Subgroups — C₂×#16840629 #21461488

Order 7808.

404 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₆₁⋊M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#14519856	C₂	1	1	yes	maximal, characteristic
3	#16840629	C₂	1	4	yes	maximal, direct product factor
4	C₂×#14519734	C₂	1	1	yes	maximal, characteristic
5	C₂×#11758213	—	2	1	no	—
6	D₆₁⋊M₁₆	—	2	1	no	—
7	D₆₁⋊M₁₆	—	2	1	no	—
8	D₆₁⋊M₁₆	C₄	1	2	yes	complemented
9	D₆₁⋊M₁₆	C₂²	1	2	yes	—
10	C₂×#11758217	—	2	1	no	—
11	C₂×D₆₁⋊C₈	C₄	1	2	yes	complemented
12	#14519856	—	2	1	no	—
13	#14519856	—	2	1	no	—
14	#14519856	C₄	1	2	yes	complemented
15	#14519856	C₂²	1	2	yes	—
16	C₂×C₄⋊#6923	—	2	1	no	—
17	C₂×C₂²⋊#6923	—	2	1	no	—
18	C₂×C₄×#6923	C₄	1	2	yes	complemented
19	C₂²×C₄×D₆₁	C₂²	1	1	yes	characteristic
20	#14519734	—	2	1	no	—
21	#14519734	—	2	1	no	—
22	#14519734	C₄	1	2	yes	complemented
23	#14519734	C₂²	1	2	yes	—
24	C₂×#11758196	—	2	1	no	—
25	C₂×#11758198	—	2	1	no	—
26	C₂×C₄×Dic₆₁	C₄	1	1	yes	characteristic, complemented
27	C₂×C₄²×C₆₁	C₄	1	1	yes	characteristic, complemented
28	C₂³×D₆₁	D₄	1	1	yes	characteristic
29	C₂²×#6923	—	2	2	no	—
30	C₂²×C₄×C₆₁	—	2	1	no	—
31	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
32	C₂²×Dic₆₁	—	2	1	no	—
33	C₂²×Dic₆₁	D₄	1	1	yes	characteristic
34	C₂×C₄×D₆₁	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×D₆₁	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×D₆₁	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×D₆₁	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×D₆₁	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×D₆₁	D₄	1	1	yes	characteristic
40	C₂×C₄×D₆₁	D₄	1	1	yes	characteristic
41	C₂×C₄×D₆₁	D₄	1	1	yes	characteristic
42	C₂×C₄×D₆₁	D₄	1	1	yes	characteristic
43	C₂×C₄×D₆₁	Q₈	1	1	yes	characteristic
44	C₂×C₄×D₆₁	Q₈	1	1	yes	characteristic
45	C₂×C₄×D₆₁	C₂³	1	1	yes	characteristic
46	C₂²⋊#6923	—	2	2	no	—
47	C₂²⋊#6923	—	2	2	no	—
48	#11758217	—	2	2	no	—
49	#11758217	—	2	2	no	—
50	C₂×#92420	—	2	2	no	—
51	C₄⋊#6923	—	2	2	no	—
52	C₄⋊#6923	—	2	2	no	—
53	C₄×#6923	—	2	2	no	—
54	C₄×#6923	C₂×C₄	1	4	yes	complemented
55	#11758213	—	2	2	no	—
56	#11758213	—	2	2	no	—
57	D₆₁⋊C₈	—	2	2	no	—
58	D₆₁⋊C₈	C₂×C₄	1	4	yes	complemented
59	C₄²×C₆₁	—	2	1	no	—
60	C₄²×C₆₁	C₂×C₄	1	2	yes	complemented
61	#11758198	—	2	2	no	—
62	#11758198	—	2	2	no	—
63	#11758196	—	2	2	no	—
64	#11758196	—	2	2	no	—
65	C₄×Dic₆₁	—	2	1	no	—
66	C₄×Dic₆₁	C₂×C₄	1	2	yes	complemented
67	C₂³×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×D₆₁	—	4	2	no	—
69	C₂²×D₆₁	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×D₆₁	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×D₆₁	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×D₆₁	C₄⋊C₄	1	1	yes	characteristic
73	C₂²×D₆₁	C₄⋊C₄	1	1	yes	characteristic
74	C₂²×D₆₁	C₂×D₄	1	1	yes	characteristic
75	C₂×#6923	—	2	4	no	—
76	C₂×#6923	—	4	4	no	—
77	C₂×C₄×C₆₁	—	2	2	no	—
78	C₂×C₄×C₆₁	—	4	2	no	—
79	C₂×C₄×C₆₁	C₄²	1	1	yes	characteristic
80	C₂×C₄×C₆₁	C₄²	1	1	yes	characteristic
81	C₂×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
85	C₂×Dic₆₁	—	2	2	no	—
86	C₂×Dic₆₁	—	4	2	no	—
87	C₂×Dic₆₁	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×Dic₆₁	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×Dic₆₁	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
91	C₂×Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
92	C₂×Dic₆₁	C₂×D₄	1	1	yes	characteristic
93	C₄×D₆₁	C₄²	1	1	yes	characteristic
94	C₄×D₆₁	C₄²	1	1	yes	characteristic
95	C₄×D₆₁	C₂²⋊C₄	1	1	yes	characteristic
96	C₄×D₆₁	C₂²⋊C₄	1	1	yes	characteristic
97	C₄×D₆₁	C₂²⋊C₄	1	1	yes	characteristic
98	C₄×D₆₁	C₂²⋊C₄	1	1	yes	characteristic
99	C₄×D₆₁	C₄⋊C₄	1	1	yes	characteristic
100	C₄×D₆₁	C₄⋊C₄	1	1	yes	characteristic
101	C₄×D₆₁	C₄⋊C₄	1	1	yes	characteristic
102	C₄×D₆₁	C₄⋊C₄	1	1	yes	characteristic
103	C₄×D₆₁	C₄⋊C₄	1	1	yes	characteristic
104	C₄×D₆₁	C₄⋊C₄	1	1	yes	characteristic
105	C₄×D₆₁	C₂²×C₄	1	1	yes	characteristic
106	C₄×D₆₁	C₂²×C₄	1	1	yes	characteristic
107	C₄×D₆₁	C₂×D₄	1	1	yes	characteristic
108	C₄×D₆₁	C₂×Q₈	1	1	yes	characteristic
109	#92420	—	2	4	no	—
110	C₂²×C₆₁	—	2	2	no	—
111	C₂²×C₆₁	#95	1	1	yes	characteristic
112	C₂²×C₆₁	#95	1	1	yes	characteristic
113	C₂²×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×D₆₁	—	2	2	no	—
115	C₂×D₆₁	—	2	2	no	—
116	C₂×D₆₁	—	4	1	no	—
117	C₂×D₆₁	—	4	1	no	—
118	C₂×D₆₁	—	4	1	no	—
119	C₂×D₆₁	—	4	1	no	—
120	C₂×D₆₁	#95	1	1	yes	characteristic
121	C₂×D₆₁	#95	1	1	yes	characteristic
122	C₂×D₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×D₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
124	#6923	—	4	4	no	—
125	C₄×C₆₁	—	4	2	no	—
126	C₄×C₆₁	#95	1	1	yes	characteristic
127	C₄×C₆₁	#95	1	1	yes	characteristic
128	C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
129	C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
130	Dic₆₁	—	4	2	no	—
131	Dic₆₁	#95	1	1	yes	characteristic
132	Dic₆₁	#95	1	1	yes	characteristic
133	Dic₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	Dic₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₂×H(3,ℤ/4)	—	61	1	no	maximal, Sylow
136	C₂×C₆₁	—	2	1	no	—
137	C₂×C₆₁	—	2	1	no	—
138	C₂×C₆₁	H(3,ℤ/4)	1	2	yes	complemented
139	C₂×C₆₁	C₂×#95	1	1	yes	characteristic
140	D₆₁	—	2	1	no	—
141	D₆₁	—	2	1	no	—
142	D₆₁	—	2	1	no	—
143	D₆₁	—	2	1	no	—
144	C₂²×M₁₆	—	61	1	no	—
145	C₂×#117	—	61	1	no	—
146	C₂×#117	—	61	1	no	—
147	H(3,ℤ/4)	—	61	4	no	—
148	C₆₁	C₂×H(3,ℤ/4)	1	1	yes	characteristic, complemented, Sylow
149	C₂³×C₄	—	61	1	no	—
150	C₂×M₁₆	—	61	2	no	—
151	C₂×M₁₆	—	61	2	no	—
152	C₂×M₁₆	—	122	1	no	—
153	C₂×M₁₆	—	122	1	no	—
154	C₂×M₁₆	—	122	1	no	—
155	C₂×M₁₆	—	122	1	no	—
156	C₂²×C₈	—	61	2	no	—
157	#117	—	61	2	no	—
158	#117	—	61	2	no	—
159	#117	—	61	2	no	—
160	#117	—	61	2	no	—
161	#117	—	122	1	no	—
162	#117	—	122	1	no	—
163	#117	—	122	1	no	—
164	#117	—	122	1	no	—
165	C₂×C₄⋊C₄	—	122	1	no	—
166	C₂×C₄⋊C₄	—	122	1	no	—
167	C₂×C₂²⋊C₄	—	122	1	no	—
168	C₂×C₂²⋊C₄	—	122	1	no	—
169	C₂×C₄²	—	61	2	no	—
170	C₂×C₄²	—	61	1	no	—
171	C₂×C₄²	#6923	1	1	yes	characteristic, complemented
172	C₂⁴	—	61	1	no	—
173	C₂²×C₄	—	2	1	no	—
174	C₂²×C₄	—	61	1	no	—
175	C₂²×C₄	—	61	1	no	—
176	C₂²×C₄	—	61	1	no	—
177	C₂²×C₄	—	61	1	no	—
178	C₂²×C₄	—	61	1	no	—
179	C₂²×C₄	—	61	1	no	—
180	C₂²×C₄	—	61	1	no	—
181	C₂²×C₄	—	61	1	no	—
182	C₂²×C₄	—	61	1	no	—
183	C₂²×C₄	—	61	1	no	—
184	C₂²×C₄	—	61	1	no	—
185	C₂²×C₄	—	61	1	no	—
186	C₂²×C₄	—	61	1	no	—
187	C₂²×C₄	—	122	2	no	—
188	C₂²×C₄	—	122	1	no	—
189	C₂²×C₄	C₂×#6923	1	1	yes	characteristic
190	M₁₆	—	122	2	no	—
191	M₁₆	—	122	2	no	—
192	M₁₆	—	122	2	no	—
193	M₁₆	—	122	2	no	—
194	C₂×C₈	—	61	4	no	—
195	C₂×C₈	—	122	2	no	—
196	C₂×C₈	—	122	2	no	—
197	C₄⋊C₄	—	122	2	no	—
198	C₄⋊C₄	—	122	2	no	—
199	C₄⋊C₄	—	122	2	no	—
200	C₄⋊C₄	—	122	2	no	—
201	C₂²⋊C₄	—	122	2	no	—
202	C₂²⋊C₄	—	122	2	no	—
203	C₂²⋊C₄	—	122	2	no	—
204	C₂²⋊C₄	—	122	2	no	—
205	C₄²	—	2	1	no	—
206	C₄²	—	61	4	no	—
207	C₄²	—	61	2	no	—
208	C₄²	—	122	2	no	—
209	C₄²	—	122	1	no	—
210	C₄²	C₂×#6923	1	2	yes	complemented
211	C₂³	—	61	1	no	—
212	C₂³	—	61	1	no	—
213	C₂³	—	61	1	no	—
214	C₂³	—	61	1	no	—
215	C₂³	—	61	1	no	—
216	C₂³	—	61	1	no	—
217	C₂³	—	244	2	no	—
218	C₂³	C₂²⋊#6923	1	1	yes	characteristic
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	—	61	1	no	—
222	C₂×C₄	—	61	1	no	—
223	C₂×C₄	—	61	1	no	—
224	C₂×C₄	—	61	1	no	—
225	C₂×C₄	—	61	1	no	—
226	C₂×C₄	—	61	1	no	—
227	C₂×C₄	—	61	1	no	—
228	C₂×C₄	—	61	1	no	—
229	C₂×C₄	—	61	1	no	—
230	C₂×C₄	—	61	1	no	—
231	C₂×C₄	—	61	1	no	—
232	C₂×C₄	—	61	1	no	—
233	C₂×C₄	—	61	1	no	—
234	C₂×C₄	—	61	1	no	—
235	C₂×C₄	—	61	1	no	—
236	C₂×C₄	—	61	1	no	—
237	C₂×C₄	—	61	1	no	—
238	C₂×C₄	—	61	1	no	—
239	C₂×C₄	—	61	1	no	—
240	C₂×C₄	—	61	1	no	—
241	C₂×C₄	—	61	1	no	—
242	C₂×C₄	—	61	1	no	—
243	C₂×C₄	—	122	4	no	—
244	C₂×C₄	—	122	2	no	—
245	C₂×C₄	—	244	2	no	—
246	C₂×C₄	—	244	4	no	—
247	C₂×C₄	C₄×#6923	1	1	yes	characteristic
248	C₂×C₄	C₄×#6923	1	1	yes	characteristic
249	C₂×C₄	C₄⋊#6923	1	1	yes	characteristic
250	C₂×C₄	C₄⋊#6923	1	1	yes	characteristic
251	C₂×C₄	C₂²⋊#6923	1	1	yes	characteristic
252	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
253	C₈	—	122	4	no	—
254	C₂²	—	2	2	no	—
255	C₂²	—	61	1	no	—
256	C₂²	—	61	1	no	—
257	C₂²	—	61	1	no	—
258	C₂²	—	61	1	no	—
259	C₂²	—	122	2	no	—
260	C₂²	—	122	2	no	—
261	C₂²	—	244	1	no	—
262	C₂²	—	244	1	no	—
263	C₂²	—	244	1	no	—
264	C₂²	—	244	1	no	—
265	C₂²	(C₂×#6923)⋊C₄	1	1	yes	characteristic
266	C₂²	(C₂×#6923)⋊C₄	1	1	yes	characteristic
267	C₂²	C₂×C₂²⋊#6923	1	1	yes	characteristic
268	C₄	—	4	2	no	—
269	C₄	—	61	1	no	—
270	C₄	—	61	1	no	—
271	C₄	—	61	1	no	—
272	C₄	—	61	1	no	—
273	C₄	—	244	2	no	—
274	C₄	—	244	4	no	—
275	C₄	(C₂×#6923)⋊C₄	1	1	yes	characteristic
276	C₄	(C₂×#6923)⋊C₄	1	1	yes	characteristic
277	C₄	C₂×C₄×#6923	1	1	yes	characteristic
278	C₄	C₂×C₄⋊#6923	1	1	yes	characteristic
279	C₂	—	2	1	no	—
280	C₂	—	2	1	no	—
281	C₂	—	122	1	no	—
282	C₂	—	122	1	no	—
283	C₂	—	122	1	no	—
284	C₂	—	122	1	no	—
285	C₂	#16840629	1	2	yes	direct product factor
286	C₂	C₂×(C₂×#6923)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₄×#6923
commutator	C₄×C₆₁	C₂×C₄²
Frattini	C₂×C₄	C₂²×#6923
Fitting	C₂×C₄²×C₆₁	C₄
socle	C₂²×C₆₁	#95
hypercenter	C₂×C₄²	#6923
hypocenter	C₆₁	C₂×H(3,ℤ/4)
LCS	G ▷ C₄×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
3. #16840629	C₂	direct (N × H)
8. D₆₁⋊M₁₆	C₄	semidirect
11. C₂×D₆₁⋊C₈	C₄	semidirect
14. #14519856	C₄	semidirect
18. C₂×C₄×#6923	C₄	semidirect
22. #14519734	C₄	semidirect
26. C₂×C₄×Dic₆₁	C₄	semidirect
27. C₂×C₄²×C₆₁	C₄	semidirect
54. C₄×#6923	C₂×C₄	semidirect
58. D₆₁⋊C₈	C₂×C₄	semidirect
60. C₄²×C₆₁	C₂×C₄	semidirect
66. C₄×Dic₆₁	C₂×C₄	semidirect
138. C₂×C₆₁	H(3,ℤ/4)	semidirect
148. C₆₁	C₂×H(3,ℤ/4)	semidirect
171. C₂×C₄²	#6923	semidirect
210. C₄²	C₂×#6923	semidirect
285. C₂	#16840629	direct (N × H)

Subgroups — C2×#16840629 #21461488

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16840629 #21461488