Subgroups — C₂×#21462176 #30624134

Order 15616.

726 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21462176	C₂	1	8	yes	maximal, direct product factor
2	C₂×C₄⋊#11758187	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊#11758187	C₂	1	2	yes	maximal
4	C₂×C₄×#11758187	C₂	1	2	yes	maximal, complemented
5	C₂²×C₄×#92418	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11758187	C₂²	1	4	yes	—
7	C₄⋊#11758187	C₂²	1	8	yes	complemented
8	C₂²⋊#11758187	C₂²	1	8	yes	—
9	C₄×#11758187	C₂²	1	8	yes	complemented
10	C₂³×#92418	C₄	1	1	yes	characteristic
11	C₂³×#92418	C₂²	1	1	yes	characteristic
12	C₂×C₄×#92418	C₄	1	4	yes	—
13	C₂×C₄×#92418	C₄	1	1	yes	characteristic
14	C₂×C₄×#92418	C₄	1	1	yes	characteristic
15	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
16	C₂×C₄×#92418	C₂²	1	4	yes	—
17	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
18	C₂²×C₄²×C₆₁	C₄	1	1	yes	characteristic
19	C₂×#11758187	—	2	8	no	—
20	C₂×#11758187	C₂³	1	8	yes	—
21	C₂²×#92418	—	2	4	no	—
22	C₂²×#92418	—	2	4	no	—
23	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
24	C₂²×#92418	C₂×C₄	1	2	yes	—
25	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
26	C₂²×#92418	C₂×C₄	1	4	yes	—
27	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
28	C₂²×#92418	C₂³	1	2	yes	—
29	C₂²×#92418	C₂³	1	1	yes	characteristic
30	C₄×#92418	C₂×C₄	1	4	yes	—
31	C₄×#92418	C₂×C₄	1	4	yes	—
32	C₄×#92418	C₂×C₄	1	2	yes	—
33	C₄×#92418	C₂×C₄	1	2	yes	—
34	C₄×#92418	C₂³	1	2	yes	—
35	C₄×#92418	C₂³	1	2	yes	—
36	C₂³×C₄×C₆₁	C₈	1	1	yes	characteristic
37	C₂³×C₄×C₆₁	C₈	1	1	yes	characteristic
38	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₆₁	C₈	1	1	yes	characteristic
40	C₂×C₄²×C₆₁	C₈	1	4	yes	—
41	C₂×C₄²×C₆₁	C₈	1	1	yes	characteristic
42	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₆₁	C₂×C₄	1	4	yes	—
44	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
45	C₂⁴×C₆₁	C₂×C₈	1	1	yes	characteristic
46	C₂²×C₄×C₆₁	—	2	4	no	—
47	C₂²×C₄×C₆₁	—	2	2	no	—
48	C₂²×C₄×C₆₁	—	2	2	no	—
49	C₂²×C₄×C₆₁	C₁₆	1	8	yes	complemented
50	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
52	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
53	C₂²×C₄×C₆₁	C₂×C₈	1	4	yes	—
54	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
56	C₂²×C₄×C₆₁	C₂×C₈	1	4	yes	—
57	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
58	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
60	C₄²×C₆₁	C₂×C₈	1	4	yes	—
61	C₄²×C₆₁	C₂×C₈	1	4	yes	—
62	C₄²×C₆₁	C₂×C₈	1	2	yes	—
63	C₄²×C₆₁	C₂×C₈	1	2	yes	—
64	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
65	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
66	C₂×#92418	—	2	2	no	—
67	C₂×#92418	—	2	2	no	—
68	C₂×#92418	—	2	4	no	—
69	C₂×#92418	—	2	8	no	—
70	C₂×#92418	—	2	4	no	—
71	C₂×#92418	M₁₆	1	4	yes	—
72	C₂×#92418	C₂²×C₄	1	2	yes	—
73	C₂×#92418	C₂²×C₄	1	2	yes	—
74	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
75	C₂×#92418	C₂²×C₄	1	2	yes	—
76	C₂×#92418	#41	1	4	yes	—
77	C₂×#92418	C₂⁴	1	1	yes	characteristic
78	#11758187	—	2	8	no	—
79	C₂³×C₆₁	—	2	2	no	—
80	C₂³×C₆₁	—	2	2	no	—
81	C₂³×C₆₁	C₂×C₁₆	1	4	yes	—
82	C₂³×C₆₁	C₂²×C₈	1	2	yes	—
83	C₂³×C₆₁	C₂²×C₈	1	1	yes	characteristic
84	C₂×C₄×C₆₁	—	2	2	no	—
85	C₂×C₄×C₆₁	—	2	2	no	—
86	C₂×C₄×C₆₁	—	2	2	no	—
87	C₂×C₄×C₆₁	—	2	2	no	—
88	C₂×C₄×C₆₁	—	2	4	no	—
89	C₂×C₄×C₆₁	—	2	8	no	—
90	C₂×C₄×C₆₁	C₂×C₁₆	1	4	yes	—
91	C₂×C₄×C₆₁	C₂×C₁₆	1	4	yes	—
92	C₂×C₄×C₆₁	C₂×C₁₆	1	16	yes	complemented
93	C₂×C₄×C₆₁	M₃₂	1	4	yes	—
94	C₂×C₄×C₆₁	#117	1	2	yes	—
95	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
96	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
97	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
98	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
99	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
100	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
101	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
102	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
103	#92418	—	2	2	no	—
104	#92418	—	2	2	no	—
105	#92418	#117	1	2	yes	—
106	#92418	C₂×M₁₆	1	4	yes	—
107	#92418	C₂×#41	1	2	yes	—
108	C₂×#2162	—	61	1	no	maximal, Sylow
109	C₂²×C₆₁	—	2	2	no	—
110	C₂²×C₆₁	—	2	2	no	—
111	C₂²×C₆₁	—	2	8	no	—
112	C₂²×C₆₁	#431	1	2	yes	—
113	C₂²×C₆₁	C₂²×C₁₆	1	4	yes	—
114	C₂²×C₆₁	C₂²×C₁₆	1	2	yes	—
115	C₂²×C₆₁	C₂×M₃₂	1	2	yes	—
116	C₂²×C₆₁	C₂³×C₈	1	1	yes	characteristic
117	C₄×C₆₁	—	2	2	no	—
118	C₄×C₆₁	—	2	2	no	—
119	C₄×C₆₁	#431	1	2	yes	—
120	C₄×C₆₁	C₂²×C₁₆	1	4	yes	—
121	C₄×C₆₁	C₂²×C₁₆	1	4	yes	—
122	C₄×C₆₁	C₂×M₃₂	1	4	yes	—
123	C₄×C₆₁	C₂×#117	1	1	yes	characteristic
124	C₄×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
125	C₂²×C₄×C₈	—	61	1	no	—
126	#2162	—	61	8	no	—
127	C₂×C₄⋊C₁₆	—	61	2	no	—
128	C₂×C₂²⋊C₁₆	—	61	2	no	—
129	C₂×C₄×C₁₆	—	61	2	no	—
130	C₂×C₆₁	—	2	4	no	—
131	C₂×C₆₁	#2162	1	4	yes	complemented
132	C₂×C₆₁	C₂×#431	1	1	yes	characteristic
133	C₂×C₆₁	C₂³×C₁₆	1	1	yes	characteristic
134	C₂×C₆₁	C₂²×M₃₂	1	1	yes	characteristic
135	C₂³×C₈	—	61	1	no	—
136	C₂³×C₈	—	61	1	no	—
137	C₂²×C₄²	#6923	1	1	yes	characteristic
138	C₂²×C₁₆	—	61	4	no	—
139	C₂×C₄×C₈	—	61	1	no	—
140	C₂×C₄×C₈	—	61	4	no	—
141	C₂×C₄×C₈	—	61	1	no	—
142	C₂×C₄×C₈	—	61	4	no	—
143	C₂×C₄×C₈	—	61	1	no	—
144	C₂×C₄×C₈	—	61	1	no	—
145	C₄⋊C₁₆	—	61	8	no	—
146	C₂²⋊C₁₆	—	61	8	no	—
147	C₄×C₁₆	—	61	8	no	—
148	C₆₁	C₂×#2162	1	1	yes	characteristic, complemented, Sylow
149	C₂³×C₄	#92420	1	1	yes	characteristic
150	C₂³×C₄	#92420	1	1	yes	characteristic
151	C₂³×C₄	C₂×#6923	1	1	yes	characteristic
152	C₂²×C₈	—	61	1	no	—
153	C₂²×C₈	—	61	1	no	—
154	C₂²×C₈	—	61	2	no	—
155	C₂²×C₈	—	61	2	no	—
156	C₂²×C₈	—	61	4	no	—
157	C₂²×C₈	—	61	1	no	—
158	C₂²×C₈	—	61	1	no	—
159	C₂²×C₈	—	122	4	no	—
160	C₂²×C₈	—	122	4	no	—
161	C₂×C₄²	#92420	1	1	yes	characteristic
162	C₂×C₄²	#92420	1	1	yes	characteristic
163	C₂×C₄²	#92420	1	4	yes	—
164	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
165	C₂×C₄²	C₂×#6923	1	4	yes	—
166	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
167	C₂×C₁₆	—	61	8	no	—
168	C₂×C₁₆	—	122	8	no	—
169	C₄×C₈	—	61	2	no	—
170	C₄×C₈	—	61	2	no	—
171	C₄×C₈	—	61	2	no	—
172	C₄×C₈	—	61	4	no	—
173	C₄×C₈	—	61	2	no	—
174	C₄×C₈	—	61	4	no	—
175	C₂⁴	C₂×#92420	1	1	yes	characteristic
176	C₂²×C₄	—	2	4	no	—
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	#11758187	1	8	yes	complemented
180	C₂²×C₄	C₂×#92420	1	1	yes	characteristic
181	C₂²×C₄	C₂×#92420	1	2	yes	—
182	C₂²×C₄	C₂×#92420	1	1	yes	characteristic
183	C₂²×C₄	C₂×#92420	1	4	yes	—
184	C₂²×C₄	C₂×#92420	1	1	yes	characteristic
185	C₂²×C₄	C₂×#92420	1	1	yes	characteristic
186	C₂²×C₄	C₂×#92420	1	4	yes	—
187	C₂²×C₄	C₂×#92420	1	1	yes	characteristic
188	C₂²×C₄	C₂²×#6923	1	2	yes	—
189	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
190	C₂×C₈	—	61	4	no	—
191	C₂×C₈	—	61	4	no	—
192	C₂×C₈	—	61	1	no	—
193	C₂×C₈	—	61	1	no	—
194	C₂×C₈	—	61	2	no	—
195	C₂×C₈	—	61	2	no	—
196	C₂×C₈	—	61	2	no	—
197	C₂×C₈	—	122	4	no	—
198	C₂×C₈	—	122	4	no	—
199	C₂×C₈	—	122	8	no	—
200	C₂×C₈	—	122	2	no	—
201	C₂×C₈	—	122	2	no	—
202	C₄²	C₂×#92420	1	4	yes	—
203	C₄²	C₂×#92420	1	4	yes	—
204	C₄²	C₂×#92420	1	2	yes	—
205	C₄²	C₂×#92420	1	2	yes	—
206	C₄²	C₂²×#6923	1	2	yes	—
207	C₄²	C₂²×#6923	1	2	yes	—
208	C₁₆	—	122	8	no	—
209	C₂³	—	2	2	no	—
210	C₂³	—	2	2	no	—
211	C₂³	C₂×#11758187	1	4	yes	—
212	C₂³	C₂²×#92420	1	2	yes	—
213	C₂³	C₂²×#92420	1	1	yes	characteristic
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	2	8	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	C₂×#11758187	1	4	yes	—
221	C₂×C₄	C₂×#11758187	1	4	yes	—
222	C₂×C₄	C₂×#11758187	1	16	yes	complemented
223	C₂×C₄	#14519851	1	4	yes	—
224	C₂×C₄	#14519856	1	2	yes	—
225	C₂×C₄	C₂²×#92420	1	2	yes	—
226	C₂×C₄	C₂²×#92420	1	2	yes	—
227	C₂×C₄	C₂²×#92420	1	1	yes	characteristic
228	C₂×C₄	C₂²×#92420	1	2	yes	—
229	C₂×C₄	C₂²×#92420	1	2	yes	—
230	C₂×C₄	C₂²×#92420	1	2	yes	—
231	C₂×C₄	D₆₁⋊M₁₆	1	2	yes	—
232	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
233	C₈	—	61	2	no	—
234	C₈	—	61	2	no	—
235	C₈	—	61	4	no	—
236	C₈	—	122	2	no	—
237	C₈	—	122	2	no	—
238	C₂²	—	2	8	no	—
239	C₂²	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	#16840778	1	2	yes	—
242	C₂²	C₂²×#11758187	1	4	yes	—
243	C₂²	C₂²×#11758187	1	2	yes	—
244	C₂²	C₂×#14519851	1	2	yes	—
245	C₂²	C₂³×#92420	1	1	yes	characteristic
246	C₄	—	2	2	no	—
247	C₄	—	2	2	no	—
248	C₄	#16840778	1	2	yes	—
249	C₄	C₂²×#11758187	1	4	yes	—
250	C₄	C₂²×#11758187	1	4	yes	—
251	C₄	C₂×#14519851	1	4	yes	—
252	C₄	C₂×#14519856	1	1	yes	characteristic
253	C₄	C₂×D₆₁⋊M₁₆	1	1	yes	characteristic
254	C₂	—	2	4	no	—
255	C₂	#21462176	1	4	yes	direct product factor
256	C₂	C₂×#16840778	1	1	yes	characteristic
257	C₂	C₂³×#11758187	1	1	yes	characteristic
258	C₂	C₂²×#14519851	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	C₂×#6923
commutator	C₂×C₆₁	C₂³×C₁₆
Frattini	C₂×C₄	C₂³×#6923
Fitting	C₂²×C₄²×C₆₁	C₄
socle	C₂³×C₆₁	C₂²×C₈
prime radical	C₂⁴×C₆₁	C₂×C₈
cosocle	C₂×#92418	C₂⁴
hypercenter	C₂²×C₄²	#6923
hypocenter	C₆₁	C₂×#2162
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
1. #21462176	C₂	direct (N × H)
2. C₂×C₄⋊#11758187	C₂	semidirect
4. C₂×C₄×#11758187	C₂	semidirect
7. C₄⋊#11758187	C₂²	semidirect
9. C₄×#11758187	C₂²	semidirect
49. C₂²×C₄×C₆₁	C₁₆	semidirect
92. C₂×C₄×C₆₁	C₂×C₁₆	semidirect
131. C₂×C₆₁	#2162	semidirect
148. C₆₁	C₂×#2162	semidirect
179. C₂²×C₄	#11758187	semidirect
222. C₂×C₄	C₂×#11758187	semidirect
255. C₂	#21462176	direct (N × H)

Maximal subgroups

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

subgroup	index
#21462176	2
C₂×C₄⋊#11758187	2
C₂×C₂²⋊#11758187	2
C₂×C₄×#11758187	2
C₂²×C₄×#92418	2
C₂×#2162	61

Subgroups — C2×#21462176 #30624134

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21462176 #30624134