Subgroups — #30598037

Order 15616.

446 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊C₄×#92420	C₂	1	2	yes	maximal
2	C₂×#16840626	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×#92420	C₂	1	1	yes	maximal, characteristic
4	#21460655	C₂	1	2	yes	maximal
5	#21445711	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#92420	C₂²	1	2	yes	—
7	C₂×C₄×#92420	—	2	2	no	—
8	C₂×C₄×#92420	C₂²	1	1	yes	characteristic
9	C₂×C₄×#92420	C₂²	1	1	yes	characteristic
10	#16840628	—	2	2	no	—
11	#16840626	C₄	1	4	yes	complemented
12	C₈×#92420	C₄	1	4	yes	complemented
13	C₂×#14519710	C₄	1	2	yes	—
14	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×Dic₆₁	C₂²	1	2	yes	—
16	#16839525	C₄	1	1	yes	characteristic
17	C₆₁×#340	C₄	1	1	yes	characteristic
18	C₂²×#92420	—	2	1	no	—
19	C₂²×#92420	—	2	2	no	—
20	C₂²×#92420	D₄	1	1	yes	characteristic
21	C₂²×#92420	D₄	1	1	yes	characteristic
22	C₄⋊#92420	—	2	2	no	—
23	C₄⋊#92420	C₂×C₄	1	4	yes	—
24	C₄×#92420	—	2	1	no	—
25	C₄×#92420	—	4	4	no	—
26	C₄×#92420	C₂×C₄	1	2	yes	—
27	C₄×#92420	C₂×C₄	1	2	yes	—
28	C₄×#92420	Q₈	1	2	yes	—
29	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
31	C₂×C₄×Dic₆₁	—	2	2	no	—
32	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
33	#14519710	C₈	1	8	yes	complemented
34	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
35	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
36	#14519684	—	2	2	no	—
37	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₆₁	—	2	2	no	—
40	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
41	C₂²×Dic₆₁	—	2	2	no	—
42	C₂²×Dic₆₁	C₂×D₄	1	1	yes	characteristic
43	C₂²×Dic₆₁	#41	1	1	yes	characteristic
44	C₂×#92420	—	2	4	no	—
45	C₂×#92420	—	2	2	no	—
46	C₂×#92420	—	2	1	no	—
47	C₂×#92420	—	2	1	no	—
48	C₂×#92420	—	2	4	no	—
49	C₂×#92420	—	4	2	no	—
50	C₂×#92420	—	4	2	no	—
51	C₂×#92420	C₄⋊C₄	1	2	yes	—
52	C₂×#92420	C₄⋊C₄	1	2	yes	—
53	C₂×#92420	D₈	1	2	yes	—
54	C₂×#92420	Q₁₆	1	2	yes	—
55	C₂×C₈×C₆₁	—	2	2	no	—
56	C₂×C₈×C₆₁	C₄²	1	2	yes	—
57	C₄⋊C₄×C₆₁	—	2	2	no	—
58	C₄⋊C₄×C₆₁	C₂×C₈	1	4	yes	—
59	C₄⋊Dic₆₁	—	2	2	no	—
60	C₄⋊Dic₆₁	C₂×C₈	1	4	yes	—
61	C₄×Dic₆₁	—	4	2	no	—
62	C₄×Dic₆₁	—	4	2	no	—
63	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
64	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
65	C₄×Dic₆₁	C₂×Q₈	1	1	yes	characteristic
66	C₄×Dic₆₁	#41	1	1	yes	characteristic
67	C₂×#92418	C₄²	1	2	yes	—
68	C₂×#92418	C₂×C₈	1	4	yes	—
69	C₂³×C₆₁	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₆₁	—	2	4	no	—
71	C₂×C₄×C₆₁	—	4	2	no	—
72	C₂×C₄×C₆₁	—	4	2	no	—
73	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
74	C₂×C₄×C₆₁	C₄×Q₈	1	1	yes	characteristic
75	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
76	C₂×C₄×C₆₁	#131	1	2	yes	—
77	C₂×Dic₆₁	—	2	2	no	—
78	C₂×Dic₆₁	—	2	4	no	—
79	C₂×Dic₆₁	—	4	2	no	—
80	C₂×Dic₆₁	—	4	2	no	—
81	C₂×Dic₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂×Dic₆₁	#117	1	1	yes	characteristic
83	C₂×Dic₆₁	C₄×D₄	1	1	yes	characteristic
84	C₂×Dic₆₁	C₄×Q₈	1	1	yes	characteristic
85	C₂×Dic₆₁	C₂×D₈	1	1	yes	characteristic
86	C₂×Dic₆₁	C₂×Q₁₆	1	1	yes	characteristic
87	C₂×Dic₆₁	#135	1	1	yes	characteristic
88	C₂×Dic₆₁	#135	1	1	yes	characteristic
89	#92420	—	2	4	no	—
90	#92420	—	2	2	no	—
91	#92420	—	4	4	no	—
92	#92420	#107	1	4	yes	complemented
93	C₈×C₆₁	—	2	2	no	—
94	#92418	C₄×C₈	1	4	yes	complemented
95	C₈×#107	—	61	1	no	maximal, Sylow
96	C₂²×C₆₁	C₄×C₄⋊C₄	1	1	yes	characteristic
97	C₂²×C₆₁	C₈×D₄	1	2	yes	—
98	C₂²×C₆₁	C₄×D₈	1	1	yes	characteristic
99	C₂²×C₆₁	C₄×Q₁₆	1	1	yes	characteristic
100	C₂²×C₆₁	#443	1	2	yes	—
101	C₄×C₆₁	—	4	4	no	—
102	C₄×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
103	C₄×C₆₁	#405	1	1	yes	characteristic
104	C₄×C₆₁	C₈×Q₈	1	2	yes	—
105	Dic₆₁	—	2	1	no	—
106	Dic₆₁	—	2	1	no	—
107	Dic₆₁	—	4	4	no	—
108	Dic₆₁	C₂×#107	1	1	yes	characteristic
109	Dic₆₁	#427	1	1	yes	characteristic
110	Dic₆₁	C₄×D₈	1	1	yes	characteristic
111	Dic₆₁	C₄×Q₁₆	1	1	yes	characteristic
112	C₄×#107	—	61	1	no	—
113	C₈×C₄⋊C₄	—	61	2	no	—
114	C₂×#335	—	61	1	no	—
115	C₂×C₈²	—	61	1	no	—
116	#1279	—	61	2	no	—
117	C₂×C₆₁	C₈×D₈	1	2	yes	—
118	C₂×C₆₁	C₈×Q₁₆	1	2	yes	—
119	C₂×C₆₁	C₈×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₆₁	#1772	1	1	yes	characteristic
121	C₂×C₆₁	C₄×#107	1	1	yes	characteristic
122	C₂×#107	—	61	2	no	—
123	C₂×C₄⋊C₈	—	61	2	no	—
124	C₂×C₄×C₈	—	61	1	no	—
125	C₂×C₄×C₈	—	61	1	no	—
126	C₂×C₄×C₈	—	61	1	no	—
127	C₂×C₄×C₈	—	122	2	no	—
128	C₄×C₄⋊C₄	—	61	2	no	—
129	#340	—	61	1	no	—
130	#340	#6923	1	1	yes	characteristic
131	#336	—	122	2	no	—
132	#335	—	61	4	no	—
133	C₈²	—	61	4	no	—
134	C₆₁	C₈×#107	1	1	yes	characteristic, complemented, Sylow
135	C₂²×C₈	—	61	1	no	—
136	C₂²×C₈	—	61	1	no	—
137	C₂²×C₈	—	61	1	no	—
138	C₂²×C₈	—	122	1	no	—
139	C₂²×C₈	—	122	2	no	—
140	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
141	C₂×C₄⋊C₄	—	61	2	no	—
142	C₂×C₄⋊C₄	C₂×#6923	1	2	yes	—
143	C₂×C₄²	—	61	1	no	—
144	C₂×C₄²	—	122	2	no	—
145	#107	—	61	8	no	—
146	C₄⋊C₈	—	61	4	no	—
147	C₄⋊C₈	—	122	2	no	—
148	C₄×C₈	—	61	2	no	—
149	C₄×C₈	—	61	2	no	—
150	C₄×C₈	—	61	2	no	—
151	C₄×C₈	—	61	2	no	—
152	C₄×C₈	—	61	2	no	—
153	C₄×C₈	—	122	1	no	—
154	C₄×C₈	—	244	4	no	—
155	#95	—	122	2	no	—
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	—	61	1	no	—
158	C₂²×C₄	—	61	1	no	—
159	C₂²×C₄	—	122	2	no	—
160	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	61	2	no	—
163	C₂×C₈	—	61	2	no	—
164	C₂×C₈	—	61	2	no	—
165	C₂×C₈	—	61	2	no	—
166	C₂×C₈	—	61	4	no	—
167	C₂×C₈	—	61	2	no	—
168	C₂×C₈	—	122	2	no	—
169	C₂×C₈	—	122	1	no	—
170	C₂×C₈	—	122	1	no	—
171	C₂×C₈	—	122	4	no	—
172	C₂×C₈	—	122	4	no	—
173	C₂×C₈	—	244	2	no	—
174	C₂×C₈	—	244	2	no	—
175	C₂×C₈	C₄×#6923	1	2	yes	—
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	61	4	no	—
178	C₄⋊C₄	—	122	2	no	—
179	C₄⋊C₄	D₆₁⋊C₈	1	4	yes	—
180	C₄²	—	61	1	no	—
181	C₄²	—	61	1	no	—
182	C₄²	—	61	1	no	—
183	C₄²	—	61	1	no	—
184	C₄²	—	244	2	no	—
185	C₄²	—	244	2	no	—
186	C₂³	D₄×#6923	1	1	yes	characteristic
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	61	1	no	—
191	C₂×C₄	—	61	1	no	—
192	C₂×C₄	—	61	1	no	—
193	C₂×C₄	—	61	1	no	—
194	C₂×C₄	—	61	1	no	—
195	C₂×C₄	—	61	1	no	—
196	C₂×C₄	—	61	1	no	—
197	C₂×C₄	—	61	1	no	—
198	C₂×C₄	—	122	2	no	—
199	C₂×C₄	—	122	4	no	—
200	C₂×C₄	—	244	2	no	—
201	C₂×C₄	—	244	2	no	—
202	C₂×C₄	C₂×C₄×#6923	1	1	yes	characteristic
203	C₂×C₄	Q₈×#6923	1	1	yes	characteristic
204	C₂×C₄	C₂×D₆₁⋊C₈	1	2	yes	—
205	C₂×C₄	#14519864	1	2	yes	—
206	C₈	—	2	2	no	—
207	C₈	—	61	4	no	—
208	C₈	—	61	4	no	—
209	C₈	—	122	2	no	—
210	C₈	—	122	4	no	—
211	C₈	—	244	4	no	—
212	C₂²	C₄⋊C₄×#6923	1	1	yes	characteristic
213	C₂²	#16840789	1	2	yes	—
214	C₂²	D₈×#6923	1	1	yes	characteristic
215	C₂²	Q₁₆×#6923	1	1	yes	characteristic
216	C₂²	#16840805	1	2	yes	—
217	C₄	—	4	4	no	—
218	C₄	—	61	1	no	—
219	C₄	—	61	1	no	—
220	C₄	—	61	1	no	—
221	C₄	—	61	1	no	—
222	C₄	—	122	1	no	—
223	C₄	—	122	1	no	—
224	C₄	—	244	4	no	—
225	C₄	C₄×D₆₁⋊C₈	1	1	yes	characteristic
226	C₄	#16840726	1	1	yes	characteristic
227	C₄	#16840811	1	2	yes	—
228	C₂	#21461399	1	2	yes	—
229	C₂	#21461406	1	2	yes	—
230	C₂	#21461611	1	1	yes	characteristic
231	C₂	#21461622	1	1	yes	characteristic
232	C₂	#107×#6923	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₄×#6923
commutator	C₄×C₆₁	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#6923
Fitting	C₆₁×#340	C₄
socle, prime radical	C₂³×C₆₁	C₄×D₄
cosocle	C₂×C₄×Dic₆₁	C₂³
hypercenter	#340	#6923
hypocenter	C₆₁	C₈×#107
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

Complements

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

N	H ≅ G/N	type
11. #16840626	C₄	semidirect
12. C₈×#92420	C₄	semidirect
33. #14519710	C₈	semidirect
92. #92420	#107	semidirect
94. #92418	C₄×C₈	semidirect
134. C₆₁	C₈×#107	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊C₄×#92420	2
C₂×#16840626	2
C₂×C₈×#92420	2
#21460655	2
#21445711	2
C₈×#107	61