Subgroups — C₂×#21461376 #30612024

Order 15616.

854 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16840765	C₂	1	2	yes	maximal
2	#21461376	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16840626	C₂	1	2	yes	maximal, complemented
4	C₂×#16840625	C₂	1	2	yes	maximal, complemented
5	C₂²×C₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11758217	—	2	2	no	—
7	#16840765	—	2	4	no	—
8	#16840765	C₂²	1	8	yes	—
9	C₂×#14519844	—	2	2	no	—
10	C₂×C₄⋊#92420	C₂²	1	4	yes	—
11	#16840626	C₂²	1	8	yes	complemented
12	#16840625	C₂²	1	8	yes	complemented
13	C₂³×#92418	C₄	1	1	yes	characteristic
14	C₂²×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
15	C₂×C₈×Dic₆₁	C₄	1	1	yes	characteristic
16	C₂×C₈×Dic₆₁	C₄	1	4	yes	—
17	C₂×C₈×Dic₆₁	C₄	1	1	yes	characteristic
18	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
19	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
20	C₂×C₈×Dic₆₁	C₂²	1	4	yes	—
21	C₂³×C₈×C₆₁	C₄	1	1	yes	characteristic
22	C₂×#11758217	—	2	8	no	—
23	C₂×#11758217	—	4	8	no	—
24	C₂²×#92420	—	2	4	no	—
25	#14519844	—	2	8	no	—
26	C₄⋊#92420	—	2	4	no	—
27	C₄⋊#92420	C₂³	1	8	yes	—
28	C₂³×Dic₆₁	D₄	1	1	yes	characteristic
29	C₂³×Dic₆₁	Q₈	1	1	yes	characteristic
30	C₂²×#92418	—	2	4	no	—
31	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
32	C₂²×#92418	C₂×C₄	1	4	yes	—
33	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×Dic₆₁	C₂×C₄	1	2	yes	—
35	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
37	C₂×C₄×Dic₆₁	D₄	1	2	yes	—
38	C₂×C₄×Dic₆₁	Q₈	1	2	yes	—
39	C₂×C₄×Dic₆₁	Q₈	1	1	yes	characteristic
40	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
41	C₂×C₄×Dic₆₁	C₂³	1	2	yes	—
42	C₈×Dic₆₁	C₂×C₄	1	4	yes	—
43	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
44	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
45	C₈×Dic₆₁	C₂×C₄	1	4	yes	—
46	C₈×Dic₆₁	C₂³	1	2	yes	—
47	C₈×Dic₆₁	C₂³	1	2	yes	—
48	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₈×C₆₁	—	2	4	no	—
50	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₈×C₆₁	C₂×C₄	1	4	yes	—
52	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
53	C₂⁴×C₆₁	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₆₁	—	2	2	no	—
55	C₂²×C₄×C₆₁	—	2	2	no	—
56	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
60	C₂²×Dic₆₁	—	2	4	no	—
61	C₂²×Dic₆₁	C₄⋊C₄	1	2	yes	—
62	C₂²×Dic₆₁	C₄⋊C₄	1	2	yes	—
63	C₂²×Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
64	C₂²×Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
65	C₂²×Dic₆₁	D₈	1	4	yes	—
66	C₂²×Dic₆₁	Q₁₆	1	4	yes	—
67	C₂²×Dic₆₁	C₂×D₄	1	1	yes	characteristic
68	C₂²×Dic₆₁	C₂×D₄	1	2	yes	—
69	C₂²×Dic₆₁	C₂×Q₈	1	1	yes	characteristic
70	C₂²×Dic₆₁	C₂×Q₈	1	2	yes	—
71	#11758217	—	4	16	no	—
72	C₂×#92420	—	2	8	no	—
73	C₂×#92420	—	4	8	no	—
74	C₂×C₈×C₆₁	—	2	2	no	—
75	C₂×C₈×C₆₁	—	2	8	no	—
76	C₂×C₈×C₆₁	M₁₆	1	4	yes	complemented
77	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
79	C₄×Dic₆₁	C₄⋊C₄	1	2	yes	—
80	C₄×Dic₆₁	C₄⋊C₄	1	2	yes	—
81	C₄×Dic₆₁	C₄⋊C₄	1	2	yes	—
82	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
83	C₄×Dic₆₁	C₂²×C₄	1	2	yes	—
84	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
85	C₄×Dic₆₁	C₂×D₄	1	2	yes	—
86	C₄×Dic₆₁	C₂×Q₈	1	2	yes	—
87	C₄×Dic₆₁	C₂×Q₈	1	1	yes	characteristic
88	C₄×Dic₆₁	C₂⁴	1	1	yes	characteristic
89	C₂×#92418	—	2	2	no	—
90	C₂×#92418	—	2	8	no	—
91	C₂×#92418	M₁₆	1	4	yes	complemented
92	C₂×#92418	C₂²×C₄	1	2	yes	—
93	C₂×#92418	C₂²×C₄	1	2	yes	—
94	C₂³×C₆₁	—	2	2	no	—
95	C₂³×C₆₁	—	2	2	no	—
96	C₂³×C₆₁	#107	1	4	yes	—
97	C₂³×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
98	C₂³×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₆₁	—	2	2	no	—
100	C₂×C₄×C₆₁	—	2	2	no	—
101	C₂×C₄×C₆₁	—	2	2	no	—
102	C₂×C₄×C₆₁	—	2	4	no	—
103	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
104	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
106	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
107	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
108	C₂×Dic₆₁	—	2	2	no	—
109	C₂×Dic₆₁	—	2	4	no	—
110	C₂×Dic₆₁	—	2	4	no	—
111	C₂×Dic₆₁	#107	1	8	yes	—
112	C₂×Dic₆₁	#107	1	4	yes	—
113	C₂×Dic₆₁	#108	1	4	yes	—
114	C₂×Dic₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂×Dic₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×Dic₆₁	C₂×C₄⋊C₄	1	2	yes	—
117	C₂×Dic₆₁	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×Dic₆₁	C₂×D₈	1	2	yes	—
119	C₂×Dic₆₁	C₂×D₈	1	4	yes	—
120	C₂×Dic₆₁	C₂×Q₁₆	1	4	yes	—
121	C₂×Dic₆₁	C₂×Q₁₆	1	2	yes	—
122	C₂×Dic₆₁	C₂²×D₄	1	1	yes	characteristic
123	C₂×Dic₆₁	C₂²×Q₈	1	1	yes	characteristic
124	#92420	—	4	8	no	—
125	C₈×C₆₁	—	2	2	no	—
126	C₈×C₆₁	C₂×M₁₆	1	4	yes	complemented
127	#92418	—	2	2	no	—
128	#92418	C₂×M₁₆	1	4	yes	complemented
129	C₂×#1567	—	61	1	no	maximal, Sylow
130	C₂²×C₆₁	—	2	8	no	—
131	C₂²×C₆₁	—	2	2	no	—
132	C₂²×C₆₁	—	2	2	no	—
133	C₂²×C₆₁	#423	1	2	yes	—
134	C₂²×C₆₁	C₂×#107	1	4	yes	—
135	C₂²×C₆₁	C₂×#107	1	2	yes	—
136	C₂²×C₆₁	C₂×#108	1	2	yes	—
137	C₂²×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
138	C₄×C₆₁	—	2	2	no	—
139	C₄×C₆₁	#423	1	2	yes	—
140	C₄×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
141	C₄×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
142	Dic₆₁	—	2	1	no	—
143	Dic₆₁	—	2	1	no	—
144	Dic₆₁	C₂×#107	1	2	yes	—
145	Dic₆₁	C₂×#107	1	4	yes	—
146	Dic₆₁	C₂×#108	1	4	yes	—
147	Dic₆₁	C₂²×D₈	1	1	yes	characteristic
148	Dic₆₁	C₂²×Q₁₆	1	1	yes	characteristic
149	C₂×#423	—	61	2	no	—
150	C₂²×C₄×C₈	—	61	1	no	—
151	#1567	—	61	8	no	—
152	C₂×#335	—	61	2	no	—
153	C₂×#335	—	61	2	no	—
154	C₂×C₆₁	—	2	4	no	—
155	C₂×C₆₁	#1567	1	4	yes	complemented
156	C₂×C₆₁	C₂×#423	1	1	yes	characteristic
157	C₂×C₆₁	C₂²×#107	1	1	yes	characteristic
158	C₂×C₆₁	C₂²×#108	1	1	yes	characteristic
159	C₂²×M₁₆	—	122	2	no	—
160	C₂³×C₈	—	61	1	no	—
161	C₂³×C₈	#6923	1	1	yes	characteristic
162	C₂²×C₄²	—	61	1	no	—
163	#423	—	61	8	no	—
164	#423	—	122	4	no	—
165	C₂×C₄⋊C₈	—	61	4	no	—
166	C₂×C₄⋊C₈	—	122	2	no	—
167	C₂×C₄×C₈	—	61	1	no	—
168	C₂×C₄×C₈	—	61	1	no	—
169	C₂×C₄×C₈	—	61	1	no	—
170	C₂×C₄×C₈	—	61	4	no	—
171	C₂×C₄×C₈	—	61	1	no	—
172	C₂×C₄×C₈	—	61	4	no	—
173	#335	—	61	8	no	—
174	#335	—	61	8	no	—
175	C₆₁	C₂×#1567	1	1	yes	characteristic, complemented, Sylow
176	C₂³×C₄	—	61	1	no	—
177	C₂³×C₄	—	61	1	no	—
178	C₂³×C₄	C₂×#6923	1	1	yes	characteristic
179	C₂×M₁₆	—	122	8	no	—
180	C₂×M₁₆	—	244	8	no	—
181	C₂²×C₈	—	2	4	no	—
182	C₂²×C₈	—	61	1	no	—
183	C₂²×C₈	—	61	4	no	—
184	C₂²×C₈	—	61	1	no	—
185	C₂²×C₈	—	122	4	no	—
186	C₂²×C₈	—	122	4	no	—
187	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
188	C₂²×C₈	C₂×#6923	1	4	yes	—
189	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
190	C₂×C₄²	—	61	1	no	—
191	C₂×C₄²	—	61	2	no	—
192	C₂×C₄²	—	61	2	no	—
193	C₂×C₄²	—	61	1	no	—
194	C₂×C₄²	—	61	1	no	—
195	C₂×C₄²	—	61	2	no	—
196	C₂×C₄²	—	61	2	no	—
197	C₂×C₄²	—	61	1	no	—
198	C₄⋊C₈	—	61	8	no	—
199	C₄⋊C₈	—	122	4	no	—
200	C₄⋊C₈	—	122	8	no	—
201	C₄×C₈	—	61	2	no	—
202	C₄×C₈	—	61	2	no	—
203	C₄×C₈	—	61	2	no	—
204	C₄×C₈	—	61	4	no	—
205	C₄×C₈	—	61	4	no	—
206	C₄×C₈	—	61	2	no	—
207	C₂⁴	C₄⋊#6923	1	1	yes	characteristic
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	61	4	no	—
211	C₂²×C₄	—	61	4	no	—
212	C₂²×C₄	—	61	2	no	—
213	C₂²×C₄	—	61	2	no	—
214	C₂²×C₄	—	61	1	no	—
215	C₂²×C₄	—	61	1	no	—
216	C₂²×C₄	—	61	1	no	—
217	C₂²×C₄	—	61	2	no	—
218	C₂²×C₄	—	61	2	no	—
219	C₂²×C₄	—	61	1	no	—
220	C₂²×C₄	—	122	4	no	—
221	C₂²×C₄	C₄⋊#6923	1	2	yes	—
222	C₂²×C₄	C₄⋊#6923	1	1	yes	characteristic
223	C₂²×C₄	C₂²×#6923	1	2	yes	—
224	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
225	M₁₆	—	244	16	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	8	no	—
228	C₂×C₈	—	61	4	no	—
229	C₂×C₈	—	61	2	no	—
230	C₂×C₈	—	61	2	no	—
231	C₂×C₈	—	122	8	no	—
232	C₂×C₈	—	122	2	no	—
233	C₂×C₈	—	122	8	no	—
234	C₂×C₈	—	244	8	no	—
235	C₂×C₈	#11758217	1	4	yes	complemented
236	C₂×C₈	C₂²×#6923	1	2	yes	—
237	C₂×C₈	C₂²×#6923	1	2	yes	—
238	C₄²	—	61	2	no	—
239	C₄²	—	61	1	no	—
240	C₄²	—	61	2	no	—
241	C₄²	—	61	2	no	—
242	C₄²	—	61	2	no	—
243	C₄²	—	61	2	no	—
244	C₄²	—	61	1	no	—
245	C₄²	—	61	1	no	—
246	C₄²	—	61	2	no	—
247	C₄²	—	61	1	no	—
248	C₂³	—	2	2	no	—
249	C₂³	—	2	2	no	—
250	C₂³	#14519846	1	4	yes	—
251	C₂³	C₂×C₄⋊#6923	1	2	yes	—
252	C₂³	C₂×C₄⋊#6923	1	1	yes	characteristic
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	61	8	no	—
258	C₂×C₄	—	61	4	no	—
259	C₂×C₄	—	61	2	no	—
260	C₂×C₄	—	61	4	no	—
261	C₂×C₄	—	61	4	no	—
262	C₂×C₄	—	61	2	no	—
263	C₂×C₄	—	61	4	no	—
264	C₂×C₄	—	61	2	no	—
265	C₂×C₄	—	61	1	no	—
266	C₂×C₄	—	61	2	no	—
267	C₂×C₄	—	61	1	no	—
268	C₂×C₄	—	61	1	no	—
269	C₂×C₄	—	61	1	no	—
270	C₂×C₄	—	122	4	no	—
271	C₂×C₄	—	122	4	no	—
272	C₂×C₄	—	122	2	no	—
273	C₂×C₄	C₂×C₄⋊#6923	1	2	yes	—
274	C₂×C₄	C₂×C₄⋊#6923	1	1	yes	characteristic
275	C₂×C₄	C₂×#11758217	1	2	yes	—
276	C₂×C₄	D₆₁⋊M₁₆	1	2	yes	—
277	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
278	C₈	—	2	2	no	—
279	C₈	—	61	4	no	—
280	C₈	—	122	2	no	—
281	C₈	—	244	8	no	—
282	C₈	C₂×#11758217	1	4	yes	complemented
283	C₂²	—	2	8	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	#16840765	1	2	yes	—
287	C₂²	C₂×#14519846	1	4	yes	—
288	C₂²	C₂×#14519846	1	2	yes	—
289	C₂²	D₆₁⋊#108	1	2	yes	—
290	C₂²	C₂²×C₄⋊#6923	1	1	yes	characteristic
291	C₄	—	2	2	no	—
292	C₄	—	61	1	no	—
293	C₄	—	61	1	no	—
294	C₄	—	61	2	no	—
295	C₄	—	61	4	no	—
296	C₄	—	61	4	no	—
297	C₄	—	122	1	no	—
298	C₄	—	122	1	no	—
299	C₄	#16840765	1	2	yes	—
300	C₄	C₂²×#11758217	1	1	yes	characteristic
301	C₄	C₂×D₆₁⋊M₁₆	1	1	yes	characteristic
302	C₂	—	2	4	no	—
303	C₂	#21461376	1	4	yes	direct product factor
304	C₂	C₂×#16840765	1	1	yes	characteristic
305	C₂	C₂²×#14519846	1	1	yes	characteristic
306	C₂	C₂×D₆₁⋊#108	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊#6923
commutator	C₂×C₄×C₆₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×#6923
Fitting	C₂³×C₈×C₆₁	C₄
socle	C₂³×C₆₁	C₂×C₄⋊C₄
prime radical	C₂⁴×C₆₁	C₄⋊C₄
cosocle	C₄×Dic₆₁	C₂⁴
hypercenter	C₂³×C₈	#6923
hypocenter	C₆₁	C₂×#1567
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
2. #21461376	C₂	direct (N × H)
3. C₂×#16840626	C₂	semidirect
4. C₂×#16840625	C₂	semidirect
11. #16840626	C₂²	semidirect
12. #16840625	C₂²	semidirect
76. C₂×C₈×C₆₁	M₁₆	semidirect
91. C₂×#92418	M₁₆	semidirect
126. C₈×C₆₁	C₂×M₁₆	semidirect
128. #92418	C₂×M₁₆	semidirect
155. C₂×C₆₁	#1567	semidirect
175. C₆₁	C₂×#1567	semidirect
235. C₂×C₈	#11758217	semidirect
282. C₈	C₂×#11758217	semidirect
303. C₂	#21461376	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×#16840765	2
#21461376	2
C₂×#16840626	2
C₂×#16840625	2
C₂²×C₈×Dic₆₁	2
C₂×#1567	61

Subgroups — C2×#21461376 #30612024

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21461376 #30612024