Subgroups — #30593923

Order 15616.

474 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	C₄×C₂²⋊#92420	C₂	1	2	yes	maximal
2	C₂×#16839516	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂²⋊#92420	C₄	1	4	yes	—
4	C₂×C₄×#92420	—	2	2	no	—
5	#16840628	—	2	2	no	—
6	C₂³×#92418	C₄	1	1	yes	characteristic
7	C₂²×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
8	#16839516	—	2	2	no	—
9	C₂³×C₈×C₆₁	C₄	1	1	yes	characteristic
10	C₂²×#92420	—	2	4	no	—
11	C₂²⋊#92420	—	2	4	no	—
12	C₂²⋊#92420	C₈	1	8	yes	complemented
13	C₄×#92420	—	4	2	no	—
14	C₄×#92420	—	4	2	no	—
15	C₂³×Dic₆₁	C₂×C₄	1	2	yes	—
16	C₂²×#92418	—	2	1	no	—
17	C₂²×#92418	—	2	1	no	—
18	C₂²×#92418	—	4	1	no	—
19	C₂²×#92418	—	4	1	no	—
20	C₂²×#92418	C₈	1	2	yes	complemented
21	C₂×C₄×Dic₆₁	—	4	1	no	—
22	C₂×C₄×Dic₆₁	—	4	1	no	—
23	C₂×C₄×Dic₆₁	D₄	1	2	yes	—
24	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
25	C₂×C₄×Dic₆₁	Q₈	1	1	yes	characteristic
26	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₆₁	—	2	1	no	—
28	C₂²×C₈×C₆₁	—	2	1	no	—
29	C₂²×C₈×C₆₁	—	4	1	no	—
30	C₂²×C₈×C₆₁	—	4	1	no	—
31	C₂²×C₈×C₆₁	C₈	1	2	yes	complemented
32	C₂⁴×C₆₁	C₄²	1	1	yes	characteristic
33	C₂²×C₄×C₆₁	—	2	1	no	—
34	C₂²×C₄×C₆₁	—	2	1	no	—
35	C₂²×C₄×C₆₁	—	2	1	no	—
36	C₂²×C₄×C₆₁	—	2	1	no	—
37	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
38	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
40	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
41	C₂²×C₄×C₆₁	M₁₆	1	1	yes	characteristic
42	C₂²×C₄×C₆₁	M₁₆	1	1	yes	characteristic
43	C₂²×Dic₆₁	—	4	2	no	—
44	C₂²×Dic₆₁	—	4	2	no	—
45	C₂²×Dic₆₁	C₂²⋊C₄	1	2	yes	—
46	C₂²×Dic₆₁	C₄⋊C₄	1	2	yes	—
47	C₂²×Dic₆₁	C₂×C₈	1	4	yes	—
48	C₂²×Dic₆₁	M₁₆	1	4	yes	—
49	C₂×#92420	—	2	8	no	—
50	C₂×#92420	—	4	4	no	—
51	C₂×#92420	—	4	4	no	—
52	C₂×C₈×C₆₁	—	2	2	no	—
53	C₂×C₈×C₆₁	—	4	2	no	—
54	C₂×C₈×C₆₁	—	4	1	no	—
55	C₂×C₈×C₆₁	—	4	1	no	—
56	C₂×C₈×C₆₁	—	4	2	no	—
57	C₄×Dic₆₁	—	2	2	no	—
58	C₄×Dic₆₁	—	2	2	no	—
59	C₄×Dic₆₁	—	4	1	no	—
60	C₄×Dic₆₁	—	4	1	no	—
61	C₂×#92418	—	2	2	no	—
62	C₂×#92418	—	4	2	no	—
63	C₂×#92418	—	4	1	no	—
64	C₂×#92418	—	4	1	no	—
65	C₂×#92418	—	4	2	no	—
66	C₂³×C₆₁	—	2	1	no	—
67	C₂³×C₆₁	—	2	1	no	—
68	C₂³×C₆₁	—	2	2	no	—
69	C₂³×C₆₁	#95	1	1	yes	characteristic
70	C₂³×C₆₁	C₄×C₈	1	2	yes	—
71	C₂³×C₆₁	C₄×C₈	1	1	yes	characteristic
72	C₂³×C₆₁	#97	1	2	yes	—
73	C₂³×C₆₁	#97	1	1	yes	characteristic
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	—	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	C₂²⋊C₈	1	1	yes	characteristic
85	C₂×C₄×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
86	C₂×C₄×C₆₁	#99	1	1	yes	characteristic
87	C₂×C₄×C₆₁	#99	1	1	yes	characteristic
88	C₂×C₄×C₆₁	#100	1	1	yes	characteristic
89	C₂×C₄×C₆₁	#101	1	1	yes	characteristic
90	C₂×C₄×C₆₁	C₄⋊C₈	1	1	yes	characteristic
91	C₂×C₄×C₆₁	C₄⋊C₈	1	1	yes	characteristic
92	C₂×Dic₆₁	—	2	2	no	—
93	C₂×Dic₆₁	—	2	2	no	—
94	C₂×Dic₆₁	—	2	2	no	—
95	C₂×Dic₆₁	—	2	2	no	—
96	C₂×Dic₆₁	—	4	4	no	—
97	C₂×Dic₆₁	—	4	4	no	—
98	C₂×Dic₆₁	C₂²⋊C₈	1	4	yes	—
99	C₂×Dic₆₁	C₄⋊C₈	1	4	yes	—
100	#92420	—	4	8	no	—
101	C₈×C₆₁	—	4	2	no	—
102	#92418	—	4	2	no	—
103	#7016	—	61	1	no	maximal, Sylow
104	C₂²×C₆₁	—	2	1	no	—
105	C₂²×C₆₁	—	2	1	no	—
106	C₂²×C₆₁	—	2	1	no	—
107	C₂²×C₆₁	—	2	1	no	—
108	C₂²×C₆₁	—	2	2	no	—
109	C₂²×C₆₁	—	2	2	no	—
110	C₂²×C₆₁	—	2	2	no	—
111	C₂²×C₆₁	—	2	2	no	—
112	C₂²×C₆₁	C₈²	1	1	yes	characteristic
113	C₂²×C₆₁	#322	1	1	yes	characteristic
114	C₂²×C₆₁	#322	1	2	yes	—
115	C₂²×C₆₁	#336	1	2	yes	—
116	C₂²×C₆₁	#336	1	1	yes	characteristic
117	C₂²×C₆₁	#342	1	1	yes	characteristic
118	C₂²×C₆₁	#343	1	1	yes	characteristic
119	C₂²×C₆₁	#344	1	2	yes	—
120	C₄×C₆₁	—	2	1	no	—
121	C₄×C₆₁	—	2	1	no	—
122	C₄×C₆₁	#323	1	1	yes	characteristic
123	C₄×C₆₁	#323	1	1	yes	characteristic
124	C₄×C₆₁	#324	1	1	yes	characteristic
125	C₄×C₆₁	#324	1	1	yes	characteristic
126	Dic₆₁	—	2	4	no	—
127	Dic₆₁	—	2	4	no	—
128	C₄×C₂²⋊C₈	—	61	2	no	—
129	C₂×#336	—	61	1	no	—
130	C₂×C₆₁	—	2	1	no	—
131	C₂×C₆₁	—	2	1	no	—
132	C₂×C₆₁	—	2	2	no	—
133	C₂×C₆₁	#1273	1	1	yes	characteristic
134	C₂×C₆₁	#1280	1	1	yes	characteristic
135	C₂×C₆₁	#1281	1	1	yes	characteristic
136	C₂×C₆₁	#1282	1	2	yes	—
137	C₂×C₆₁	#1283	1	2	yes	—
138	C₂³×C₈	—	61	1	no	—
139	C₂³×C₈	#6923	1	1	yes	characteristic
140	C₂²×C₄²	—	61	1	no	—
141	C₂×C₂²⋊C₈	—	61	4	no	—
142	C₂×C₄×C₈	—	122	2	no	—
143	#336	—	122	2	no	—
144	#336	—	122	2	no	—
145	C₆₁	#7016	1	1	yes	characteristic, complemented, Sylow
146	C₂³×C₄	—	61	2	no	—
147	C₂³×C₄	C₂×#6923	1	1	yes	characteristic
148	C₂²×C₈	—	2	1	no	—
149	C₂²×C₈	—	2	1	no	—
150	C₂²×C₈	—	4	1	no	—
151	C₂²×C₈	—	4	1	no	—
152	C₂²×C₈	—	61	2	no	—
153	C₂²×C₈	—	122	4	no	—
154	C₂²×C₈	—	122	1	no	—
155	C₂²×C₈	—	122	1	no	—
156	C₂²×C₈	—	244	1	no	—
157	C₂²×C₈	—	244	1	no	—
158	C₂²×C₈	#92420	1	2	yes	complemented
159	C₂×C₄²	—	61	2	no	—
160	C₂×C₄²	—	61	1	no	—
161	C₂×C₄²	—	61	1	no	—
162	C₂×C₄²	—	244	1	no	—
163	C₂×C₄²	—	244	1	no	—
164	C₂²⋊C₈	—	61	8	no	—
165	C₂²⋊C₈	—	122	4	no	—
166	C₄×C₈	—	244	2	no	—
167	C₄×C₈	—	244	2	no	—
168	C₂⁴	C₄×#6923	1	1	yes	characteristic
169	C₂²×C₄	—	2	1	no	—
170	C₂²×C₄	—	2	1	no	—
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	—	61	2	no	—
174	C₂²×C₄	—	61	2	no	—
175	C₂²×C₄	—	61	4	no	—
176	C₂²×C₄	—	61	4	no	—
177	C₂²×C₄	—	244	2	no	—
178	C₂²×C₄	—	244	2	no	—
179	C₂²×C₄	D₆₁⋊C₈	1	1	yes	characteristic
180	C₂²×C₄	#11758213	1	1	yes	characteristic
181	C₂²×C₄	C₄⋊#6923	1	1	yes	characteristic
182	C₂²×C₄	C₂×#92420	1	1	yes	characteristic
183	C₂²×C₄	#11758217	1	1	yes	characteristic
184	C₂²×C₄	C₂²⋊#6923	1	1	yes	characteristic
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	4	2	no	—
187	C₂×C₈	—	4	1	no	—
188	C₂×C₈	—	4	1	no	—
189	C₂×C₈	—	4	2	no	—
190	C₂×C₈	—	122	8	no	—
191	C₂×C₈	—	122	2	no	—
192	C₂×C₈	—	244	2	no	—
193	C₂×C₈	—	244	4	no	—
194	C₂×C₈	—	244	4	no	—
195	C₂×C₈	—	244	1	no	—
196	C₂×C₈	—	244	1	no	—
197	C₂×C₈	—	244	2	no	—
198	C₄²	—	122	2	no	—
199	C₄²	—	122	2	no	—
200	C₄²	—	244	1	no	—
201	C₄²	—	244	1	no	—
202	C₂³	—	2	1	no	—
203	C₂³	—	2	1	no	—
204	C₂³	—	2	2	no	—
205	C₂³	(C₂×#6923)⋊C₄	1	1	yes	characteristic
206	C₂³	C₈×#6923	1	2	yes	—
207	C₂³	C₄×#92420	1	1	yes	characteristic
208	C₂³	#14519831	1	2	yes	—
209	C₂³	#14519832	1	1	yes	characteristic
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	—	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	—	61	4	no	—
221	C₂×C₄	—	61	4	no	—
222	C₂×C₄	—	122	2	no	—
223	C₂×C₄	—	122	2	no	—
224	C₂×C₄	—	122	2	no	—
225	C₂×C₄	—	122	2	no	—
226	C₂×C₄	—	244	4	no	—
227	C₂×C₄	—	244	4	no	—
228	C₂×C₄	C₂²⋊#92420	1	1	yes	characteristic
229	C₂×C₄	#14519834	1	1	yes	characteristic
230	C₂×C₄	#14519835	1	1	yes	characteristic
231	C₂×C₄	#14519836	1	1	yes	characteristic
232	C₂×C₄	#14519837	1	1	yes	characteristic
233	C₂×C₄	#14519838	1	1	yes	characteristic
234	C₂×C₄	C₄⋊#92420	1	1	yes	characteristic
235	C₂×C₄	#14519844	1	1	yes	characteristic
236	C₈	—	4	2	no	—
237	C₈	—	244	8	no	—
238	C₈	—	244	2	no	—
239	C₂²	—	2	1	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	2	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	2	2	no	—
247	C₂²	C₈×#92420	1	1	yes	characteristic
248	C₂²	#16840603	1	1	yes	characteristic
249	C₂²	#92420⋊C₈	1	2	yes	—
250	C₂²	(C₂×#6923)⋊C₈	1	2	yes	—
251	C₂²	#16840628	1	1	yes	characteristic
252	C₂²	#16840640	1	1	yes	characteristic
253	C₂²	#16840642	1	1	yes	characteristic
254	C₂²	#16840644	1	2	yes	—
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	122	4	no	—
258	C₄	—	122	4	no	—
259	C₄	#16840605	1	1	yes	characteristic
260	C₄	#16840606	1	1	yes	characteristic
261	C₄	#16840607	1	1	yes	characteristic
262	C₄	#16840608	1	1	yes	characteristic
263	C₂	—	2	1	no	—
264	C₂	—	2	1	no	—
265	C₂	—	2	2	no	—
266	C₂	#21460641	1	1	yes	characteristic
267	C₂	#21460657	1	1	yes	characteristic
268	C₂	#21460659	1	1	yes	characteristic
269	C₂	#21460661	1	2	yes	—
270	C₂	#21460664	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²⋊#6923
commutator	C₂²×C₆₁	C₈²
Frattini	C₂³×C₄	C₂×#6923
Fitting	C₂³×C₈×C₆₁	C₄
socle	C₂³×C₆₁	#95
prime radical	C₂⁴×C₆₁	C₄²
cosocle	C₂²×C₄×Dic₆₁	C₂²
hypercenter	C₂³×C₈	#6923
hypocenter	C₆₁	#7016
LCS	G ▷ C₂²×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
12. C₂²⋊#92420	C₈	semidirect
20. C₂²×#92418	C₈	semidirect
31. C₂²×C₈×C₆₁	C₈	semidirect
145. C₆₁	#7016	semidirect
158. C₂²×C₈	#92420	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₂²⋊#92420	2
C₂×#16839516	2
#7016	61