Subgroups — #29595235

Order 15616.

462 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₂₄₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#16839565	C₂	1	2	yes	maximal
3	C₂×#16839538	C₂	1	1	yes	maximal, characteristic
4	#21443910	C₂	1	2	yes	maximal
5	C₄×#107×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₄₄	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₁₂₂	—	2	1	no	—
8	C₂²×#11758185	C₂²	1	2	yes	—
9	C₄×Dic₂₄₄	—	2	2	no	—
10	C₄×Dic₂₄₄	C₄	1	4	yes	complemented
11	C₂×#14519711	C₂²	1	1	yes	characteristic
12	C₂×#14519702	—	2	1	no	—
13	#16839565	C₄	1	8	yes	complemented
14	#16839538	—	2	2	no	—
15	C₂×#107×C₆₁	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₄×C₄⋊C₄×C₆₁	—	2	1	no	—
18	C₆₁×#340	—	2	1	no	—
19	C₂²×Dic₁₂₂	—	2	1	no	—
20	C₂×Dic₂₄₄	—	2	1	no	—
21	C₂×Dic₂₄₄	—	2	2	no	—
22	C₂×Dic₂₄₄	C₂×C₄	1	4	yes	—
23	C₂×Dic₂₄₄	C₂×C₄	1	2	yes	—
24	C₄×Dic₁₂₂	—	2	4	no	—
25	C₄×Dic₁₂₂	—	4	2	no	—
26	C₂×C₄⋊Dic₆₁	—	2	1	no	—
27	C₂×#11758196	—	4	1	no	—
28	C₂×C₄×Dic₆₁	—	4	1	no	—
29	C₂×#11758185	—	2	2	no	—
30	C₂×#11758185	—	2	2	no	—
31	C₂×#11758185	C₂×C₄	1	4	yes	—
32	#14519711	—	2	1	no	—
33	#14519711	C₂×C₄	1	2	yes	—
34	#14519702	—	2	4	no	—
35	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
37	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
38	C₂×C₄²×C₆₁	—	4	1	no	—
39	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
40	#107×C₆₁	—	2	2	no	—
41	#107×C₆₁	C₂×C₄	1	4	yes	—
42	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
43	C₄×C₈×C₆₁	D₄	1	2	yes	—
44	#95×C₆₁	—	4	1	no	—
45	C₂²×C₄×C₆₁	—	4	2	no	—
46	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
47	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
48	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
49	C₂²×Dic₆₁	—	4	1	no	—
50	C₂²×Dic₆₁	—	4	1	no	—
51	C₂×Dic₁₂₂	—	2	2	no	—
52	C₂×Dic₁₂₂	—	2	4	no	—
53	C₂×Dic₁₂₂	—	4	1	no	—
54	C₂×Dic₁₂₂	—	4	2	no	—
55	C₂×C₈×C₆₁	—	2	2	no	—
56	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
57	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
59	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₆₁	#41	1	2	yes	—
61	C₄⋊C₄×C₆₁	—	2	4	no	—
62	C₄⋊C₄×C₆₁	—	4	2	no	—
63	C₄²×C₆₁	—	8	2	no	—
64	C₄²×C₆₁	C₂²⋊C₄	1	2	yes	—
65	C₄²×C₆₁	SD₁₆	1	2	yes	—
66	C₄⋊Dic₆₁	—	2	2	no	—
67	C₄⋊Dic₆₁	—	4	1	no	—
68	#11758196	—	4	4	no	—
69	C₄×Dic₆₁	—	4	4	no	—
70	Dic₂₄₄	—	2	2	no	—
71	Dic₂₄₄	—	2	4	no	—
72	Dic₂₄₄	C₄²	1	4	yes	complemented
73	#11758185	—	2	4	no	—
74	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
75	C₂×C₄×C₆₁	—	2	1	no	—
76	C₂×C₄×C₆₁	—	2	1	no	—
77	C₂×C₄×C₆₁	—	2	2	no	—
78	C₂×C₄×C₆₁	—	4	4	no	—
79	C₂×C₄×C₆₁	—	8	2	no	—
80	C₂×C₄×C₆₁	—	8	2	no	—
81	C₂×C₄×C₆₁	#102	1	2	yes	—
82	C₂×C₄×C₆₁	#102	1	2	yes	—
83	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
85	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
86	C₂×C₄×C₆₁	#135	1	2	yes	—
87	C₂×Dic₆₁	—	4	2	no	—
88	C₂×Dic₆₁	—	4	4	no	—
89	C₂×Dic₆₁	—	4	2	no	—
90	C₂×Dic₆₁	—	8	1	no	—
91	C₂×Dic₆₁	—	8	1	no	—
92	Dic₁₂₂	—	2	4	no	—
93	Dic₁₂₂	—	4	2	no	—
94	Dic₁₂₂	—	4	4	no	—
95	C₈×C₆₁	—	2	2	no	—
96	C₈×C₆₁	C₂×C₄²	1	1	yes	characteristic
97	C₈×C₆₁	#117	1	1	yes	characteristic
98	C₈×C₆₁	C₄×D₄	1	2	yes	—
99	#12638	—	61	1	no	maximal, Sylow
100	C₂²×C₆₁	C₂×#102	1	1	yes	characteristic
101	C₂²×C₆₁	C₄×D₈	1	2	yes	—
102	C₂²×C₆₁	#509	1	1	yes	characteristic
103	C₂²×C₆₁	#510	1	2	yes	—
104	C₂²×C₆₁	#510	1	1	yes	characteristic
105	C₄×C₆₁	—	2	2	no	—
106	C₄×C₆₁	—	2	2	no	—
107	C₄×C₆₁	—	8	4	no	—
108	C₄×C₆₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
109	C₄×C₆₁	#416	1	1	yes	characteristic
110	C₄×C₆₁	C₄×SD₁₆	1	2	yes	—
111	Dic₆₁	—	4	4	no	—
112	Dic₆₁	—	8	2	no	—
113	C₂×C₄×Q₁₆	—	61	1	no	—
114	C₂×#358	—	61	2	no	—
115	C₂×#346	—	61	1	no	—
116	C₄×#107	D₆₁	1	1	yes	characteristic
117	#1380	—	61	2	no	—
118	C₂×C₆₁	C₄×#102	1	1	yes	characteristic
119	C₂×C₆₁	#2139	1	1	yes	characteristic
120	C₂×C₆₁	#2141	1	1	yes	characteristic
121	C₂×C₆₁	#2175	1	2	yes	—
122	C₂×C₆₁	#2176	1	2	yes	—
123	C₂²×Q₁₆	—	61	1	no	—
124	C₂×C₄×Q₈	—	122	1	no	—
125	C₂²×C₁₆	—	61	2	no	—
126	C₄×Q₁₆	—	61	4	no	—
127	C₄×Q₁₆	—	122	2	no	—
128	C₂×#107	—	61	1	no	—
129	C₂×#107	C₂×D₆₁	1	2	yes	—
130	C₂×#103	—	122	1	no	—
131	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
132	C₄×C₄⋊C₄	—	2	1	no	—
133	#358	—	61	8	no	—
134	#346	—	122	2	no	—
135	#340	—	2	1	no	—
136	C₆₁	#12638	1	1	yes	characteristic, complemented, Sylow
137	C₂²×Q₈	—	122	1	no	—
138	C₂×Q₁₆	—	61	4	no	—
139	C₂×Q₁₆	—	61	2	no	—
140	C₂×Q₁₆	—	122	1	no	—
141	C₂×Q₁₆	—	122	2	no	—
142	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
143	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
144	C₄×Q₈	—	122	4	no	—
145	C₄×Q₈	—	244	2	no	—
146	C₂×C₄⋊C₄	—	2	2	no	—
147	C₂×C₄⋊C₄	—	122	1	no	—
148	C₂×C₄⋊C₄	—	244	1	no	—
149	C₂×C₄²	—	4	1	no	—
150	C₂×C₄²	—	244	1	no	—
151	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
152	C₂×C₁₆	—	61	4	no	—
153	C₂×C₁₆	—	122	2	no	—
154	C₂×C₁₆	—	122	2	no	—
155	#107	—	2	2	no	—
156	#107	—	61	2	no	—
157	#107	—	122	1	no	—
158	#107	C₄×D₆₁	1	4	yes	—
159	#103	—	122	4	no	—
160	C₄×C₈	C₄×D₆₁	1	2	yes	—
161	C₄×C₈	D₂₄₄	1	2	yes	—
162	#95	—	4	1	no	—
163	C₂×Q₈	—	122	2	no	—
164	C₂×Q₈	—	122	4	no	—
165	C₂×Q₈	—	244	1	no	—
166	C₂×Q₈	—	244	2	no	—
167	C₂²×C₄	—	4	2	no	—
168	C₂²×C₄	—	244	1	no	—
169	C₂²×C₄	—	244	1	no	—
170	C₂²×C₄	#11758199	1	1	yes	characteristic
171	C₂²×C₄	#11758199	1	1	yes	characteristic
172	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
173	Q₁₆	—	61	4	no	—
174	Q₁₆	—	122	2	no	—
175	Q₁₆	—	122	4	no	—
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	#11758198	1	2	yes	—
178	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
179	C₂×C₈	C₂×C₄×D₆₁	1	2	yes	—
180	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
181	C₂×C₈	#11758222	1	2	yes	—
182	C₄⋊C₄	—	2	4	no	—
183	C₄⋊C₄	—	4	2	no	—
184	C₄⋊C₄	—	122	2	no	—
185	C₄⋊C₄	—	244	4	no	—
186	C₄⋊C₄	—	244	1	no	—
187	C₄²	—	8	2	no	—
188	C₄²	—	244	4	no	—
189	C₄²	#11758198	1	2	yes	—
190	C₄²	#11758201	1	2	yes	—
191	C₁₆	—	122	4	no	—
192	C₂³	C₂×#11758199	1	1	yes	characteristic
193	Q₈	—	122	4	no	—
194	Q₈	—	244	2	no	—
195	Q₈	—	244	4	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	4	4	no	—
200	C₂×C₄	—	8	2	no	—
201	C₂×C₄	—	8	2	no	—
202	C₂×C₄	—	488	1	no	—
203	C₂×C₄	—	488	1	no	—
204	C₂×C₄	—	244	2	no	—
205	C₂×C₄	—	244	2	no	—
206	C₂×C₄	—	244	4	no	—
207	C₂×C₄	#14519697	1	2	yes	—
208	C₂×C₄	#14519697	1	2	yes	—
209	C₂×C₄	C₂×#11758198	1	1	yes	characteristic
210	C₂×C₄	C₄×C₂²⋊D₆₁	1	2	yes	—
211	C₂×C₄	C₂×#11758201	1	1	yes	characteristic
212	C₂×C₄	#14519797	1	2	yes	—
213	C₈	—	2	2	no	—
214	C₈	C₄²×D₆₁	1	1	yes	characteristic
215	C₈	#14519734	1	1	yes	characteristic
216	C₈	C₄×D₂₄₄	1	2	yes	—
217	C₂²	C₂×#14519697	1	1	yes	characteristic
218	C₂²	C₄×#11758199	1	2	yes	—
219	C₂²	#16840198	1	1	yes	characteristic
220	C₂²	#16840202	1	2	yes	—
221	C₂²	#16840203	1	1	yes	characteristic
222	C₄	—	2	2	no	—
223	C₄	—	2	2	no	—
224	C₄	—	8	4	no	—
225	C₄	—	488	2	no	—
226	C₄	—	244	4	no	—
227	C₄	C₄×#11758198	1	1	yes	characteristic
228	C₄	#16839756	1	1	yes	characteristic
229	C₄	C₄×#11758201	1	2	yes	—
230	C₂	C₄×#14519697	1	1	yes	characteristic
231	C₂	#21447529	1	1	yes	characteristic
232	C₂	#21447538	1	1	yes	characteristic
233	C₂	#21447683	1	2	yes	—
234	C₂	#21447688	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#11758199
commutator	C₈×C₆₁	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₄×#107×C₆₁	C₂
socle, prime radical	C₂³×C₆₁	C₂×D₈
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	C₄×#107	D₆₁
hypocenter	C₆₁	#12638
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#107

Complements

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

N	H ≅ G/N	type
10. C₄×Dic₂₄₄	C₄	semidirect
13. #16839565	C₄	semidirect
72. Dic₂₄₄	C₄²	semidirect
136. C₆₁	#12638	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₂₄₄	2
C₂×#16839565	2
C₂×#16839538	2
#21443910	2
C₄×#107×C₆₁	2
#12638	61