Subgroups — C₂×#21445436 #29667707

Order 15616.

878 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16840167	C₂	1	1	yes	maximal, characteristic
2	C₂×#16840165	C₂	1	2	yes	maximal
3	C₂×#16840141	C₂	1	2	yes	maximal
4	#21445436	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₈×#92418	C₂	1	1	yes	maximal, characteristic
6	C₂×C₆₁×#500	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11758202	—	2	4	no	—
8	C₂×#14519777	C₂²	1	2	yes	—
9	#16840167	—	2	2	no	—
10	#16840167	C₂²	1	4	yes	—
11	#16840165	—	2	4	no	—
12	#16840165	C₂²	1	8	yes	—
13	#16840141	—	2	4	no	—
14	#16840141	C₂²	1	8	yes	—
15	C₂×#14519711	—	2	2	no	—
16	C₂×#14519710	—	2	4	no	—
17	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
18	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
19	C₈×#92418	C₂²	1	4	yes	—
20	C₂×C₄⋊Q₈×C₆₁	C₂²	1	2	yes	—
21	C₆₁×#500	—	2	2	no	—
22	C₆₁×#500	C₂²	1	4	yes	—
23	C₂×#107×C₆₁	—	2	2	no	—
24	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
25	C₂²×Dic₁₂₂	—	2	2	no	—
26	C₂×#11758202	—	2	16	no	—
27	C₂×#11758202	—	4	16	no	—
28	C₂²×#92418	D₄	1	2	yes	—
29	C₂²×#92418	D₄	1	2	yes	—
30	#14519777	—	2	2	no	—
31	#14519777	—	2	2	no	—
32	#14519777	—	2	2	no	—
33	#14519777	C₂³	1	4	yes	—
34	C₂×C₄⋊Dic₆₁	—	2	2	no	—
35	C₂×C₄⋊Dic₆₁	—	2	2	no	—
36	#14519711	—	2	8	no	—
37	#14519710	—	2	16	no	—
38	C₄×#92418	D₄	1	2	yes	—
39	C₄×#92418	D₄	1	2	yes	—
40	C₄×#92418	C₂³	1	2	yes	—
41	C₄×#92418	C₂³	1	2	yes	—
42	C₂²×Q₈×C₆₁	—	2	2	no	—
43	C₂²×C₈×C₆₁	D₄	1	2	yes	—
44	C₄⋊Q₈×C₆₁	—	2	2	no	—
45	C₄⋊Q₈×C₆₁	—	2	2	no	—
46	C₄⋊Q₈×C₆₁	—	2	2	no	—
47	C₄⋊Q₈×C₆₁	C₂³	1	4	yes	—
48	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
49	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
50	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
51	#107×C₆₁	—	2	8	no	—
52	C₄×C₈×C₆₁	D₄	1	2	yes	—
53	C₄×C₈×C₆₁	C₂³	1	2	yes	—
54	C₂×Q₈×C₆₁	—	2	4	no	—
55	C₂×Q₈×C₆₁	—	4	2	no	—
56	C₂×Q₈×C₆₁	—	4	8	no	—
57	C₂²×C₄×C₆₁	—	4	2	no	—
58	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
61	C₂²×Dic₆₁	—	4	2	no	—
62	C₂×Dic₁₂₂	—	2	4	no	—
63	C₂×Dic₁₂₂	—	4	8	no	—
64	C₂×Dic₁₂₂	—	4	2	no	—
65	C₂×C₈×C₆₁	Q₁₆	1	8	yes	complemented
66	C₂×C₈×C₆₁	C₂×D₄	1	4	yes	—
67	C₄⋊C₄×C₆₁	—	2	4	no	—
68	C₄⋊C₄×C₆₁	—	2	4	no	—
69	C₄⋊C₄×C₆₁	—	4	2	no	—
70	C₄⋊C₄×C₆₁	—	4	2	no	—
71	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
75	#11758202	—	4	32	no	—
76	C₄⋊Dic₆₁	—	2	4	no	—
77	C₄⋊Dic₆₁	—	2	4	no	—
78	C₄⋊Dic₆₁	—	4	2	no	—
79	C₄⋊Dic₆₁	—	4	2	no	—
80	C₂×#92418	D₈	1	8	yes	—
81	C₂×#92418	Q₁₆	1	8	yes	complemented
82	C₂×#92418	C₂×D₄	1	4	yes	—
83	C₂×#92418	C₂×D₄	1	4	yes	—
84	C₂³×C₆₁	#127	1	1	yes	characteristic
85	Q₈×C₆₁	—	4	8	no	—
86	Q₈×C₆₁	—	8	4	no	—
87	C₂×C₄×C₆₁	—	4	4	no	—
88	C₂×C₄×C₆₁	—	8	4	no	—
89	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
90	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
91	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
92	C₂×C₄×C₆₁	C₂×D₈	1	4	yes	—
93	C₂×C₄×C₆₁	C₂×Q₁₆	1	4	yes	—
94	C₂×C₄×C₆₁	C₂×Q₁₆	1	4	yes	—
95	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
96	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
97	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
98	C₂×Dic₆₁	—	4	4	no	—
99	C₂×Dic₆₁	—	8	4	no	—
100	Dic₁₂₂	—	4	8	no	—
101	Dic₁₂₂	—	8	4	no	—
102	C₈×C₆₁	C₂×Q₁₆	1	8	yes	complemented
103	#92418	C₂×D₈	1	8	yes	—
104	#92418	C₂×Q₁₆	1	8	yes	complemented
105	C₂×#1713	—	61	1	no	maximal, Sylow
106	C₂²×C₆₁	#493	1	2	yes	—
107	C₂²×C₆₁	#494	1	2	yes	—
108	C₂²×C₆₁	#494	1	2	yes	—
109	C₂²×C₆₁	C₂×#127	1	1	yes	characteristic
110	C₄×C₆₁	—	8	4	no	—
111	C₄×C₆₁	#493	1	2	yes	—
112	C₄×C₆₁	#494	1	2	yes	—
113	C₄×C₆₁	#494	1	2	yes	—
114	C₄×C₆₁	C₂²×D₈	1	2	yes	—
115	C₄×C₆₁	C₂²×Q₁₆	1	2	yes	—
116	C₄×C₆₁	C₂²×Q₁₆	1	2	yes	—
117	Dic₆₁	—	8	4	no	—
118	C₂×#500	—	61	1	no	—
119	C₂×#500	—	61	2	no	—
120	C₂×#500	D₆₁	1	1	yes	characteristic
121	C₂×#494	—	61	2	no	—
122	#1713	—	61	8	no	—
123	C₂×C₈²	—	61	1	no	—
124	C₂×C₆₁	#1713	1	4	yes	complemented
125	C₂×C₆₁	C₂×#493	1	1	yes	characteristic
126	C₂×C₆₁	C₂×#494	1	1	yes	characteristic
127	C₂×C₆₁	C₂×#494	1	1	yes	characteristic
128	C₂²×Q₁₆	—	122	4	no	—
129	C₂×C₄⋊Q₈	—	61	2	no	—
130	C₂×C₄⋊Q₈	C₂×D₆₁	1	2	yes	—
131	#500	—	2	2	no	—
132	#500	—	61	8	no	—
133	#500	—	61	4	no	—
134	#500	—	122	4	no	—
135	#500	—	122	2	no	—
136	#500	C₂×D₆₁	1	4	yes	—
137	#494	—	61	8	no	—
138	#494	—	122	4	no	—
139	C₂×#107	—	2	2	no	—
140	C₂×#107	—	122	4	no	—
141	C₂×#107	—	122	2	no	—
142	C₂×C₄×C₈	—	61	1	no	—
143	C₂×C₄×C₈	—	61	1	no	—
144	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
145	C₈²	—	61	4	no	—
146	C₆₁	C₂×#1713	1	1	yes	characteristic, complemented, Sylow
147	C₂²×Q₈	—	2	2	no	—
148	C₂²×Q₈	—	122	2	no	—
149	C₂×Q₁₆	—	122	16	no	—
150	C₂×Q₁₆	—	244	16	no	—
151	C₂²×C₈	—	61	2	no	—
152	C₂²×C₈	—	61	2	no	—
153	C₂²×C₈	C₂²⋊D₆₁	1	2	yes	—
154	C₄⋊Q₈	—	2	2	no	—
155	C₄⋊Q₈	—	2	2	no	—
156	C₄⋊Q₈	—	2	2	no	—
157	C₄⋊Q₈	—	61	4	no	—
158	C₄⋊Q₈	—	122	2	no	—
159	C₄⋊Q₈	—	122	2	no	—
160	C₄⋊Q₈	—	122	2	no	—
161	C₄⋊Q₈	C₂²×D₆₁	1	4	yes	—
162	C₂×C₄⋊C₄	—	2	2	no	—
163	C₂×C₄⋊C₄	—	2	2	no	—
164	C₂×C₄⋊C₄	—	122	2	no	—
165	C₂×C₄⋊C₄	—	122	2	no	—
166	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
167	#107	—	2	8	no	—
168	#107	—	122	16	no	—
169	#107	—	122	8	no	—
170	C₄×C₈	—	61	2	no	—
171	C₄×C₈	—	61	2	no	—
172	C₄×C₈	—	61	2	no	—
173	C₄×C₈	—	61	2	no	—
174	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
175	C₄×C₈	C₂²×D₆₁	1	2	yes	—
176	C₂×Q₈	—	2	4	no	—
177	C₂×Q₈	—	4	2	no	—
178	C₂×Q₈	—	4	8	no	—
179	C₂×Q₈	—	122	4	no	—
180	C₂×Q₈	—	244	8	no	—
181	C₂×Q₈	—	244	2	no	—
182	C₂²×C₄	—	4	2	no	—
183	C₂²×C₄	—	244	2	no	—
184	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
185	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
186	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
187	Q₁₆	—	244	32	no	—
188	C₂×C₈	—	61	4	no	—
189	C₂×C₈	—	61	4	no	—
190	C₂×C₈	—	61	8	no	—
191	C₂×C₈	—	61	8	no	—
192	C₂×C₈	#11758202	1	8	yes	complemented
193	C₂×C₈	C₂×C₂²⋊D₆₁	1	4	yes	—
194	C₄⋊C₄	—	2	4	no	—
195	C₄⋊C₄	—	2	4	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₄⋊C₄	—	4	2	no	—
198	C₄⋊C₄	—	122	4	no	—
199	C₄⋊C₄	—	122	4	no	—
200	C₄⋊C₄	—	244	2	no	—
201	C₄⋊C₄	—	244	2	no	—
202	C₄²	D₄×D₆₁	1	1	yes	characteristic
203	C₄²	D₄×D₆₁	1	1	yes	characteristic
204	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
205	C₄²	C₂³×D₆₁	1	1	yes	characteristic
206	C₂³	#14519773	1	1	yes	characteristic
207	Q₈	—	4	8	no	—
208	Q₈	—	8	4	no	—
209	Q₈	—	488	4	no	—
210	Q₈	—	244	8	no	—
211	C₂×C₄	—	4	4	no	—
212	C₂×C₄	—	8	4	no	—
213	C₂×C₄	—	488	4	no	—
214	C₂×C₄	—	244	4	no	—
215	C₂×C₄	#14519773	1	1	yes	characteristic
216	C₂×C₄	#14519773	1	1	yes	characteristic
217	C₂×C₄	#14519773	1	1	yes	characteristic
218	C₂×C₄	D₈×D₆₁	1	4	yes	—
219	C₂×C₄	Q₁₆×D₆₁	1	4	yes	—
220	C₂×C₄	C₂×#11758202	1	4	yes	—
221	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
222	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
223	C₂×C₄	C₂²×C₂²⋊D₆₁	1	1	yes	characteristic
224	C₈	—	61	8	no	—
225	C₈	—	61	8	no	—
226	C₈	C₂×#11758202	1	8	yes	complemented
227	C₂²	#16840137	1	2	yes	—
228	C₂²	#16840140	1	2	yes	—
229	C₂²	#16840141	1	2	yes	—
230	C₂²	C₂×#14519773	1	1	yes	characteristic
231	C₄	—	8	4	no	—
232	C₄	—	488	4	no	—
233	C₄	#16840137	1	2	yes	—
234	C₄	#16840140	1	2	yes	—
235	C₄	#16840141	1	2	yes	—
236	C₄	C₂×D₈×D₆₁	1	2	yes	—
237	C₄	C₂×Q₁₆×D₆₁	1	2	yes	—
238	C₄	C₂²×#11758202	1	2	yes	—
239	C₂	#21445436	1	4	yes	direct product factor
240	C₂	C₂×#16840137	1	1	yes	characteristic
241	C₂	C₂×#16840140	1	1	yes	characteristic
242	C₂	C₂×#16840141	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14519773
commutator, cosocle	C₄²×C₆₁	C₂⁴
Frattini	C₄²	C₂³×D₆₁
Fitting	C₂×C₆₁×#500	C₂
socle, prime radical	C₂³×C₆₁	#127
hypercenter	C₂×#500	D₆₁
hypocenter	C₆₁	C₂×#1713
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
4. #21445436	C₂	direct (N × H)
65. C₂×C₈×C₆₁	Q₁₆	semidirect
81. C₂×#92418	Q₁₆	semidirect
102. C₈×C₆₁	C₂×Q₁₆	semidirect
104. #92418	C₂×Q₁₆	semidirect
124. C₂×C₆₁	#1713	semidirect
146. C₆₁	C₂×#1713	semidirect
192. C₂×C₈	#11758202	semidirect
226. C₈	C₂×#11758202	semidirect
239. C₂	#21445436	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16840167	2
C₂×#16840165	2
C₂×#16840141	2
#21445436	2
C₂×C₈×#92418	2
C₂×C₆₁×#500	2
C₂×#1713	61

Subgroups — C2×#21445436 #29667707

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21445436 #29667707