Subgroups — C₂×#21448105 #29835510

Order 15616.

518 conjugacy classes of subgroups.

The subgroup lattice has 220 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×#16840164	C₂	1	1	yes	maximal, characteristic
3	#21448105	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16839590	C₂	1	2	yes	maximal
5	C₂×#16839588	C₂	1	2	yes	maximal
6	C₂×C₆₁×#346	C₂	1	1	yes	maximal, characteristic
7	C₂×#14519777	—	2	1	no	—
8	C₂×#14519769	—	2	1	no	—
9	#16840167	—	2	2	no	—
10	#16840167	C₂²	1	4	yes	—
11	#16840164	—	2	2	no	—
12	#16840164	C₂²	1	4	yes	—
13	C₂×#14519711	—	2	1	no	—
14	C₂×#14519711	C₂²	1	1	yes	characteristic
15	C₂×#14519711	C₂²	1	2	yes	—
16	C₂×#14519711	C₂²	1	1	yes	characteristic
17	C₂×#14519709	—	2	1	no	—
18	#16839590	C₂²	1	8	yes	—
19	#16839588	C₂²	1	8	yes	—
20	C₂²×C₁₆×C₆₁	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
22	C₆₁×#346	C₂²	1	4	yes	—
23	C₂²×Dic₁₂₂	—	4	1	no	—
24	#14519777	—	2	4	no	—
25	#14519777	—	4	2	no	—
26	#14519769	—	2	4	no	—
27	#14519769	—	4	2	no	—
28	C₂×C₄⋊Dic₆₁	—	2	1	no	—
29	C₂×C₄⋊Dic₆₁	—	2	1	no	—
30	C₂×C₄⋊Dic₆₁	—	2	2	no	—
31	C₂×C₄⋊Dic₆₁	—	4	1	no	—
32	C₂×#11758196	—	4	1	no	—
33	C₂×#11758196	—	4	1	no	—
34	#14519711	—	2	4	no	—
35	#14519711	—	2	1	no	—
36	#14519711	—	2	1	no	—
37	#14519711	—	2	2	no	—
38	#14519711	C₂³	1	2	yes	—
39	#14519711	C₂³	1	4	yes	—
40	#14519711	C₂³	1	2	yes	—
41	#14519709	—	2	4	no	—
42	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
45	C₂×C₁₆×C₆₁	Q₈	1	8	yes	complemented
46	C₂×C₁₆×C₆₁	C₂³	1	4	yes	—
47	C₄×C₈×C₆₁	D₄	1	2	yes	—
48	C₄×C₈×C₆₁	C₂³	1	2	yes	—
49	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
51	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
52	C₂²×Dic₆₁	—	4	2	no	—
53	C₂²×Dic₆₁	—	4	1	no	—
54	C₂²×Dic₆₁	—	4	1	no	—
55	C₂×Dic₁₂₂	—	4	4	no	—
56	C₂×Dic₁₂₂	—	8	4	no	—
57	C₂×C₈×C₆₁	Q₁₆	1	4	yes	—
58	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
60	C₂×C₈×C₆₁	C₂×Q₈	1	4	yes	—
61	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
62	C₄²×C₆₁	D₈	1	2	yes	—
63	C₄²×C₆₁	C₂×D₄	1	2	yes	—
64	C₄⋊Dic₆₁	—	2	2	no	—
65	C₄⋊Dic₆₁	—	2	2	no	—
66	C₄⋊Dic₆₁	—	2	4	no	—
67	C₄⋊Dic₆₁	—	4	1	no	—
68	C₄⋊Dic₆₁	—	4	1	no	—
69	C₄⋊Dic₆₁	—	4	2	no	—
70	C₄⋊Dic₆₁	—	4	4	no	—
71	#11758196	—	4	4	no	—
72	#11758196	—	4	4	no	—
73	C₁₆×C₆₁	C₂×Q₈	1	8	yes	complemented
74	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
75	C₂×C₄×C₆₁	—	2	2	no	—
76	C₂×C₄×C₆₁	—	2	2	no	—
77	C₂×C₄×C₆₁	C₄⋊Q₈	1	2	yes	—
78	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
79	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
80	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
81	C₂×C₄×C₆₁	C₂×Q₁₆	1	2	yes	—
82	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
83	C₂×Dic₆₁	—	4	2	no	—
84	C₂×Dic₆₁	—	4	2	no	—
85	C₂×Dic₆₁	—	4	4	no	—
86	C₂×Dic₆₁	—	8	4	no	—
87	C₂×Dic₆₁	—	8	2	no	—
88	C₂×Dic₆₁	—	8	2	no	—
89	Dic₁₂₂	—	8	8	no	—
90	C₈×C₆₁	C₄⋊Q₈	1	2	yes	—
91	C₈×C₆₁	C₂×Q₁₆	1	4	yes	—
92	C₈×C₆₁	C₂²×Q₈	1	2	yes	—
93	C₂×#2255	—	61	1	no	maximal, Sylow
94	C₂²×C₆₁	#500	1	2	yes	—
95	C₂²×C₆₁	#509	1	2	yes	—
96	C₂²×C₆₁	#510	1	2	yes	—
97	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
98	C₄×C₆₁	—	2	2	no	—
99	C₄×C₆₁	—	2	2	no	—
100	C₄×C₆₁	#500	1	2	yes	—
101	C₄×C₆₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
102	C₄×C₆₁	C₂²×Q₁₆	1	1	yes	characteristic
103	Dic₆₁	—	8	4	no	—
104	Dic₆₁	—	8	2	no	—
105	Dic₆₁	—	8	2	no	—
106	C₂×#500	—	61	1	no	—
107	C₂×#499	—	61	1	no	—
108	#2255	—	61	8	no	—
109	C₂×#367	—	61	2	no	—
110	C₂×#366	—	61	2	no	—
111	C₂×#346	D₆₁	1	1	yes	characteristic
112	C₂×C₆₁	#2255	1	4	yes	complemented
113	C₂×C₆₁	C₂×#500	1	1	yes	characteristic
114	C₂×C₆₁	C₂×#509	1	1	yes	characteristic
115	C₂×C₆₁	C₂×#510	1	1	yes	characteristic
116	C₂×C₄⋊Q₈	—	122	1	no	—
117	C₂×#125	—	122	1	no	—
118	C₂²×C₁₆	C₂×D₆₁	1	2	yes	—
119	#500	—	61	4	no	—
120	#500	—	122	2	no	—
121	#499	—	61	4	no	—
122	#499	—	122	2	no	—
123	C₂×#107	—	61	1	no	—
124	C₂×#107	—	61	1	no	—
125	C₂×#107	—	61	2	no	—
126	C₂×#107	—	122	1	no	—
127	C₂×#106	—	122	1	no	—
128	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
129	#367	—	61	8	no	—
130	#366	—	61	8	no	—
131	#346	C₂×D₆₁	1	4	yes	—
132	C₆₁	C₂×#2255	1	1	yes	characteristic, complemented, Sylow
133	C₂²×Q₈	—	244	1	no	—
134	C₂²×C₈	D₂₄₄	1	1	yes	characteristic
135	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
136	C₄⋊Q₈	—	122	4	no	—
137	C₄⋊Q₈	—	244	2	no	—
138	#125	—	122	4	no	—
139	#125	—	244	2	no	—
140	C₂×C₄⋊C₄	—	122	1	no	—
141	C₂×C₄⋊C₄	—	122	1	no	—
142	C₂×C₄⋊C₄	—	122	2	no	—
143	C₂×C₄⋊C₄	—	244	1	no	—
144	C₂×C₄⋊C₄	—	244	1	no	—
145	C₂×C₄⋊C₄	—	244	1	no	—
146	C₂×C₄²	D₂₄₄	1	1	yes	characteristic
147	C₂×C₁₆	Dic₁₂₂	1	8	yes	complemented
148	C₂×C₁₆	C₂²×D₆₁	1	4	yes	—
149	#107	—	61	2	no	—
150	#107	—	61	4	no	—
151	#107	—	61	2	no	—
152	#107	—	122	1	no	—
153	#107	—	122	1	no	—
154	#107	—	122	2	no	—
155	#107	—	122	4	no	—
156	#106	—	122	4	no	—
157	C₄×C₈	D₂₄₄	1	2	yes	—
158	C₄×C₈	C₂²×D₆₁	1	2	yes	—
159	C₂×Q₈	—	488	4	no	—
160	C₂×Q₈	—	244	4	no	—
161	C₂²×C₄	—	244	1	no	—
162	C₂²×C₄	—	244	1	no	—
163	C₂²×C₄	—	244	2	no	—
164	C₂²×C₄	D₄₈₈	1	1	yes	characteristic
165	C₂²×C₄	D₄₈₈	1	1	yes	characteristic
166	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
167	C₂×C₈	Dic₂₄₄	1	4	yes	—
168	C₂×C₈	C₂×Dic₁₂₂	1	4	yes	—
169	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
170	C₂×C₈	C₂×D₂₄₄	1	2	yes	—
171	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
172	C₄⋊C₄	—	122	2	no	—
173	C₄⋊C₄	—	122	2	no	—
174	C₄⋊C₄	—	122	4	no	—
175	C₄⋊C₄	—	244	1	no	—
176	C₄⋊C₄	—	244	1	no	—
177	C₄⋊C₄	—	244	2	no	—
178	C₄⋊C₄	—	244	4	no	—
179	C₄⋊C₄	—	244	4	no	—
180	C₄⋊C₄	—	244	4	no	—
181	C₄²	D₄₈₈	1	2	yes	—
182	C₄²	C₂×D₂₄₄	1	2	yes	—
183	C₁₆	C₂×Dic₁₂₂	1	8	yes	complemented
184	C₂³	C₂×D₄₈₈	1	1	yes	characteristic
185	Q₈	—	488	8	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	488	2	no	—
189	C₂×C₄	—	488	4	no	—
190	C₂×C₄	—	488	2	no	—
191	C₂×C₄	—	244	2	no	—
192	C₂×C₄	—	244	2	no	—
193	C₂×C₄	—	244	4	no	—
194	C₂×C₄	#14519777	1	2	yes	—
195	C₂×C₄	C₂×D₄₈₈	1	2	yes	—
196	C₂×C₄	C₂×D₄₈₈	1	2	yes	—
197	C₂×C₄	C₂×D₄₈₈	1	1	yes	characteristic
198	C₂×C₄	C₂×Dic₂₄₄	1	2	yes	—
199	C₂×C₄	C₂²×D₂₄₄	1	1	yes	characteristic
200	C₈	#14519777	1	2	yes	—
201	C₈	C₂×Dic₂₄₄	1	4	yes	—
202	C₈	C₂²×Dic₁₂₂	1	2	yes	—
203	C₂²	#16840167	1	2	yes	—
204	C₂²	#16840200	1	2	yes	—
205	C₂²	#16840205	1	2	yes	—
206	C₂²	C₂²×D₄₈₈	1	1	yes	characteristic
207	C₄	—	2	2	no	—
208	C₄	—	2	2	no	—
209	C₄	—	488	2	no	—
210	C₄	—	488	4	no	—
211	C₄	—	488	2	no	—
212	C₄	#16840167	1	2	yes	—
213	C₄	C₂×#14519777	1	1	yes	characteristic
214	C₄	C₂²×Dic₂₄₄	1	1	yes	characteristic
215	C₂	#21448105	1	4	yes	direct product factor
216	C₂	C₂×#16840167	1	1	yes	characteristic
217	C₂	C₂×#16840200	1	1	yes	characteristic
218	C₂	C₂×#16840205	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄₈₈
commutator, cosocle	C₂×C₈×C₆₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₁
Fitting	C₂×C₆₁×#346	C₂
socle, prime radical	C₂³×C₆₁	C₂×D₈
hypercenter	C₂×#346	D₆₁
hypocenter	C₆₁	C₂×#2255
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
3. #21448105	C₂	direct (N × H)
45. C₂×C₁₆×C₆₁	Q₈	semidirect
73. C₁₆×C₆₁	C₂×Q₈	semidirect
112. C₂×C₆₁	#2255	semidirect
132. C₆₁	C₂×#2255	semidirect
147. C₂×C₁₆	Dic₁₂₂	semidirect
183. C₁₆	C₂×Dic₁₂₂	semidirect
215. C₂	#21448105	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×#16840164	2
#21448105	2
C₂×#16839590	2
C₂×#16839588	2
C₂×C₆₁×#346	2
C₂×#2255	61

Subgroups — C2×#21448105 #29835510

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21448105 #29835510