Subgroups — C₂×#21443913 #29594900

Order 15616.

574 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14519713	C₂	1	2	yes	maximal
2	#21443913	C₂	1	4	yes	maximal, direct product factor
3	C₂³×C₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11758194	—	2	2	no	—
5	C₂×#14519713	—	2	2	no	—
6	C₂×#14519713	—	2	4	no	—
7	C₂×#14519713	C₄	1	8	yes	—
8	C₂×#14519713	C₂²	1	4	yes	—
9	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆×C₆₁	—	2	4	no	—
11	C₂²×C₁₆×C₆₁	C₄	1	2	yes	—
12	C₂²×C₁₆×C₆₁	C₄	1	2	yes	—
13	C₂²×C₁₆×C₆₁	C₂²	1	2	yes	—
14	C₂×#11758194	—	2	8	no	—
15	C₂×#11758194	—	2	4	no	—
16	C₂×#11758194	—	4	2	no	—
17	C₂×#11758194	—	4	4	no	—
18	C₂²×#92418	—	4	2	no	—
19	#14519713	—	2	8	no	—
20	#14519713	—	2	4	no	—
21	#14519713	C₂×C₄	1	8	yes	—
22	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₆₁	C₂×C₄	1	4	yes	—
26	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₆₁	D₄	1	4	yes	—
28	C₂²×C₈×C₆₁	Q₈	1	1	yes	characteristic
29	C₂²×C₈×C₆₁	Q₈	1	1	yes	characteristic
30	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
31	C₂×C₁₆×C₆₁	—	2	4	no	—
32	C₂×C₁₆×C₆₁	—	2	4	no	—
33	C₂×C₁₆×C₆₁	—	2	4	no	—
34	C₂×C₁₆×C₆₁	C₂×C₄	1	4	yes	—
35	C₂⁴×C₆₁	Q₁₆	1	1	yes	characteristic
36	C₂²×C₄×C₆₁	C₂²⋊C₄	1	4	yes	—
37	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
38	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
40	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
41	C₂²×C₄×C₆₁	SD₁₆	1	4	yes	—
42	C₂²×C₄×C₆₁	Q₁₆	1	1	yes	characteristic
43	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₆₁	C₄²	1	2	yes	—
45	C₂×C₈×C₆₁	C₄²	1	2	yes	—
46	C₂×C₈×C₆₁	C₂²⋊C₄	1	4	yes	—
47	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
48	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
49	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₆₁	C₄⋊C₄	1	4	yes	—
52	C₂×C₈×C₆₁	C₄⋊C₄	1	4	yes	—
53	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
55	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
56	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
57	#11758194	—	2	8	no	—
58	#11758194	—	4	8	no	—
59	#11758194	—	4	4	no	—
60	C₂×#92418	—	4	8	no	—
61	C₂×#92418	—	4	4	no	—
62	C₁₆×C₆₁	—	2	4	no	—
63	C₂³×C₆₁	—	2	4	no	—
64	C₂³×C₆₁	#103	1	4	yes	—
65	C₂³×C₆₁	#107	1	1	yes	characteristic
66	C₂³×C₆₁	#107	1	1	yes	characteristic
67	C₂³×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
68	C₂×C₄×C₆₁	#95	1	2	yes	—
69	C₂×C₄×C₆₁	#95	1	2	yes	—
70	C₂×C₄×C₆₁	#102	1	4	yes	—
71	C₂×C₄×C₆₁	#102	1	4	yes	—
72	C₂×C₄×C₆₁	#103	1	4	yes	—
73	C₂×C₄×C₆₁	#106	1	2	yes	—
74	C₂×C₄×C₆₁	#106	1	2	yes	—
75	C₂×C₄×C₆₁	#107	1	1	yes	characteristic
76	C₂×C₄×C₆₁	#107	1	1	yes	characteristic
77	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	2	yes	—
78	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
80	C₂×C₄×C₆₁	C₂×SD₁₆	1	2	yes	—
81	C₈×C₆₁	#95	1	2	yes	—
82	C₈×C₆₁	#95	1	2	yes	—
83	C₈×C₆₁	C₂×C₄²	1	1	yes	characteristic
84	C₈×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₈×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
86	#92418	—	4	8	no	—
87	C₂×#1381	—	61	1	no	maximal, Sylow
88	C₂²×C₆₁	—	2	8	no	—
89	C₂²×C₆₁	—	2	2	no	—
90	C₂²×C₆₁	—	2	2	no	—
91	C₂²×C₆₁	#340	1	2	yes	—
92	C₂²×C₆₁	#340	1	2	yes	—
93	C₂²×C₆₁	#368	1	4	yes	—
94	C₂²×C₆₁	C₂×#103	1	2	yes	—
95	C₂²×C₆₁	C₂×#107	1	1	yes	characteristic
96	C₄×C₆₁	#340	1	2	yes	—
97	C₄×C₆₁	#340	1	2	yes	—
98	C₄×C₆₁	C₂×#95	1	1	yes	characteristic
99	C₄×C₆₁	C₂×#102	1	2	yes	—
100	C₄×C₆₁	C₂×#106	1	1	yes	characteristic
101	C₂³×C₁₆	D₆₁	1	1	yes	characteristic
102	C₂²×#108	—	61	2	no	—
103	#1381	—	61	4	no	—
104	C₂×C₆₁	—	2	2	no	—
105	C₂×C₆₁	—	2	2	no	—
106	C₂×C₆₁	#1381	1	4	yes	complemented
107	C₂×C₆₁	C₂×#340	1	1	yes	characteristic
108	C₂×C₆₁	C₂×#368	1	2	yes	—
109	C₂²×M₁₆	—	122	2	no	—
110	C₂³×C₈	C₂×D₆₁	1	1	yes	characteristic
111	C₂²×C₁₆	—	2	4	no	—
112	C₂²×C₁₆	Dic₆₁	1	2	yes	—
113	C₂²×C₁₆	Dic₆₁	1	2	yes	—
114	C₂²×C₁₆	C₂×D₆₁	1	2	yes	—
115	C₂×#108	—	61	8	no	—
116	C₂×#108	—	61	4	no	—
117	C₂×#108	—	122	2	no	—
118	C₂×#108	—	122	4	no	—
119	C₆₁	C₂×#1381	1	1	yes	characteristic, complemented, Sylow
120	C₂³×C₄	D₂₄₄	1	1	yes	characteristic
121	C₂×M₁₆	—	122	8	no	—
122	C₂×M₁₆	—	122	4	no	—
123	C₂×M₁₆	—	244	4	no	—
124	C₂×M₁₆	—	244	2	no	—
125	C₂²×C₈	—	244	2	no	—
126	C₂²×C₈	Dic₁₂₂	1	1	yes	characteristic
127	C₂²×C₈	Dic₁₂₂	1	1	yes	characteristic
128	C₂²×C₈	C₄×D₆₁	1	4	yes	—
129	C₂²×C₈	D₂₄₄	1	1	yes	characteristic
130	C₂²×C₈	C₂×Dic₆₁	1	1	yes	characteristic
131	C₂²×C₈	C₂×Dic₆₁	1	1	yes	characteristic
132	C₂²×C₈	C₂²⋊D₆₁	1	4	yes	—
133	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
134	C₂×C₁₆	—	2	4	no	—
135	C₂×C₁₆	—	2	4	no	—
136	C₂×C₁₆	—	2	4	no	—
137	C₂×C₁₆	C₂×Dic₆₁	1	4	yes	—
138	#108	—	61	8	no	—
139	#108	—	122	8	no	—
140	#108	—	122	4	no	—
141	C₂⁴	Dic₂₄₄	1	1	yes	characteristic
142	C₂²×C₄	#11758190	1	4	yes	—
143	C₂²×C₄	D₄₈₈	1	1	yes	characteristic
144	C₂²×C₄	D₄₈₈	1	1	yes	characteristic
145	C₂²×C₄	Dic₂₄₄	1	1	yes	characteristic
146	C₂²×C₄	C₄⋊Dic₆₁	1	1	yes	characteristic
147	C₂²×C₄	C₄⋊Dic₆₁	1	1	yes	characteristic
148	C₂²×C₄	#11758198	1	4	yes	—
149	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
150	M₁₆	—	122	8	no	—
151	M₁₆	—	244	8	no	—
152	M₁₆	—	244	4	no	—
153	C₂×C₈	—	244	8	no	—
154	C₂×C₈	—	244	4	no	—
155	C₂×C₈	C₄×Dic₆₁	1	2	yes	—
156	C₂×C₈	C₄×Dic₆₁	1	2	yes	—
157	C₂×C₈	#11758196	1	4	yes	—
158	C₂×C₈	#11758196	1	4	yes	—
159	C₂×C₈	C₄⋊Dic₆₁	1	1	yes	characteristic
160	C₂×C₈	C₄⋊Dic₆₁	1	1	yes	characteristic
161	C₂×C₈	#11758198	1	4	yes	—
162	C₂×C₈	C₂²⋊Dic₆₁	1	2	yes	—
163	C₂×C₈	C₂²⋊Dic₆₁	1	2	yes	—
164	C₂×C₈	C₂×Dic₁₂₂	1	1	yes	characteristic
165	C₂×C₈	C₂×C₄×D₆₁	1	2	yes	—
166	C₂×C₈	C₂²×Dic₆₁	1	1	yes	characteristic
167	C₂×C₈	C₂×C₂²⋊D₆₁	1	2	yes	—
168	C₁₆	—	2	4	no	—
169	C₂³	—	2	4	no	—
170	C₂³	#14519702	1	4	yes	—
171	C₂³	#14519711	1	1	yes	characteristic
172	C₂³	#14519711	1	1	yes	characteristic
173	C₂³	C₂×Dic₂₄₄	1	1	yes	characteristic
174	C₂×C₄	#14519684	1	2	yes	—
175	C₂×C₄	#14519684	1	2	yes	—
176	C₂×C₄	#14519699	1	4	yes	—
177	C₂×C₄	#14519699	1	4	yes	—
178	C₂×C₄	#14519702	1	4	yes	—
179	C₂×C₄	#14519709	1	2	yes	—
180	C₂×C₄	#14519709	1	2	yes	—
181	C₂×C₄	#14519711	1	1	yes	characteristic
182	C₂×C₄	#14519711	1	1	yes	characteristic
183	C₂×C₄	C₂×#11758198	1	2	yes	—
184	C₂×C₄	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
185	C₂×C₄	C₂×D₄₈₈	1	1	yes	characteristic
186	C₂×C₄	C₂×#11758190	1	2	yes	—
187	C₈	—	244	8	no	—
188	C₈	#14519684	1	2	yes	—
189	C₈	#14519684	1	2	yes	—
190	C₈	C₂×C₄×Dic₆₁	1	1	yes	characteristic
191	C₈	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
192	C₈	C₂×#11758196	1	2	yes	—
193	C₂²	—	2	8	no	—
194	C₂²	—	2	2	no	—
195	C₂²	—	2	2	no	—
196	C₂²	#16839525	1	2	yes	—
197	C₂²	#16839525	1	2	yes	—
198	C₂²	#16839592	1	4	yes	—
199	C₂²	C₂×#14519702	1	2	yes	—
200	C₂²	C₂×#14519711	1	1	yes	characteristic
201	C₄	#16839525	1	2	yes	—
202	C₄	#16839525	1	2	yes	—
203	C₄	C₂×#14519684	1	1	yes	characteristic
204	C₄	C₂×#14519699	1	2	yes	—
205	C₄	C₂×#14519709	1	1	yes	characteristic
206	C₂	—	2	2	no	—
207	C₂	—	2	2	no	—
208	C₂	#21443913	1	4	yes	direct product factor
209	C₂	C₂×#16839525	1	1	yes	characteristic
210	C₂	C₂×#16839592	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄₈₈
commutator	C₈×C₆₁	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₂³×C₁₆×C₆₁	C₂
socle	C₂³×C₆₁	#107
prime radical	C₂⁴×C₆₁	Q₁₆
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	C₂³×C₁₆	D₆₁
hypocenter	C₆₁	C₂×#1381
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
2. #21443913	C₂	direct (N × H)
106. C₂×C₆₁	#1381	semidirect
119. C₆₁	C₂×#1381	semidirect
208. C₂	#21443913	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14519713	2
#21443913	2
C₂³×C₁₆×C₆₁	2
C₂×#1381	61

Subgroups — C2×#21443913 #29594900

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21443913 #29594900