Subgroups — #30607047

Order 15616.

478 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#11758187	C₂	1	2	yes	maximal, complemented
2	C₂×C₂²⋊#11758187	C₂	1	2	yes	maximal
3	#21460857	C₂	1	2	yes	maximal, complemented
4	C₂×#16839718	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11758187	C₂²	1	4	yes	—
6	C₄⋊#11758187	—	2	4	no	—
7	C₂²⋊#11758187	—	2	4	no	—
8	C₂²×#11758194	C₄	1	1	yes	characteristic
9	C₂³×#92418	C₂²	1	1	yes	characteristic
10	#16839718	—	2	4	no	—
11	C₂×#14519688	C₄	1	1	yes	characteristic
12	C₂×#14519688	C₄	1	1	yes	characteristic
13	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
14	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
15	C₂×#117×C₆₁	C₄	1	1	yes	characteristic
16	C₂×#11758187	—	2	4	no	—
17	C₂×#11758187	—	2	4	no	—
18	C₂×#11758187	—	2	4	no	—
19	C₂×#11758194	—	2	2	no	—
20	C₂×#11758194	—	2	2	no	—
21	C₂×#11758194	—	2	2	no	—
22	C₂²×#92418	—	2	2	no	—
23	C₂²×#92418	—	2	2	no	—
24	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
25	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
26	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
27	C₂²×#92418	D₄	1	2	yes	—
28	C₂²×#92418	Q₈	1	2	yes	—
29	C₂²×#92418	C₂³	1	1	yes	characteristic
30	#14519688	—	2	2	no	—
31	#14519688	—	2	2	no	—
32	C₄×#92418	—	2	2	no	—
33	C₄×#92418	D₄	1	4	yes	—
34	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
35	#117×C₆₁	—	2	2	no	—
36	#117×C₆₁	—	2	2	no	—
37	C₂×C₄⋊C₄×C₆₁	C₈	1	2	yes	—
38	C₂×C₂²⋊C₄×C₆₁	C₈	1	2	yes	—
39	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
41	C₂⁴×C₆₁	C₂×C₈	1	1	yes	characteristic
42	C₂²×C₄×C₆₁	—	2	1	no	—
43	C₂²×C₄×C₆₁	—	2	1	no	—
44	C₂²×C₄×C₆₁	—	2	1	no	—
45	C₂²×C₄×C₆₁	—	2	1	no	—
46	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
48	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₆₁	M₁₆	1	2	yes	—
51	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
52	C₄⋊C₄×C₆₁	—	2	4	no	—
53	C₂²⋊C₄×C₆₁	—	2	4	no	—
54	C₄²×C₆₁	—	2	1	no	—
55	C₄²×C₆₁	—	2	1	no	—
56	C₄²×C₆₁	C₂²⋊C₄	1	2	yes	—
57	C₄²×C₆₁	M₁₆	1	2	yes	—
58	#11758194	—	2	8	no	—
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	—	2	1	no	—
64	C₂×#92418	—	2	1	no	—
65	C₂×#92418	—	2	4	no	—
66	C₂×#92418	—	2	4	no	—
67	C₂×#92418	C₂²⋊C₄	1	2	yes	—
68	C₂×#92418	C₄⋊C₄	1	2	yes	—
69	C₂×#92418	M₁₆	1	2	yes	—
70	C₂×#92418	C₂×D₄	1	1	yes	characteristic
71	C₂×#92418	C₂×D₄	1	2	yes	—
72	C₂×#92418	C₂×Q₈	1	1	yes	characteristic
73	C₂×#92418	#41	1	2	yes	—
74	#11758187	—	4	4	no	—
75	C₂³×C₆₁	—	2	1	no	—
76	C₂³×C₆₁	—	2	1	no	—
77	C₂³×C₆₁	—	2	4	no	—
78	C₂³×C₆₁	C₄⋊C₈	1	2	yes	—
79	C₂³×C₆₁	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄×C₆₁	—	2	1	no	—
81	C₂×C₄×C₆₁	—	2	1	no	—
82	C₂×C₄×C₆₁	—	2	4	no	—
83	C₂×C₄×C₆₁	—	2	2	no	—
84	C₂×C₄×C₆₁	—	2	2	no	—
85	C₂×C₄×C₆₁	—	2	2	no	—
86	C₂×C₄×C₆₁	—	2	2	no	—
87	C₂×C₄×C₆₁	—	2	2	no	—
88	C₂×C₄×C₆₁	—	2	4	no	—
89	C₂×C₄×C₆₁	C₂²⋊C₈	1	4	yes	—
90	C₂×C₄×C₆₁	C₄⋊C₈	1	2	yes	—
91	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
94	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
95	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
96	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
97	#92418	—	2	2	no	—
98	#92418	—	2	2	no	—
99	#92418	—	2	2	no	—
100	#92418	#121	1	2	yes	—
101	#92418	C₂²⋊Q₈	1	2	yes	—
102	#12784	—	61	1	no	maximal, Sylow
103	C₂²×C₆₁	—	2	2	no	—
104	C₂²×C₆₁	—	2	4	no	—
105	C₂²×C₆₁	—	4	2	no	—
106	C₂²×C₆₁	—	4	2	no	—
107	C₂²×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
108	C₂²×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
109	C₂²×C₆₁	#431	1	1	yes	characteristic
110	C₂²×C₆₁	#504	1	4	yes	—
111	C₄×C₆₁	—	2	2	no	—
112	C₄×C₆₁	—	2	4	no	—
113	C₄×C₆₁	—	4	2	no	—
114	C₄×C₆₁	#380	1	1	yes	characteristic
115	C₄×C₆₁	#407	1	1	yes	characteristic
116	C₄×C₆₁	#423	1	1	yes	characteristic
117	C₄×C₆₁	#432	1	1	yes	characteristic
118	C₂×#405	—	61	1	no	—
119	C₂×C₄⋊C₁₆	—	61	2	no	—
120	C₂×C₂²⋊C₁₆	—	61	2	no	—
121	#1374	—	61	2	no	—
122	C₂×C₆₁	—	4	2	no	—
123	C₂×C₆₁	#1797	1	1	yes	characteristic
124	C₂×C₆₁	#2113	1	2	yes	—
125	C₂×C₆₁	#2151	1	2	yes	—
126	C₂×C₆₁	#2164	1	2	yes	—
127	C₂²×M₁₆	—	61	1	no	—
128	C₂³×C₈	—	61	1	no	—
129	C₂×#117	#6923	1	1	yes	characteristic
130	C₂²×C₁₆	—	61	4	no	—
131	#405	—	122	4	no	—
132	C₂×#97	—	61	1	no	—
133	C₂×#97	—	61	1	no	—
134	C₂×C₄×C₈	—	61	1	no	—
135	C₂×C₄×C₈	—	61	1	no	—
136	C₄⋊C₁₆	—	122	4	no	—
137	C₂²⋊C₁₆	—	122	4	no	—
138	C₆₁	#12784	1	1	yes	characteristic, complemented, Sylow
139	C₂³×C₄	C₂×#6923	1	1	yes	characteristic
140	C₂×M₁₆	—	122	2	no	—
141	C₂×M₁₆	—	122	2	no	—
142	C₂×M₁₆	—	122	2	no	—
143	C₂²×C₈	—	61	1	no	—
144	C₂²×C₈	—	61	2	no	—
145	C₂²×C₈	—	61	2	no	—
146	C₂²×C₈	—	61	1	no	—
147	C₂²×C₈	—	61	1	no	—
148	C₂²×C₈	—	61	1	no	—
149	C₂²×C₈	—	122	2	no	—
150	C₂²×C₈	—	122	2	no	—
151	#117	—	2	2	no	—
152	#117	—	2	2	no	—
153	C₂×C₄⋊C₄	#92420	1	2	yes	—
154	C₂×C₂²⋊C₄	#92420	1	2	yes	—
155	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
156	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
157	C₂×C₁₆	—	122	4	no	—
158	C₂×C₁₆	—	122	4	no	—
159	C₂×C₁₆	—	122	4	no	—
160	#97	—	122	2	no	—
161	#97	—	122	2	no	—
162	C₄×C₈	—	61	4	no	—
163	C₄×C₈	—	122	2	no	—
164	C₂⁴	C₂×#92420	1	1	yes	characteristic
165	C₂²×C₄	—	2	1	no	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	—	2	1	no	—
168	C₂²×C₄	—	2	1	no	—
169	C₂²×C₄	#11758213	1	2	yes	—
170	C₂²×C₄	C₄⋊#6923	1	2	yes	—
171	C₂²×C₄	C₂×#92420	1	2	yes	—
172	C₂²×C₄	C₂×#92420	1	2	yes	—
173	C₂²×C₄	C₂×#92420	1	1	yes	characteristic
174	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
175	M₁₆	—	122	8	no	—
176	C₂×C₈	—	61	2	no	—
177	C₂×C₈	—	61	1	no	—
178	C₂×C₈	—	61	1	no	—
179	C₂×C₈	—	61	2	no	—
180	C₂×C₈	—	61	2	no	—
181	C₂×C₈	—	61	2	no	—
182	C₂×C₈	—	61	2	no	—
183	C₂×C₈	—	122	4	no	—
184	C₂×C₈	—	122	4	no	—
185	C₂×C₈	—	122	1	no	—
186	C₂×C₈	—	122	1	no	—
187	C₂×C₈	—	122	1	no	—
188	C₂×C₈	—	122	1	no	—
189	C₂×C₈	—	122	1	no	—
190	C₂×C₈	—	122	1	no	—
191	C₄⋊C₄	—	2	4	no	—
192	C₂²⋊C₄	—	2	4	no	—
193	C₄²	—	2	1	no	—
194	C₄²	—	2	1	no	—
195	C₄²	#11758217	1	2	yes	—
196	C₄²	C₂²⋊#6923	1	2	yes	—
197	C₁₆	—	244	4	no	—
198	C₂³	—	2	1	no	—
199	C₂³	—	2	1	no	—
200	C₂³	—	2	4	no	—
201	C₂³	C₄⋊#92420	1	2	yes	—
202	C₂³	C₂²×#92420	1	1	yes	characteristic
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	4	no	—
212	C₂×C₄	C₂²⋊#92420	1	4	yes	—
213	C₂×C₄	C₄⋊#92420	1	2	yes	—
214	C₂×C₄	C₂×C₂²⋊#6923	1	1	yes	characteristic
215	C₂×C₄	C₂×C₄⋊#6923	1	1	yes	characteristic
216	C₂×C₄	#14519856	1	1	yes	characteristic
217	C₂×C₄	C₂×#11758217	1	1	yes	characteristic
218	C₂×C₄	D₆₁⋊M₁₆	1	1	yes	characteristic
219	C₂×C₄	C₂×#11758213	1	1	yes	characteristic
220	C₈	—	61	2	no	—
221	C₈	—	61	2	no	—
222	C₈	—	122	2	no	—
223	C₈	—	122	2	no	—
224	C₈	—	122	2	no	—
225	C₂²	—	2	4	no	—
226	C₂²	—	2	2	no	—
227	C₂²	—	4	2	no	—
228	C₂²	—	4	2	no	—
229	C₂²	C₂×C₂²⋊#92420	1	1	yes	characteristic
230	C₂²	C₂×C₄⋊#92420	1	1	yes	characteristic
231	C₂²	#16840778	1	1	yes	characteristic
232	C₂²	#16840820	1	4	yes	—
233	C₄	—	2	4	no	—
234	C₄	—	2	2	no	—
235	C₄	—	4	2	no	—
236	C₄	#16840704	1	1	yes	characteristic
237	C₄	#16840733	1	1	yes	characteristic
238	C₄	#16840765	1	1	yes	characteristic
239	C₄	#16840783	1	1	yes	characteristic
240	C₂	—	4	2	no	—
241	C₂	#21461673	1	1	yes	characteristic
242	C₂	#21462049	1	2	yes	—
243	C₂	#21462154	1	2	yes	—
244	C₂	#21462184	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#6923
commutator, socle	C₂³×C₆₁	C₂²×C₈
Fitting	C₂×#117×C₆₁	C₄
prime radical	C₂⁴×C₆₁	C₂×C₈
cosocle	C₂²×#92418	C₂³
hypercenter	C₂×#117	#6923
hypocenter	C₆₁	#12784
LCS	G ▷ C₂³×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
1. C₂×C₄⋊#11758187	C₂	semidirect
3. #21460857	C₂	semidirect
138. C₆₁	#12784	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#11758187	2
C₂×C₂²⋊#11758187	2
#21460857	2
C₂×#16839718	2
#12784	61