Subgroups — #30596090

Order 15616.

718 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16840778	C₂	1	2	yes	maximal
2	#21460649	C₂	1	4	yes	maximal, complemented
3	C₂²×C₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic
4	#16840778	—	2	4	no	—
5	#16840778	C₄	1	8	yes	—
6	C₂×C₄⋊#92420	C₂²	1	4	yes	—
7	C₂×C₂²⋊#92420	—	2	2	no	—
8	C₂×C₄×#92420	—	2	2	no	—
9	C₂³×#92418	C₄	1	1	yes	characteristic
10	C₂²×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₆₁	—	2	4	no	—
12	C₂×C₈×Dic₆₁	C₄	1	2	yes	—
13	C₂×C₈×Dic₆₁	C₂²	1	2	yes	—
14	C₂³×C₈×C₆₁	C₄	1	1	yes	characteristic
15	C₂²×#92420	—	2	4	no	—
16	C₄⋊#92420	—	2	4	no	—
17	C₄⋊#92420	C₂×C₄	1	8	yes	—
18	C₂²⋊#92420	—	2	8	no	—
19	C₄×#92420	—	2	8	no	—
20	C₂³×Dic₆₁	D₄	1	1	yes	characteristic
21	C₂³×Dic₆₁	Q₈	1	1	yes	characteristic
22	C₂²×#92418	—	2	2	no	—
23	C₂²×#92418	—	2	2	no	—
24	C₂²×#92418	C₈	1	4	yes	complemented
25	C₂²×#92418	C₂×C₄	1	2	yes	—
26	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×Dic₆₁	C₂×C₄	1	4	yes	—
28	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
29	C₂×C₄×Dic₆₁	D₄	1	4	yes	—
30	C₂×C₄×Dic₆₁	Q₈	1	1	yes	characteristic
31	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
32	C₈×Dic₆₁	—	2	4	no	—
33	C₈×Dic₆₁	—	2	4	no	—
34	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₆₁	—	2	2	no	—
36	C₂²×C₈×C₆₁	—	2	2	no	—
37	C₂²×C₈×C₆₁	C₈	1	4	yes	complemented
38	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
39	C₂⁴×C₆₁	C₄⋊C₄	1	1	yes	characteristic
40	C₂²×C₄×C₆₁	C₄²	1	2	yes	—
41	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
42	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
44	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
45	C₂²×C₄×C₆₁	M₁₆	1	2	yes	—
46	C₂²×C₄×C₆₁	M₁₆	1	2	yes	—
47	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
48	C₂²×Dic₆₁	—	2	4	no	—
49	C₂²×Dic₆₁	—	2	4	no	—
50	C₂²×Dic₆₁	C₂²⋊C₄	1	4	yes	—
51	C₂²×Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×Dic₆₁	C₄⋊C₄	1	4	yes	—
53	C₂²×Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×Dic₆₁	C₂×D₄	1	1	yes	characteristic
55	C₂²×Dic₆₁	C₂×Q₈	1	1	yes	characteristic
56	C₂×#92420	—	2	8	no	—
57	C₂×#92420	—	4	8	no	—
58	C₂×C₈×C₆₁	—	2	8	no	—
59	C₂×C₈×C₆₁	—	2	2	no	—
60	C₂×C₈×C₆₁	—	2	2	no	—
61	C₂×C₈×C₆₁	C₂×C₈	1	4	yes	—
62	C₄×Dic₆₁	C₄²	1	2	yes	—
63	C₄×Dic₆₁	C₂²⋊C₄	1	2	yes	—
64	C₄×Dic₆₁	C₂²⋊C₄	1	4	yes	—
65	C₄×Dic₆₁	C₄⋊C₄	1	4	yes	—
66	C₄×Dic₆₁	C₂²×C₄	1	2	yes	—
67	C₄×Dic₆₁	C₂×D₄	1	2	yes	—
68	C₂×#92418	—	2	8	no	—
69	C₂×#92418	—	2	2	no	—
70	C₂×#92418	—	2	2	no	—
71	C₂×#92418	C₂×C₈	1	4	yes	—
72	C₂³×C₆₁	—	2	2	no	—
73	C₂³×C₆₁	—	2	2	no	—
74	C₂³×C₆₁	#95	1	2	yes	—
75	C₂³×C₆₁	C₄⋊C₈	1	2	yes	—
76	C₂³×C₆₁	C₄⋊C₈	1	2	yes	—
77	C₂³×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₆₁	#95	1	2	yes	—
79	C₂×C₄×C₆₁	C₄×C₈	1	4	yes	—
80	C₂×C₄×C₆₁	#97	1	4	yes	—
81	C₂×C₄×C₆₁	C₂²⋊C₈	1	4	yes	—
82	C₂×C₄×C₆₁	C₂²⋊C₈	1	4	yes	—
83	C₂×C₄×C₆₁	C₄⋊C₈	1	2	yes	—
84	C₂×C₄×C₆₁	C₄⋊C₈	1	2	yes	—
85	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
86	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
88	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
89	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
90	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
91	C₂×Dic₆₁	—	2	2	no	—
92	C₂×Dic₆₁	—	2	2	no	—
93	C₂×Dic₆₁	—	2	8	no	—
94	C₂×Dic₆₁	—	2	8	no	—
95	C₂×Dic₆₁	#95	1	2	yes	—
96	C₂×Dic₆₁	#95	1	2	yes	—
97	C₂×Dic₆₁	#108	1	4	yes	—
98	C₂×Dic₆₁	C₂×C₂²⋊C₄	1	2	yes	—
99	C₂×Dic₆₁	C₂×C₄⋊C₄	1	2	yes	—
100	C₂×Dic₆₁	#135	1	4	yes	—
101	#92420	—	4	8	no	—
102	C₈×C₆₁	—	2	4	no	—
103	#92418	—	2	4	no	—
104	#7872	—	61	1	no	maximal, Sylow
105	C₂²×C₆₁	—	2	4	no	—
106	C₂²×C₆₁	—	2	4	no	—
107	C₂²×C₆₁	—	2	2	no	—
108	C₂²×C₆₁	—	2	2	no	—
109	C₂²×C₆₁	#336	1	4	yes	—
110	C₂²×C₆₁	C₂×#95	1	1	yes	characteristic
111	C₂²×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
112	C₂²×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
113	C₂²×C₆₁	#427	1	2	yes	—
114	C₂²×C₆₁	C₂×#108	1	2	yes	—
115	C₄×C₆₁	#336	1	4	yes	—
116	C₄×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
117	C₄×C₆₁	C₂×#97	1	1	yes	characteristic
118	C₄×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
119	C₄×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
120	Dic₆₁	—	2	2	no	—
121	Dic₆₁	—	2	2	no	—
122	Dic₆₁	#341	1	4	yes	—
123	Dic₆₁	#416	1	4	yes	—
124	C₂×#431	—	61	2	no	—
125	C₂²×C₄×C₈	—	61	1	no	—
126	#1277	—	61	4	no	—
127	C₂×C₆₁	—	2	4	no	—
128	C₂×C₆₁	#1564	1	2	yes	—
129	C₂×C₆₁	#1564	1	2	yes	—
130	C₂×C₆₁	C₂×#336	1	1	yes	characteristic
131	C₂×C₆₁	#1735	1	1	yes	characteristic
132	C₂×C₆₁	C₂×#341	1	1	yes	characteristic
133	C₂³×C₈	—	61	1	no	—
134	C₂³×C₈	#6923	1	1	yes	characteristic
135	C₂²×C₄²	—	61	1	no	—
136	#431	—	61	8	no	—
137	#431	—	122	4	no	—
138	C₂×C₄⋊C₈	—	61	4	no	—
139	C₂×C₂²⋊C₈	—	122	2	no	—
140	C₂×C₄×C₈	—	61	2	no	—
141	C₂×C₄×C₈	—	61	2	no	—
142	C₂×C₄×C₈	—	122	4	no	—
143	C₂×C₄×C₈	—	122	2	no	—
144	C₆₁	#7872	1	1	yes	characteristic, complemented, Sylow
145	C₂³×C₄	—	61	1	no	—
146	C₂³×C₄	—	61	1	no	—
147	C₂³×C₄	C₂×#6923	1	1	yes	characteristic
148	C₂²×C₈	—	2	2	no	—
149	C₂²×C₈	—	2	2	no	—
150	C₂²×C₈	—	61	4	no	—
151	C₂²×C₈	—	61	2	no	—
152	C₂²×C₈	—	122	2	no	—
153	C₂²×C₈	—	122	2	no	—
154	C₂²×C₈	—	122	4	no	—
155	C₂²×C₈	#92420	1	4	yes	complemented
156	C₂²×C₈	C₂×#6923	1	2	yes	—
157	C₂×C₄²	—	61	4	no	—
158	C₂×C₄²	—	61	1	no	—
159	C₂×C₄²	—	61	1	no	—
160	C₂×C₄²	—	61	4	no	—
161	C₂×C₄²	—	61	1	no	—
162	C₂×C₄²	—	61	1	no	—
163	C₄⋊C₈	—	61	8	no	—
164	C₄⋊C₈	—	122	4	no	—
165	C₂²⋊C₈	—	122	8	no	—
166	C₄×C₈	—	122	4	no	—
167	C₄×C₈	—	122	4	no	—
168	C₄×C₈	—	122	8	no	—
169	C₂⁴	C₄⋊#6923	1	1	yes	characteristic
170	C₂²×C₄	—	61	1	no	—
171	C₂²×C₄	—	61	1	no	—
172	C₂²×C₄	—	61	4	no	—
173	C₂²×C₄	—	61	4	no	—
174	C₂²×C₄	—	61	1	no	—
175	C₂²×C₄	—	61	1	no	—
176	C₂²×C₄	—	122	4	no	—
177	C₂²×C₄	—	122	4	no	—
178	C₂²×C₄	D₆₁⋊C₈	1	2	yes	—
179	C₂²×C₄	#11758213	1	2	yes	—
180	C₂²×C₄	C₄×#6923	1	2	yes	—
181	C₂²×C₄	C₄⋊#6923	1	1	yes	characteristic
182	C₂²×C₄	C₂×#92420	1	2	yes	—
183	C₂²×C₄	#11758217	1	2	yes	—
184	C₂²×C₄	C₂²⋊#6923	1	2	yes	—
185	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
186	C₂×C₈	—	2	8	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	61	4	no	—
190	C₂×C₈	—	122	8	no	—
191	C₂×C₈	—	122	2	no	—
192	C₂×C₈	—	122	2	no	—
193	C₂×C₈	—	122	8	no	—
194	C₂×C₈	—	244	8	no	—
195	C₂×C₈	C₂×#92420	1	4	yes	—
196	C₄²	—	61	4	no	—
197	C₄²	—	61	4	no	—
198	C₄²	—	61	2	no	—
199	C₄²	—	61	2	no	—
200	C₄²	—	61	2	no	—
201	C₄²	—	61	2	no	—
202	C₂³	—	2	2	no	—
203	C₂³	—	2	2	no	—
204	C₂³	(C₂×#6923)⋊C₄	1	2	yes	—
205	C₂³	C₄⋊#92420	1	2	yes	—
206	C₂³	#14519844	1	2	yes	—
207	C₂³	C₂×C₄⋊#6923	1	1	yes	characteristic
208	C₂×C₄	—	61	4	no	—
209	C₂×C₄	—	61	4	no	—
210	C₂×C₄	—	61	2	no	—
211	C₂×C₄	—	61	2	no	—
212	C₂×C₄	—	61	2	no	—
213	C₂×C₄	—	61	2	no	—
214	C₂×C₄	—	122	8	no	—
215	C₂×C₄	—	122	8	no	—
216	C₂×C₄	—	122	2	no	—
217	C₂×C₄	—	122	2	no	—
218	C₂×C₄	(C₂×#6923)⋊C₄	1	2	yes	—
219	C₂×C₄	C₄×#92420	1	4	yes	—
220	C₂×C₄	#14519832	1	4	yes	—
221	C₂×C₄	C₂²⋊#92420	1	4	yes	—
222	C₂×C₄	#14519834	1	4	yes	—
223	C₂×C₄	C₄⋊#92420	1	2	yes	—
224	C₂×C₄	#14519844	1	2	yes	—
225	C₂×C₄	C₂×C₄×#6923	1	1	yes	characteristic
226	C₂×C₄	C₂×C₂²⋊#6923	1	1	yes	characteristic
227	C₂×C₄	C₂²×#92420	1	1	yes	characteristic
228	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
229	C₂×C₄	C₂×#11758217	1	1	yes	characteristic
230	C₂×C₄	C₂×#11758213	1	1	yes	characteristic
231	C₈	—	2	4	no	—
232	C₈	—	122	4	no	—
233	C₈	—	244	8	no	—
234	C₂²	—	2	2	no	—
235	C₂²	—	2	2	no	—
236	C₂²	—	2	4	no	—
237	C₂²	—	2	4	no	—
238	C₂²	#16840628	1	4	yes	—
239	C₂²	C₂×(C₂×#6923)⋊C₄	1	1	yes	characteristic
240	C₂²	C₂×C₄⋊#92420	1	1	yes	characteristic
241	C₂²	C₂×#14519844	1	1	yes	characteristic
242	C₂²	#16840772	1	2	yes	—
243	C₂²	D₆₁⋊#108	1	2	yes	—
244	C₄	—	61	4	no	—
245	C₄	—	61	4	no	—
246	C₄	—	122	2	no	—
247	C₄	—	122	2	no	—
248	C₄	#16840628	1	4	yes	—
249	C₄	C₂×C₄×#92420	1	1	yes	characteristic
250	C₄	C₂×#14519832	1	1	yes	characteristic
251	C₄	C₂×C₂²⋊#92420	1	1	yes	characteristic
252	C₄	C₂×#14519834	1	1	yes	characteristic
253	C₂	—	2	4	no	—
254	C₂	#21461366	1	2	yes	—
255	C₂	#21461368	1	2	yes	—
256	C₂	C₂×#16840628	1	1	yes	characteristic
257	C₂	#21461505	1	1	yes	characteristic
258	C₂	#21461511	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₄⋊#6923
commutator	C₄×C₆₁	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#6923
Fitting	C₂³×C₈×C₆₁	C₄
socle	C₂³×C₆₁	C₂×C₄⋊C₄
prime radical	C₂⁴×C₆₁	C₄⋊C₄
cosocle	C₂×C₄×Dic₆₁	C₂³
hypercenter	C₂³×C₈	#6923
hypocenter	C₆₁	#7872
LCS	G ▷ C₄×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
2. #21460649	C₂	semidirect
24. C₂²×#92418	C₈	semidirect
37. C₂²×C₈×C₆₁	C₈	semidirect
144. C₆₁	#7872	semidirect
155. 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
C₂×#16840778	2
#21460649	2
C₂²×C₈×Dic₆₁	2
#7872	61