Subgroups — #30598101

Order 15616.

334 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21461366	C₂	1	1	yes	maximal, characteristic
2	#21461026	C₂	1	2	yes	maximal
3	C₂×#16840603	C₂	1	1	yes	maximal, characteristic
4	#21460652	C₂	1	2	yes	maximal
5	#108×Dic₆₁	C₂	1	1	yes	maximal, characteristic
6	#16840778	C₂²	1	2	yes	—
7	C₂×C₄×#92420	C₂²	1	1	yes	characteristic
8	C₂×C₄×#92420	C₂²	1	1	yes	characteristic
9	#16840625	C₄	1	2	yes	—
10	#16840623	C₄	1	2	yes	—
11	#92420⋊C₈	—	2	4	no	—
12	#16840603	C₄	1	2	yes	—
13	#16840603	C₄	1	2	yes	—
14	C₂×#14519713	C₄	1	1	yes	characteristic
15	M₁₆×Dic₆₁	C₂²	1	2	yes	—
16	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
17	#16839526	C₄	1	2	yes	—
18	C₂×#108×C₆₁	C₄	1	1	yes	characteristic
19	C₂²×#92420	—	2	1	no	—
20	C₂²×#92420	Q₈	1	2	yes	—
21	C₄⋊#92420	C₂×C₄	1	4	yes	—
22	C₂²⋊#92420	—	2	2	no	—
23	C₄×#92420	—	2	2	no	—
24	C₄×#92420	—	2	1	no	—
25	C₄×#92420	C₂×C₄	1	2	yes	—
26	C₄×#92420	C₂×C₄	1	2	yes	—
27	C₄×#92420	D₄	1	2	yes	—
28	C₂×#11758194	C₂×C₄	1	2	yes	—
29	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
31	#14519713	—	2	2	no	—
32	#14519687	—	2	2	no	—
33	C₈×Dic₆₁	—	2	2	no	—
34	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
35	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
36	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
37	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
38	C₂×M₁₆×C₆₁	C₂×C₄	1	2	yes	—
39	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
40	#108×C₆₁	—	2	2	no	—
41	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
42	C₂²×Dic₆₁	C₂×Q₈	1	1	yes	characteristic
43	C₂²×Dic₆₁	#41	1	1	yes	characteristic
44	C₂×#92420	—	2	2	no	—
45	C₂×#92420	—	2	4	no	—
46	C₂×#92420	—	4	2	no	—
47	C₂×#92420	—	4	2	no	—
48	C₂×#92420	C₄⋊C₄	1	4	yes	—
49	M₁₆×C₆₁	—	2	2	no	—
50	M₁₆×C₆₁	—	2	2	no	—
51	C₂×C₈×C₆₁	—	2	2	no	—
52	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
53	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
54	C₂×C₈×C₆₁	M₁₆	1	2	yes	—
55	C₂×C₈×C₆₁	M₁₆	1	2	yes	—
56	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
57	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
58	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
59	C₄×Dic₆₁	#41	1	1	yes	characteristic
60	#11758194	—	2	2	no	—
61	#11758194	—	2	2	no	—
62	C₂×#92418	—	2	1	no	—
63	C₂×#92418	—	2	1	no	—
64	C₂×#92418	—	2	2	no	—
65	C₂×#92418	C₄²	1	1	yes	characteristic
66	C₂×#92418	C₄²	1	1	yes	characteristic
67	C₂³×C₆₁	C₄×Q₈	1	1	yes	characteristic
68	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
69	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
71	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
72	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
73	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
74	C₂×Dic₆₁	—	2	2	no	—
75	C₂×Dic₆₁	—	2	2	no	—
76	C₂×Dic₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×Dic₆₁	#117	1	1	yes	characteristic
78	C₂×Dic₆₁	C₄×D₄	1	1	yes	characteristic
79	C₂×Dic₆₁	C₄×Q₈	1	1	yes	characteristic
80	#92420	—	2	4	no	—
81	#92420	—	4	2	no	—
82	#92420	—	4	4	no	—
83	C₈×C₆₁	—	4	2	no	—
84	C₈×C₆₁	#97	1	2	yes	—
85	C₈×C₆₁	#97	1	2	yes	—
86	#92418	—	2	1	no	—
87	#92418	—	2	1	no	—
88	#92418	—	4	2	no	—
89	#8295	—	61	1	no	maximal, Sylow
90	C₂²×C₆₁	—	2	1	no	—
91	C₂²×C₆₁	—	2	1	no	—
92	C₂²×C₆₁	C₄×C₄⋊C₄	1	1	yes	characteristic
93	C₂²×C₆₁	#446	1	1	yes	characteristic
94	C₂²×C₆₁	#446	1	1	yes	characteristic
95	C₄×C₆₁	C₂×#97	1	1	yes	characteristic
96	C₄×C₆₁	#405	1	1	yes	characteristic
97	C₄×C₆₁	#436	1	1	yes	characteristic
98	C₄×C₆₁	#436	1	1	yes	characteristic
99	Dic₆₁	—	2	1	no	—
100	Dic₆₁	—	2	1	no	—
101	Dic₆₁	#430	1	2	yes	—
102	Dic₆₁	#444	1	2	yes	—
103	C₄×#108	—	61	1	no	—
104	#1564	—	61	1	no	—
105	#1451	—	61	2	no	—
106	C₂×#322	—	61	1	no	—
107	#1278	—	61	2	no	—
108	C₂×C₆₁	—	2	1	no	—
109	C₂×C₆₁	—	2	1	no	—
110	C₂×C₆₁	#1770	1	1	yes	characteristic
111	C₂×C₆₁	#1773	1	1	yes	characteristic
112	C₂×C₆₁	#1778	1	1	yes	characteristic
113	#431	—	61	2	no	—
114	C₂×#108	—	61	1	no	—
115	C₂×#108	#6923	1	1	yes	characteristic
116	C₄×M₁₆	—	61	2	no	—
117	C₂×C₄×C₈	—	61	1	no	—
118	C₂×C₄×C₈	—	61	1	no	—
119	C₂×C₄×C₈	—	61	1	no	—
120	#341	—	61	2	no	—
121	#335	—	61	2	no	—
122	#334	—	61	2	no	—
123	#322	—	61	2	no	—
124	#322	—	61	2	no	—
125	#322	—	122	4	no	—
126	C₆₁	#8295	1	1	yes	characteristic, complemented, Sylow
127	C₂×M₁₆	—	61	2	no	—
128	C₂×M₁₆	C₂×#6923	1	2	yes	—
129	C₂²×C₈	—	61	1	no	—
130	C₂²×C₈	—	61	2	no	—
131	C₂²×C₈	—	122	1	no	—
132	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
133	C₂×C₄²	—	61	1	no	—
134	#108	—	2	2	no	—
135	#108	—	122	2	no	—
136	C₄⋊C₈	—	61	4	no	—
137	C₂²⋊C₈	—	122	2	no	—
138	#97	—	122	2	no	—
139	C₄×C₈	—	61	2	no	—
140	C₄×C₈	—	61	2	no	—
141	C₄×C₈	—	61	1	no	—
142	C₄×C₈	—	61	1	no	—
143	C₄×C₈	—	61	1	no	—
144	C₄×C₈	—	61	1	no	—
145	C₄×C₈	—	61	2	no	—
146	C₄×C₈	—	122	2	no	—
147	C₄×C₈	—	122	2	no	—
148	C₄×C₈	—	122	1	no	—
149	C₂²×C₄	—	61	1	no	—
150	C₂²×C₄	—	61	1	no	—
151	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
152	M₁₆	—	2	2	no	—
153	M₁₆	—	2	2	no	—
154	M₁₆	—	122	2	no	—
155	M₁₆	—	122	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	61	4	no	—
158	C₂×C₈	—	61	1	no	—
159	C₂×C₈	—	61	1	no	—
160	C₂×C₈	—	122	1	no	—
161	C₂×C₈	—	122	1	no	—
162	C₂×C₈	—	122	2	no	—
163	C₂×C₈	—	122	4	no	—
164	C₂×C₈	—	122	2	no	—
165	C₂×C₈	—	244	2	no	—
166	C₂×C₈	—	244	2	no	—
167	C₂×C₈	#11758213	1	2	yes	—
168	C₂×C₈	C₄×#6923	1	1	yes	characteristic
169	C₂×C₈	C₄×#6923	1	1	yes	characteristic
170	C₂×C₈	#11758217	1	2	yes	—
171	C₄²	—	61	1	no	—
172	C₄²	—	61	1	no	—
173	C₄²	—	61	1	no	—
174	C₄²	—	61	1	no	—
175	C₂³	Q₈×#6923	1	1	yes	characteristic
176	C₂×C₄	—	61	1	no	—
177	C₂×C₄	—	61	1	no	—
178	C₂×C₄	—	61	1	no	—
179	C₂×C₄	—	61	1	no	—
180	C₂×C₄	—	122	2	no	—
181	C₂×C₄	—	122	2	no	—
182	C₂×C₄	C₂×C₄×#6923	1	1	yes	characteristic
183	C₂×C₄	D₄×#6923	1	1	yes	characteristic
184	C₂×C₄	C₂×#11758217	1	1	yes	characteristic
185	C₂×C₄	C₂×#11758213	1	1	yes	characteristic
186	C₂×C₄	#14519864	1	1	yes	characteristic
187	C₂×C₄	#14519865	1	1	yes	characteristic
188	C₈	—	4	2	no	—
189	C₈	—	122	4	no	—
190	C₈	—	122	1	no	—
191	C₈	—	122	1	no	—
192	C₈	—	244	4	no	—
193	C₈	—	244	2	no	—
194	C₈	—	244	2	no	—
195	C₈	#14519832	1	2	yes	—
196	C₈	#14519832	1	2	yes	—
197	C₂²	—	2	1	no	—
198	C₂²	—	2	1	no	—
199	C₂²	C₄⋊C₄×#6923	1	1	yes	characteristic
200	C₂²	#16840812	1	1	yes	characteristic
201	C₂²	#16840813	1	1	yes	characteristic
202	C₄	—	61	2	no	—
203	C₄	—	61	2	no	—
204	C₄	—	122	1	no	—
205	C₄	—	122	1	no	—
206	C₄	C₂×#14519832	1	1	yes	characteristic
207	C₄	#16840727	1	1	yes	characteristic
208	C₄	#16840796	1	1	yes	characteristic
209	C₄	#16840797	1	1	yes	characteristic
210	C₂	—	2	1	no	—
211	C₂	—	2	1	no	—
212	C₂	#21461615	1	1	yes	characteristic
213	C₂	#21461628	1	1	yes	characteristic
214	C₂	#21461640	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#6923
commutator	C₂×C₄×C₆₁	C₂×C₄²
Frattini	C₂²×C₄	C₂²×#6923
Fitting	C₂×#108×C₆₁	C₄
socle	C₂²×C₆₁	C₄×C₄⋊C₄
prime radical	C₂³×C₆₁	C₄×Q₈
cosocle	C₂×C₄×Dic₆₁	C₂³
hypercenter	C₂×#108	#6923
hypocenter	C₆₁	#8295
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

Complements

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

N	H ≅ G/N	type
126. C₆₁	#8295	semidirect

Maximal subgroups

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

subgroup	index
#21461366	2
#21461026	2
C₂×#16840603	2
#21460652	2
#108×Dic₆₁	2
#8295	61