#11758187	C₂²	1	1	yes	characteristic
15	C₂²⋊#11758187	C₂²	1	2	yes	—
16	#16840651	C₂²	1	4	yes	—
17	#16840650	C₂²	1	2	yes	—
18	#16840648	C₂²	1	4	yes	—
19	C₄×#11758187	C₂²	1	2	yes	—
20	C₂×C₄×Dic₁₂₂	C₄	1	1	yes	characteristic
21	#16839886	C₄	1	1	yes	characteristic
22	#16839886	C₄	1	1	yes	characteristic
23	#16839886	C₄	1	2	yes	—
24	#16839886	C₂²	1	1	yes	characteristic
25	#16839886	C₂²	1	1	yes	characteristic
26	#16839886	C₂²	1	2	yes	—
27	C₂×#14519706	C₄	1	2	yes	—
28	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
29	C₂×C₈×Dic₆₁	C₂²	1	2	yes	—
30	C₂×C₄⋊C₈×C₆₁	C₄	1	1	yes	characteristic
31	C₂×#11758187	C₂³	1	2	yes	—
32	C₂×#11758187	C₂³	1	2	yes	—
33	C₂×#11758187	C₂³	1	4	yes	—
34	C₂²×Dic₁₂₂	C₈	1	1	yes	characteristic
35	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
36	C₂²×#92418	C₂³	1	1	yes	characteristic
37	C₄×Dic₁₂₂	C₈	1	1	yes	characteristic
38	C₄×Dic₁₂₂	C₈	1	2	yes	—
39	C₄×Dic₁₂₂	C₈	1	1	yes	characteristic
40	C₄×Dic₁₂₂	C₂×C₄	1	1	yes	characteristic
41	C₄×Dic₁₂₂	C₂×C₄	1	2	yes	—
42	C₄×Dic₁₂₂	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄⋊Dic₆₁	C₈	1	1	yes	characteristic
44	C₂×#11758196	C₈	1	2	yes	—
45	C₂×C₄×Dic₆₁	C₂×C₄	1	2	yes	—
46	#14519706	C₂×C₄	1	2	yes	—
47	#14519706	C₂×C₄	1	2	yes	—
48	#14519706	C₂×C₄	1	2	yes	—
49	#14519706	C₂×C₄	1	2	yes	—
50	C₄×#92418	C₂×C₄	1	1	yes	characteristic
51	C₄×#92418	C₂×C₄	1	1	yes	characteristic
52	C₄×#92418	C₂³	1	1	yes	characteristic
53	C₄×#92418	C₂³	1	1	yes	characteristic
54	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
55	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
56	C₈×Dic₆₁	C₂³	1	2	yes	—
57	C₈×Dic₆₁	C₂³	1	2	yes	—
58	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
59	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
60	C₄⋊C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
61	C₄⋊C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
62	C₄⋊C₈×C₆₁	C₂×C₄	1	2	yes	—
63	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
64	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
65	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
66	C₂²×Dic₆₁	C₂×C₈	1	2	yes	—
67	C₂²×Dic₆₁	C₂×C₈	1	2	yes	—
68	C₂×Dic₁₂₂	—	2	4	no	—
69	C₂×Dic₁₂₂	C₂×C₈	1	1	yes	characteristic
70	C₂×Dic₁₂₂	C₂×C₈	1	2	yes	—
71	C₂×Dic₁₂₂	C₂×C₈	1	1	yes	characteristic
72	C₂×C₈×C₆₁	—	4	2	no	—
73	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
75	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
76	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
77	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
78	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
79	C₄⋊Dic₆₁	C₂×C₈	1	2	yes	—
80	C₄⋊Dic₆₁	C₂×C₈	1	1	yes	characteristic
81	C₄⋊Dic₆₁	C₂×C₈	1	1	yes	characteristic
82	#11758196	C₂×C₈	1	2	yes	—
83	#11758196	C₂×C₈	1	2	yes	—
84	#11758196	C₂×C₈	1	2	yes	—
85	#11758196	C₂×C₈	1	2	yes	—
86	C₄×Dic₆₁	C₂×C₈	1	2	yes	—
87	C₄×Dic₆₁	C₂×C₈	1	2	yes	—
88	C₄×Dic₆₁	C₂²×C₄	1	2	yes	—
89	C₄×Dic₆₁	C₂²×C₄	1	2	yes	—
90	C₂×#92418	—	2	2	no	—
91	C₂×#92418	—	4	1	no	—
92	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
93	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
94	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
95	C₂×#92418	C₂⁴	1	1	yes	characteristic
96	#11758187	—	2	2	no	—
97	#11758187	—	2	2	no	—
98	#11758187	—	2	4	no	—
99	C₂³×C₆₁	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₄×C₆₁	—	2	2	no	—
101	C₂×C₄×C₆₁	—	2	1	no	—
102	C₂×C₄×C₆₁	—	2	1	no	—
103	C₂×C₄×C₆₁	—	2	2	no	—
104	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
105	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
106	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
107	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
110	C₂×Dic₆₁	—	2	4	no	—
111	C₂×Dic₆₁	—	2	4	no	—
112	C₂×Dic₆₁	C₂²×C₈	1	2	yes	—
113	C₂×Dic₆₁	C₂²×C₈	1	2	yes	—
114	C₂×Dic₆₁	C₂²×C₈	1	2	yes	—
115	C₂×Dic₆₁	C₂²×C₈	1	2	yes	—
116	Dic₁₂₂	—	2	2	no	—
117	Dic₁₂₂	—	2	4	no	—
118	Dic₁₂₂	M₃₂	1	4	yes	—
119	C₈×C₆₁	—	2	2	no	—
120	C₈×C₆₁	—	2	2	no	—
121	#92418	—	2	1	no	—
122	#92418	—	2	1	no	—
123	#92418	—	2	1	no	—
124	#92418	2₋¹⁺⁴	1	2	yes	—
125	#33498	—	61	1	no	maximal, Sylow
126	C₂²×C₆₁	—	2	1	no	—
127	C₂²×C₆₁	—	2	1	no	—
128	C₂²×C₆₁	#504	1	2	yes	—
129	C₂²×C₆₁	C₂³×C₈	1	1	yes	characteristic
130	C₄×C₆₁	—	2	1	no	—
131	C₄×C₆₁	—	2	1	no	—
132	C₄×C₆₁	—	2	1	no	—
133	C₄×C₆₁	—	2	1	no	—
134	C₄×C₆₁	C₂×M₃₂	1	2	yes	—
135	C₄×C₆₁	#519	1	1	yes	characteristic
136	C₄×C₆₁	#568	1	1	yes	characteristic
137	Dic₆₁	—	2	2	no	—
138	Dic₆₁	—	2	2	no	—
139	Dic₆₁	—	2	2	no	—
140	Dic₆₁	C₂×M₃₂	1	4	yes	—
141	C₂×C₈×Q₈	—	61	1	no	—
142	#2183	—	61	2	no	—
143	#2183	—	61	4	no	—
144	Q₈×C₁₆	—	61	2	no	—
145	#2163	—	61	1	no	—
146	#2163	—	61	2	no	—
147	#2162	—	61	1	no	—
148	#2162	—	61	2	no	—
149	C₂×C₆₁	—	2	2	no	—
150	C₂×C₆₁	#2982	1	1	yes	characteristic
151	C₂×C₆₁	C₂²×M₃₂	1	1	yes	characteristic
152	C₂×C₆₁	C₂×#504	1	1	yes	characteristic
153	C₂×C₄×Q₈	—	61	1	no	—
154	C₈×Q₈	—	61	1	no	—
155	C₈×Q₈	—	61	2	no	—
156	C₈×Q₈	—	61	1	no	—
157	C₈×Q₈	—	61	2	no	—
158	C₈×Q₈	—	61	1	no	—
159	C₈×Q₈	—	61	1	no	—
160	C₂×C₄⋊C₈	—	61	2	no	—
161	C₂×C₄⋊C₈	#6923	1	1	yes	characteristic
162	C₂×C₄×C₈	—	61	2	no	—
163	C₂×C₄×C₈	—	61	1	no	—
164	C₄⋊C₁₆	—	61	2	no	—
165	C₄⋊C₁₆	—	61	4	no	—
166	C₄⋊C₁₆	—	61	2	no	—
167	C₄⋊C₁₆	—	61	4	no	—
168	C₂²⋊C₁₆	—	61	1	no	—
169	C₂²⋊C₁₆	—	61	1	no	—
170	C₂²⋊C₁₆	—	61	2	no	—
171	#346	—	61	2	no	—
172	#346	—	61	4	no	—
173	C₄×C₁₆	—	61	2	no	—
174	C₄×C₁₆	—	61	4	no	—
175	C₆₁	#33498	1	1	yes	characteristic, complemented, Sylow
176	C₂²×Q₈	—	61	1	no	—
177	C₂²×C₈	—	61	1	no	—
178	C₂²×C₈	—	61	1	no	—
179	C₂²×C₈	C₂×#6923	1	2	yes	—
180	C₄×Q₈	—	61	1	no	—
181	C₄×Q₈	—	61	2	no	—
182	C₄×Q₈	—	61	1	no	—
183	C₄×Q₈	—	61	1	no	—
184	C₄×Q₈	—	61	2	no	—
185	C₄×Q₈	—	61	1	no	—
186	C₂×C₄⋊C₄	—	61	2	no	—
187	C₂×C₄⋊C₄	—	61	1	no	—
188	C₂×C₄²	—	61	2	no	—
189	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
190	C₂×C₁₆	—	61	2	no	—
191	C₂×C₁₆	—	61	2	no	—
192	C₂×C₁₆	—	61	4	no	—
193	C₄⋊C₈	—	61	2	no	—
194	C₄⋊C₈	—	61	2	no	—
195	C₄⋊C₈	—	61	2	no	—
196	C₄⋊C₈	—	61	2	no	—
197	C₄⋊C₈	C₂×#6923	1	1	yes	characteristic
198	C₄⋊C₈	C₂×#6923	1	2	yes	—
199	C₄⋊C₈	C₂×#6923	1	1	yes	characteristic
200	C₄×C₈	—	61	2	no	—
201	C₄×C₈	—	61	2	no	—
202	C₄×C₈	—	61	2	no	—
203	C₄×C₈	—	61	2	no	—
204	C₄×C₈	—	61	1	no	—
205	C₄×C₈	—	61	1	no	—
206	C₄×C₈	—	61	1	no	—
207	C₄×C₈	—	61	1	no	—
208	C₂×Q₈	—	61	1	no	—
209	C₂×Q₈	—	61	2	no	—
210	C₂×Q₈	—	61	1	no	—
211	C₂×Q₈	—	122	4	no	—
212	C₂²×C₄	—	61	2	no	—
213	C₂²×C₄	—	61	2	no	—
214	C₂²×C₄	D₆₁⋊C₈	1	1	yes	characteristic
215	C₂²×C₄	D₆₁⋊C₈	1	1	yes	characteristic
216	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
217	C₂×C₈	—	4	2	no	—
218	C₂×C₈	—	61	1	no	—
219	C₂×C₈	—	61	1	no	—
220	C₂×C₈	—	61	1	no	—
221	C₂×C₈	—	61	1	no	—
222	C₂×C₈	—	122	2	no	—
223	C₂×C₈	—	244	1	no	—
224	C₂×C₈	C₂²×#6923	1	2	yes	—
225	C₂×C₈	C₂²×#6923	1	2	yes	—
226	C₄⋊C₄	—	61	2	no	—
227	C₄⋊C₄	—	61	2	no	—
228	C₄⋊C₄	—	61	2	no	—
229	C₄⋊C₄	—	61	2	no	—
230	C₄⋊C₄	—	61	2	no	—
231	C₄⋊C₄	—	61	1	no	—
232	C₄⋊C₄	—	61	1	no	—
233	C₄²	—	61	2	no	—
234	C₄²	—	61	2	no	—
235	C₄²	—	61	2	no	—
236	C₄²	—	61	2	no	—
237	C₄²	D₆₁⋊C₈	1	1	yes	characteristic
238	C₄²	D₆₁⋊C₈	1	1	yes	characteristic
239	C₄²	C₂²×#6923	1	1	yes	characteristic
240	C₄²	C₂²×#6923	1	1	yes	characteristic
241	C₁₆	—	122	2	no	—
242	C₁₆	—	122	2	no	—
243	C₁₆	—	122	4	no	—
244	C₂³	C₂×D₆₁⋊C₈	1	1	yes	characteristic
245	Q₈	—	61	4	no	—
246	Q₈	—	122	4	no	—
247	Q₈	—	122	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	61	2	no	—
253	C₂×C₄	—	61	2	no	—
254	C₂×C₄	—	61	2	no	—
255	C₂×C₄	—	61	2	no	—
256	C₂×C₄	—	122	4	no	—
257	C₂×C₄	—	122	4	no	—
258	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
259	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
260	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
261	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
262	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
263	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
264	C₈	—	2	2	no	—
265	C₈	—	2	2	no	—
266	C₈	—	61	2	no	—
267	C₈	—	122	1	no	—
268	C₈	—	122	1	no	—
269	C₈	—	122	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	2	1	no	—
272	C₂²	#16840821	1	2	yes	—
273	C₂²	C₂²×D₆₁⋊C₈	1	1	yes	characteristic
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	61	4	no	—
279	C₄	—	122	2	no	—
280	C₄	—	122	2	no	—
281	C₄	—	122	2	no	—
282	C₄	D₆₁⋊M₃₂	1	2	yes	—
283	C₄	#16840830	1	1	yes	characteristic
284	C₄	#16840840	1	1	yes	characteristic
285	C₂	—	2	2	no	—
286	C₂	#21462714	1	1	yes	characteristic
287	C₂	C₂×D₆₁⋊M₃₂	1	1	yes	characteristic
288	C₂	C₂×#16840821	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#6923
commutator, socle	C₂²×C₆₁	C₂³×C₈
Fitting	C₂×C₄⋊C₈×C₆₁	C₄
prime radical	C₂³×C₆₁	C₂²×C₈
cosocle	C₂×#92418	C₂⁴
hypercenter	C₂×C₄⋊C₈	#6923
hypocenter	C₆₁	#33498
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
1. #21462236	C₂	semidirect
2. #21462235	C₂	semidirect
3. #21462233	C₂	semidirect
175. C₆₁	#33498	semidirect

Maximal subgroups

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

subgroup	index
#21462236	2
#21462235	2
#21462233	2
#21462181	2
#21462180	2
#21462177	2
#21462176	2
C₂×#16839886	2
#33498	61