Subgroups — C₂×#21462028 #30623817

Order 15616.

854 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14519851	C₂	1	1	yes	maximal, characteristic
2	C₂³×#11758187	C₂	1	1	yes	maximal, characteristic
3	#21462028	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16840650	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×#11758187	C₂	1	2	yes	maximal, complemented
6	C₂×#16839718	C₂	1	1	yes	maximal, characteristic
7	C₂×#14519851	C₄	1	8	yes	complemented
8	C₂×#14519851	C₂²	1	4	yes	—
9	C₂²×#11758187	C₄	1	8	yes	complemented
10	C₂²×#11758187	C₂²	1	1	yes	characteristic
11	C₂²×#11758187	C₂²	1	2	yes	—
12	C₂²×#11758187	C₂²	1	1	yes	characteristic
13	C₂²×#11758187	C₂²	1	4	yes	—
14	#16840650	C₂²	1	8	yes	complemented
15	C₄×#11758187	C₂²	1	8	yes	complemented
16	C₂²×#11758194	C₄	1	1	yes	characteristic
17	C₂³×#92418	C₂²	1	1	yes	characteristic
18	#16839718	C₄	1	4	yes	—
19	#16839718	C₂²	1	4	yes	—
20	C₂×#14519688	C₄	1	2	yes	—
21	C₂×C₄×#92418	C₂²	1	2	yes	—
22	C₂×#117×C₆₁	C₄	1	1	yes	characteristic
23	#14519851	C₂×C₄	1	16	yes	complemented
24	C₂×#11758187	C₂×C₄	1	4	yes	—
25	C₂×#11758187	C₂×C₄	1	4	yes	—
26	C₂×#11758187	C₂×C₄	1	8	yes	—
27	C₂×#11758187	C₂×C₄	1	16	yes	complemented
28	C₂×#11758187	C₂³	1	2	yes	—
29	C₂×#11758187	C₂³	1	2	yes	—
30	C₂×#11758187	C₂³	1	4	yes	—
31	C₂×#11758194	C₈	1	8	yes	—
32	C₂×#11758194	C₂×C₄	1	4	yes	—
33	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
34	C₂²×#92418	C₂×C₄	1	2	yes	—
35	C₂²×#92418	C₂×C₄	1	4	yes	—
36	C₂²×#92418	C₂×C₄	1	4	yes	—
37	C₂²×#92418	C₂×C₄	1	2	yes	—
38	C₂²×#92418	C₂³	1	1	yes	characteristic
39	C₂²×#92418	C₂³	1	2	yes	—
40	#14519688	C₂×C₄	1	8	yes	—
41	C₄×#92418	C₂×C₄	1	4	yes	—
42	C₄×#92418	C₂³	1	4	yes	—
43	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
44	#117×C₆₁	C₈	1	4	yes	—
45	#117×C₆₁	C₂×C₄	1	4	yes	—
46	C₂×C₄⋊C₄×C₆₁	C₈	1	2	yes	—
47	C₂×C₂²⋊C₄×C₆₁	C₈	1	2	yes	—
48	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
49	C₂⁴×C₆₁	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₆₁	C₄²	1	4	yes	—
51	C₂²×C₄×C₆₁	C₂×C₈	1	4	yes	—
52	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
53	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
54	C₂²×C₄×C₆₁	C₂×C₈	1	4	yes	—
55	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
57	C₄⋊C₄×C₆₁	C₂×C₈	1	8	yes	—
58	C₂²⋊C₄×C₆₁	C₂×C₈	1	8	yes	—
59	C₄²×C₆₁	C₂×C₈	1	4	yes	—
60	C₄²×C₆₁	C₂²×C₄	1	4	yes	—
61	#11758194	C₂×C₈	1	16	yes	—
62	C₂×#92418	C₄²	1	4	yes	—
63	C₂×#92418	C₄²	1	8	yes	—
64	C₂×#92418	C₂×C₈	1	8	yes	—
65	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
66	C₂×#92418	C₂²×C₄	1	2	yes	—
67	C₂×#92418	C₂²×C₄	1	2	yes	—
68	C₂×#92418	C₂²×C₄	1	2	yes	—
69	C₂×#92418	C₂²×C₄	1	4	yes	—
70	C₂×#92418	C₂²×C₄	1	4	yes	—
71	C₂×#92418	C₂²×C₄	1	4	yes	—
72	C₂×#92418	C₂⁴	1	1	yes	characteristic
73	#11758187	C₂²×C₄	1	4	yes	—
74	#11758187	C₂²×C₄	1	4	yes	—
75	#11758187	C₂²×C₄	1	8	yes	—
76	C₂³×C₆₁	—	2	4	no	—
77	C₂³×C₆₁	C₄×C₈	1	4	yes	—
78	C₂³×C₆₁	C₂²×C₈	1	2	yes	—
79	C₂³×C₆₁	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄×C₆₁	—	2	8	no	—
81	C₂×C₄×C₆₁	C₄×C₈	1	8	yes	—
82	C₂×C₄×C₆₁	C₄×C₈	1	4	yes	—
83	C₂×C₄×C₆₁	C₂×C₄²	1	4	yes	—
84	C₂×C₄×C₆₁	C₂×C₄²	1	2	yes	—
85	C₂×C₄×C₆₁	C₂²×C₈	1	8	yes	—
86	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
87	C₂×C₄×C₆₁	C₂²×C₈	1	4	yes	—
88	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
89	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
90	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
91	#92418	C₂×C₄²	1	2	yes	—
92	#92418	C₂×C₄²	1	4	yes	—
93	#92418	C₂²×C₈	1	8	yes	—
94	#92418	C₂³×C₄	1	1	yes	characteristic
95	#92418	C₂³×C₄	1	1	yes	characteristic
96	C₂×#2108	—	61	1	no	maximal, Sylow
97	C₂²×C₆₁	—	2	8	no	—
98	C₂²×C₆₁	—	2	4	no	—
99	C₂²×C₆₁	C₂×C₄×C₈	1	4	yes	—
100	C₂²×C₆₁	C₂×C₄×C₈	1	2	yes	—
101	C₂²×C₆₁	#504	1	4	yes	—
102	C₂²×C₆₁	C₂³×C₈	1	1	yes	characteristic
103	C₄×C₆₁	—	2	8	no	—
104	C₄×C₆₁	C₂×C₄×C₈	1	2	yes	—
105	C₄×C₆₁	C₂×C₄×C₈	1	4	yes	—
106	C₄×C₆₁	C₂²×C₄²	1	1	yes	characteristic
107	C₄×C₆₁	C₂³×C₈	1	1	yes	characteristic
108	C₂²×M₃₂	—	61	1	no	—
109	C₂³×C₁₆	—	61	1	no	—
110	C₂×#405	—	61	1	no	—
111	#2108	—	61	8	no	—
112	C₂×#346	—	61	2	no	—
113	C₂×C₄×C₁₆	—	61	2	no	—
114	C₂×C₆₁	—	2	4	no	—
115	C₂×C₆₁	#2108	1	4	yes	complemented
116	C₂×C₆₁	C₂²×C₄×C₈	1	1	yes	characteristic
117	C₂×C₆₁	C₂×#504	1	2	yes	—
118	C₂²×M₁₆	—	61	1	no	—
119	C₂³×C₈	—	61	1	no	—
120	C₂×#117	#6923	1	1	yes	characteristic
121	C₂×M₃₂	—	61	4	no	—
122	C₂×M₃₂	—	61	8	no	—
123	C₂²×C₁₆	—	61	1	no	—
124	C₂²×C₁₆	—	61	2	no	—
125	C₂²×C₁₆	—	61	1	no	—
126	C₂²×C₁₆	—	61	4	no	—
127	C₂²×C₁₆	—	61	8	no	—
128	#405	—	61	4	no	—
129	#405	—	61	4	no	—
130	C₂×#97	—	61	2	no	—
131	C₂×C₄×C₈	—	61	2	no	—
132	#346	—	61	8	no	—
133	C₄×C₁₆	—	61	8	no	—
134	C₆₁	C₂×#2108	1	1	yes	characteristic, complemented, Sylow
135	C₂³×C₄	C₂×#6923	1	1	yes	characteristic
136	C₂×M₁₆	—	61	8	no	—
137	C₂×M₁₆	—	61	4	no	—
138	C₂²×C₈	—	61	4	no	—
139	C₂²×C₈	—	61	4	no	—
140	C₂²×C₈	—	61	2	no	—
141	C₂²×C₈	—	61	2	no	—
142	C₂²×C₈	—	61	1	no	—
143	C₂²×C₈	—	61	1	no	—
144	C₂²×C₈	—	61	2	no	—
145	#117	#92420	1	4	yes	—
146	#117	C₂×#6923	1	4	yes	—
147	C₂×C₄⋊C₄	#92420	1	2	yes	—
148	C₂×C₂²⋊C₄	#92420	1	2	yes	—
149	C₂×C₄²	C₂×#6923	1	2	yes	—
150	M₃₂	—	61	16	no	—
151	C₂×C₁₆	—	61	4	no	—
152	C₂×C₁₆	—	61	4	no	—
153	C₂×C₁₆	—	61	8	no	—
154	C₂×C₁₆	—	61	16	no	—
155	C₂×C₁₆	—	61	2	no	—
156	C₂×C₁₆	—	61	4	no	—
157	C₂×C₁₆	—	61	2	no	—
158	#97	—	61	8	no	—
159	C₄×C₈	—	61	4	no	—
160	C₄×C₈	—	61	4	no	—
161	C₂⁴	C₂×#92420	1	1	yes	characteristic
162	C₂²×C₄	D₆₁⋊C₈	1	4	yes	—
163	C₂²×C₄	C₄×#6923	1	4	yes	—
164	C₂²×C₄	C₂×#92420	1	4	yes	—
165	C₂²×C₄	C₂×#92420	1	1	yes	characteristic
166	C₂²×C₄	C₂×#92420	1	2	yes	—
167	C₂²×C₄	C₂²×#6923	1	2	yes	—
168	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
169	M₁₆	—	61	16	no	—
170	C₂×C₈	—	61	4	no	—
171	C₂×C₈	—	61	2	no	—
172	C₂×C₈	—	61	2	no	—
173	C₂×C₈	—	61	4	no	—
174	C₂×C₈	—	61	8	no	—
175	C₂×C₈	—	61	4	no	—
176	C₂×C₈	—	61	8	no	—
177	C₂×C₈	—	61	1	no	—
178	C₂×C₈	—	61	1	no	—
179	C₂×C₈	—	61	4	no	—
180	C₂×C₈	—	61	2	no	—
181	C₄⋊C₄	C₂×#92420	1	8	yes	—
182	C₂²⋊C₄	C₂×#92420	1	8	yes	—
183	C₄²	C₂×#92420	1	4	yes	—
184	C₄²	C₂²×#6923	1	4	yes	—
185	C₁₆	—	61	4	no	—
186	C₁₆	—	61	4	no	—
187	C₁₆	—	61	8	no	—
188	C₂³	—	2	4	no	—
189	C₂³	C₄×#92420	1	4	yes	—
190	C₂³	C₂²×#92420	1	2	yes	—
191	C₂³	C₂²×#92420	1	1	yes	characteristic
192	C₂×C₄	—	2	8	no	—
193	C₂×C₄	C₄×#92420	1	8	yes	—
194	C₂×C₄	C₄×#92420	1	4	yes	—
195	C₂×C₄	C₂×C₄×#6923	1	4	yes	—
196	C₂×C₄	C₂×C₄×#6923	1	2	yes	—
197	C₂×C₄	C₂²×#92420	1	8	yes	—
198	C₂×C₄	C₂²×#92420	1	1	yes	characteristic
199	C₂×C₄	C₂²×#92420	1	2	yes	—
200	C₂×C₄	C₂×D₆₁⋊C₈	1	2	yes	—
201	C₂×C₄	C₂×D₆₁⋊C₈	1	4	yes	—
202	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
203	C₈	—	61	2	no	—
204	C₈	—	61	4	no	—
205	C₈	—	61	1	no	—
206	C₈	—	61	1	no	—
207	C₈	—	61	8	no	—
208	C₂²	—	2	8	no	—
209	C₂²	—	2	4	no	—
210	C₂²	C₂×C₄×#92420	1	4	yes	—
211	C₂²	C₂×C₄×#92420	1	2	yes	—
212	C₂²	#16840820	1	4	yes	—
213	C₂²	C₂³×#92420	1	1	yes	characteristic
214	C₄	—	2	8	no	—
215	C₄	C₂×C₄×#92420	1	4	yes	—
216	C₄	C₂×C₄×#92420	1	2	yes	—
217	C₄	C₂²×C₄×#6923	1	1	yes	characteristic
218	C₄	C₂²×D₆₁⋊C₈	1	1	yes	characteristic
219	C₂	—	2	4	no	—
220	C₂	#21462028	1	4	yes	direct product factor
221	C₂	C₂²×C₄×#92420	1	1	yes	characteristic
222	C₂	C₂×#16840820	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×#6923
commutator	C₂×C₆₁	C₂²×C₄×C₈
Frattini	C₂×C₄	C₂³×#6923
Fitting	C₂×#117×C₆₁	C₄
socle	C₂³×C₆₁	C₂²×C₈
prime radical	C₂⁴×C₆₁	C₂×C₈
cosocle	C₂×#92418	C₂⁴
hypercenter	C₂×#117	#6923
hypocenter	C₆₁	C₂×#2108
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
3. #21462028	C₂	direct (N × H)
4. C₂×#16840650	C₂	semidirect
5. C₂×C₄×#11758187	C₂	semidirect
7. C₂×#14519851	C₄	semidirect
9. C₂²×#11758187	C₄	semidirect
14. #16840650	C₂²	semidirect
15. C₄×#11758187	C₂²	semidirect
23. #14519851	C₂×C₄	semidirect
27. C₂×#11758187	C₂×C₄	semidirect
115. C₂×C₆₁	#2108	semidirect
134. C₆₁	C₂×#2108	semidirect
220. C₂	#21462028	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×#14519851	2
C₂³×#11758187	2
#21462028	2
C₂×#16840650	2
C₂×C₄×#11758187	2
C₂×#16839718	2
C₂×#2108	61

Subgroups — C2×#21462028 #30623817

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21462028 #30623817