Subgroups — #29578627

Order 15616.

306 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	C₄×#14519713	C₂	1	1	yes	maximal, characteristic
2	#21444569	C₂	1	2	yes	maximal
3	#21444102	C₂	1	1	yes	maximal, characteristic
4	#21443669	C₂	1	2	yes	maximal
5	C₂×C₆₁×#335	C₂	1	1	yes	maximal, characteristic
6	#16839815	C₂²	1	1	yes	characteristic
7	#16839815	C₂²	1	1	yes	characteristic
8	C₂×#14519713	C₄	1	1	yes	characteristic
9	C₂×#14519713	C₄	1	1	yes	characteristic
10	C₄×#11758194	C₂²	1	2	yes	—
11	#16839527	C₄	1	2	yes	—
12	#16839514	—	2	2	no	—
13	#16839512	—	2	2	no	—
14	#16839483	—	2	1	no	—
15	C₈×#92418	—	2	1	no	—
16	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
18	C₆₁×#335	—	2	1	no	—
19	C₆₁×#335	—	2	1	no	—
20	C₂×#11758194	C₂×C₄	1	2	yes	—
21	C₂×#11758194	C₂×C₄	1	2	yes	—
22	#14519713	—	2	2	no	—
23	#14519713	C₈	1	4	yes	complemented
24	C₄⋊#92418	—	2	1	no	—
25	C₄⋊#92418	D₄	1	1	yes	characteristic
26	C₄⋊#92418	D₄	1	1	yes	characteristic
27	C₂²⋊#92418	—	2	1	no	—
28	C₂²⋊#92418	Q₈	1	2	yes	—
29	#14519688	—	2	2	no	—
30	C₄×#92418	—	2	2	no	—
31	C₄×#92418	—	2	1	no	—
32	C₄×#92418	—	2	1	no	—
33	C₂²×C₈×C₆₁	—	2	2	no	—
34	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
37	C₄⋊C₈×C₆₁	—	2	2	no	—
38	C₄⋊C₈×C₆₁	—	2	2	no	—
39	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
40	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
41	C₄×C₈×C₆₁	Q₈	1	1	yes	characteristic
42	C₄×C₈×C₆₁	Q₈	1	1	yes	characteristic
43	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
45	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
46	C₂×C₈×C₆₁	—	2	1	no	—
47	C₂×C₈×C₆₁	—	2	1	no	—
48	C₂×C₈×C₆₁	—	4	2	no	—
49	C₂×C₈×C₆₁	—	8	2	no	—
50	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₆₁	C₂×C₈	1	2	yes	—
55	C₂×C₈×C₆₁	M₁₆	1	2	yes	—
56	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
59	C₄²×C₆₁	#41	1	1	yes	characteristic
60	#11758194	—	2	2	no	—
61	#11758194	—	2	2	no	—
62	#11758194	—	2	2	no	—
63	#11758194	C₂×C₈	1	4	yes	—
64	C₂×#92418	—	2	2	no	—
65	C₂×#92418	—	2	2	no	—
66	C₂×#92418	—	2	1	no	—
67	C₂×#92418	—	2	1	no	—
68	C₂×#92418	—	2	2	no	—
69	C₂³×C₆₁	C₄×Q₈	1	1	yes	characteristic
70	C₂×C₄×C₆₁	—	2	1	no	—
71	C₂×C₄×C₆₁	—	2	1	no	—
72	C₂×C₄×C₆₁	—	4	1	no	—
73	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
75	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
76	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
77	C₂×C₄×C₆₁	#125	1	1	yes	characteristic
78	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
79	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
81	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
82	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
83	C₈×C₆₁	—	2	1	no	—
84	C₈×C₆₁	—	2	1	no	—
85	C₈×C₆₁	—	8	2	no	—
86	C₈×C₆₁	C₄⋊C₈	1	2	yes	—
87	C₈×C₆₁	C₄⋊C₈	1	2	yes	—
88	#92418	—	2	4	no	—
89	#92418	—	4	2	no	—
90	#92418	—	4	2	no	—
91	#92418	—	4	1	no	—
92	#92418	—	4	1	no	—
93	#9608	—	61	1	no	maximal, Sylow
94	C₂²×C₆₁	—	2	1	no	—
95	C₂²×C₆₁	—	2	1	no	—
96	C₂²×C₆₁	#389	1	1	yes	characteristic
97	C₂²×C₆₁	C₈×Q₈	1	1	yes	characteristic
98	C₂²×C₆₁	#446	1	1	yes	characteristic
99	C₄×C₆₁	—	2	1	no	—
100	C₄×C₆₁	—	2	1	no	—
101	C₄×C₆₁	—	2	1	no	—
102	C₄×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
103	C₄×C₆₁	#424	1	1	yes	characteristic
104	C₄×C₆₁	C₈×D₄	1	1	yes	characteristic
105	C₄×C₆₁	#436	1	1	yes	characteristic
106	C₄×C₆₁	#472	1	1	yes	characteristic
107	C₄×C₆₁	#473	1	1	yes	characteristic
108	C₄×#108	—	61	1	no	—
109	#1565	—	61	2	no	—
110	C₂×#335	D₆₁	1	1	yes	characteristic
111	#1450	—	61	1	no	—
112	#1283	—	61	2	no	—
113	C₂×C₆₁	—	2	1	no	—
114	C₂×C₆₁	—	2	1	no	—
115	C₂×C₆₁	#1939	1	1	yes	characteristic
116	C₂×C₆₁	#1952	1	1	yes	characteristic
117	C₂×C₆₁	#1953	1	1	yes	characteristic
118	#431	—	61	1	no	—
119	#431	—	61	1	no	—
120	C₂×#108	—	61	1	no	—
121	C₂×#108	—	61	1	no	—
122	C₂×C₄⋊C₈	C₂×D₆₁	1	2	yes	—
123	C₄×M₁₆	—	61	2	no	—
124	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
125	#341	—	61	2	no	—
126	#335	—	2	1	no	—
127	#335	—	2	1	no	—
128	#335	—	122	2	no	—
129	#334	—	122	2	no	—
130	#322	—	122	1	no	—
131	C₈²	—	122	1	no	—
132	C₆₁	#9608	1	1	yes	characteristic, complemented, Sylow
133	C₂×M₁₆	—	61	2	no	—
134	C₂×M₁₆	—	61	2	no	—
135	C₂²×C₈	—	2	2	no	—
136	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
137	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
138	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
139	#108	—	61	4	no	—
140	#108	—	122	2	no	—
141	C₄⋊C₈	—	2	2	no	—
142	C₄⋊C₈	—	2	2	no	—
143	C₄⋊C₈	—	61	1	no	—
144	C₄⋊C₈	—	61	1	no	—
145	C₄⋊C₈	—	122	1	no	—
146	C₂²⋊C₈	—	61	2	no	—
147	C₂²⋊C₈	—	122	1	no	—
148	#97	—	122	2	no	—
149	C₄×C₈	—	122	2	no	—
150	C₄×C₈	—	122	1	no	—
151	C₄×C₈	—	122	1	no	—
152	C₄×C₈	Dic₁₂₂	1	1	yes	characteristic
153	C₄×C₈	Dic₁₂₂	1	1	yes	characteristic
154	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
155	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
156	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
157	C₂²×C₄	#11758224	1	1	yes	characteristic
158	C₂²×C₄	Q₈×D₆₁	1	1	yes	characteristic
159	M₁₆	—	61	4	no	—
160	M₁₆	—	122	2	no	—
161	M₁₆	—	122	2	no	—
162	M₁₆	—	122	2	no	—
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	4	2	no	—
166	C₂×C₈	—	8	2	no	—
167	C₂×C₈	—	122	2	no	—
168	C₂×C₈	—	122	2	no	—
169	C₂×C₈	—	122	1	no	—
170	C₂×C₈	—	122	1	no	—
171	C₂×C₈	—	122	2	no	—
172	C₂×C₈	C₈×D₆₁	1	2	yes	—
173	C₂×C₈	#11758189	1	2	yes	—
174	C₂×C₈	#11758196	1	1	yes	characteristic
175	C₂×C₈	#11758196	1	1	yes	characteristic
176	C₂×C₈	#11758196	1	1	yes	characteristic
177	C₂×C₈	#11758196	1	1	yes	characteristic
178	C₄²	C₂×Dic₁₂₂	1	1	yes	characteristic
179	C₄²	D₄×D₆₁	1	1	yes	characteristic
180	C₄²	#11758226	1	1	yes	characteristic
181	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
182	C₂³	#14519742	1	1	yes	characteristic
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	4	1	no	—
186	C₂×C₄	C₂×#11758196	1	1	yes	characteristic
187	C₂×C₄	#14519739	1	1	yes	characteristic
188	C₂×C₄	#14519748	1	1	yes	characteristic
189	C₂×C₄	#14519753	1	1	yes	characteristic
190	C₂×C₄	#14519768	1	1	yes	characteristic
191	C₂×C₄	#14519776	1	1	yes	characteristic
192	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
193	C₂×C₄	C₂×#11758189	1	1	yes	characteristic
194	C₂×C₄	#14519785	1	1	yes	characteristic
195	C₂×C₄	#14519785	1	1	yes	characteristic
196	C₈	—	2	1	no	—
197	C₈	—	2	1	no	—
198	C₈	—	8	2	no	—
199	C₈	—	122	4	no	—
200	C₈	—	244	2	no	—
201	C₈	—	244	1	no	—
202	C₈	—	244	1	no	—
203	C₈	—	244	2	no	—
204	C₈	#14519706	1	2	yes	—
205	C₈	#14519706	1	2	yes	—
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	#16839658	1	1	yes	characteristic
209	C₂²	#16839886	1	1	yes	characteristic
210	C₂²	#16839889	1	1	yes	characteristic
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	C₂×#14519706	1	1	yes	characteristic
215	C₄	#16839790	1	1	yes	characteristic
216	C₄	#16839827	1	1	yes	characteristic
217	C₄	#16839839	1	1	yes	characteristic
218	C₄	#16840029	1	1	yes	characteristic
219	C₄	#16840034	1	1	yes	characteristic
220	C₂	—	2	1	no	—
221	C₂	—	2	1	no	—
222	C₂	#21446535	1	1	yes	characteristic
223	C₂	#21446602	1	1	yes	characteristic
224	C₂	#21446607	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14519748
commutator	C₂×C₄×C₆₁	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₂×C₆₁×#335	C₂
socle	C₂²×C₆₁	#389
prime radical	C₂³×C₆₁	C₄×Q₈
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₂×#335	D₆₁
hypocenter	C₆₁	#9608
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
23. #14519713	C₈	semidirect
132. C₆₁	#9608	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14519713	2
#21444569	2
#21444102	2
#21443669	2
C₂×C₆₁×#335	2
#9608	61