Subgroups — C₂×#21444577 #29649014

Order 15616.

854 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16839786	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×#92418	C₂	1	1	yes	maximal, characteristic
3	#21444577	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16839514	C₂	1	4	yes	maximal, complemented
5	C₂×C₆₁×#423	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11758194	—	2	1	no	—
7	C₂³×#92418	C₄	1	2	yes	—
8	#16839786	—	2	2	no	—
9	#16839786	C₂²	1	4	yes	—
10	C₂×C₄⋊#92418	—	2	1	no	—
11	C₂×C₄⋊#92418	C₂²	1	2	yes	—
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	2	yes	—
16	C₂×C₄×#92418	C₂²	1	4	yes	—
17	#16839514	C₂²	1	16	yes	complemented
18	C₂²×M₁₆×C₆₁	—	2	1	no	—
19	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
20	C₆₁×#423	—	2	2	no	—
21	C₆₁×#423	C₂²	1	4	yes	—
22	C₂×C₄⋊C₈×C₆₁	—	2	1	no	—
23	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
24	C₂×#11758194	—	2	4	no	—
25	C₂×#11758194	—	4	4	no	—
26	C₂²×#92418	—	2	8	no	—
27	C₂²×#92418	—	2	2	no	—
28	C₂²×#92418	C₂×C₄	1	2	yes	—
29	C₂²×#92418	C₂×C₄	1	8	yes	—
30	C₂²×#92418	C₂×C₄	1	2	yes	—
31	C₄⋊#92418	—	2	4	no	—
32	C₄⋊#92418	—	2	2	no	—
33	C₄⋊#92418	C₂³	1	4	yes	—
34	C₄×#92418	C₂×C₄	1	4	yes	—
35	C₄×#92418	C₂×C₄	1	4	yes	—
36	C₄×#92418	C₂×C₄	1	4	yes	—
37	C₄×#92418	C₂³	1	4	yes	—
38	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
40	C₂³×C₄×C₆₁	Q₈	1	1	yes	characteristic
41	C₂×M₁₆×C₆₁	—	2	4	no	—
42	C₂×M₁₆×C₆₁	—	4	4	no	—
43	C₂²×C₈×C₆₁	—	2	2	no	—
44	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₆₁	D₄	1	2	yes	—
48	C₂×C₄²×C₆₁	Q₈	1	1	yes	characteristic
49	C₂×C₄²×C₆₁	Q₈	1	2	yes	—
50	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
52	C₄⋊C₈×C₆₁	—	2	4	no	—
53	C₄⋊C₈×C₆₁	—	2	2	no	—
54	C₄⋊C₈×C₆₁	C₂³	1	4	yes	—
55	C₂⁴×C₆₁	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₆₁	—	2	4	no	—
57	C₂²×C₄×C₆₁	—	2	4	no	—
58	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
59	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
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	1	yes	characteristic
64	C₂²×C₄×C₆₁	D₈	1	4	yes	—
65	C₂²×C₄×C₆₁	Q₁₆	1	4	yes	—
66	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
71	C₂²×C₄×C₆₁	C₂×Q₈	1	2	yes	—
72	M₁₆×C₆₁	—	4	8	no	—
73	C₂×C₈×C₆₁	—	2	4	no	—
74	C₂×C₈×C₆₁	—	4	4	no	—
75	C₄²×C₆₁	C₄⋊C₄	1	2	yes	—
76	C₄²×C₆₁	C₄⋊C₄	1	2	yes	—
77	C₄²×C₆₁	C₄⋊C₄	1	2	yes	—
78	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
79	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
80	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₆₁	C₂×D₄	1	2	yes	—
82	C₄²×C₆₁	C₂×Q₈	1	2	yes	—
83	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
84	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
85	#11758194	—	4	8	no	—
86	C₂×#92418	—	2	4	no	—
87	C₂×#92418	—	2	16	no	—
88	C₂×#92418	—	2	4	no	—
89	C₂×#92418	—	4	4	no	—
90	C₂×#92418	M₁₆	1	8	yes	complemented
91	C₂×#92418	C₂²×C₄	1	4	yes	—
92	C₂×#92418	C₂²×C₄	1	4	yes	—
93	C₂³×C₆₁	—	2	2	no	—
94	C₂³×C₆₁	—	2	2	no	—
95	C₂³×C₆₁	#107	1	4	yes	—
96	C₂³×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
97	C₂³×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
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	4	no	—
102	C₂×C₄×C₆₁	—	2	4	no	—
103	C₂×C₄×C₆₁	—	2	4	no	—
104	C₂×C₄×C₆₁	#107	1	8	yes	—
105	C₂×C₄×C₆₁	#107	1	4	yes	—
106	C₂×C₄×C₆₁	#108	1	4	yes	—
107	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
109	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
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	2	yes	—
113	C₂×C₄×C₆₁	C₂×M₁₆	1	4	yes	—
114	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
115	C₂×C₄×C₆₁	C₂×D₈	1	4	yes	—
116	C₂×C₄×C₆₁	C₂×Q₁₆	1	4	yes	—
117	C₂×C₄×C₆₁	C₂×Q₁₆	1	2	yes	—
118	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
119	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
121	C₈×C₆₁	—	4	4	no	—
122	#92418	—	2	4	no	—
123	#92418	—	4	4	no	—
124	#92418	C₂×M₁₆	1	8	yes	complemented
125	C₂×#1567	—	61	1	no	maximal, Sylow
126	C₂²×C₆₁	—	2	8	no	—
127	C₂²×C₆₁	—	2	2	no	—
128	C₂²×C₆₁	—	2	2	no	—
129	C₂²×C₆₁	#423	1	2	yes	—
130	C₂²×C₆₁	C₂×#107	1	4	yes	—
131	C₂²×C₆₁	C₂×#107	1	2	yes	—
132	C₂²×C₆₁	C₂×#108	1	2	yes	—
133	C₂²×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
134	C₄×C₆₁	—	2	2	no	—
135	C₄×C₆₁	—	2	1	no	—
136	C₄×C₆₁	—	2	1	no	—
137	C₄×C₆₁	#423	1	2	yes	—
138	C₄×C₆₁	C₂×#107	1	2	yes	—
139	C₄×C₆₁	C₂×#107	1	4	yes	—
140	C₄×C₆₁	C₂×#108	1	4	yes	—
141	C₄×C₆₁	C₂²×M₁₆	1	2	yes	—
142	C₄×C₆₁	C₂²×D₈	1	1	yes	characteristic
143	C₄×C₆₁	C₂²×Q₁₆	1	1	yes	characteristic
144	C₂×#423	—	61	1	no	—
145	C₂×#423	D₆₁	1	1	yes	characteristic
146	C₂²×C₄×C₈	—	61	1	no	—
147	#1567	—	61	8	no	—
148	C₂×#335	—	61	4	no	—
149	C₂×C₆₁	—	2	4	no	—
150	C₂×C₆₁	#1567	1	4	yes	complemented
151	C₂×C₆₁	C₂×#423	1	1	yes	characteristic
152	C₂×C₆₁	C₂²×#107	1	1	yes	characteristic
153	C₂×C₆₁	C₂²×#108	1	1	yes	characteristic
154	C₂²×M₁₆	—	2	1	no	—
155	C₂²×M₁₆	—	122	1	no	—
156	C₂³×C₈	—	61	2	no	—
157	C₂²×C₄²	C₂×D₆₁	1	1	yes	characteristic
158	#423	—	2	2	no	—
159	#423	—	61	4	no	—
160	#423	—	122	2	no	—
161	#423	C₂×D₆₁	1	4	yes	—
162	C₂×C₄⋊C₈	—	2	1	no	—
163	C₂×C₄⋊C₈	—	61	2	no	—
164	C₂×C₄⋊C₈	—	122	1	no	—
165	C₂×C₄⋊C₈	C₂×D₆₁	1	2	yes	—
166	C₂×C₄×C₈	—	61	1	no	—
167	C₂×C₄×C₈	—	61	4	no	—
168	C₂×C₄×C₈	—	61	1	no	—
169	C₂×C₄×C₈	—	61	2	no	—
170	C₂×C₄×C₈	—	61	4	no	—
171	#335	—	61	16	no	—
172	C₆₁	C₂×#1567	1	1	yes	characteristic, complemented, Sylow
173	C₂³×C₄	Dic₁₂₂	1	1	yes	characteristic
174	C₂³×C₄	C₄×D₆₁	1	1	yes	characteristic
175	C₂³×C₄	C₂²⋊D₆₁	1	1	yes	characteristic
176	C₂×M₁₆	—	2	4	no	—
177	C₂×M₁₆	—	4	4	no	—
178	C₂×M₁₆	—	122	4	no	—
179	C₂×M₁₆	—	244	4	no	—
180	C₂²×C₈	—	2	2	no	—
181	C₂²×C₈	—	61	2	no	—
182	C₂²×C₈	—	61	8	no	—
183	C₂²×C₈	—	61	2	no	—
184	C₂²×C₈	—	122	8	no	—
185	C₂²×C₈	—	122	2	no	—
186	C₂×C₄²	Dic₁₂₂	1	1	yes	characteristic
187	C₂×C₄²	Dic₁₂₂	1	2	yes	—
188	C₂×C₄²	C₄×D₆₁	1	2	yes	—
189	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
190	C₂×C₄²	C₂²⋊D₆₁	1	2	yes	—
191	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
192	C₂×C₄²	C₂²×D₆₁	1	2	yes	—
193	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
194	C₄⋊C₈	—	2	4	no	—
195	C₄⋊C₈	—	2	2	no	—
196	C₄⋊C₈	—	61	4	no	—
197	C₄⋊C₈	—	122	2	no	—
198	C₄⋊C₈	—	122	4	no	—
199	C₄⋊C₈	C₂²×D₆₁	1	4	yes	—
200	C₄×C₈	—	61	4	no	—
201	C₄×C₈	—	61	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	#11758196	1	1	yes	characteristic
208	C₂²×C₄	#11758196	1	2	yes	—
209	C₂²×C₄	#11758196	1	2	yes	—
210	C₂²×C₄	#11758196	1	1	yes	characteristic
211	C₂²×C₄	#11758196	1	2	yes	—
212	C₂²×C₄	#11758196	1	1	yes	characteristic
213	C₂²×C₄	#11758199	1	4	yes	—
214	C₂²×C₄	#11758202	1	4	yes	—
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	—	4	8	no	—
222	M₁₆	—	244	8	no	—
223	C₂×C₈	—	2	4	no	—
224	C₂×C₈	—	4	4	no	—
225	C₂×C₈	—	61	4	no	—
226	C₂×C₈	—	61	8	no	—
227	C₂×C₈	—	61	4	no	—
228	C₂×C₈	—	122	16	no	—
229	C₂×C₈	—	122	4	no	—
230	C₂×C₈	—	122	4	no	—
231	C₂×C₈	—	244	4	no	—
232	C₄²	#11758196	1	2	yes	—
233	C₄²	#11758196	1	2	yes	—
234	C₄²	#11758196	1	2	yes	—
235	C₄²	C₂×Dic₁₂₂	1	2	yes	—
236	C₄²	C₂×Dic₁₂₂	1	1	yes	characteristic
237	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
238	C₄²	C₂×C₄×D₆₁	1	2	yes	—
239	C₄²	C₂×C₂²⋊D₆₁	1	2	yes	—
240	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
241	C₄²	C₂³×D₆₁	1	1	yes	characteristic
242	C₂³	—	2	2	no	—
243	C₂³	—	2	2	no	—
244	C₂³	#14519710	1	4	yes	—
245	C₂³	C₂×#11758196	1	2	yes	—
246	C₂³	C₂×#11758196	1	1	yes	characteristic
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	#14519710	1	8	yes	—
254	C₂×C₄	#14519710	1	4	yes	—
255	C₂×C₄	#14519712	1	4	yes	—
256	C₂×C₄	C₂×#11758196	1	2	yes	—
257	C₂×C₄	C₂×#11758196	1	1	yes	characteristic
258	C₂×C₄	C₂×#11758196	1	1	yes	characteristic
259	C₂×C₄	C₂×#11758196	1	2	yes	—
260	C₂×C₄	C₂×#11758196	1	1	yes	characteristic
261	C₂×C₄	C₂×#11758196	1	2	yes	—
262	C₂×C₄	M₁₆×D₆₁	1	4	yes	—
263	C₂×C₄	C₂×#11758199	1	2	yes	—
264	C₂×C₄	C₂×#11758199	1	4	yes	—
265	C₂×C₄	C₂×#11758202	1	4	yes	—
266	C₂×C₄	C₂×#11758202	1	2	yes	—
267	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
268	C₂×C₄	C₂²×C₂²⋊D₆₁	1	1	yes	characteristic
269	C₂×C₄	C₂²×Dic₁₂₂	1	1	yes	characteristic
270	C₈	—	4	4	no	—
271	C₈	—	61	8	no	—
272	C₈	—	122	4	no	—
273	C₈	—	244	4	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	2	8	no	—
277	C₂²	#16839785	1	2	yes	—
278	C₂²	C₂×#14519710	1	4	yes	—
279	C₂²	C₂×#14519710	1	2	yes	—
280	C₂²	C₂×#14519712	1	2	yes	—
281	C₂²	C₂²×#11758196	1	1	yes	characteristic
282	C₄	—	2	2	no	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	#16839785	1	2	yes	—
286	C₄	C₂×#14519710	1	4	yes	—
287	C₄	C₂×#14519710	1	2	yes	—
288	C₄	C₂×#14519712	1	4	yes	—
289	C₄	C₂×M₁₆×D₆₁	1	2	yes	—
290	C₄	C₂²×#11758199	1	1	yes	characteristic
291	C₄	C₂²×#11758202	1	1	yes	characteristic
292	C₂	—	2	4	no	—
293	C₂	#21444577	1	4	yes	direct product factor
294	C₂	C₂×#16839785	1	1	yes	characteristic
295	C₂	C₂²×#14519710	1	1	yes	characteristic
296	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×#423	C₂
socle	C₂³×C₆₁	C₂×C₄⋊C₄
prime radical	C₂⁴×C₆₁	C₄⋊C₄
cosocle	C₄²×C₆₁	C₂⁴
hypercenter	C₂×#423	D₆₁
hypocenter	C₆₁	C₂×#1567
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
3. #21444577	C₂	direct (N × H)
4. C₂×#16839514	C₂	semidirect
17. #16839514	C₂²	semidirect
90. C₂×#92418	M₁₆	semidirect
124. #92418	C₂×M₁₆	semidirect
150. C₂×C₆₁	#1567	semidirect
172. C₆₁	C₂×#1567	semidirect
293. C₂	#21444577	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×#16839786	2
C₂²×C₄×#92418	2
#21444577	2
C₂×#16839514	2
C₂×C₆₁×#423	2
C₂×#1567	61

Subgroups — C2×#21444577 #29649014

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21444577 #29649014