Subgroups — #29590040

Order 15616.

430 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21447618	C₂	1	1	yes	maximal, characteristic, complemented
2	#21444578	C₂	1	1	yes	maximal, characteristic, complemented
3	#21443780	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×D₈×C₆₁	C₂	1	1	yes	maximal, characteristic
5	#16839786	—	2	1	no	—
6	C₄⋊#11758185	—	2	1	no	—
7	C₂²⋊#11758185	—	2	1	no	—
8	C₄×#11758185	C₂²	1	2	yes	complemented
9	#16839515	—	2	1	no	—
10	#16839515	C₂²	1	2	yes	complemented
11	C₂²×D₈×C₆₁	C₄	1	1	yes	characteristic
12	C₂×C₄×D₄×C₆₁	—	2	1	no	—
13	C₄×D₈×C₆₁	—	2	1	no	—
14	C₄×D₈×C₆₁	—	2	1	no	—
15	C₄×D₈×C₆₁	C₄	1	2	yes	—
16	C₄×D₈×C₆₁	C₂²	1	2	yes	—
17	C₂×#107×C₆₁	C₄	1	1	yes	characteristic
18	C₂×#102×C₆₁	—	2	1	no	—
19	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
20	C₂×#11758194	—	4	1	no	—
21	C₂×#11758185	—	2	2	no	—
22	C₄⋊#92418	—	2	2	no	—
23	C₄⋊#92418	—	4	1	no	—
24	C₂²×D₄×C₆₁	—	2	1	no	—
25	C₂³×C₄×C₆₁	—	4	1	no	—
26	C₂×D₈×C₆₁	—	2	4	no	—
27	C₂×D₈×C₆₁	—	2	1	no	—
28	C₂×D₈×C₆₁	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
31	C₄×D₄×C₆₁	—	2	2	no	—
32	C₄×D₄×C₆₁	—	2	2	no	—
33	C₄×D₄×C₆₁	—	4	1	no	—
34	C₄×D₄×C₆₁	—	4	1	no	—
35	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
36	C₂×C₂²⋊C₄×C₆₁	—	4	1	no	—
37	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
38	#107×C₆₁	—	2	1	no	—
39	#107×C₆₁	C₂×C₄	1	2	yes	—
40	#102×C₆₁	—	2	2	no	—
41	#102×C₆₁	—	2	2	no	—
42	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
44	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
45	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
46	C₂⁴×C₆₁	—	4	1	no	—
47	C₂×D₄×C₆₁	—	2	4	no	—
48	C₂×D₄×C₆₁	—	2	2	no	—
49	C₂×D₄×C₆₁	—	4	2	no	—
50	C₂×D₄×C₆₁	—	4	1	no	—
51	C₂²×C₄×C₆₁	—	4	1	no	—
52	C₂²×C₄×C₆₁	—	4	2	no	—
53	C₂²×C₄×C₆₁	—	4	2	no	—
54	C₂²×C₄×C₆₁	—	8	2	no	—
55	C₂²×C₄×C₆₁	—	8	2	no	—
56	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
58	C₂²×C₄×C₆₁	SD₁₆	1	1	yes	characteristic
59	D₈×C₆₁	—	2	4	no	—
60	D₈×C₆₁	—	2	2	no	—
61	D₈×C₆₁	M₁₆	1	4	yes	complemented
62	C₂×C₈×C₆₁	—	2	2	no	—
63	C₂×C₈×C₆₁	—	4	1	no	—
64	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
68	C₄⋊C₄×C₆₁	—	2	2	no	—
69	C₄⋊C₄×C₆₁	—	4	1	no	—
70	C₂²⋊C₄×C₆₁	—	4	2	no	—
71	C₂²⋊C₄×C₆₁	—	4	2	no	—
72	C₄²×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
73	C₄²×C₆₁	D₈	1	1	yes	characteristic
74	C₄²×C₆₁	SD₁₆	1	1	yes	characteristic
75	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
76	#11758194	—	8	2	no	—
77	C₂×#92418	—	4	2	no	—
78	#11758185	—	4	2	no	—
79	C₂³×C₆₁	—	4	4	no	—
80	C₂³×C₆₁	—	4	2	no	—
81	C₂³×C₆₁	—	8	2	no	—
82	C₂³×C₆₁	—	8	2	no	—
83	C₂³×C₆₁	#102	1	1	yes	characteristic
84	D₄×C₆₁	—	2	4	no	—
85	D₄×C₆₁	—	4	4	no	—
86	D₄×C₆₁	—	4	2	no	—
87	C₂×C₄×C₆₁	—	2	1	no	—
88	C₂×C₄×C₆₁	—	2	1	no	—
89	C₂×C₄×C₆₁	—	4	1	no	—
90	C₂×C₄×C₆₁	—	4	2	no	—
91	C₂×C₄×C₆₁	—	8	2	no	—
92	C₂×C₄×C₆₁	—	8	4	no	—
93	C₂×C₄×C₆₁	—	8	2	no	—
94	C₂×C₄×C₆₁	—	8	2	no	—
95	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
96	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
97	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
98	C₂×C₄×C₆₁	D₁₆	1	2	yes	—
99	C₂×C₄×C₆₁	SD₃₂	1	2	yes	—
100	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
102	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
103	C₈×C₆₁	—	2	1	no	—
104	C₈×C₆₁	—	2	1	no	—
105	C₈×C₆₁	—	2	1	no	—
106	C₈×C₆₁	C₂×M₁₆	1	2	yes	—
107	#92418	—	8	2	no	—
108	#11657	—	61	1	no	maximal, Sylow
109	C₂²×C₆₁	—	2	1	no	—
110	C₂²×C₆₁	—	2	1	no	—
111	C₂²×C₆₁	—	4	4	no	—
112	C₂²×C₆₁	—	8	8	no	—
113	C₂²×C₆₁	—	8	2	no	—
114	C₂²×C₆₁	—	8	2	no	—
115	C₂²×C₆₁	#357	1	2	yes	—
116	C₂²×C₆₁	C₂×#102	1	1	yes	characteristic
117	C₄×C₆₁	—	2	1	no	—
118	C₄×C₆₁	—	2	1	no	—
119	C₄×C₆₁	—	2	1	no	—
120	C₄×C₆₁	—	8	2	no	—
121	C₄×C₆₁	#357	1	2	yes	—
122	C₄×C₆₁	#407	1	1	yes	characteristic
123	C₄×C₆₁	#421	1	1	yes	characteristic
124	C₄×C₆₁	C₂×D₁₆	1	1	yes	characteristic
125	C₄×C₆₁	C₂×SD₃₂	1	1	yes	characteristic
126	C₂×C₄×D₈	D₆₁	1	1	yes	characteristic
127	#2162	—	61	1	no	—
128	#1567	—	61	1	no	—
129	#1331	—	61	4	no	—
130	C₂×C₆₁	—	2	2	no	—
131	C₂×C₆₁	—	8	4	no	—
132	C₂×C₆₁	#1486	1	1	yes	characteristic
133	C₂×C₆₁	C₂×#357	1	1	yes	characteristic
134	C₂×C₆₁	#2143	1	1	yes	characteristic
135	C₂²×D₈	Dic₆₁	1	1	yes	characteristic
136	C₂×C₄×D₄	—	2	1	no	—
137	C₄×D₈	—	2	1	no	—
138	C₄×D₈	—	2	1	no	—
139	C₄×D₈	Dic₆₁	1	2	yes	—
140	C₄×D₈	C₂×D₆₁	1	2	yes	—
141	C₂×#107	Dic₆₁	1	1	yes	characteristic
142	#423	—	122	1	no	—
143	C₂×#102	—	2	1	no	—
144	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
145	C₄⋊C₁₆	—	122	1	no	—
146	C₂²⋊C₁₆	—	122	1	no	—
147	C₄×C₁₆	—	61	2	no	—
148	#335	—	61	2	no	—
149	#335	—	122	1	no	—
150	C₆₁	#11657	1	1	yes	characteristic, complemented, Sylow
151	C₂²×D₄	—	2	1	no	—
152	C₂³×C₄	—	4	1	no	—
153	C₂×D₈	—	2	4	no	—
154	C₂×D₈	—	2	1	no	—
155	C₂×D₈	C₂×Dic₆₁	1	2	yes	—
156	C₂×M₁₆	—	244	1	no	—
157	C₂²×C₈	C₂×Dic₆₁	1	1	yes	characteristic
158	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
159	C₄×D₄	—	2	2	no	—
160	C₄×D₄	—	2	2	no	—
161	C₄×D₄	—	4	1	no	—
162	C₄×D₄	—	4	1	no	—
163	C₂×C₄⋊C₄	—	2	1	no	—
164	C₂×C₂²⋊C₄	—	4	1	no	—
165	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
166	C₂×C₁₆	—	122	2	no	—
167	#107	—	2	1	no	—
168	#107	C₂×Dic₆₁	1	2	yes	—
169	C₄⋊C₈	—	122	2	no	—
170	C₄⋊C₈	—	244	1	no	—
171	#102	—	2	2	no	—
172	#102	—	2	2	no	—
173	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
174	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
175	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
176	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
177	C₂⁴	—	4	1	no	—
178	C₂×D₄	—	2	4	no	—
179	C₂×D₄	—	2	2	no	—
180	C₂×D₄	—	4	2	no	—
181	C₂×D₄	—	4	1	no	—
182	C₂²×C₄	—	4	1	no	—
183	C₂²×C₄	—	4	2	no	—
184	C₂²×C₄	—	4	2	no	—
185	C₂²×C₄	—	8	2	no	—
186	C₂²×C₄	—	8	2	no	—
187	C₂²×C₄	#11758199	1	1	yes	characteristic
188	C₂²×C₄	#11758200	1	1	yes	characteristic
189	C₂²×C₄	C₂²⋊Dic₆₁	1	1	yes	characteristic
190	D₈	—	2	4	no	—
191	D₈	—	2	2	no	—
192	D₈	#11758194	1	4	yes	complemented
193	M₁₆	—	488	2	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	4	1	no	—
196	C₂×C₈	—	244	2	no	—
197	C₂×C₈	C₂²⋊Dic₆₁	1	1	yes	characteristic
198	C₂×C₈	C₂²⋊Dic₆₁	1	1	yes	characteristic
199	C₂×C₈	C₂²×Dic₆₁	1	1	yes	characteristic
200	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
201	C₄⋊C₄	—	2	2	no	—
202	C₄⋊C₄	—	4	1	no	—
203	C₂²⋊C₄	—	4	2	no	—
204	C₂²⋊C₄	—	4	2	no	—
205	C₄²	#11758199	1	1	yes	characteristic
206	C₄²	#11758200	1	1	yes	characteristic
207	C₄²	C₂²⋊Dic₆₁	1	1	yes	characteristic
208	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
209	C₁₆	—	244	2	no	—
210	C₂³	—	4	4	no	—
211	C₂³	—	4	2	no	—
212	C₂³	—	8	2	no	—
213	C₂³	—	8	2	no	—
214	C₂³	#14519698	1	1	yes	characteristic
215	D₄	—	2	4	no	—
216	D₄	—	4	4	no	—
217	D₄	—	4	2	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	4	1	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	8	2	no	—
223	C₂×C₄	—	8	4	no	—
224	C₂×C₄	—	8	2	no	—
225	C₂×C₄	—	8	2	no	—
226	C₂×C₄	#14519698	1	1	yes	characteristic
227	C₂×C₄	#14519698	1	1	yes	characteristic
228	C₂×C₄	#14519698	1	1	yes	characteristic
229	C₂×C₄	#14519718	1	2	yes	—
230	C₂×C₄	#14519721	1	2	yes	—
231	C₂×C₄	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
232	C₂×C₄	C₂×#11758199	1	1	yes	characteristic
233	C₂×C₄	C₂×#11758200	1	1	yes	characteristic
234	C₈	—	2	1	no	—
235	C₈	—	2	1	no	—
236	C₈	—	2	1	no	—
237	C₈	—	488	2	no	—
238	C₈	C₂×#11758194	1	2	yes	—
239	C₂²	—	2	1	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	4	4	no	—
242	C₂²	—	8	8	no	—
243	C₂²	—	8	2	no	—
244	C₂²	—	8	2	no	—
245	C₂²	#16839563	1	2	yes	—
246	C₂²	C₂×#14519698	1	1	yes	characteristic
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	2	1	no	—
250	C₄	—	8	2	no	—
251	C₄	#16839563	1	2	yes	—
252	C₄	#16839726	1	1	yes	characteristic
253	C₄	#16839779	1	1	yes	characteristic
254	C₄	C₂×#14519718	1	1	yes	characteristic
255	C₄	C₂×#14519721	1	1	yes	characteristic
256	C₂	—	2	2	no	—
257	C₂	—	8	4	no	—
258	C₂	#21444244	1	1	yes	characteristic
259	C₂	C₂×#16839563	1	1	yes	characteristic
260	C₂	#21447547	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#11758199
commutator	C₂×C₈×C₆₁	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₆₁
Fitting	C₂×C₄×D₈×C₆₁	C₂
socle	C₂²×C₆₁	C₂×#102
prime radical	C₂²×D₈×C₆₁	C₄
cosocle	C₄×C₈×C₆₁	C₂³
hypercenter	C₂×C₄×D₈	D₆₁
hypocenter	C₆₁	#11657
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
1. #21447618	C₂	semidirect
2. #21444578	C₂	semidirect
3. #21443780	C₂	semidirect
8. C₄×#11758185	C₂²	semidirect
10. #16839515	C₂²	semidirect
61. D₈×C₆₁	M₁₆	semidirect
150. C₆₁	#11657	semidirect
192. D₈	#11758194	semidirect

Maximal subgroups

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

subgroup	index
#21447618	2
#21444578	2
#21443780	2
C₂×C₄×D₈×C₆₁	2
#11657	61