Subgroups — C₂×#21462039 #30623845

Order 15616.

738 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14519852	C₂	1	1	yes	maximal, characteristic
2	C₂×#16840725	C₂	1	1	yes	maximal, characteristic
3	#21462039	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16840653	C₂	1	2	yes	maximal, complemented
5	C₂×#16840652	C₂	1	2	yes	maximal, complemented
6	C₂×M₃₂×D₆₁	C₂	1	1	yes	maximal, characteristic
7	C₂×D₆₁⋊M₁₆	C₄	1	1	yes	characteristic
8	C₂×#14519856	C₄	1	1	yes	characteristic
9	C₂×#14519852	C₄	1	2	yes	complemented
10	C₂×#14519852	C₄	1	4	yes	complemented
11	C₂×#14519852	C₂²	1	1	yes	characteristic
12	C₂×#14519852	C₂²	1	1	yes	characteristic
13	C₂×#14519852	C₂²	1	4	yes	—
14	C₂²×#11758187	C₄	1	2	yes	complemented
15	#16840725	C₄	1	4	yes	—
16	#16840725	C₂²	1	4	yes	—
17	C₂×#14519831	C₂²	1	2	yes	—
18	C₂×C₈×#6923	C₄	1	2	yes	—
19	#16840653	C₂²	1	8	yes	complemented
20	#16840652	C₂²	1	8	yes	complemented
21	C₂²×C₈×D₆₁	C₂²	1	1	yes	characteristic
22	C₂×#14519717	C₄	1	1	yes	characteristic, complemented
23	C₂×#14519716	C₂²	1	2	yes	—
24	M₃₂×D₆₁	C₄	1	4	yes	complemented
25	M₃₂×D₆₁	C₂²	1	4	yes	—
26	C₂×C₁₆×D₆₁	C₄	1	2	yes	complemented
27	C₂×M₃₂×C₆₁	C₄	1	1	yes	characteristic, complemented
28	C₂×#11758213	—	2	1	no	—
29	D₆₁⋊M₁₆	—	2	2	no	—
30	D₆₁⋊M₁₆	C₂×C₄	1	4	yes	—
31	C₂×#11758217	—	2	1	no	—
32	C₂×D₆₁⋊C₈	C₂×C₄	1	2	yes	—
33	#14519856	—	2	2	no	—
34	#14519856	C₂×C₄	1	4	yes	—
35	C₂×C₄⋊#6923	—	2	1	no	—
36	C₂×C₂²⋊#6923	—	2	1	no	—
37	C₂×C₄×#6923	C₂×C₄	1	2	yes	—
38	#14519852	C₂×C₄	1	2	yes	—
39	#14519852	C₂×C₄	1	2	yes	—
40	#14519852	C₂×C₄	1	8	yes	complemented
41	#14519852	C₂³	1	2	yes	—
42	#14519852	C₂³	1	2	yes	—
43	C₂×#11758187	C₂×C₄	1	2	yes	—
44	C₂×#11758187	C₂×C₄	1	2	yes	—
45	C₂×#11758187	C₂×C₄	1	8	yes	complemented
46	#14519831	C₂×C₄	1	4	yes	—
47	#14519831	C₂³	1	4	yes	—
48	C₈×#6923	C₂×C₄	1	8	yes	—
49	C₂²×C₄×D₆₁	C₂×C₄	1	1	yes	characteristic
50	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
51	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
52	C₂×C₈×D₆₁	C₂×C₄	1	2	yes	—
53	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
54	C₂×C₈×D₆₁	C₂×C₄	1	2	yes	—
55	C₂×C₈×D₆₁	C₂×C₄	1	2	yes	—
56	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
57	C₂×C₈×D₆₁	C₂³	1	1	yes	characteristic
58	C₂×C₈×D₆₁	C₂³	1	2	yes	—
59	#14519717	C₂×C₄	1	4	yes	complemented
60	#14519716	C₂×C₄	1	4	yes	—
61	#14519716	C₂³	1	4	yes	—
62	C₂×#11758185	C₂×C₄	1	2	yes	—
63	C₁₆×D₆₁	C₂×C₄	1	8	yes	complemented
64	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
65	M₃₂×C₆₁	C₂×C₄	1	4	yes	complemented
66	C₂×C₁₆×C₆₁	C₂×C₄	1	2	yes	—
67	C₂³×D₆₁	M₁₆	1	1	yes	characteristic
68	C₂²×#6923	—	2	2	no	—
69	C₂²×C₄×C₆₁	C₄²	1	1	yes	characteristic
70	C₂²×Dic₆₁	M₁₆	1	1	yes	characteristic
71	C₂×C₄×D₆₁	C₄²	1	2	yes	—
72	C₂×C₄×D₆₁	C₄²	1	1	yes	characteristic
73	C₂×C₄×D₆₁	M₁₆	1	1	yes	characteristic
74	C₂×C₄×D₆₁	M₁₆	1	2	yes	—
75	C₂×C₄×D₆₁	M₁₆	1	2	yes	—
76	C₂×C₄×D₆₁	M₁₆	1	1	yes	characteristic
77	C₂×C₄×D₆₁	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×D₆₁	C₂²×C₄	1	2	yes	—
79	C₂²⋊#6923	—	2	4	no	—
80	#11758217	—	2	4	no	—
81	C₂×#92420	—	2	2	no	—
82	C₄⋊#6923	—	2	4	no	—
83	C₄×#6923	—	2	2	no	—
84	C₄×#6923	C₂²×C₄	1	4	yes	—
85	#11758213	—	2	4	no	—
86	D₆₁⋊C₈	—	2	2	no	—
87	D₆₁⋊C₈	C₂²×C₄	1	4	yes	—
88	C₂×C₈×C₆₁	C₄²	1	2	yes	—
89	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
90	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
92	C₂×#92418	C₄²	1	2	yes	—
93	C₂×#92418	C₄²	1	1	yes	characteristic
94	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
95	C₂×#92418	C₂²×C₄	1	2	yes	—
96	C₈×D₆₁	C₄²	1	2	yes	—
97	C₈×D₆₁	C₄²	1	2	yes	—
98	C₈×D₆₁	C₄²	1	2	yes	—
99	C₈×D₆₁	C₂²×C₄	1	2	yes	—
100	C₈×D₆₁	C₂²×C₄	1	1	yes	characteristic
101	C₈×D₆₁	C₂²×C₄	1	2	yes	—
102	C₈×D₆₁	C₂²×C₄	1	1	yes	characteristic
103	C₈×D₆₁	C₂²×C₄	1	1	yes	characteristic
104	C₈×D₆₁	C₂²×C₄	1	2	yes	—
105	C₈×D₆₁	C₂⁴	1	1	yes	characteristic
106	#11758187	C₂²×C₄	1	4	yes	—
107	#11758187	C₂²×C₄	1	4	yes	—
108	C₁₆×C₆₁	C₂²×C₄	1	4	yes	—
109	#11758185	C₂²×C₄	1	4	yes	—
110	C₂³×C₆₁	#97	1	1	yes	characteristic
111	C₂²×D₆₁	—	4	2	no	—
112	C₂²×D₆₁	#97	1	1	yes	characteristic
113	C₂²×D₆₁	#97	1	2	yes	—
114	C₂²×D₆₁	C₂×M₁₆	1	1	yes	characteristic
115	C₂²×D₆₁	C₂×M₁₆	1	2	yes	—
116	C₂×#6923	—	2	4	no	—
117	C₂×#6923	—	4	4	no	—
118	C₂×C₄×C₆₁	#97	1	2	yes	—
119	C₂×C₄×C₆₁	#97	1	1	yes	characteristic
120	C₂×C₄×C₆₁	C₂×C₄²	1	2	yes	—
121	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
122	C₂×Dic₆₁	#97	1	2	yes	—
123	C₂×Dic₆₁	#97	1	1	yes	characteristic
124	C₂×Dic₆₁	C₂×M₁₆	1	1	yes	characteristic
125	C₂×Dic₆₁	C₂×M₁₆	1	2	yes	—
126	C₄×D₆₁	#97	1	2	yes	—
127	C₄×D₆₁	#97	1	2	yes	—
128	C₄×D₆₁	#97	1	2	yes	—
129	C₄×D₆₁	C₂×C₄²	1	2	yes	—
130	C₄×D₆₁	C₂×C₄²	1	1	yes	characteristic
131	C₄×D₆₁	C₂×M₁₆	1	1	yes	characteristic
132	C₄×D₆₁	C₂×M₁₆	1	2	yes	—
133	C₄×D₆₁	C₂×M₁₆	1	1	yes	characteristic
134	C₄×D₆₁	C₂×M₁₆	1	2	yes	—
135	C₄×D₆₁	C₂³×C₄	1	1	yes	characteristic
136	#92420	—	2	4	no	—
137	C₈×C₆₁	C₂×C₄²	1	2	yes	—
138	C₈×C₆₁	C₂×C₄²	1	1	yes	characteristic
139	C₈×C₆₁	C₂³×C₄	1	1	yes	characteristic
140	#92418	C₂×C₄²	1	2	yes	—
141	#92418	C₂×C₄²	1	1	yes	characteristic
142	#92418	C₂³×C₄	1	1	yes	characteristic
143	C₂×#2110	—	61	1	no	maximal, Sylow
144	C₂²×C₆₁	—	2	2	no	—
145	C₂²×C₆₁	C₂×#97	1	2	yes	—
146	C₂²×C₆₁	C₂×#97	1	1	yes	characteristic
147	C₂×D₆₁	—	2	2	no	—
148	C₂×D₆₁	—	2	2	no	—
149	C₂×D₆₁	—	4	2	no	—
150	C₂×D₆₁	—	4	2	no	—
151	C₂×D₆₁	C₂×#97	1	1	yes	characteristic
152	C₂×D₆₁	C₂×#97	1	2	yes	—
153	C₂×D₆₁	C₂²×M₁₆	1	1	yes	characteristic
154	#6923	—	4	4	no	—
155	C₄×C₆₁	C₂×#97	1	2	yes	—
156	C₄×C₆₁	C₂×#97	1	1	yes	characteristic
157	C₄×C₆₁	C₂²×C₄²	1	1	yes	characteristic
158	Dic₆₁	C₂×#97	1	1	yes	characteristic
159	Dic₆₁	C₂×#97	1	2	yes	—
160	Dic₆₁	C₂²×M₁₆	1	1	yes	characteristic
161	C₂²×M₃₂	—	61	1	no	—
162	C₂²×M₃₂	—	61	1	no	—
163	C₂×#405	—	61	1	no	—
164	#2110	—	61	8	no	—
165	C₂×#347	—	61	2	no	—
166	C₂×#347	—	61	2	no	—
167	C₂×C₆₁	—	2	2	no	—
168	C₂×C₆₁	#2110	1	2	yes	complemented
169	C₂×C₆₁	C₂²×#97	1	1	yes	characteristic
170	D₆₁	—	2	1	no	—
171	D₆₁	—	2	1	no	—
172	D₆₁	—	2	2	no	—
173	C₂²×M₁₆	—	61	1	no	—
174	C₂³×C₈	—	61	1	no	—
175	C₂×#117	—	61	1	no	—
176	C₂×M₃₂	—	61	1	no	—
177	C₂×M₃₂	—	61	1	no	—
178	C₂×M₃₂	—	61	4	no	—
179	C₂×M₃₂	—	61	2	no	—
180	C₂×M₃₂	—	61	4	no	—
181	C₂×M₃₂	—	61	1	no	—
182	C₂×M₃₂	—	61	4	no	—
183	C₂×M₃₂	—	61	4	no	—
184	C₂×M₃₂	—	61	2	no	—
185	C₂×M₃₂	#6923	1	1	yes	characteristic, complemented
186	C₂²×C₁₆	—	61	2	no	—
187	C₂²×C₁₆	—	61	2	no	—
188	#405	—	61	4	no	—
189	#405	—	61	4	no	—
190	C₂×#97	—	61	2	no	—
191	C₂×C₄×C₈	—	61	2	no	—
192	#347	—	61	8	no	—
193	#347	—	61	8	no	—
194	C₆₁	C₂×#2110	1	1	yes	characteristic, complemented, Sylow
195	C₂³×C₄	—	61	1	no	—
196	C₂×M₁₆	—	61	4	no	—
197	C₂×M₁₆	—	122	1	no	—
198	C₂×M₁₆	—	122	2	no	—
199	C₂×M₁₆	—	122	1	no	—
200	C₂²×C₈	—	61	1	no	—
201	C₂²×C₈	—	61	1	no	—
202	C₂²×C₈	—	61	2	no	—
203	C₂²×C₈	—	61	2	no	—
204	C₂²×C₈	—	61	1	no	—
205	C₂²×C₈	—	61	1	no	—
206	C₂²×C₈	—	61	2	no	—
207	C₂²×C₈	—	61	1	no	—
208	C₂²×C₈	—	61	2	no	—
209	C₂²×C₈	—	61	2	no	—
210	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
211	#117	—	61	4	no	—
212	#117	—	122	2	no	—
213	C₂×C₄⋊C₄	—	122	1	no	—
214	C₂×C₂²⋊C₄	—	122	1	no	—
215	C₂×C₄²	—	61	2	no	—
216	M₃₂	—	61	2	no	—
217	M₃₂	—	61	2	no	—
218	M₃₂	—	61	8	no	—
219	M₃₂	—	61	2	no	—
220	M₃₂	—	61	2	no	—
221	M₃₂	—	61	4	no	—
222	M₃₂	—	61	4	no	—
223	M₃₂	—	61	4	no	—
224	M₃₂	C₂×#6923	1	4	yes	complemented
225	C₂×C₁₆	—	61	2	no	—
226	C₂×C₁₆	—	61	2	no	—
227	C₂×C₁₆	—	61	8	no	—
228	C₂×C₁₆	—	61	2	no	—
229	C₂×C₁₆	—	61	8	no	—
230	C₂×C₁₆	C₂×#6923	1	2	yes	—
231	#97	—	61	4	no	—
232	#97	—	61	4	no	—
233	C₄×C₈	—	61	8	no	—
234	C₂⁴	—	61	1	no	—
235	C₂²×C₄	—	61	1	no	—
236	C₂²×C₄	—	61	1	no	—
237	C₂²×C₄	—	61	2	no	—
238	C₂²×C₄	—	61	1	no	—
239	C₂²×C₄	—	61	2	no	—
240	C₂²×C₄	—	61	2	no	—
241	C₂²×C₄	—	61	2	no	—
242	C₂²×C₄	—	61	1	no	—
243	C₂²×C₄	—	61	1	no	—
244	C₂²×C₄	—	122	2	no	—
245	C₂²×C₄	C₄×#6923	1	1	yes	characteristic
246	M₁₆	—	122	4	no	—
247	M₁₆	—	122	4	no	—
248	C₂×C₈	—	61	2	no	—
249	C₂×C₈	—	61	2	no	—
250	C₂×C₈	—	61	1	no	—
251	C₂×C₈	—	61	1	no	—
252	C₂×C₈	—	61	2	no	—
253	C₂×C₈	—	61	1	no	—
254	C₂×C₈	—	61	2	no	—
255	C₂×C₈	—	61	2	no	—
256	C₂×C₈	—	61	2	no	—
257	C₂×C₈	—	61	1	no	—
258	C₂×C₈	—	61	2	no	—
259	C₂×C₈	—	61	1	no	—
260	C₂×C₈	—	61	2	no	—
261	C₂×C₈	—	61	1	no	—
262	C₂×C₈	—	61	4	no	—
263	C₂×C₈	—	122	2	no	—
264	C₂×C₈	—	122	2	no	—
265	C₂×C₈	C₄×#6923	1	2	yes	—
266	C₂×C₈	C₄×#6923	1	1	yes	characteristic
267	C₂×C₈	C₂²×#6923	1	2	yes	—
268	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
269	C₄⋊C₄	—	122	4	no	—
270	C₂²⋊C₄	—	122	4	no	—
271	C₄²	—	61	4	no	—
272	C₄²	—	122	2	no	—
273	C₁₆	—	61	4	no	—
274	C₁₆	—	61	4	no	—
275	C₁₆	—	61	4	no	—
276	C₁₆	C₂²×#6923	1	4	yes	—
277	C₂³	—	61	1	no	—
278	C₂³	—	61	1	no	—
279	C₂³	—	61	2	no	—
280	C₂³	—	61	2	no	—
281	C₂³	—	244	2	no	—
282	C₂³	#14519831	1	1	yes	characteristic
283	C₂×C₄	—	61	2	no	—
284	C₂×C₄	—	61	1	no	—
285	C₂×C₄	—	61	1	no	—
286	C₂×C₄	—	61	2	no	—
287	C₂×C₄	—	61	2	no	—
288	C₂×C₄	—	61	2	no	—
289	C₂×C₄	—	61	2	no	—
290	C₂×C₄	—	61	1	no	—
291	C₂×C₄	—	61	2	no	—
292	C₂×C₄	—	61	1	no	—
293	C₂×C₄	—	61	2	no	—
294	C₂×C₄	—	61	2	no	—
295	C₂×C₄	—	61	1	no	—
296	C₂×C₄	—	61	1	no	—
297	C₂×C₄	—	122	4	no	—
298	C₂×C₄	—	244	4	no	—
299	C₂×C₄	#14519831	1	1	yes	characteristic
300	C₂×C₄	#14519831	1	2	yes	—
301	C₂×C₄	C₂×C₄×#6923	1	2	yes	—
302	C₂×C₄	C₂×C₄×#6923	1	1	yes	characteristic
303	C₈	—	61	2	no	—
304	C₈	—	61	1	no	—
305	C₈	—	61	1	no	—
306	C₈	—	122	4	no	—
307	C₈	C₂×C₄×#6923	1	2	yes	—
308	C₈	C₂×C₄×#6923	1	1	yes	characteristic
309	C₈	C₂³×#6923	1	1	yes	characteristic
310	C₂²	—	2	2	no	—
311	C₂²	—	61	1	no	—
312	C₂²	—	61	1	no	—
313	C₂²	—	61	2	no	—
314	C₂²	—	122	2	no	—
315	C₂²	—	122	2	no	—
316	C₂²	—	244	2	no	—
317	C₂²	—	244	2	no	—
318	C₂²	C₂×#14519831	1	2	yes	—
319	C₂²	C₂×#14519831	1	1	yes	characteristic
320	C₄	—	61	1	no	—
321	C₄	—	61	1	no	—
322	C₄	—	61	2	no	—
323	C₄	—	244	4	no	—
324	C₄	C₂×#14519831	1	2	yes	—
325	C₄	C₂×#14519831	1	1	yes	characteristic
326	C₄	C₂²×C₄×#6923	1	1	yes	characteristic
327	C₂	—	2	2	no	—
328	C₂	—	122	1	no	—
329	C₂	—	122	1	no	—
330	C₂	—	122	2	no	—
331	C₂	#21462039	1	2	yes	direct product factor
332	C₂	C₂²×#14519831	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄×#6923
commutator	C₄×C₆₁	C₂²×C₄²
Frattini	C₈	C₂³×#6923
Fitting	C₂×M₃₂×C₆₁	C₄
socle	C₂²×C₆₁	C₂×#97
prime radical	C₂³×D₆₁	M₁₆
cosocle	C₈×D₆₁	C₂⁴
hypercenter	C₂×M₃₂	#6923
hypocenter	C₆₁	C₂×#2110
LCS	G ▷ C₄×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
3. #21462039	C₂	direct (N × H)
4. C₂×#16840653	C₂	semidirect
5. C₂×#16840652	C₂	semidirect
9. C₂×#14519852	C₄	semidirect
10. C₂×#14519852	C₄	semidirect
14. C₂²×#11758187	C₄	semidirect
19. #16840653	C₂²	semidirect
20. #16840652	C₂²	semidirect
22. C₂×#14519717	C₄	semidirect
24. M₃₂×D₆₁	C₄	semidirect
26. C₂×C₁₆×D₆₁	C₄	semidirect
27. C₂×M₃₂×C₆₁	C₄	semidirect
40. #14519852	C₂×C₄	semidirect
45. C₂×#11758187	C₂×C₄	semidirect
59. #14519717	C₂×C₄	semidirect
63. C₁₆×D₆₁	C₂×C₄	semidirect
65. M₃₂×C₆₁	C₂×C₄	semidirect
168. C₂×C₆₁	#2110	semidirect
185. C₂×M₃₂	#6923	semidirect
194. C₆₁	C₂×#2110	semidirect
224. M₃₂	C₂×#6923	semidirect
331. C₂	#21462039	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×#14519852	2
C₂×#16840725	2
#21462039	2
C₂×#16840653	2
C₂×#16840652	2
C₂×M₃₂×D₆₁	2
C₂×#2110	61

Subgroups — C2×#21462039 #30623845

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21462039 #30623845