Subgroups — C₂×#21447866 #29834260

Order 15616.

702 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×#14519723	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₂₄₄	C₂	1	1	yes	maximal, characteristic
3	#21447866	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄⋊#11758185	C₂	1	1	yes	maximal, characteristic
5	C₂×#16839566	C₂	1	2	yes	maximal
6	C₂×C₆₁×#494	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₂₄₄	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₂₂	—	2	1	no	—
9	C₂×#14519723	—	2	8	no	—
10	C₂×#14519723	C₂²	1	8	yes	—
11	C₂²×#11758185	C₂²	1	2	yes	—
12	C₄×Dic₂₄₄	—	2	2	no	—
13	C₄×Dic₂₄₄	C₂²	1	4	yes	—
14	C₂×#14519711	C₂²	1	1	yes	characteristic
15	C₂×#14519702	—	2	1	no	—
16	C₄⋊#11758185	C₂²	1	4	yes	—
17	#16839566	C₂²	1	8	yes	—
18	C₂²×Q₁₆×C₆₁	—	2	1	no	—
19	C₂²×Q₁₆×C₆₁	C₂²	1	2	yes	—
20	C₂×C₄⋊Q₈×C₆₁	—	2	1	no	—
21	C₆₁×#494	—	2	2	no	—
22	C₆₁×#494	C₂²	1	4	yes	—
23	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
24	C₂²×Dic₁₂₂	—	2	1	no	—
25	C₂×Dic₂₄₄	—	2	1	no	—
26	C₂×Dic₂₄₄	—	2	2	no	—
27	C₂×Dic₂₄₄	D₄	1	4	yes	—
28	C₂×Dic₂₄₄	C₂³	1	2	yes	—
29	C₄×Dic₁₂₂	—	2	4	no	—
30	C₄×Dic₁₂₂	—	4	2	no	—
31	C₂×C₄⋊Dic₆₁	—	2	1	no	—
32	C₂×#11758196	—	4	1	no	—
33	C₂×C₄×Dic₆₁	—	4	1	no	—
34	#14519723	—	2	16	no	—
35	C₂×#11758185	—	2	4	no	—
36	C₂×#11758185	C₂³	1	4	yes	—
37	#14519711	—	2	1	no	—
38	#14519711	C₂³	1	2	yes	—
39	#14519702	—	2	4	no	—
40	C₂²×Q₈×C₆₁	—	2	2	no	—
41	C₂×Q₁₆×C₆₁	—	2	4	no	—
42	C₂×Q₁₆×C₆₁	—	2	4	no	—
43	C₂×Q₁₆×C₆₁	—	2	2	no	—
44	C₂×Q₁₆×C₆₁	—	2	4	no	—
45	C₂×Q₁₆×C₆₁	—	4	4	no	—
46	C₂×Q₁₆×C₆₁	C₂³	1	4	yes	—
47	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
49	C₄⋊Q₈×C₆₁	—	2	4	no	—
50	C₄⋊Q₈×C₆₁	—	4	2	no	—
51	C₂×C₄⋊C₄×C₆₁	—	4	1	no	—
52	C₂×C₄⋊C₄×C₆₁	—	4	1	no	—
53	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
54	C₄×C₈×C₆₁	D₄	1	2	yes	—
55	C₄×C₈×C₆₁	C₂³	1	2	yes	—
56	C₂×Q₈×C₆₁	—	2	4	no	—
57	C₂×Q₈×C₆₁	—	4	2	no	—
58	C₂×Q₈×C₆₁	—	4	4	no	—
59	C₂×Q₈×C₆₁	—	4	4	no	—
60	C₂²×C₄×C₆₁	—	4	2	no	—
61	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
62	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
63	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
64	C₂²×Dic₆₁	—	4	1	no	—
65	C₂²×Dic₆₁	—	4	1	no	—
66	C₂×Dic₁₂₂	—	2	4	no	—
67	C₂×Dic₁₂₂	—	2	2	no	—
68	C₂×Dic₁₂₂	—	4	1	no	—
69	C₂×Dic₁₂₂	—	4	2	no	—
70	Q₁₆×C₆₁	—	2	4	no	—
71	Q₁₆×C₆₁	—	2	4	no	—
72	Q₁₆×C₆₁	—	4	8	no	—
73	Q₁₆×C₆₁	—	4	4	no	—
74	C₂×C₈×C₆₁	—	2	2	no	—
75	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
78	C₂×C₈×C₆₁	#41	1	2	yes	—
79	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
80	C₄⋊C₄×C₆₁	—	4	4	no	—
81	C₄⋊C₄×C₆₁	—	4	4	no	—
82	C₄²×C₆₁	D₈	1	2	yes	—
83	C₄²×C₆₁	C₂×D₄	1	2	yes	—
84	C₄⋊Dic₆₁	—	2	2	no	—
85	C₄⋊Dic₆₁	—	4	1	no	—
86	#11758196	—	4	4	no	—
87	C₄×Dic₆₁	—	4	4	no	—
88	Dic₂₄₄	—	2	4	no	—
89	Dic₂₄₄	—	2	2	no	—
90	Dic₂₄₄	C₂×D₄	1	4	yes	—
91	#11758185	—	2	4	no	—
92	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
93	Q₈×C₆₁	—	4	4	no	—
94	Q₈×C₆₁	—	4	4	no	—
95	Q₈×C₆₁	—	8	4	no	—
96	C₂×C₄×C₆₁	—	2	2	no	—
97	C₂×C₄×C₆₁	—	4	4	no	—
98	C₂×C₄×C₆₁	—	8	4	no	—
99	C₂×C₄×C₆₁	Q₃₂	1	4	yes	complemented
100	C₂×C₄×C₆₁	#121	1	2	yes	—
101	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
102	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
103	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
104	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
105	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
106	C₂×Dic₆₁	—	4	2	no	—
107	C₂×Dic₆₁	—	4	4	no	—
108	C₂×Dic₆₁	—	4	2	no	—
109	C₂×Dic₆₁	—	8	2	no	—
110	Dic₁₂₂	—	2	4	no	—
111	Dic₁₂₂	—	4	2	no	—
112	Dic₁₂₂	—	4	4	no	—
113	C₈×C₆₁	—	2	2	no	—
114	C₈×C₆₁	#121	1	2	yes	—
115	C₈×C₆₁	C₂²×D₄	1	1	yes	characteristic
116	C₈×C₆₁	C₂×#41	1	1	yes	characteristic
117	C₂×#2209	—	61	1	no	maximal, Sylow
118	C₂²×C₆₁	#459	1	2	yes	—
119	C₂²×C₆₁	C₂×Q₃₂	1	2	yes	—
120	C₂²×C₆₁	#510	1	2	yes	—
121	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
122	C₄×C₆₁	—	2	2	no	—
123	C₄×C₆₁	—	8	4	no	—
124	C₄×C₆₁	#459	1	2	yes	—
125	C₄×C₆₁	C₂×Q₃₂	1	4	yes	complemented
126	C₄×C₆₁	C₂×#121	1	1	yes	characteristic
127	C₄×C₆₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
128	Dic₆₁	—	4	4	no	—
129	Dic₆₁	—	8	2	no	—
130	C₂²×Q₃₂	—	61	2	no	—
131	C₂×#494	D₆₁	1	1	yes	characteristic
132	C₂×C₄×Q₁₆	—	61	1	no	—
133	#2209	—	61	8	no	—
134	C₂×C₄⋊C₁₆	—	61	1	no	—
135	C₂×#358	—	61	2	no	—
136	C₂×C₆₁	#2209	1	4	yes	complemented
137	C₂×C₆₁	C₂×#459	1	1	yes	characteristic
138	C₂×C₆₁	C₂²×Q₃₂	1	1	yes	characteristic
139	C₂×C₆₁	C₂×#510	1	1	yes	characteristic
140	C₂²×Q₁₆	—	2	1	no	—
141	C₂²×Q₁₆	—	61	1	no	—
142	C₂²×Q₁₆	C₂×D₆₁	1	2	yes	—
143	C₂×C₄⋊Q₈	—	2	1	no	—
144	C₂×C₄×Q₈	—	122	1	no	—
145	C₂×Q₃₂	—	61	8	no	—
146	C₂×Q₃₂	—	122	8	no	—
147	C₂²×C₁₆	—	61	2	no	—
148	#494	—	2	2	no	—
149	#494	C₂×D₆₁	1	4	yes	—
150	C₄×Q₁₆	—	61	4	no	—
151	C₄×Q₁₆	—	122	2	no	—
152	C₂×#107	—	61	1	no	—
153	C₂×#103	—	122	1	no	—
154	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
155	C₄⋊C₁₆	—	61	4	no	—
156	#358	—	61	8	no	—
157	C₆₁	C₂×#2209	1	1	yes	characteristic, complemented, Sylow
158	C₂²×Q₈	—	2	2	no	—
159	C₂²×Q₈	—	122	1	no	—
160	C₂×Q₁₆	—	2	4	no	—
161	C₂×Q₁₆	—	2	4	no	—
162	C₂×Q₁₆	—	2	4	no	—
163	C₂×Q₁₆	—	2	2	no	—
164	C₂×Q₁₆	—	4	4	no	—
165	C₂×Q₁₆	—	61	4	no	—
166	C₂×Q₁₆	—	61	2	no	—
167	C₂×Q₁₆	—	122	2	no	—
168	C₂×Q₁₆	—	122	1	no	—
169	C₂×Q₁₆	C₂²×D₆₁	1	4	yes	—
170	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
171	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
172	C₄⋊Q₈	—	2	4	no	—
173	C₄⋊Q₈	—	4	2	no	—
174	C₄×Q₈	—	122	4	no	—
175	C₄×Q₈	—	244	2	no	—
176	C₂×C₄⋊C₄	—	4	1	no	—
177	C₂×C₄⋊C₄	—	4	1	no	—
178	C₂×C₄⋊C₄	—	122	1	no	—
179	C₂×C₄⋊C₄	—	244	1	no	—
180	C₂×C₄²	—	244	1	no	—
181	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
182	Q₃₂	—	122	16	no	—
183	C₂×C₁₆	—	61	4	no	—
184	C₂×C₁₆	—	122	4	no	—
185	#107	—	61	2	no	—
186	#107	—	122	1	no	—
187	#103	—	122	4	no	—
188	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
189	C₄×C₈	C₂²×D₆₁	1	2	yes	—
190	C₂×Q₈	—	2	4	no	—
191	C₂×Q₈	—	4	2	no	—
192	C₂×Q₈	—	4	4	no	—
193	C₂×Q₈	—	4	4	no	—
194	C₂×Q₈	—	122	2	no	—
195	C₂×Q₈	—	122	4	no	—
196	C₂×Q₈	—	244	1	no	—
197	C₂×Q₈	—	244	2	no	—
198	C₂²×C₄	—	4	2	no	—
199	C₂²×C₄	—	244	1	no	—
200	C₂²×C₄	—	244	1	no	—
201	C₂²×C₄	#11758199	1	1	yes	characteristic
202	C₂²×C₄	#11758199	1	1	yes	characteristic
203	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
204	Q₁₆	—	2	4	no	—
205	Q₁₆	—	2	4	no	—
206	Q₁₆	—	4	8	no	—
207	Q₁₆	—	4	4	no	—
208	Q₁₆	—	61	4	no	—
209	Q₁₆	—	122	2	no	—
210	Q₁₆	—	122	4	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	D₄×D₆₁	1	2	yes	—
213	C₂×C₈	#11758224	1	2	yes	—
214	C₂×C₈	C₂×C₂²⋊D₆₁	1	2	yes	—
215	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
216	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
217	C₄⋊C₄	—	4	4	no	—
218	C₄⋊C₄	—	4	4	no	—
219	C₄⋊C₄	—	122	2	no	—
220	C₄⋊C₄	—	244	1	no	—
221	C₄⋊C₄	—	244	4	no	—
222	C₄²	—	244	4	no	—
223	C₄²	#11758199	1	2	yes	—
224	C₄²	C₂×C₂²⋊D₆₁	1	2	yes	—
225	C₁₆	—	122	4	no	—
226	C₂³	C₂×#11758199	1	1	yes	characteristic
227	Q₈	—	4	4	no	—
228	Q₈	—	4	4	no	—
229	Q₈	—	8	4	no	—
230	Q₈	—	122	4	no	—
231	Q₈	—	244	4	no	—
232	Q₈	—	244	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	8	4	no	—
236	C₂×C₄	—	488	2	no	—
237	C₂×C₄	—	244	2	no	—
238	C₂×C₄	—	244	4	no	—
239	C₂×C₄	—	244	2	no	—
240	C₂×C₄	#14519723	1	4	yes	complemented
241	C₂×C₄	#14519748	1	2	yes	—
242	C₂×C₄	C₂×#11758199	1	2	yes	—
243	C₂×C₄	C₂×#11758199	1	1	yes	characteristic
244	C₂×C₄	C₂×#11758199	1	2	yes	—
245	C₂×C₄	#14519803	1	2	yes	—
246	C₂×C₄	C₂²×C₂²⋊D₆₁	1	1	yes	characteristic
247	C₈	—	2	2	no	—
248	C₈	#14519748	1	2	yes	—
249	C₈	C₂×D₄×D₆₁	1	1	yes	characteristic
250	C₈	C₂×#11758224	1	1	yes	characteristic
251	C₂²	#16839954	1	2	yes	—
252	C₂²	C₂×#14519723	1	2	yes	—
253	C₂²	#16840203	1	2	yes	—
254	C₂²	C₂²×#11758199	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	8	4	no	—
257	C₄	—	488	2	no	—
258	C₄	—	244	4	no	—
259	C₄	#16839954	1	2	yes	—
260	C₄	C₂×#14519723	1	4	yes	complemented
261	C₄	C₂×#14519748	1	1	yes	characteristic
262	C₄	C₂×#14519803	1	1	yes	characteristic
263	C₂	#21447866	1	4	yes	direct product factor
264	C₂	C₂×#16839954	1	1	yes	characteristic
265	C₂	C₂²×#14519723	1	1	yes	characteristic
266	C₂	C₂×#16840203	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11758199
commutator, cosocle	C₂×C₈×C₆₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₁
Fitting	C₂×C₆₁×#494	C₂
socle, prime radical	C₂³×C₆₁	C₂×D₈
hypercenter	C₂×#494	D₆₁
hypocenter	C₆₁	C₂×#2209
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
3. #21447866	C₂	direct (N × H)
99. C₂×C₄×C₆₁	Q₃₂	semidirect
125. C₄×C₆₁	C₂×Q₃₂	semidirect
136. C₂×C₆₁	#2209	semidirect
157. C₆₁	C₂×#2209	semidirect
240. C₂×C₄	#14519723	semidirect
260. C₄	C₂×#14519723	semidirect
263. C₂	#21447866	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×#14519723	2
C₂×C₄×Dic₂₄₄	2
#21447866	2
C₂×C₄⋊#11758185	2
C₂×#16839566	2
C₂×C₆₁×#494	2
C₂×#2209	61

Subgroups — C2×#21447866 #29834260

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21447866 #29834260