Subgroups — #29573756

Order 15616.

422 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21445936	C₂	1	1	yes	maximal, characteristic
2	C₄²×#92418	C₂	1	1	yes	maximal, characteristic
3	C₂×#16839514	C₂	1	2	yes	maximal
4	C₂×#16839512	C₂	1	2	yes	maximal
5	C₆₁×#1831	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#92418	C₂²	1	2	yes	—
7	C₂×C₄×#92418	C₄	1	1	yes	characteristic
8	C₂×C₄×#92418	C₄	1	1	yes	characteristic
9	C₂×C₄×#92418	C₄	1	1	yes	characteristic
10	C₂×C₄×#92418	C₄	1	1	yes	characteristic
11	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
12	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
13	#16839517	—	2	2	no	—
14	#16839514	—	2	4	no	—
15	#16839512	—	2	4	no	—
16	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
17	C₄³×C₆₁	C₂²	1	1	yes	characteristic
18	C₆₁×#336	—	2	2	no	—
19	C₂²×#92418	—	2	2	no	—
20	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
21	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
22	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
23	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
24	C₄⋊#92418	—	2	2	no	—
25	C₄⋊#92418	—	2	2	no	—
26	C₄×#92418	—	2	2	no	—
27	C₄×#92418	—	2	2	no	—
28	C₄×#92418	—	2	2	no	—
29	C₄×#92418	C₈	1	4	yes	complemented
30	C₄×#92418	D₄	1	2	yes	—
31	C₄×#92418	D₄	1	2	yes	—
32	C₄×#92418	Q₈	1	2	yes	—
33	C₄×#92418	Q₈	1	2	yes	—
34	C₂²×C₈×C₆₁	—	2	2	no	—
35	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₆₁	D₄	1	2	yes	—
38	C₂×C₄²×C₆₁	Q₈	1	2	yes	—
39	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
40	C₄⋊C₈×C₆₁	—	2	2	no	—
41	C₄⋊C₈×C₆₁	—	2	2	no	—
42	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
43	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
44	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₆₁	—	4	2	no	—
48	C₂×C₈×C₆₁	—	4	4	no	—
49	C₄²×C₆₁	—	2	4	no	—
50	C₄²×C₆₁	—	2	1	no	—
51	C₄²×C₆₁	—	2	1	no	—
52	C₄²×C₆₁	—	2	4	no	—
53	C₄²×C₆₁	C₂×C₈	1	2	yes	—
54	C₄²×C₆₁	M₁₆	1	2	yes	—
55	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
56	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
58	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
59	C₂×#92418	—	2	1	no	—
60	C₂×#92418	—	2	1	no	—
61	C₂×#92418	—	2	1	no	—
62	C₂×#92418	—	2	1	no	—
63	C₂×#92418	—	4	2	no	—
64	C₂×#92418	—	4	4	no	—
65	C₂×#92418	C₄⋊C₄	1	2	yes	—
66	C₂×#92418	C₄⋊C₄	1	2	yes	—
67	C₂×#92418	C₄⋊C₄	1	2	yes	—
68	C₂×#92418	C₄⋊C₄	1	2	yes	—
69	C₂×#92418	C₂×C₈	1	2	yes	—
70	C₂×#92418	C₂×C₈	1	2	yes	—
71	C₂×#92418	M₁₆	1	2	yes	—
72	C₂×#92418	M₁₆	1	2	yes	—
73	C₂³×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₆₁	—	2	2	no	—
75	C₂×C₄×C₆₁	—	2	2	no	—
76	C₂×C₄×C₆₁	—	2	2	no	—
77	C₂×C₄×C₆₁	—	2	2	no	—
78	C₂×C₄×C₆₁	—	2	2	no	—
79	C₂×C₄×C₆₁	C₄⋊C₈	1	4	yes	—
80	C₂×C₄×C₆₁	#108	1	4	yes	—
81	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
84	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
85	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
86	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
87	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
88	C₂×C₄×C₆₁	C₂×M₁₆	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×C₄×C₆₁	#135	1	4	yes	—
92	C₈×C₆₁	—	8	2	no	—
93	#92418	—	2	2	no	—
94	#92418	—	2	2	no	—
95	#92418	—	8	2	no	—
96	#92418	C₄⋊C₈	1	4	yes	—
97	#92418	C₄⋊C₈	1	4	yes	—
98	#8807	—	61	1	no	maximal, Sylow
99	C₂²×C₆₁	#384	1	1	yes	characteristic
100	C₂²×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
101	C₂²×C₆₁	#423	1	1	yes	characteristic
102	C₂²×C₆₁	#427	1	2	yes	—
103	C₂²×C₆₁	C₂×#108	1	2	yes	—
104	C₄×C₆₁	—	2	4	no	—
105	C₄×C₆₁	—	2	4	no	—
106	C₄×C₆₁	—	2	1	no	—
107	C₄×C₆₁	—	2	1	no	—
108	C₄×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
109	C₄×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
110	C₄×C₆₁	#423	1	1	yes	characteristic
111	C₄×C₆₁	#423	1	1	yes	characteristic
112	C₄×C₆₁	#495	1	2	yes	—
113	C₄×C₆₁	#499	1	2	yes	—
114	#1831	—	61	1	no	—
115	#1831	D₆₁	1	1	yes	characteristic
116	C₄²×C₈	—	61	1	no	—
117	C₂×#335	—	61	2	no	—
118	C₂×#334	—	61	2	no	—
119	C₂×C₆₁	#1564	1	2	yes	—
120	C₂×C₆₁	#1568	1	2	yes	—
121	C₂×C₆₁	#1842	1	1	yes	characteristic
122	C₂×C₆₁	#1846	1	1	yes	characteristic
123	C₂×C₆₁	#1848	1	1	yes	characteristic
124	C₂×C₄⋊C₈	—	61	2	no	—
125	C₂×C₄⋊C₈	C₂×D₆₁	1	2	yes	—
126	C₂×C₄×C₈	—	61	1	no	—
127	C₂×C₄×C₈	—	61	1	no	—
128	C₂×C₄×C₈	—	61	1	no	—
129	C₂×C₄×C₈	—	61	1	no	—
130	C₂×C₄×C₈	—	61	1	no	—
131	C₂×C₄×C₈	—	61	1	no	—
132	C₄³	C₂×D₆₁	1	1	yes	characteristic
133	#336	—	2	2	no	—
134	#336	—	122	2	no	—
135	#335	—	122	4	no	—
136	#334	—	122	4	no	—
137	C₆₁	#8807	1	1	yes	characteristic, complemented, Sylow
138	C₂²×C₈	—	2	2	no	—
139	C₂²×C₈	—	61	1	no	—
140	C₂²×C₈	—	61	1	no	—
141	C₂²×C₈	—	61	1	no	—
142	C₂²×C₈	—	61	1	no	—
143	C₂²×C₈	—	122	2	no	—
144	C₂×C₄²	Dic₁₂₂	1	2	yes	—
145	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
146	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
147	C₂×C₄²	C₂²⋊D₆₁	1	2	yes	—
148	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
149	C₄⋊C₈	—	2	2	no	—
150	C₄⋊C₈	—	2	2	no	—
151	C₄⋊C₈	—	122	2	no	—
152	C₄⋊C₈	—	122	2	no	—
153	C₄×C₈	—	61	4	no	—
154	C₄×C₈	—	61	2	no	—
155	C₄×C₈	—	61	2	no	—
156	C₄×C₈	—	61	2	no	—
157	C₄×C₈	—	61	2	no	—
158	C₄×C₈	—	122	2	no	—
159	C₄×C₈	—	122	2	no	—
160	C₄×C₈	—	122	2	no	—
161	C₂²×C₄	#11758196	1	2	yes	—
162	C₂²×C₄	#11758196	1	2	yes	—
163	C₂²×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
164	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
165	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
166	C₂×C₈	—	4	2	no	—
167	C₂×C₈	—	4	4	no	—
168	C₂×C₈	—	61	2	no	—
169	C₂×C₈	—	61	2	no	—
170	C₂×C₈	—	61	2	no	—
171	C₂×C₈	—	61	2	no	—
172	C₂×C₈	—	61	2	no	—
173	C₂×C₈	—	61	2	no	—
174	C₂×C₈	—	61	2	no	—
175	C₂×C₈	—	61	2	no	—
176	C₂×C₈	—	122	1	no	—
177	C₂×C₈	—	122	1	no	—
178	C₂×C₈	—	122	1	no	—
179	C₂×C₈	—	122	1	no	—
180	C₂×C₈	—	244	2	no	—
181	C₂×C₈	—	244	4	no	—
182	C₄²	—	2	4	no	—
183	C₄²	—	2	1	no	—
184	C₄²	—	2	1	no	—
185	C₄²	—	2	4	no	—
186	C₄²	C₈×D₆₁	1	2	yes	—
187	C₄²	#11758189	1	2	yes	—
188	C₄²	D₄×D₆₁	1	1	yes	characteristic
189	C₄²	D₄×D₆₁	1	1	yes	characteristic
190	C₄²	Q₈×D₆₁	1	1	yes	characteristic
191	C₄²	Q₈×D₆₁	1	1	yes	characteristic
192	C₂³	C₂×#11758196	1	1	yes	characteristic
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	#14519706	1	4	yes	—
199	C₂×C₄	#14519712	1	4	yes	—
200	C₂×C₄	C₄⋊C₄×D₆₁	1	1	yes	characteristic
201	C₂×C₄	C₄⋊C₄×D₆₁	1	1	yes	characteristic
202	C₂×C₄	#14519773	1	1	yes	characteristic
203	C₂×C₄	#14519775	1	1	yes	characteristic
204	C₂×C₄	#14519776	1	1	yes	characteristic
205	C₂×C₄	#14519776	1	1	yes	characteristic
206	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
207	C₂×C₄	M₁₆×D₆₁	1	1	yes	characteristic
208	C₂×C₄	M₁₆×D₆₁	1	1	yes	characteristic
209	C₂×C₄	C₂×#11758189	1	1	yes	characteristic
210	C₂×C₄	#14519797	1	4	yes	—
211	C₈	—	8	2	no	—
212	C₈	—	61	4	no	—
213	C₈	—	61	4	no	—
214	C₈	—	122	2	no	—
215	C₈	—	122	2	no	—
216	C₈	—	488	2	no	—
217	C₂²	#16839631	1	1	yes	characteristic
218	C₂²	C₂×#14519706	1	1	yes	characteristic
219	C₂²	#16839785	1	1	yes	characteristic
220	C₂²	#16839798	1	2	yes	—
221	C₂²	C₂×#14519712	1	2	yes	—
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	2	4	no	—
225	C₄	—	2	4	no	—
226	C₄	C₄⋊C₈×D₆₁	1	1	yes	characteristic
227	C₄	C₄⋊C₈×D₆₁	1	1	yes	characteristic
228	C₄	#16839784	1	1	yes	characteristic
229	C₄	#16839784	1	1	yes	characteristic
230	C₄	#16840144	1	2	yes	—
231	C₄	#16840161	1	2	yes	—
232	C₂	#21444565	1	2	yes	—
233	C₂	#21444580	1	2	yes	—
234	C₂	#21445970	1	1	yes	characteristic
235	C₂	#21445981	1	1	yes	characteristic
236	C₂	#21445990	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₆₁
commutator	C₂×C₄×C₆₁	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1831	C₂
socle, prime radical	C₂³×C₆₁	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#1831	D₆₁
hypocenter	C₆₁	#8807
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1831

Complements

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

N	H ≅ G/N	type
29. C₄×#92418	C₈	semidirect
137. C₆₁	#8807	semidirect

Maximal subgroups

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

subgroup	index
#21445936	2
C₄²×#92418	2
C₂×#16839514	2
C₂×#16839512	2
C₆₁×#1831	2
#8807	61