Subgroups — #30606398

Order 15616.

490 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16840658	C₂	1	2	yes	maximal
2	C₄×#14519851	C₂	1	2	yes	maximal, complemented
3	#21460866	C₂	1	2	yes	maximal, complemented
4	C₂×#16839718	C₂	1	1	yes	maximal, characteristic
5	C₂×#14519851	C₂²	1	4	yes	—
6	#16840658	C₄	1	8	yes	complemented
7	#16840650	—	2	2	no	—
8	C₄×#11758187	—	2	2	no	—
9	C₂²×#11758194	C₂²	1	1	yes	characteristic
10	C₂³×#92418	C₄	1	1	yes	characteristic
11	#16839718	—	2	2	no	—
12	#16839718	—	2	2	no	—
13	C₂×#14519688	C₄	1	1	yes	characteristic
14	C₂×#14519688	C₂²	1	1	yes	characteristic
15	C₂×C₄×#92418	C₄	1	1	yes	characteristic
16	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
17	C₂×#117×C₆₁	C₄	1	1	yes	characteristic
18	#14519851	—	2	4	no	—
19	#14519851	C₂×C₄	1	8	yes	—
20	C₂×#11758187	—	2	4	no	—
21	C₂×#11758194	—	2	2	no	—
22	C₂×#11758194	—	2	2	no	—
23	C₂×#11758194	C₂×C₄	1	4	yes	—
24	C₂²×#92418	—	2	1	no	—
25	C₂²×#92418	—	2	1	no	—
26	C₂²×#92418	—	2	2	no	—
27	C₂²×#92418	C₈	1	4	yes	—
28	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
29	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
30	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
31	C₂²×#92418	C₂³	1	1	yes	characteristic
32	#14519688	—	2	2	no	—
33	#14519688	—	2	2	no	—
34	C₄×#92418	—	2	1	no	—
35	C₄×#92418	—	2	1	no	—
36	C₄×#92418	D₄	1	2	yes	—
37	C₄×#92418	D₄	1	2	yes	—
38	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
39	#117×C₆₁	—	2	2	no	—
40	#117×C₆₁	—	2	2	no	—
41	C₂×C₄⋊C₄×C₆₁	C₈	1	2	yes	—
42	C₂×C₂²⋊C₄×C₆₁	C₈	1	2	yes	—
43	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
45	C₂⁴×C₆₁	C₂×C₈	1	1	yes	characteristic
46	C₂²×C₄×C₆₁	—	2	1	no	—
47	C₂²×C₄×C₆₁	—	2	1	no	—
48	C₂²×C₄×C₆₁	—	2	1	no	—
49	C₂²×C₄×C₆₁	—	2	1	no	—
50	C₂²×C₄×C₆₁	C₄²	1	2	yes	—
51	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
52	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
53	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
54	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
55	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
56	C₄⋊C₄×C₆₁	—	2	4	no	—
57	C₂²⋊C₄×C₆₁	—	2	4	no	—
58	C₄²×C₆₁	—	2	1	no	—
59	C₄²×C₆₁	—	2	1	no	—
60	C₄²×C₆₁	C₂²⋊C₄	1	2	yes	—
61	C₄²×C₆₁	M₁₆	1	2	yes	—
62	#11758194	—	2	8	no	—
63	C₂×#92418	—	2	1	no	—
64	C₂×#92418	—	2	1	no	—
65	C₂×#92418	—	2	1	no	—
66	C₂×#92418	—	2	1	no	—
67	C₂×#92418	—	2	4	no	—
68	C₂×#92418	—	2	4	no	—
69	C₂×#92418	C₄²	1	2	yes	—
70	C₂×#92418	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×#92418	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×#92418	C₂×C₈	1	2	yes	—
73	C₂×#92418	C₂×C₈	1	2	yes	—
74	C₂×#92418	M₁₆	1	2	yes	—
75	C₂×#92418	C₂²×C₄	1	2	yes	—
76	C₂×#92418	C₂×D₄	1	2	yes	—
77	C₂×#92418	#41	1	2	yes	—
78	#11758187	—	2	8	no	—
79	C₂³×C₆₁	—	2	1	no	—
80	C₂³×C₆₁	—	2	1	no	—
81	C₂³×C₆₁	—	4	2	no	—
82	C₂³×C₆₁	C₄×C₈	1	2	yes	—
83	C₂³×C₆₁	C₂²×C₈	1	1	yes	characteristic
84	C₂×C₄×C₆₁	—	2	1	no	—
85	C₂×C₄×C₆₁	—	2	1	no	—
86	C₂×C₄×C₆₁	—	2	2	no	—
87	C₂×C₄×C₆₁	—	2	2	no	—
88	C₂×C₄×C₆₁	—	2	2	no	—
89	C₂×C₄×C₆₁	—	2	2	no	—
90	C₂×C₄×C₆₁	—	2	2	no	—
91	C₂×C₄×C₆₁	—	4	2	no	—
92	C₂×C₄×C₆₁	—	4	2	no	—
93	C₂×C₄×C₆₁	C₄×C₈	1	2	yes	—
94	C₂×C₄×C₆₁	C₂²⋊C₈	1	4	yes	—
95	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
96	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
98	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
99	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
100	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
101	#92418	—	2	2	no	—
102	#92418	—	2	2	no	—
103	#92418	C₂²⋊C₈	1	2	yes	—
104	#92418	C₂²⋊C₈	1	2	yes	—
105	#92418	C₄×D₄	1	4	yes	—
106	#12602	—	61	1	no	maximal, Sylow
107	C₂²×C₆₁	—	2	2	no	—
108	C₂²×C₆₁	—	2	2	no	—
109	C₂²×C₆₁	—	4	2	no	—
110	C₂²×C₆₁	—	4	2	no	—
111	C₂²×C₆₁	—	4	2	no	—
112	C₂²×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
113	C₂²×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
114	C₂²×C₆₁	#431	1	1	yes	characteristic
115	C₄×C₆₁	—	2	4	no	—
116	C₄×C₆₁	—	2	2	no	—
117	C₄×C₆₁	—	4	2	no	—
118	C₄×C₆₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
119	C₄×C₆₁	C₄×M₁₆	1	1	yes	characteristic
120	C₄×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
121	C₄×C₆₁	#431	1	1	yes	characteristic
122	C₂×#405	—	61	1	no	—
123	C₂×#349	—	61	2	no	—
124	C₄×M₃₂	—	61	2	no	—
125	#1377	—	61	2	no	—
126	C₂×C₆₁	—	2	1	no	—
127	C₂×C₆₁	—	2	1	no	—
128	C₂×C₆₁	—	4	2	no	—
129	C₂×C₆₁	C₄×C₂²⋊C₈	1	1	yes	characteristic
130	C₂×C₆₁	#2115	1	2	yes	—
131	C₂²×M₁₆	—	61	1	no	—
132	C₂³×C₈	—	61	1	no	—
133	C₂×#117	#6923	1	1	yes	characteristic
134	C₂×M₃₂	—	61	4	no	—
135	#405	—	122	2	no	—
136	#405	—	122	2	no	—
137	C₂×#97	—	61	1	no	—
138	C₂×#97	—	61	1	no	—
139	C₂×C₄×C₈	—	61	1	no	—
140	C₂×C₄×C₈	—	61	1	no	—
141	#349	—	61	8	no	—
142	#346	—	122	2	no	—
143	C₄×C₁₆	—	122	2	no	—
144	C₆₁	#12602	1	1	yes	characteristic, complemented, Sylow
145	C₂³×C₄	C₂×#6923	1	1	yes	characteristic
146	C₂×M₁₆	—	61	4	no	—
147	C₂×M₁₆	—	122	2	no	—
148	C₂×M₁₆	—	122	2	no	—
149	C₂²×C₈	—	61	1	no	—
150	C₂²×C₈	—	61	1	no	—
151	C₂²×C₈	—	61	4	no	—
152	C₂²×C₈	—	61	1	no	—
153	C₂²×C₈	—	61	1	no	—
154	C₂²×C₈	—	122	1	no	—
155	C₂²×C₈	—	122	1	no	—
156	C₂²×C₈	—	122	2	no	—
157	#117	—	2	2	no	—
158	#117	—	2	2	no	—
159	C₂×C₄⋊C₄	#92420	1	2	yes	—
160	C₂×C₂²⋊C₄	#92420	1	2	yes	—
161	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
162	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
163	M₃₂	—	61	8	no	—
164	M₃₂	—	122	4	no	—
165	C₂×C₁₆	—	122	4	no	—
166	#97	—	122	2	no	—
167	#97	—	122	2	no	—
168	C₄×C₈	—	61	2	no	—
169	C₄×C₈	—	61	2	no	—
170	C₄×C₈	—	122	1	no	—
171	C₄×C₈	—	122	1	no	—
172	C₂⁴	C₂×#92420	1	1	yes	characteristic
173	C₂²×C₄	—	2	1	no	—
174	C₂²×C₄	—	2	1	no	—
175	C₂²×C₄	—	2	1	no	—
176	C₂²×C₄	—	2	1	no	—
177	C₂²×C₄	D₆₁⋊C₈	1	2	yes	—
178	C₂²×C₄	C₄×#6923	1	2	yes	—
179	C₂²×C₄	C₂×#92420	1	1	yes	characteristic
180	C₂²×C₄	C₂×#92420	1	2	yes	—
181	C₂²×C₄	C₂×#92420	1	2	yes	—
182	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
183	M₁₆	—	122	8	no	—
184	C₂×C₈	—	61	2	no	—
185	C₂×C₈	—	61	1	no	—
186	C₂×C₈	—	61	1	no	—
187	C₂×C₈	—	61	2	no	—
188	C₂×C₈	—	61	2	no	—
189	C₂×C₈	—	61	2	no	—
190	C₂×C₈	—	61	2	no	—
191	C₂×C₈	—	61	2	no	—
192	C₂×C₈	—	61	2	no	—
193	C₂×C₈	—	122	4	no	—
194	C₂×C₈	—	122	4	no	—
195	C₂×C₈	—	122	1	no	—
196	C₂×C₈	—	122	1	no	—
197	C₂×C₈	—	122	1	no	—
198	C₂×C₈	—	122	1	no	—
199	C₄⋊C₄	—	2	4	no	—
200	C₂²⋊C₄	—	2	4	no	—
201	C₄²	—	2	1	no	—
202	C₄²	—	2	1	no	—
203	C₄²	#11758217	1	2	yes	—
204	C₄²	C₂²⋊#6923	1	2	yes	—
205	C₁₆	—	122	8	no	—
206	C₂³	—	2	1	no	—
207	C₂³	—	2	1	no	—
208	C₂³	—	4	2	no	—
209	C₂³	C₄×#92420	1	2	yes	—
210	C₂³	C₂²×#92420	1	1	yes	characteristic
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	C₄×#92420	1	2	yes	—
221	C₂×C₄	C₂²⋊#92420	1	4	yes	—
222	C₂×C₄	C₂×C₄×#6923	1	1	yes	characteristic
223	C₂×C₄	C₂×C₂²⋊#6923	1	1	yes	characteristic
224	C₂×C₄	#14519856	1	1	yes	characteristic
225	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
226	C₂×C₄	C₂×#11758217	1	1	yes	characteristic
227	C₂×C₄	D₆₁⋊M₁₆	1	1	yes	characteristic
228	C₈	—	61	2	no	—
229	C₈	—	61	2	no	—
230	C₈	—	61	4	no	—
231	C₈	—	122	2	no	—
232	C₈	—	122	2	no	—
233	C₂²	—	2	2	no	—
234	C₂²	—	2	2	no	—
235	C₂²	—	4	2	no	—
236	C₂²	—	4	2	no	—
237	C₂²	—	4	2	no	—
238	C₂²	C₂×C₄×#92420	1	1	yes	characteristic
239	C₂²	C₂×C₂²⋊#92420	1	1	yes	characteristic
240	C₂²	#16840778	1	1	yes	characteristic
241	C₄	—	2	2	no	—
242	C₄	—	2	4	no	—
243	C₄	—	4	2	no	—
244	C₄	C₄×C₂²⋊#6923	1	1	yes	characteristic
245	C₄	C₄×#11758217	1	1	yes	characteristic
246	C₄	D₆₁⋊(C₂²⋊C₈)	1	1	yes	characteristic
247	C₄	#16840779	1	1	yes	characteristic
248	C₂	—	2	1	no	—
249	C₂	—	2	1	no	—
250	C₂	—	4	2	no	—
251	C₂	C₄×C₂²⋊#92420	1	1	yes	characteristic
252	C₂	#21462055	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊#6923
commutator	C₂²×C₆₁	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#6923
Fitting	C₂×#117×C₆₁	C₄
socle	C₂²×C₆₁	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₆₁	C₂×C₈
cosocle	C₂²×#92418	C₂³
hypercenter	C₂×#117	#6923
hypocenter	C₆₁	#12602
LCS	G ▷ C₂²×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
2. C₄×#14519851	C₂	semidirect
3. #21460866	C₂	semidirect
6. #16840658	C₄	semidirect
144. C₆₁	#12602	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16840658	2
C₄×#14519851	2
#21460866	2
C₂×#16839718	2
#12602	61