Subgroups — C₂×#21462046 #30623859

Order 15616.

1406 conjugacy classes of subgroups.

The subgroup lattice has 268 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	2	yes	maximal, complemented
2	#21462046	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₂²⋊#11758187	C₂	1	4	yes	maximal, complemented
4	C₂⁴×#92418	C₂	1	1	yes	maximal, characteristic
5	C₂×#14519851	—	2	8	no	—
6	C₂×#14519851	C₂²	1	8	yes	complemented
7	C₂²×#11758187	C₂²	1	4	yes	complemented
8	C₂²⋊#11758187	C₂²	1	16	yes	complemented
9	C₂³×#92418	—	2	8	no	—
10	C₂³×#92418	C₄	1	4	yes	—
11	C₂³×#92418	C₄	1	2	yes	—
12	C₂³×#92418	C₄	1	1	yes	characteristic
13	C₂³×#92418	C₂²	1	2	yes	—
14	C₂³×#92418	C₂²	1	4	yes	—
15	C₂³×#92418	C₂²	1	1	yes	characteristic
16	C₂⁴×C₄×C₆₁	C₄	1	1	yes	characteristic
17	#14519851	—	2	16	no	—
18	C₂×#11758187	—	2	8	no	—
19	C₂×#11758187	C₂³	1	8	yes	complemented
20	C₂²×#92418	—	2	4	no	—
21	C₂²×#92418	—	2	16	no	—
22	C₂²×#92418	—	2	8	no	—
23	C₂²×#92418	—	2	8	no	—
24	C₂²×#92418	—	2	16	no	—
25	C₂²×#92418	C₂×C₄	1	4	yes	—
26	C₂²×#92418	C₂×C₄	1	2	yes	—
27	C₂²×#92418	C₂×C₄	1	4	yes	—
28	C₂²×#92418	C₂×C₄	1	8	yes	—
29	C₂²×#92418	C₂×C₄	1	2	yes	—
30	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
31	C₂²×#92418	D₄	1	8	yes	—
32	C₂²×#92418	C₂³	1	4	yes	—
33	C₂²×#92418	C₂³	1	1	yes	characteristic
34	C₂²×#92418	C₂³	1	2	yes	—
35	C₂⁵×C₆₁	C₈	1	1	yes	characteristic
36	C₂³×C₄×C₆₁	—	2	4	no	—
37	C₂³×C₄×C₆₁	—	2	4	no	—
38	C₂³×C₄×C₆₁	C₈	1	1	yes	characteristic
39	C₂³×C₄×C₆₁	C₈	1	4	yes	—
40	C₂³×C₄×C₆₁	C₈	1	2	yes	—
41	C₂³×C₄×C₆₁	C₂×C₄	1	4	yes	—
42	C₂³×C₄×C₆₁	C₂×C₄	1	2	yes	—
43	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
44	C₂⁴×C₆₁	—	2	4	no	—
45	C₂⁴×C₆₁	—	2	8	no	—
46	C₂⁴×C₆₁	C₂×C₈	1	4	yes	—
47	C₂⁴×C₆₁	C₂×C₈	1	1	yes	characteristic
48	C₂⁴×C₆₁	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₆₁	—	2	4	no	—
50	C₂²×C₄×C₆₁	—	2	16	no	—
51	C₂²×C₄×C₆₁	—	2	8	no	—
52	C₂²×C₄×C₆₁	—	2	8	no	—
53	C₂²×C₄×C₆₁	—	2	4	no	—
54	C₂²×C₄×C₆₁	—	2	8	no	—
55	C₂²×C₄×C₆₁	—	2	4	no	—
56	C₂²×C₄×C₆₁	C₂²⋊C₄	1	4	yes	—
57	C₂²×C₄×C₆₁	C₂×C₈	1	4	yes	—
58	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
59	C₂²×C₄×C₆₁	C₂×C₈	1	8	yes	—
60	C₂²×C₄×C₆₁	C₂×C₈	1	4	yes	—
61	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
62	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
63	C₂²×C₄×C₆₁	M₁₆	1	4	yes	—
64	C₂²×C₄×C₆₁	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
67	C₂×#92418	—	2	8	no	—
68	C₂×#92418	—	2	4	no	—
69	C₂×#92418	—	2	8	no	—
70	C₂×#92418	—	2	16	no	—
71	C₂×#92418	—	2	8	no	—
72	C₂×#92418	C₂²⋊C₄	1	8	yes	—
73	C₂×#92418	C₂²⋊C₄	1	4	yes	—
74	C₂×#92418	C₂²×C₄	1	4	yes	—
75	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
76	C₂×#92418	C₂²×C₄	1	2	yes	—
77	C₂×#92418	C₂×D₄	1	4	yes	—
78	C₂×#92418	C₂×D₄	1	8	yes	—
79	C₂×#92418	C₂⁴	1	1	yes	characteristic
80	#11758187	—	2	8	no	—
81	C₂³×C₆₁	—	2	32	no	—
82	C₂³×C₆₁	—	2	8	no	—
83	C₂³×C₆₁	—	2	4	no	—
84	C₂³×C₆₁	—	2	4	no	—
85	C₂³×C₆₁	—	2	4	no	—
86	C₂³×C₆₁	—	2	16	no	—
87	C₂³×C₆₁	C₂²⋊C₈	1	4	yes	—
88	C₂³×C₆₁	M₃₂	1	8	yes	complemented
89	C₂³×C₆₁	C₂²×C₈	1	2	yes	—
90	C₂³×C₆₁	C₂²×C₈	1	4	yes	—
91	C₂³×C₆₁	C₂²×C₈	1	1	yes	characteristic
92	C₂×C₄×C₆₁	—	2	8	no	—
93	C₂×C₄×C₆₁	—	2	4	no	—
94	C₂×C₄×C₆₁	—	2	4	no	—
95	C₂×C₄×C₆₁	—	2	4	no	—
96	C₂×C₄×C₆₁	—	2	16	no	—
97	C₂×C₄×C₆₁	—	2	8	no	—
98	C₂×C₄×C₆₁	C₂²⋊C₈	1	8	yes	—
99	C₂×C₄×C₆₁	C₂²⋊C₈	1	4	yes	—
100	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	4	yes	—
101	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	2	yes	—
102	C₂×C₄×C₆₁	C₂²×C₈	1	4	yes	—
103	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
104	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
105	C₂×C₄×C₆₁	C₂×M₁₆	1	4	yes	—
106	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
107	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
108	#92418	—	2	4	no	—
109	#92418	C₂×C₂²⋊C₄	1	2	yes	—
110	#92418	C₂×C₂²⋊C₄	1	4	yes	—
111	#92418	C₂²×D₄	1	2	yes	—
112	C₂×#2112	—	61	1	no	maximal, Sylow
113	C₂²×C₆₁	—	2	32	no	—
114	C₂²×C₆₁	—	2	4	no	—
115	C₂²×C₆₁	—	2	8	no	—
116	C₂²×C₆₁	—	2	16	no	—
117	C₂²×C₆₁	—	2	4	no	—
118	C₂²×C₆₁	—	2	4	no	—
119	C₂²×C₆₁	C₂×C₂²⋊C₈	1	4	yes	—
120	C₂²×C₆₁	C₂×C₂²⋊C₈	1	2	yes	—
121	C₂²×C₆₁	C₂×M₃₂	1	4	yes	—
122	C₂²×C₆₁	C₂×M₃₂	1	8	yes	complemented
123	C₂²×C₆₁	C₂³×C₈	1	1	yes	characteristic
124	C₄×C₆₁	—	2	4	no	—
125	C₄×C₆₁	C₂×C₂²⋊C₈	1	4	yes	—
126	C₄×C₆₁	C₂×C₂²⋊C₈	1	2	yes	—
127	C₄×C₆₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
128	C₄×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
129	C₂⁴×C₈	—	61	1	no	—
130	C₂²×M₃₂	—	61	2	no	—
131	#2112	—	61	8	no	—
132	C₂×C₂²⋊C₁₆	—	61	4	no	—
133	C₂×C₆₁	—	2	4	no	—
134	C₂×C₆₁	—	2	8	no	—
135	C₂×C₆₁	#2112	1	4	yes	complemented
136	C₂×C₆₁	C₂²×C₂²⋊C₈	1	1	yes	characteristic
137	C₂×C₆₁	C₂²×M₃₂	1	2	yes	—
138	C₂⁴×C₄	#6923	1	1	yes	characteristic
139	C₂³×C₈	—	61	1	no	—
140	C₂³×C₈	—	61	2	no	—
141	C₂³×C₈	—	61	4	no	—
142	C₂³×C₈	—	61	4	no	—
143	C₂³×C₈	—	61	2	no	—
144	C₂³×C₈	—	61	1	no	—
145	C₂³×C₈	—	122	8	no	—
146	C₂×M₃₂	—	61	8	no	—
147	C₂×M₃₂	—	122	8	no	—
148	C₂²×C₁₆	—	61	4	no	—
149	C₂²⋊C₁₆	—	61	16	no	—
150	C₆₁	C₂×#2112	1	1	yes	characteristic, complemented, Sylow
151	C₂⁵	#92420	1	1	yes	characteristic
152	C₂³×C₄	—	2	4	no	—
153	C₂³×C₄	—	2	4	no	—
154	C₂³×C₄	#92420	1	1	yes	characteristic
155	C₂³×C₄	#92420	1	4	yes	—
156	C₂³×C₄	#92420	1	2	yes	—
157	C₂³×C₄	C₂×#6923	1	4	yes	—
158	C₂³×C₄	C₂×#6923	1	2	yes	—
159	C₂³×C₄	C₂×#6923	1	1	yes	characteristic
160	C₂²×C₈	—	61	8	no	—
161	C₂²×C₈	—	61	2	no	—
162	C₂²×C₈	—	61	2	no	—
163	C₂²×C₈	—	61	1	no	—
164	C₂²×C₈	—	61	1	no	—
165	C₂²×C₈	—	61	4	no	—
166	C₂²×C₈	—	61	4	no	—
167	C₂²×C₈	—	61	8	no	—
168	C₂²×C₈	—	61	4	no	—
169	C₂²×C₈	—	61	2	no	—
170	C₂²×C₈	—	122	8	no	—
171	C₂²×C₈	—	122	16	no	—
172	C₂²×C₈	—	122	16	no	—
173	C₂²×C₈	—	122	8	no	—
174	C₂²×C₈	—	122	4	no	—
175	M₃₂	—	122	16	no	—
176	C₂×C₁₆	—	61	8	no	—
177	C₂×C₁₆	—	122	8	no	—
178	C₂⁴	—	2	4	no	—
179	C₂⁴	—	2	8	no	—
180	C₂⁴	C₂×#92420	1	4	yes	—
181	C₂⁴	C₂×#92420	1	1	yes	characteristic
182	C₂⁴	C₂×#92420	1	2	yes	—
183	C₂²×C₄	—	2	4	no	—
184	C₂²×C₄	—	2	16	no	—
185	C₂²×C₄	—	2	8	no	—
186	C₂²×C₄	—	2	8	no	—
187	C₂²×C₄	—	2	4	no	—
188	C₂²×C₄	—	2	8	no	—
189	C₂²×C₄	—	2	4	no	—
190	C₂²×C₄	C₂×#92420	1	4	yes	—
191	C₂²×C₄	C₂×#92420	1	2	yes	—
192	C₂²×C₄	C₂×#92420	1	8	yes	—
193	C₂²×C₄	C₂×#92420	1	4	yes	—
194	C₂²×C₄	C₂×#92420	1	1	yes	characteristic
195	C₂²×C₄	C₂×#92420	1	2	yes	—
196	C₂²×C₄	#11758217	1	4	yes	—
197	C₂²×C₄	C₂²⋊#6923	1	4	yes	—
198	C₂²×C₄	C₂²×#6923	1	4	yes	—
199	C₂²×C₄	C₂²×#6923	1	2	yes	—
200	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
201	C₂×C₈	—	61	4	no	—
202	C₂×C₈	—	61	8	no	—
203	C₂×C₈	—	61	8	no	—
204	C₂×C₈	—	61	4	no	—
205	C₂×C₈	—	61	1	no	—
206	C₂×C₈	—	61	1	no	—
207	C₂×C₈	—	61	2	no	—
208	C₂×C₈	—	61	4	no	—
209	C₂×C₈	—	122	8	no	—
210	C₂×C₈	—	122	16	no	—
211	C₂×C₈	—	122	8	no	—
212	C₂×C₈	—	122	8	no	—
213	C₂×C₈	—	122	4	no	—
214	C₁₆	—	122	8	no	—
215	C₂³	—	2	32	no	—
216	C₂³	—	2	8	no	—
217	C₂³	—	2	4	no	—
218	C₂³	—	2	4	no	—
219	C₂³	—	2	4	no	—
220	C₂³	—	2	16	no	—
221	C₂³	C₂²⋊#92420	1	4	yes	—
222	C₂³	#14519851	1	8	yes	complemented
223	C₂³	C₂²×#92420	1	2	yes	—
224	C₂³	C₂²×#92420	1	4	yes	—
225	C₂³	C₂²×#92420	1	1	yes	characteristic
226	C₂×C₄	—	2	8	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	16	no	—
229	C₂×C₄	—	2	8	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	C₂²⋊#92420	1	8	yes	—
233	C₂×C₄	C₂²⋊#92420	1	4	yes	—
234	C₂×C₄	C₂×C₂²⋊#6923	1	4	yes	—
235	C₂×C₄	C₂×C₂²⋊#6923	1	2	yes	—
236	C₂×C₄	C₂²×#92420	1	4	yes	—
237	C₂×C₄	C₂²×#92420	1	2	yes	—
238	C₂×C₄	C₂²×#92420	1	1	yes	characteristic
239	C₂×C₄	C₂×#11758217	1	4	yes	—
240	C₂×C₄	C₂×#11758217	1	2	yes	—
241	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
242	C₈	—	61	2	no	—
243	C₈	—	61	4	no	—
244	C₈	—	61	2	no	—
245	C₈	—	122	4	no	—
246	C₂²	—	2	32	no	—
247	C₂²	—	2	8	no	—
248	C₂²	—	2	16	no	—
249	C₂²	—	2	4	no	—
250	C₂²	—	2	4	no	—
251	C₂²	—	2	4	no	—
252	C₂²	C₂×C₂²⋊#92420	1	4	yes	—
253	C₂²	C₂×C₂²⋊#92420	1	2	yes	—
254	C₂²	C₂×#14519851	1	4	yes	—
255	C₂²	C₂×#14519851	1	8	yes	complemented
256	C₂²	C₂³×#92420	1	1	yes	characteristic
257	C₄	—	2	4	no	—
258	C₄	C₂×C₂²⋊#92420	1	4	yes	—
259	C₄	C₂×C₂²⋊#92420	1	2	yes	—
260	C₄	C₂²×C₂²⋊#6923	1	1	yes	characteristic
261	C₄	C₂²×#11758217	1	1	yes	characteristic
262	C₂	—	2	4	no	—
263	C₂	—	2	8	no	—
264	C₂	#21462046	1	4	yes	direct product factor
265	C₂	C₂²×C₂²⋊#92420	1	1	yes	characteristic
266	C₂	C₂²×#14519851	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×#6923
commutator	C₂²×C₆₁	C₂³×C₈
Frattini	C₂×C₄	C₂³×#6923
Fitting	C₂⁴×C₄×C₆₁	C₄
socle	C₂³×C₆₁	C₂²×C₈
prime radical	C₂⁵×C₆₁	C₈
cosocle	C₂×#92418	C₂⁴
hypercenter	C₂⁴×C₄	#6923
hypocenter	C₆₁	C₂×#2112
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#14519851	C₂	semidirect
2. #21462046	C₂	direct (N × H)
3. C₂×C₂²⋊#11758187	C₂	semidirect
6. C₂×#14519851	C₂²	semidirect
7. C₂²×#11758187	C₂²	semidirect
8. C₂²⋊#11758187	C₂²	semidirect
19. C₂×#11758187	C₂³	semidirect
88. C₂³×C₆₁	M₃₂	semidirect
122. C₂²×C₆₁	C₂×M₃₂	semidirect
135. C₂×C₆₁	#2112	semidirect
150. C₆₁	C₂×#2112	semidirect
222. C₂³	#14519851	semidirect
255. C₂²	C₂×#14519851	semidirect
264. C₂	#21462046	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
#21462046	2
C₂×C₂²⋊#11758187	2
C₂⁴×#92418	2
C₂×#2112	61

Subgroups — C2×#21462046 #30623859

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21462046 #30623859