Subgroups — C₂×#21444607 #29649097

Order 15616.

714 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16839815	C₂	1	1	yes	maximal, characteristic
2	C₂×#16839786	C₂	1	1	yes	maximal, characteristic
3	#21444607	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16839514	C₂	1	2	yes	maximal, complemented
5	C₂×#16839512	C₂	1	2	yes	maximal, complemented
6	C₂×C₆₁×#431	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11758194	—	2	1	no	—
8	#16839815	C₄	1	4	yes	—
9	#16839815	C₂²	1	4	yes	—
10	#16839786	—	2	2	no	—
11	#16839786	C₂²	1	4	yes	—
12	C₂×C₄⋊#92418	—	2	1	no	—
13	C₂×C₄⋊#92418	C₄	1	2	yes	—
14	C₂×C₄⋊#92418	C₂²	1	2	yes	—
15	C₂×C₂²⋊#92418	C₄	1	2	yes	—
16	C₂×C₄×#92418	C₂²	1	2	yes	—
17	#16839514	C₂²	1	8	yes	complemented
18	#16839512	C₂²	1	8	yes	complemented
19	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
20	C₆₁×#431	—	2	2	no	—
21	C₆₁×#431	C₂²	1	4	yes	—
22	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊C₈×C₆₁	—	2	1	no	—
25	C₂×C₄×C₈×C₆₁	—	2	1	no	—
26	C₂×#11758194	—	2	4	no	—
27	C₂×#11758194	—	4	4	no	—
28	C₂²×#92418	—	2	2	no	—
29	C₂²×#92418	C₂×C₄	1	4	yes	—
30	C₄⋊#92418	—	2	4	no	—
31	C₄⋊#92418	—	2	2	no	—
32	C₄⋊#92418	C₂×C₄	1	8	yes	—
33	C₄⋊#92418	C₂³	1	4	yes	—
34	C₂²⋊#92418	C₂×C₄	1	8	yes	—
35	C₄×#92418	C₂×C₄	1	4	yes	—
36	C₄×#92418	C₂³	1	4	yes	—
37	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
38	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
39	C₂³×C₄×C₆₁	Q₈	1	1	yes	characteristic
40	C₂²×C₈×C₆₁	—	2	1	no	—
41	C₂²×C₈×C₆₁	—	2	1	no	—
42	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₆₁	D₄	1	2	yes	—
46	C₂×C₄²×C₆₁	Q₈	1	1	yes	characteristic
47	C₂×C₄²×C₆₁	Q₈	1	2	yes	—
48	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
50	C₄⋊C₈×C₆₁	—	2	1	no	—
51	C₄⋊C₈×C₆₁	—	2	1	no	—
52	C₄⋊C₈×C₆₁	C₂³	1	2	yes	—
53	C₄⋊C₈×C₆₁	C₂³	1	2	yes	—
54	C₂²⋊C₈×C₆₁	—	2	4	no	—
55	C₄×C₈×C₆₁	—	2	4	no	—
56	C₂⁴×C₆₁	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₆₁	—	2	4	no	—
58	C₂²×C₄×C₆₁	—	2	4	no	—
59	C₂²×C₄×C₆₁	—	2	2	no	—
60	C₂²×C₄×C₆₁	—	2	2	no	—
61	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
62	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
72	C₂²×C₄×C₆₁	C₂×Q₈	1	2	yes	—
73	C₂×C₈×C₆₁	—	2	2	no	—
74	C₂×C₈×C₆₁	—	2	2	no	—
75	C₂×C₈×C₆₁	—	4	2	no	—
76	C₂×C₈×C₆₁	—	4	2	no	—
77	C₄²×C₆₁	C₄⋊C₄	1	2	yes	—
78	C₄²×C₆₁	C₄⋊C₄	1	2	yes	—
79	C₄²×C₆₁	C₄⋊C₄	1	2	yes	—
80	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
82	C₄²×C₆₁	C₂×D₄	1	2	yes	—
83	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₆₁	C₂×Q₈	1	2	yes	—
85	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
86	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
87	#11758194	—	4	8	no	—
88	C₂×#92418	—	2	8	no	—
89	C₂×#92418	—	2	4	no	—
90	C₂×#92418	—	4	4	no	—
91	C₂×#92418	C₂²×C₄	1	8	yes	—
92	C₂³×C₆₁	—	2	2	no	—
93	C₂³×C₆₁	—	2	2	no	—
94	C₂³×C₆₁	—	2	2	no	—
95	C₂³×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
96	C₂³×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₆₁	—	2	2	no	—
98	C₂×C₄×C₆₁	—	2	2	no	—
99	C₂×C₄×C₆₁	—	2	2	no	—
100	C₂×C₄×C₆₁	—	2	4	no	—
101	C₂×C₄×C₆₁	—	2	2	no	—
102	C₂×C₄×C₆₁	—	2	2	no	—
103	C₂×C₄×C₆₁	—	2	4	no	—
104	C₂×C₄×C₆₁	—	2	4	no	—
105	C₂×C₄×C₆₁	—	2	4	no	—
106	C₂×C₄×C₆₁	—	2	4	no	—
107	C₂×C₄×C₆₁	#108	1	4	yes	—
108	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
110	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
114	C₂×C₄×C₆₁	#131	1	4	yes	—
115	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
116	C₂×C₄×C₆₁	#137	1	2	yes	—
117	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
120	C₈×C₆₁	—	4	2	no	—
121	C₈×C₆₁	—	4	2	no	—
122	#92418	—	2	8	no	—
123	#92418	—	4	4	no	—
124	C₂×#1574	—	61	1	no	maximal, Sylow
125	C₂²×C₆₁	—	2	2	no	—
126	C₂²×C₆₁	—	2	2	no	—
127	C₂²×C₆₁	—	2	2	no	—
128	C₂²×C₆₁	—	4	4	no	—
129	C₂²×C₆₁	#424	1	2	yes	—
130	C₂²×C₆₁	#428	1	2	yes	—
131	C₂²×C₆₁	C₂×#108	1	2	yes	—
132	C₂²×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
133	C₄×C₆₁	—	2	2	no	—
134	C₄×C₆₁	—	2	2	no	—
135	C₄×C₆₁	—	2	1	no	—
136	C₄×C₆₁	—	2	1	no	—
137	C₄×C₆₁	#424	1	2	yes	—
138	C₄×C₆₁	#428	1	2	yes	—
139	C₄×C₆₁	C₂×#108	1	4	yes	—
140	C₄×C₆₁	C₂×#131	1	2	yes	—
141	C₄×C₆₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
142	C₄×C₆₁	C₂×#137	1	1	yes	characteristic
143	C₂×#431	—	61	1	no	—
144	C₂×#431	D₆₁	1	1	yes	characteristic
145	C₂×#423	—	61	1	no	—
146	#1574	—	61	8	no	—
147	C₂×#335	—	61	2	no	—
148	C₂×#334	—	61	2	no	—
149	C₂×C₆₁	—	4	2	no	—
150	C₂×C₆₁	#1574	1	4	yes	complemented
151	C₂×C₆₁	C₂×#424	1	1	yes	characteristic
152	C₂×C₆₁	C₂×#428	1	1	yes	characteristic
153	C₂×C₆₁	C₂²×#108	1	1	yes	characteristic
154	C₂²×M₁₆	—	122	1	no	—
155	C₂²×C₄²	C₂×D₆₁	1	1	yes	characteristic
156	#431	—	2	2	no	—
157	#431	—	61	4	no	—
158	#431	—	61	4	no	—
159	#431	C₂×D₆₁	1	4	yes	—
160	#423	—	61	4	no	—
161	#423	—	122	2	no	—
162	C₂×C₄⋊C₈	—	61	2	no	—
163	C₂×C₄⋊C₈	—	61	2	no	—
164	C₂×C₄⋊C₈	—	122	1	no	—
165	C₂×C₄⋊C₈	C₂×D₆₁	1	1	yes	characteristic
166	C₂×C₄⋊C₈	C₂×D₆₁	1	1	yes	characteristic
167	C₂×C₂²⋊C₈	—	2	1	no	—
168	C₂×C₂²⋊C₈	—	61	2	no	—
169	C₂×C₄×C₈	—	2	1	no	—
170	C₂×C₄×C₈	—	61	2	no	—
171	#335	—	61	8	no	—
172	#334	—	61	8	no	—
173	C₆₁	C₂×#1574	1	1	yes	characteristic, complemented, Sylow
174	C₂³×C₄	Dic₁₂₂	1	1	yes	characteristic
175	C₂³×C₄	C₄×D₆₁	1	1	yes	characteristic
176	C₂³×C₄	C₂²⋊D₆₁	1	1	yes	characteristic
177	C₂×M₁₆	—	122	4	no	—
178	C₂×M₁₆	—	244	4	no	—
179	C₂²×C₈	—	2	1	no	—
180	C₂²×C₈	—	2	1	no	—
181	C₂²×C₈	—	61	4	no	—
182	C₂²×C₈	—	122	2	no	—
183	C₂×C₄²	Dic₁₂₂	1	1	yes	characteristic
184	C₂×C₄²	Dic₁₂₂	1	2	yes	—
185	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
186	C₂×C₄²	C₄×D₆₁	1	2	yes	—
187	C₂×C₄²	C₂²⋊D₆₁	1	2	yes	—
188	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
189	C₂×C₄²	C₂²×D₆₁	1	2	yes	—
190	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
191	C₄⋊C₈	—	2	1	no	—
192	C₄⋊C₈	—	2	1	no	—
193	C₄⋊C₈	—	61	8	no	—
194	C₄⋊C₈	—	61	4	no	—
195	C₄⋊C₈	—	122	2	no	—
196	C₄⋊C₈	—	122	4	no	—
197	C₄⋊C₈	C₂²×D₆₁	1	2	yes	—
198	C₄⋊C₈	C₂²×D₆₁	1	2	yes	—
199	C₂²⋊C₈	—	2	4	no	—
200	C₂²⋊C₈	—	61	8	no	—
201	C₄×C₈	—	2	4	no	—
202	C₄×C₈	—	61	4	no	—
203	C₄×C₈	—	61	4	no	—
204	C₂⁴	#11758196	1	1	yes	characteristic
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	#11758196	1	1	yes	characteristic
210	C₂²×C₄	#11758196	1	2	yes	—
211	C₂²×C₄	#11758196	1	1	yes	characteristic
212	C₂²×C₄	#11758196	1	2	yes	—
213	C₂²×C₄	#11758196	1	2	yes	—
214	C₂²×C₄	#11758196	1	1	yes	characteristic
215	C₂²×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
216	C₂²×C₄	C₂×Dic₁₂₂	1	2	yes	—
217	C₂²×C₄	C₂×C₄×D₆₁	1	2	yes	—
218	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₂²⋊D₆₁	1	2	yes	—
220	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
221	M₁₆	—	244	8	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	4	2	no	—
225	C₂×C₈	—	4	2	no	—
226	C₂×C₈	—	61	8	no	—
227	C₂×C₈	—	122	8	no	—
228	C₂×C₈	—	122	4	no	—
229	C₂×C₈	—	244	4	no	—
230	C₄²	#11758196	1	2	yes	—
231	C₄²	#11758196	1	2	yes	—
232	C₄²	#11758196	1	2	yes	—
233	C₄²	C₂×Dic₁₂₂	1	2	yes	—
234	C₄²	C₂×Dic₁₂₂	1	1	yes	characteristic
235	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
236	C₄²	C₂×C₄×D₆₁	1	2	yes	—
237	C₄²	C₂×C₂²⋊D₆₁	1	2	yes	—
238	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
239	C₄²	C₂³×D₆₁	1	1	yes	characteristic
240	C₂³	—	2	2	no	—
241	C₂³	—	2	2	no	—
242	C₂³	—	2	2	no	—
243	C₂³	C₂×#11758196	1	2	yes	—
244	C₂³	C₂×#11758196	1	1	yes	characteristic
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	#14519712	1	4	yes	—
256	C₂×C₄	C₂×#11758196	1	2	yes	—
257	C₂×C₄	C₂×#11758196	1	2	yes	—
258	C₂×C₄	C₂×#11758196	1	1	yes	characteristic
259	C₂×C₄	C₂×#11758196	1	1	yes	characteristic
260	C₂×C₄	C₂×#11758196	1	1	yes	characteristic
261	C₂×C₄	C₂×#11758196	1	2	yes	—
262	C₂×C₄	#14519783	1	4	yes	—
263	C₂×C₄	#14519802	1	2	yes	—
264	C₂×C₄	#14519807	1	2	yes	—
265	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
266	C₂×C₄	C₂²×C₂²⋊D₆₁	1	1	yes	characteristic
267	C₂×C₄	C₂²×Dic₁₂₂	1	1	yes	characteristic
268	C₈	—	4	2	no	—
269	C₈	—	4	2	no	—
270	C₈	—	122	8	no	—
271	C₈	—	244	4	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	4	4	no	—
276	C₂²	#16839788	1	2	yes	—
277	C₂²	#16839805	1	2	yes	—
278	C₂²	C₂×#14519712	1	2	yes	—
279	C₂²	C₂²×#11758196	1	1	yes	characteristic
280	C₄	—	2	2	no	—
281	C₄	—	2	2	no	—
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	#16839788	1	2	yes	—
285	C₄	#16839805	1	2	yes	—
286	C₄	C₂×#14519712	1	4	yes	—
287	C₄	C₂×#14519783	1	2	yes	—
288	C₄	C₂×#14519802	1	1	yes	characteristic
289	C₄	C₂×#14519807	1	1	yes	characteristic
290	C₂	—	4	2	no	—
291	C₂	#21444607	1	4	yes	direct product factor
292	C₂	C₂×#16839788	1	1	yes	characteristic
293	C₂	C₂×#16839805	1	1	yes	characteristic
294	C₂	C₂²×#14519712	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₆₁
commutator	C₂×C₄×C₆₁	C₂³×C₄
Frattini	C₄²	C₂³×D₆₁
Fitting	C₂×C₆₁×#431	C₂
socle	C₂³×C₆₁	C₂×C₄⋊C₄
prime radical	C₂⁴×C₆₁	C₄⋊C₄
cosocle	C₄²×C₆₁	C₂⁴
hypercenter	C₂×#431	D₆₁
hypocenter	C₆₁	C₂×#1574
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #21444607	C₂	direct (N × H)
4. C₂×#16839514	C₂	semidirect
5. C₂×#16839512	C₂	semidirect
17. #16839514	C₂²	semidirect
18. #16839512	C₂²	semidirect
150. C₂×C₆₁	#1574	semidirect
173. C₆₁	C₂×#1574	semidirect
291. C₂	#21444607	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16839815	2
C₂×#16839786	2
#21444607	2
C₂×#16839514	2
C₂×#16839512	2
C₂×C₆₁×#431	2
C₂×#1574	61

Subgroups — C2×#21444607 #29649097

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21444607 #29649097