Subgroups — #29567200

Order 15616.

262 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#11758185	C₂	1	1	yes	maximal, characteristic
2	#21445979	C₂	1	1	yes	maximal, characteristic
3	C₂×C₆₁×#335	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11758185	—	2	1	no	—
5	C₂×#14519711	—	2	1	no	—
6	C₂×#14519711	C₄	1	2	yes	—
7	#16839632	—	2	1	no	—
8	C₄⋊#11758185	—	2	1	no	—
9	C₄⋊#11758185	C₄	1	2	yes	complemented
10	C₂×C₄⋊C₈×C₆₁	—	2	1	no	—
11	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
12	C₆₁×#335	—	2	1	no	—
13	C₆₁×#335	C₄	1	2	yes	complemented
14	C₂×C₄⋊Dic₆₁	—	2	2	no	—
15	C₂×C₄⋊Dic₆₁	—	4	1	no	—
16	C₂×C₄⋊Dic₆₁	—	4	1	no	—
17	C₂×#11758185	—	2	2	no	—
18	C₂×#11758185	—	4	1	no	—
19	C₂×#11758185	—	4	1	no	—
20	#14519711	—	4	1	no	—
21	#14519711	—	4	1	no	—
22	#14519711	—	4	2	no	—
23	C₂²×C₈×C₆₁	—	4	1	no	—
24	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
27	C₄⋊C₈×C₆₁	—	2	2	no	—
28	C₄⋊C₈×C₆₁	—	4	1	no	—
29	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
32	C₄×C₈×C₆₁	Q₈	1	1	yes	characteristic
33	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
35	C₂²×C₄×C₆₁	SD₁₆	1	1	yes	characteristic
36	C₂²×Dic₆₁	—	4	2	no	—
37	C₂×C₈×C₆₁	—	2	1	no	—
38	C₂×C₈×C₆₁	—	2	1	no	—
39	C₂×C₈×C₆₁	—	4	2	no	—
40	C₂×C₈×C₆₁	—	8	1	no	—
41	C₂×C₈×C₆₁	—	8	1	no	—
42	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
43	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₆₁	D₈	1	1	yes	characteristic
47	C₂×C₈×C₆₁	SD₁₆	1	1	yes	characteristic
48	C₂×C₈×C₆₁	SD₁₆	1	1	yes	characteristic
49	C₂×C₈×C₆₁	Q₁₆	1	1	yes	characteristic
50	C₄²×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
51	C₄²×C₆₁	C₄⋊C₄	1	1	yes	characteristic
52	C₄²×C₆₁	SD₁₆	1	1	yes	characteristic
53	C₄²×C₆₁	Q₁₆	1	1	yes	characteristic
54	C₄⋊Dic₆₁	—	4	2	no	—
55	C₄⋊Dic₆₁	—	4	2	no	—
56	C₄⋊Dic₆₁	—	8	2	no	—
57	C₄⋊Dic₆₁	—	8	1	no	—
58	C₄⋊Dic₆₁	—	8	1	no	—
59	#11758185	—	4	2	no	—
60	C₂³×C₆₁	#102	1	1	yes	characteristic
61	C₂×C₄×C₆₁	—	2	1	no	—
62	C₂×C₄×C₆₁	—	2	1	no	—
63	C₂×C₄×C₆₁	#95	1	1	yes	characteristic
64	C₂×C₄×C₆₁	#100	1	1	yes	characteristic
65	C₂×C₄×C₆₁	#101	1	1	yes	characteristic
66	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
67	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
68	C₂×C₄×C₆₁	#103	1	1	yes	characteristic
69	C₂×C₄×C₆₁	#103	1	1	yes	characteristic
70	C₂×C₄×C₆₁	#103	1	1	yes	characteristic
71	C₂×C₄×C₆₁	#106	1	1	yes	characteristic
72	C₂×C₄×C₆₁	#107	1	1	yes	characteristic
73	C₂×C₄×C₆₁	D₁₆	1	1	yes	characteristic
74	C₂×C₄×C₆₁	SD₃₂	1	1	yes	characteristic
75	C₂×C₄×C₆₁	SD₃₂	1	1	yes	characteristic
76	C₂×C₄×C₆₁	Q₃₂	1	1	yes	characteristic
77	C₂×Dic₆₁	—	8	2	no	—
78	C₂×Dic₆₁	—	8	2	no	—
79	C₂×Dic₆₁	—	8	2	no	—
80	C₈×C₆₁	—	2	1	no	—
81	C₈×C₆₁	—	2	1	no	—
82	C₈×C₆₁	—	8	2	no	—
83	C₈×C₆₁	#102	1	1	yes	characteristic
84	C₈×C₆₁	#103	1	1	yes	characteristic
85	C₈×C₆₁	#106	1	1	yes	characteristic
86	C₈×C₆₁	#107	1	1	yes	characteristic
87	#7274	—	61	1	no	maximal, Sylow
88	C₂²×C₆₁	#331	1	1	yes	characteristic
89	C₂²×C₆₁	#332	1	1	yes	characteristic
90	C₂²×C₆₁	#340	1	1	yes	characteristic
91	C₂²×C₆₁	#357	1	1	yes	characteristic
92	C₂²×C₆₁	#358	1	1	yes	characteristic
93	C₂²×C₆₁	#361	1	1	yes	characteristic
94	C₂²×C₆₁	#362	1	1	yes	characteristic
95	C₄×C₆₁	—	2	1	no	—
96	C₄×C₆₁	—	2	1	no	—
97	C₄×C₆₁	#332	1	1	yes	characteristic
98	C₄×C₆₁	#333	1	1	yes	characteristic
99	C₄×C₆₁	#340	1	1	yes	characteristic
100	C₄×C₆₁	#343	1	1	yes	characteristic
101	C₄×C₆₁	#357	1	1	yes	characteristic
102	C₄×C₆₁	#358	1	1	yes	characteristic
103	C₄×C₆₁	#366	1	1	yes	characteristic
104	C₄×C₆₁	#367	1	1	yes	characteristic
105	Dic₆₁	—	16	2	no	—
106	C₂×C₄⋊C₁₆	—	61	1	no	—
107	#1845	—	61	1	no	—
108	C₂×#335	D₆₁	1	1	yes	characteristic
109	C₂×C₆₁	#1296	1	1	yes	characteristic
110	C₂×C₆₁	#1361	1	1	yes	characteristic
111	C₂×C₆₁	#1362	1	1	yes	characteristic
112	C₂×C₆₁	#1364	1	1	yes	characteristic
113	C₂×C₆₁	#1365	1	1	yes	characteristic
114	C₂×C₆₁	#1380	1	1	yes	characteristic
115	C₂×C₆₁	#1387	1	1	yes	characteristic
116	C₂²×C₁₆	—	122	1	no	—
117	C₂×#107	—	61	2	no	—
118	C₂×#107	—	122	1	no	—
119	C₂×C₄⋊C₈	—	2	1	no	—
120	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
121	#384	—	122	1	no	—
122	C₄⋊C₁₆	—	61	2	no	—
123	C₄⋊C₁₆	—	122	1	no	—
124	#335	—	2	1	no	—
125	#335	Dic₆₁	1	2	yes	complemented
126	C₆₁	#7274	1	1	yes	characteristic, complemented, Sylow
127	C₂²×C₈	—	4	1	no	—
128	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
129	C₂²×C₈	D₂₄₄	1	1	yes	characteristic
130	C₂×C₄⋊C₄	—	122	2	no	—
131	C₂×C₄⋊C₄	—	244	1	no	—
132	C₂×C₄⋊C₄	—	244	1	no	—
133	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
134	C₂×C₁₆	—	122	2	no	—
135	C₂×C₁₆	—	244	1	no	—
136	C₂×C₁₆	—	244	1	no	—
137	#107	—	244	1	no	—
138	#107	—	244	1	no	—
139	#107	—	244	2	no	—
140	C₄⋊C₈	—	2	2	no	—
141	C₄⋊C₈	—	4	1	no	—
142	C₄×C₈	Dic₁₂₂	1	1	yes	characteristic
143	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
144	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
145	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
146	C₂²×C₄	—	244	2	no	—
147	C₂²×C₄	#11758198	1	1	yes	characteristic
148	C₂²×C₄	#11758199	1	1	yes	characteristic
149	C₂²×C₄	#11758201	1	1	yes	characteristic
150	C₂×C₈	—	2	1	no	—
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	4	2	no	—
153	C₂×C₈	—	8	1	no	—
154	C₂×C₈	—	8	1	no	—
155	C₂×C₈	#11758190	1	1	yes	characteristic
156	C₂×C₈	#11758190	1	1	yes	characteristic
157	C₂×C₈	D₄₈₈	1	1	yes	characteristic
158	C₂×C₈	Dic₂₄₄	1	1	yes	characteristic
159	C₂×C₈	C₄×Dic₆₁	1	1	yes	characteristic
160	C₂×C₈	#11758196	1	1	yes	characteristic
161	C₂×C₈	C₄⋊Dic₆₁	1	1	yes	characteristic
162	C₂×C₈	#11758198	1	1	yes	characteristic
163	C₄⋊C₄	—	488	2	no	—
164	C₄⋊C₄	—	488	1	no	—
165	C₄⋊C₄	—	488	1	no	—
166	C₄⋊C₄	—	244	2	no	—
167	C₄⋊C₄	—	244	2	no	—
168	C₄²	#11758196	1	1	yes	characteristic
169	C₄²	#11758200	1	1	yes	characteristic
170	C₄²	#11758202	1	1	yes	characteristic
171	C₄²	C₂²⋊Dic₆₁	1	1	yes	characteristic
172	C₁₆	—	244	2	no	—
173	C₂³	#14519697	1	1	yes	characteristic
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	488	2	no	—
177	C₂×C₄	—	488	2	no	—
178	C₂×C₄	—	488	2	no	—
179	C₂×C₄	#14519684	1	1	yes	characteristic
180	C₂×C₄	#14519693	1	1	yes	characteristic
181	C₂×C₄	#14519695	1	1	yes	characteristic
182	C₂×C₄	#14519698	1	1	yes	characteristic
183	C₂×C₄	#14519699	1	1	yes	characteristic
184	C₂×C₄	#14519700	1	1	yes	characteristic
185	C₂×C₄	#14519701	1	1	yes	characteristic
186	C₂×C₄	#14519702	1	1	yes	characteristic
187	C₂×C₄	#14519708	1	1	yes	characteristic
188	C₂×C₄	#14519710	1	1	yes	characteristic
189	C₂×C₄	#14519718	1	1	yes	characteristic
190	C₂×C₄	#14519720	1	1	yes	characteristic
191	C₂×C₄	#14519721	1	1	yes	characteristic
192	C₂×C₄	#14519723	1	1	yes	characteristic
193	C₈	—	2	1	no	—
194	C₈	—	2	1	no	—
195	C₈	—	8	2	no	—
196	C₈	#14519699	1	1	yes	characteristic
197	C₈	#14519702	1	1	yes	characteristic
198	C₈	#14519709	1	1	yes	characteristic
199	C₈	#14519711	1	1	yes	characteristic
200	C₂²	#16839505	1	1	yes	characteristic
201	C₂²	#16839509	1	1	yes	characteristic
202	C₂²	#16839524	1	1	yes	characteristic
203	C₂²	#16839562	1	1	yes	characteristic
204	C₂²	#16839565	1	1	yes	characteristic
205	C₂²	#16839574	1	1	yes	characteristic
206	C₂²	#16839577	1	1	yes	characteristic
207	C₄	—	2	1	no	—
208	C₄	—	2	1	no	—
209	C₄	—	976	2	no	—
210	C₄	#16839507	1	1	yes	characteristic
211	C₄	#16839510	1	1	yes	characteristic
212	C₄	#16839525	1	1	yes	characteristic
213	C₄	#16839530	1	1	yes	characteristic
214	C₄	#16839563	1	1	yes	characteristic
215	C₄	#16839566	1	1	yes	characteristic
216	C₄	#16839587	1	1	yes	characteristic
217	C₄	#16839589	1	1	yes	characteristic
218	C₂	#21443703	1	1	yes	characteristic
219	C₂	#21443864	1	1	yes	characteristic
220	C₂	#21443867	1	1	yes	characteristic
221	C₂	#21443875	1	1	yes	characteristic
222	C₂	#21443878	1	1	yes	characteristic
223	C₂	#21443910	1	1	yes	characteristic
224	C₂	#21443929	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14519697
commutator	C₂×C₈×C₆₁	C₄²
Frattini	C₂×C₄×C₈	C₂×D₆₁
Fitting	C₂×C₆₁×#335	C₂
socle, prime radical	C₂³×C₆₁	#102
cosocle	C₂×C₄×C₈×C₆₁	C₂²
hypercenter	C₂×#335	D₆₁
hypocenter	C₆₁	#7274
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
9. C₄⋊#11758185	C₄	semidirect
13. C₆₁×#335	C₄	semidirect
125. #335	Dic₆₁	semidirect
126. C₆₁	#7274	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#11758185	2
#21445979	2
C₂×C₆₁×#335	2
#7274	61