Subgroups — #30593962

Order 15616.

352 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21461752	C₂	1	1	yes	maximal, characteristic
2	C₄×#14519844	C₂	1	1	yes	maximal, characteristic
3	#21445667	C₂	1	1	yes	maximal, characteristic
4	C₂×#14519844	—	2	1	no	—
5	C₂×#14519844	—	2	1	no	—
6	C₂×#14519844	C₄	1	2	yes	—
7	C₂×C₄⋊#92420	C₄	1	2	yes	—
8	C₂×C₄×#92420	—	2	1	no	—
9	#16840628	—	2	1	no	—
10	C₂×#14519706	—	2	1	no	—
11	C₂×C₈×Dic₆₁	—	2	1	no	—
12	C₄²×Dic₆₁	C₂²	1	1	yes	characteristic
13	#16839517	C₄	1	1	yes	characteristic
14	C₆₁×#336	C₄	1	1	yes	characteristic
15	C₂²×#92420	—	2	2	no	—
16	C₂²×#92420	—	2	2	no	—
17	#14519844	—	2	2	no	—
18	#14519844	—	4	1	no	—
19	#14519844	—	4	1	no	—
20	#14519844	—	4	1	no	—
21	#14519844	—	4	1	no	—
22	#14519844	C₈	1	4	yes	complemented
23	C₄⋊#92420	—	4	2	no	—
24	C₄×#92420	—	4	1	no	—
25	C₄×#92420	—	4	1	no	—
26	C₂²×#92418	—	2	1	no	—
27	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
30	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
31	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
32	C₂×C₄×Dic₆₁	Q₈	1	1	yes	characteristic
33	#14519706	—	2	4	no	—
34	C₈×Dic₆₁	—	4	1	no	—
35	C₈×Dic₆₁	—	4	1	no	—
36	C₂²×C₈×C₆₁	—	2	1	no	—
37	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₆₁	C₄²	1	1	yes	characteristic
39	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
40	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
41	C₂²×Dic₆₁	C₂²⋊C₄	1	1	yes	characteristic
42	C₂²×Dic₆₁	C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
45	C₂×#92420	—	2	4	no	—
46	C₂×#92420	—	4	2	no	—
47	C₂×#92420	—	4	2	no	—
48	C₂×#92420	—	4	2	no	—
49	C₂×#92420	—	4	2	no	—
50	C₂×#92420	—	4	2	no	—
51	C₂×C₈×C₆₁	—	2	2	no	—
52	C₂×C₈×C₆₁	—	4	1	no	—
53	C₂×C₈×C₆₁	—	4	1	no	—
54	C₄²×C₆₁	—	4	1	no	—
55	C₄×Dic₆₁	—	2	1	no	—
56	C₄×Dic₆₁	—	2	1	no	—
57	C₄×Dic₆₁	—	2	2	no	—
58	C₄×Dic₆₁	—	2	2	no	—
59	C₄×Dic₆₁	C₂×C₈	1	2	yes	—
60	C₄×Dic₆₁	M₁₆	1	2	yes	—
61	C₄×Dic₆₁	D₈	1	1	yes	characteristic
62	C₄×Dic₆₁	D₈	1	1	yes	characteristic
63	C₄×Dic₆₁	SD₁₆	1	1	yes	characteristic
64	C₄×Dic₆₁	SD₁₆	1	1	yes	characteristic
65	C₄×Dic₆₁	SD₁₆	1	1	yes	characteristic
66	C₄×Dic₆₁	SD₁₆	1	1	yes	characteristic
67	C₄×Dic₆₁	Q₁₆	1	1	yes	characteristic
68	C₄×Dic₆₁	Q₁₆	1	1	yes	characteristic
69	C₂×#92418	—	2	2	no	—
70	C₂×#92418	—	4	1	no	—
71	C₂×#92418	—	4	1	no	—
72	C₂³×C₆₁	#95	1	1	yes	characteristic
73	C₂×C₄×C₆₁	—	2	1	no	—
74	C₂×C₄×C₆₁	—	2	1	no	—
75	C₂×C₄×C₆₁	—	2	1	no	—
76	C₂×C₄×C₆₁	—	2	1	no	—
77	C₂×C₄×C₆₁	—	2	1	no	—
78	C₂×C₄×C₆₁	—	2	1	no	—
79	C₂×C₄×C₆₁	C₄×C₈	1	1	yes	characteristic
80	C₂×C₄×C₆₁	#97	1	1	yes	characteristic
81	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
82	C₂×C₄×C₆₁	#103	1	1	yes	characteristic
83	C₂×C₄×C₆₁	#106	1	1	yes	characteristic
84	C₂×C₄×C₆₁	#107	1	1	yes	characteristic
85	C₂×Dic₆₁	—	2	1	no	—
86	C₂×Dic₆₁	—	2	1	no	—
87	C₂×Dic₆₁	—	2	1	no	—
88	C₂×Dic₆₁	C₂²⋊C₈	1	2	yes	—
89	C₂×Dic₆₁	#100	1	1	yes	characteristic
90	C₂×Dic₆₁	#101	1	1	yes	characteristic
91	C₂×Dic₆₁	#102	1	1	yes	characteristic
92	C₂×Dic₆₁	#102	1	1	yes	characteristic
93	C₂×Dic₆₁	#102	1	1	yes	characteristic
94	C₂×Dic₆₁	#103	1	1	yes	characteristic
95	C₂×Dic₆₁	#103	1	1	yes	characteristic
96	C₂×Dic₆₁	#103	1	1	yes	characteristic
97	C₂×Dic₆₁	#104	1	2	yes	—
98	C₂×Dic₆₁	C₄⋊C₈	1	2	yes	—
99	C₂×Dic₆₁	#106	1	1	yes	characteristic
100	C₂×Dic₆₁	#107	1	1	yes	characteristic
101	C₂×Dic₆₁	#108	1	2	yes	—
102	#92420	—	4	4	no	—
103	#92420	—	8	2	no	—
104	C₈×C₆₁	—	4	2	no	—
105	#92418	—	4	2	no	—
106	#7031	—	61	1	no	maximal, Sylow
107	C₂²×C₆₁	#336	1	1	yes	characteristic
108	C₂²×C₆₁	#339	1	1	yes	characteristic
109	C₂²×C₆₁	#340	1	1	yes	characteristic
110	C₂²×C₆₁	#340	1	1	yes	characteristic
111	C₂²×C₆₁	#341	1	1	yes	characteristic
112	C₂²×C₆₁	#343	1	1	yes	characteristic
113	C₂²×C₆₁	#344	1	1	yes	characteristic
114	C₄×C₆₁	—	2	1	no	—
115	C₄×C₆₁	—	2	1	no	—
116	C₄×C₆₁	—	2	1	no	—
117	C₄×C₆₁	—	2	1	no	—
118	C₄×C₆₁	—	2	1	no	—
119	C₄×C₆₁	—	2	1	no	—
120	Dic₆₁	—	4	1	no	—
121	Dic₆₁	D₄⋊C₈	1	2	yes	—
122	Dic₆₁	Q₈⋊C₈	1	2	yes	—
123	Dic₆₁	#331	1	1	yes	characteristic
124	Dic₆₁	#332	1	1	yes	characteristic
125	Dic₆₁	#332	1	1	yes	characteristic
126	Dic₆₁	#333	1	1	yes	characteristic
127	Dic₆₁	#334	1	2	yes	—
128	Dic₆₁	#335	1	2	yes	—
129	#1842	—	61	1	no	—
130	C₄×C₄⋊C₈	—	61	1	no	—
131	C₄×C₄⋊C₈	—	61	1	no	—
132	C₂×C₆₁	#1277	1	1	yes	characteristic
133	C₂×C₆₁	#1279	1	1	yes	characteristic
134	C₂×C₆₁	#1283	1	1	yes	characteristic
135	C₂×C₆₁	#1296	1	1	yes	characteristic
136	C₂×C₆₁	#1299	1	1	yes	characteristic
137	C₂×C₆₁	#1301	1	1	yes	characteristic
138	C₂×C₆₁	#1303	1	1	yes	characteristic
139	C₂×C₄⋊C₈	—	61	2	no	—
140	C₂×C₄⋊C₈	—	61	2	no	—
141	C₂×C₄⋊C₈	—	122	1	no	—
142	C₂×C₄⋊C₈	—	122	1	no	—
143	C₂×C₄⋊C₈	—	122	1	no	—
144	C₂×C₄×C₈	—	122	1	no	—
145	C₂×C₄×C₈	—	122	1	no	—
146	C₄³	—	61	1	no	—
147	#336	—	61	1	no	—
148	#336	—	122	1	no	—
149	#336	#6923	1	1	yes	characteristic
150	C₆₁	#7031	1	1	yes	characteristic, complemented, Sylow
151	C₂²×C₈	—	2	1	no	—
152	C₂²×C₈	—	122	2	no	—
153	C₂²×C₈	—	122	1	no	—
154	C₂²×C₈	—	122	2	no	—
155	C₂×C₄²	—	61	1	no	—
156	C₂×C₄²	—	61	1	no	—
157	C₂×C₄²	—	61	1	no	—
158	C₂×C₄²	—	61	1	no	—
159	C₂×C₄²	—	61	1	no	—
160	C₂×C₄²	—	61	1	no	—
161	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
162	C₄⋊C₈	—	61	4	no	—
163	C₄⋊C₈	—	122	2	no	—
164	C₄⋊C₈	—	122	4	no	—
165	C₄⋊C₈	—	244	1	no	—
166	C₄⋊C₈	—	244	1	no	—
167	C₄⋊C₈	—	244	1	no	—
168	C₄⋊C₈	—	244	1	no	—
169	C₄⋊C₈	—	244	2	no	—
170	C₄×C₈	—	244	1	no	—
171	C₄×C₈	—	244	1	no	—
172	C₄×C₈	—	244	1	no	—
173	C₄×C₈	—	244	1	no	—
174	C₂²×C₄	—	61	1	no	—
175	C₂²×C₄	—	61	1	no	—
176	C₂²×C₄	—	61	1	no	—
177	C₂²×C₄	—	61	1	no	—
178	C₂²×C₄	C₄×#6923	1	1	yes	characteristic
179	C₂²×C₄	C₄⋊#6923	1	1	yes	characteristic
180	C₂²×C₄	C₂²⋊#6923	1	1	yes	characteristic
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	4	1	no	—
183	C₂×C₈	—	4	1	no	—
184	C₂×C₈	—	122	2	no	—
185	C₂×C₈	—	122	4	no	—
186	C₂×C₈	—	244	2	no	—
187	C₂×C₈	—	244	2	no	—
188	C₂×C₈	—	244	2	no	—
189	C₂×C₈	—	244	1	no	—
190	C₂×C₈	—	244	2	no	—
191	C₂×C₈	—	244	2	no	—
192	C₂×C₈	—	244	1	no	—
193	C₄²	—	4	1	no	—
194	C₄²	—	61	2	no	—
195	C₄²	—	61	2	no	—
196	C₄²	—	61	1	no	—
197	C₄²	—	61	1	no	—
198	C₄²	—	61	1	no	—
199	C₄²	—	61	1	no	—
200	C₄²	—	61	1	no	—
201	C₄²	—	61	1	no	—
202	C₄²	—	61	1	no	—
203	C₄²	—	61	1	no	—
204	C₄²	—	122	2	no	—
205	C₄²	—	122	1	no	—
206	C₄²	—	122	2	no	—
207	C₄²	—	122	1	no	—
208	C₂³	(C₂×#6923)⋊C₄	1	1	yes	characteristic
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
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	—	61	2	no	—
216	C₂×C₄	—	61	2	no	—
217	C₂×C₄	—	61	2	no	—
218	C₂×C₄	—	61	1	no	—
219	C₂×C₄	—	61	1	no	—
220	C₂×C₄	—	61	1	no	—
221	C₂×C₄	—	61	1	no	—
222	C₂×C₄	—	61	1	no	—
223	C₂×C₄	—	61	1	no	—
224	C₂×C₄	—	61	1	no	—
225	C₂×C₄	—	61	1	no	—
226	C₂×C₄	—	61	2	no	—
227	C₂×C₄	—	61	1	no	—
228	C₂×C₄	—	61	1	no	—
229	C₂×C₄	—	122	1	no	—
230	C₂×C₄	—	122	1	no	—
231	C₂×C₄	—	122	1	no	—
232	C₂×C₄	C₈×#6923	1	1	yes	characteristic
233	C₂×C₄	#14519831	1	1	yes	characteristic
234	C₂×C₄	#14519839	1	1	yes	characteristic
235	C₂×C₄	#14519840	1	1	yes	characteristic
236	C₂×C₄	#14519845	1	1	yes	characteristic
237	C₂×C₄	#14519846	1	1	yes	characteristic
238	C₈	—	4	2	no	—
239	C₈	—	488	2	no	—
240	C₈	—	244	2	no	—
241	C₈	—	244	4	no	—
242	C₂²	(C₂×#6923)⋊C₈	1	1	yes	characteristic
243	C₂²	#16840633	1	1	yes	characteristic
244	C₂²	#16840635	1	1	yes	characteristic
245	C₂²	#16840636	1	1	yes	characteristic
246	C₂²	#16840637	1	1	yes	characteristic
247	C₂²	#16840641	1	1	yes	characteristic
248	C₂²	#16840643	1	1	yes	characteristic
249	C₄	—	2	1	no	—
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	2	1	no	—
255	C₄	—	61	2	no	—
256	C₄	—	61	2	no	—
257	C₄	—	61	2	no	—
258	C₄	—	61	2	no	—
259	C₄	—	61	1	no	—
260	C₄	—	61	1	no	—
261	C₄	—	61	1	no	—
262	C₄	—	61	1	no	—
263	C₄	—	244	1	no	—
264	C₂	#21460648	1	1	yes	characteristic
265	C₂	#21460654	1	1	yes	characteristic
266	C₂	#21460663	1	1	yes	characteristic
267	C₂	#21460698	1	1	yes	characteristic
268	C₂	#21460704	1	1	yes	characteristic
269	C₂	#21460710	1	1	yes	characteristic
270	C₂	#21460717	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	(C₂×#6923)⋊C₄
commutator	C₂×C₄×C₆₁	C₄×C₈
Frattini	C₂×C₄²	C₂×#6923
Fitting	C₆₁×#336	C₄
socle, prime radical	C₂³×C₆₁	#95
cosocle	C₄²×Dic₆₁	C₂²
hypercenter	#336	#6923
hypocenter	C₆₁	#7031
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

Complements

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

N	H ≅ G/N	type
22. #14519844	C₈	semidirect
150. C₆₁	#7031	semidirect

Maximal subgroups

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

subgroup	index
#21461752	2
C₄×#14519844	2
#21445667	2
#7031	61