Subgroups — C₂×#21462027 #30623816

Order 15616.

854 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₆₁⋊M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#16840725	C₂	1	1	yes	maximal, characteristic
3	#21462027	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16840649	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₆×#6923	C₂	1	2	yes	maximal, complemented
6	C₂²×C₁₆×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	D₆₁⋊M₃₂	C₄	1	4	yes	complemented
11	D₆₁⋊M₃₂	C₂²	1	4	yes	—
12	C₂×#14519851	C₄	1	2	yes	complemented
13	C₂×D₆₁⋊C₁₆	C₂²	1	2	yes	—
14	#16840725	C₄	1	4	yes	—
15	#16840725	C₂²	1	4	yes	—
16	C₂×#14519831	C₄	1	2	yes	—
17	C₂×C₈×#6923	C₂²	1	2	yes	—
18	#16840649	C₂²	1	8	yes	complemented
19	C₁₆×#6923	C₂²	1	8	yes	complemented
20	C₂²×C₈×D₆₁	C₂²	1	1	yes	characteristic
21	C₂²×#11758185	C₄	1	1	yes	characteristic, complemented
22	C₂×C₁₆×D₆₁	C₄	1	1	yes	characteristic, complemented
23	C₂×C₁₆×D₆₁	C₄	1	1	yes	characteristic, complemented
24	C₂×C₁₆×D₆₁	C₄	1	4	yes	complemented
25	C₂×C₁₆×D₆₁	C₂²	1	1	yes	characteristic
26	C₂×C₁₆×D₆₁	C₂²	1	1	yes	characteristic
27	C₂×C₁₆×D₆₁	C₂²	1	4	yes	—
28	C₂²×C₁₆×C₆₁	C₄	1	1	yes	characteristic, complemented
29	C₂×#11758213	C₈	1	2	yes	—
30	D₆₁⋊M₁₆	C₈	1	4	yes	—
31	D₆₁⋊M₁₆	C₂×C₄	1	4	yes	—
32	C₂×#11758217	C₈	1	2	yes	—
33	C₂×D₆₁⋊C₈	C₂×C₄	1	2	yes	—
34	#14519856	C₈	1	4	yes	—
35	#14519856	C₂×C₄	1	4	yes	—
36	C₂×C₄⋊#6923	C₈	1	2	yes	—
37	C₂×C₂²⋊#6923	C₈	1	2	yes	—
38	C₂×C₄×#6923	C₂×C₄	1	2	yes	—
39	#14519852	C₂×C₄	1	8	yes	complemented
40	#14519851	C₂×C₄	1	8	yes	complemented
41	D₆₁⋊C₁₆	C₂×C₄	1	4	yes	—
42	D₆₁⋊C₁₆	C₂³	1	4	yes	—
43	C₂×#11758187	C₂×C₄	1	4	yes	—
44	#14519831	C₂×C₄	1	8	yes	—
45	C₈×#6923	C₂×C₄	1	4	yes	—
46	C₈×#6923	C₂³	1	4	yes	—
47	C₂²×C₄×D₆₁	C₂×C₄	1	1	yes	characteristic
48	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
49	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
50	C₂×C₈×D₆₁	C₂×C₄	1	2	yes	—
51	C₂×C₈×D₆₁	C₂×C₄	1	2	yes	—
52	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
53	C₂×C₈×D₆₁	C₂×C₄	1	2	yes	—
54	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
55	C₂×C₈×D₆₁	C₂³	1	1	yes	characteristic
56	C₂×C₈×D₆₁	C₂³	1	2	yes	—
57	C₂×#11758185	C₂×C₄	1	1	yes	characteristic
58	C₂×#11758185	C₂×C₄	1	1	yes	characteristic
59	C₂×#11758185	C₂×C₄	1	4	yes	complemented
60	C₁₆×D₆₁	C₂×C₄	1	2	yes	—
61	C₁₆×D₆₁	C₂×C₄	1	2	yes	—
62	C₁₆×D₆₁	C₂×C₄	1	4	yes	complemented
63	C₁₆×D₆₁	C₂×C₄	1	4	yes	complemented
64	C₁₆×D₆₁	C₂³	1	2	yes	—
65	C₁₆×D₆₁	C₂³	1	2	yes	—
66	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
67	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
68	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
69	C₂×C₁₆×C₆₁	C₂×C₄	1	4	yes	complemented
70	C₂³×D₆₁	C₂×C₈	1	1	yes	characteristic
71	C₂²×#6923	C₂×C₈	1	4	yes	—
72	C₂²×C₄×C₆₁	C₄²	1	1	yes	characteristic
73	C₂²×Dic₆₁	C₂×C₈	1	1	yes	characteristic
74	C₂×C₄×D₆₁	C₄²	1	2	yes	—
75	C₂×C₄×D₆₁	C₄²	1	1	yes	characteristic
76	C₂×C₄×D₆₁	C₂×C₈	1	1	yes	characteristic
77	C₂×C₄×D₆₁	C₂×C₈	1	2	yes	—
78	C₂×C₄×D₆₁	C₂×C₈	1	2	yes	—
79	C₂×C₄×D₆₁	C₂×C₈	1	1	yes	characteristic
80	C₂×C₄×D₆₁	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×D₆₁	C₂²×C₄	1	2	yes	—
82	C₂²⋊#6923	C₂×C₈	1	8	yes	—
83	#11758217	C₂×C₈	1	8	yes	—
84	C₂×#92420	C₂×C₈	1	4	yes	—
85	C₄⋊#6923	C₂×C₈	1	8	yes	—
86	C₄×#6923	C₂×C₈	1	4	yes	—
87	C₄×#6923	C₂²×C₄	1	4	yes	—
88	#11758213	C₂×C₈	1	8	yes	—
89	D₆₁⋊C₈	C₂×C₈	1	4	yes	—
90	D₆₁⋊C₈	C₂²×C₄	1	4	yes	—
91	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
92	C₂×C₈×C₆₁	C₄²	1	2	yes	—
93	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
95	C₂×#92418	C₄²	1	1	yes	characteristic
96	C₂×#92418	C₄²	1	2	yes	—
97	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
98	C₂×#92418	C₂²×C₄	1	2	yes	—
99	C₈×D₆₁	C₄²	1	2	yes	—
100	C₈×D₆₁	C₄²	1	2	yes	—
101	C₈×D₆₁	C₄²	1	2	yes	—
102	C₈×D₆₁	C₂²×C₄	1	1	yes	characteristic
103	C₈×D₆₁	C₂²×C₄	1	2	yes	—
104	C₈×D₆₁	C₂²×C₄	1	1	yes	characteristic
105	C₈×D₆₁	C₂²×C₄	1	2	yes	—
106	C₈×D₆₁	C₂²×C₄	1	1	yes	characteristic
107	C₈×D₆₁	C₂²×C₄	1	2	yes	—
108	C₈×D₆₁	C₂⁴	1	1	yes	characteristic
109	#11758187	C₂²×C₄	1	8	yes	—
110	C₁₆×C₆₁	C₂²×C₄	1	2	yes	—
111	C₁₆×C₆₁	C₂²×C₄	1	2	yes	—
112	#11758185	C₂²×C₄	1	2	yes	—
113	#11758185	C₂²×C₄	1	2	yes	—
114	C₂³×C₆₁	C₄×C₈	1	1	yes	characteristic
115	C₂²×D₆₁	—	2	4	no	—
116	C₂²×D₆₁	C₄×C₈	1	1	yes	characteristic
117	C₂²×D₆₁	C₄×C₈	1	2	yes	—
118	C₂²×D₆₁	C₂²×C₈	1	2	yes	—
119	C₂²×D₆₁	C₂²×C₈	1	1	yes	characteristic
120	C₂×#6923	—	2	8	no	—
121	C₂×#6923	C₂²×C₈	1	8	yes	—
122	C₂×C₄×C₆₁	C₄×C₈	1	2	yes	—
123	C₂×C₄×C₆₁	C₄×C₈	1	1	yes	characteristic
124	C₂×C₄×C₆₁	C₂×C₄²	1	2	yes	—
125	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
126	C₂×Dic₆₁	C₄×C₈	1	2	yes	—
127	C₂×Dic₆₁	C₄×C₈	1	1	yes	characteristic
128	C₂×Dic₆₁	C₂²×C₈	1	1	yes	characteristic
129	C₂×Dic₆₁	C₂²×C₈	1	2	yes	—
130	C₄×D₆₁	C₄×C₈	1	2	yes	—
131	C₄×D₆₁	C₄×C₈	1	2	yes	—
132	C₄×D₆₁	C₄×C₈	1	2	yes	—
133	C₄×D₆₁	C₂×C₄²	1	2	yes	—
134	C₄×D₆₁	C₂×C₄²	1	1	yes	characteristic
135	C₄×D₆₁	C₂²×C₈	1	1	yes	characteristic
136	C₄×D₆₁	C₂²×C₈	1	2	yes	—
137	C₄×D₆₁	C₂²×C₈	1	1	yes	characteristic
138	C₄×D₆₁	C₂²×C₈	1	2	yes	—
139	C₄×D₆₁	C₂³×C₄	1	1	yes	characteristic
140	#92420	C₂²×C₈	1	8	yes	—
141	C₈×C₆₁	C₂×C₄²	1	1	yes	characteristic
142	C₈×C₆₁	C₂×C₄²	1	2	yes	—
143	C₈×C₆₁	C₂³×C₄	1	1	yes	characteristic
144	#92418	C₂×C₄²	1	1	yes	characteristic
145	#92418	C₂×C₄²	1	2	yes	—
146	#92418	C₂³×C₄	1	1	yes	characteristic
147	C₂×#2108	—	61	1	no	maximal, Sylow
148	C₂²×C₆₁	—	2	2	no	—
149	C₂²×C₆₁	C₂×C₄×C₈	1	2	yes	—
150	C₂²×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
151	C₂×D₆₁	—	2	2	no	—
152	C₂×D₆₁	—	2	4	no	—
153	C₂×D₆₁	—	2	4	no	—
154	C₂×D₆₁	C₂×C₄×C₈	1	2	yes	—
155	C₂×D₆₁	C₂×C₄×C₈	1	1	yes	characteristic
156	C₂×D₆₁	#504	1	4	yes	—
157	C₂×D₆₁	C₂³×C₈	1	1	yes	characteristic
158	#6923	—	2	8	no	—
159	C₄×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
160	C₄×C₆₁	C₂×C₄×C₈	1	2	yes	—
161	C₄×C₆₁	C₂²×C₄²	1	1	yes	characteristic
162	Dic₆₁	C₂×C₄×C₈	1	1	yes	characteristic
163	Dic₆₁	C₂×C₄×C₈	1	2	yes	—
164	Dic₆₁	C₂³×C₈	1	1	yes	characteristic
165	C₂²×M₃₂	—	61	1	no	—
166	C₂³×C₁₆	—	61	1	no	—
167	C₂×#405	—	61	1	no	—
168	#2108	—	61	8	no	—
169	C₂×#346	—	61	2	no	—
170	C₂×C₄×C₁₆	—	61	2	no	—
171	C₂×C₆₁	—	2	2	no	—
172	C₂×C₆₁	#2108	1	2	yes	complemented
173	C₂×C₆₁	C₂²×C₄×C₈	1	1	yes	characteristic
174	D₆₁	—	2	2	no	—
175	D₆₁	#2108	1	2	yes	complemented
176	D₆₁	C₂×#504	1	2	yes	—
177	C₂²×M₁₆	—	61	1	no	—
178	C₂³×C₈	—	61	1	no	—
179	C₂×#117	—	61	1	no	—
180	C₂×M₃₂	—	61	4	no	—
181	C₂×M₃₂	—	61	2	no	—
182	C₂×M₃₂	—	61	4	no	—
183	C₂×M₃₂	—	61	2	no	—
184	C₂²×C₁₆	—	61	2	no	—
185	C₂²×C₁₆	—	61	1	no	—
186	C₂²×C₁₆	—	61	1	no	—
187	C₂²×C₁₆	—	61	1	no	—
188	C₂²×C₁₆	—	61	1	no	—
189	C₂²×C₁₆	—	61	1	no	—
190	C₂²×C₁₆	—	61	4	no	—
191	C₂²×C₁₆	—	61	4	no	—
192	C₂²×C₁₆	#6923	1	1	yes	characteristic, complemented
193	#405	—	61	4	no	—
194	#405	—	61	4	no	—
195	C₂×#97	—	61	2	no	—
196	C₂×C₄×C₈	—	61	2	no	—
197	#346	—	61	8	no	—
198	C₄×C₁₆	—	61	8	no	—
199	C₆₁	C₂×#2108	1	1	yes	characteristic, complemented, Sylow
200	C₂³×C₄	—	61	1	no	—
201	C₂×M₁₆	—	61	2	no	—
202	C₂×M₁₆	—	61	4	no	—
203	C₂×M₁₆	—	61	4	no	—
204	C₂×M₁₆	—	61	2	no	—
205	C₂²×C₈	—	61	2	no	—
206	C₂²×C₈	—	61	1	no	—
207	C₂²×C₈	—	61	1	no	—
208	C₂²×C₈	—	61	2	no	—
209	C₂²×C₈	—	61	1	no	—
210	C₂²×C₈	—	61	2	no	—
211	C₂²×C₈	—	61	1	no	—
212	C₂²×C₈	—	61	2	no	—
213	C₂²×C₈	—	61	1	no	—
214	C₂²×C₈	—	61	2	no	—
215	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
216	#117	—	61	4	no	—
217	#117	—	61	4	no	—
218	C₂×C₄⋊C₄	—	61	2	no	—
219	C₂×C₂²⋊C₄	—	61	2	no	—
220	C₂×C₄²	—	61	2	no	—
221	M₃₂	—	61	8	no	—
222	M₃₂	—	61	8	no	—
223	C₂×C₁₆	—	61	4	no	—
224	C₂×C₁₆	—	61	4	no	—
225	C₂×C₁₆	—	61	4	no	—
226	C₂×C₁₆	—	61	1	no	—
227	C₂×C₁₆	—	61	1	no	—
228	C₂×C₁₆	—	61	4	no	—
229	C₂×C₁₆	—	61	2	no	—
230	C₂×C₁₆	—	61	2	no	—
231	C₂×C₁₆	—	61	2	no	—
232	C₂×C₁₆	—	61	2	no	—
233	C₂×C₁₆	—	61	4	no	—
234	C₂×C₁₆	—	61	4	no	—
235	C₂×C₁₆	C₂×#6923	1	1	yes	characteristic
236	C₂×C₁₆	C₂×#6923	1	1	yes	characteristic
237	C₂×C₁₆	C₂×#6923	1	4	yes	complemented
238	#97	—	61	8	no	—
239	C₄×C₈	—	61	4	no	—
240	C₄×C₈	—	61	4	no	—
241	C₂⁴	—	61	1	no	—
242	C₂²×C₄	—	61	1	no	—
243	C₂²×C₄	—	61	1	no	—
244	C₂²×C₄	—	61	2	no	—
245	C₂²×C₄	—	61	1	no	—
246	C₂²×C₄	—	61	2	no	—
247	C₂²×C₄	—	61	1	no	—
248	C₂²×C₄	—	61	2	no	—
249	C₂²×C₄	—	61	2	no	—
250	C₂²×C₄	—	61	1	no	—
251	C₂²×C₄	—	61	4	no	—
252	C₂²×C₄	C₄×#6923	1	1	yes	characteristic
253	M₁₆	—	61	8	no	—
254	M₁₆	—	61	8	no	—
255	C₂×C₈	—	61	4	no	—
256	C₂×C₈	—	61	4	no	—
257	C₂×C₈	—	61	4	no	—
258	C₂×C₈	—	61	1	no	—
259	C₂×C₈	—	61	2	no	—
260	C₂×C₈	—	61	1	no	—
261	C₂×C₈	—	61	2	no	—
262	C₂×C₈	—	61	1	no	—
263	C₂×C₈	—	61	2	no	—
264	C₂×C₈	—	61	2	no	—
265	C₂×C₈	—	61	2	no	—
266	C₂×C₈	—	61	2	no	—
267	C₂×C₈	—	61	1	no	—
268	C₂×C₈	—	61	2	no	—
269	C₂×C₈	—	61	1	no	—
270	C₂×C₈	—	61	1	no	—
271	C₂×C₈	—	61	2	no	—
272	C₂×C₈	C₄×#6923	1	1	yes	characteristic
273	C₂×C₈	C₄×#6923	1	2	yes	—
274	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
275	C₂×C₈	C₂²×#6923	1	2	yes	—
276	C₄⋊C₄	—	61	8	no	—
277	C₂²⋊C₄	—	61	8	no	—
278	C₄²	—	61	4	no	—
279	C₄²	—	61	4	no	—
280	C₁₆	—	61	8	no	—
281	C₁₆	—	61	2	no	—
282	C₁₆	—	61	2	no	—
283	C₁₆	C₂²×#6923	1	2	yes	—
284	C₁₆	C₂²×#6923	1	2	yes	—
285	C₂³	—	61	1	no	—
286	C₂³	—	61	2	no	—
287	C₂³	—	61	1	no	—
288	C₂³	—	61	2	no	—
289	C₂³	—	122	4	no	—
290	C₂³	C₈×#6923	1	1	yes	characteristic
291	C₂×C₄	—	61	2	no	—
292	C₂×C₄	—	61	1	no	—
293	C₂×C₄	—	61	1	no	—
294	C₂×C₄	—	61	2	no	—
295	C₂×C₄	—	61	2	no	—
296	C₂×C₄	—	61	2	no	—
297	C₂×C₄	—	61	2	no	—
298	C₂×C₄	—	61	1	no	—
299	C₂×C₄	—	61	2	no	—
300	C₂×C₄	—	61	1	no	—
301	C₂×C₄	—	61	2	no	—
302	C₂×C₄	—	61	1	no	—
303	C₂×C₄	—	61	2	no	—
304	C₂×C₄	—	61	1	no	—
305	C₂×C₄	—	61	8	no	—
306	C₂×C₄	—	122	8	no	—
307	C₂×C₄	C₈×#6923	1	2	yes	—
308	C₂×C₄	C₈×#6923	1	1	yes	characteristic
309	C₂×C₄	C₂×C₄×#6923	1	2	yes	—
310	C₂×C₄	C₂×C₄×#6923	1	1	yes	characteristic
311	C₈	—	61	8	no	—
312	C₈	—	61	1	no	—
313	C₈	—	61	2	no	—
314	C₈	—	61	1	no	—
315	C₈	C₂×C₄×#6923	1	1	yes	characteristic
316	C₈	C₂×C₄×#6923	1	2	yes	—
317	C₈	C₂³×#6923	1	1	yes	characteristic
318	C₂²	—	2	2	no	—
319	C₂²	—	61	4	no	—
320	C₂²	—	61	1	no	—
321	C₂²	—	61	1	no	—
322	C₂²	—	61	2	no	—
323	C₂²	—	122	4	no	—
324	C₂²	—	122	4	no	—
325	C₂²	—	122	2	no	—
326	C₂²	C₂×C₈×#6923	1	2	yes	—
327	C₂²	C₂×C₈×#6923	1	1	yes	characteristic
328	C₄	—	61	1	no	—
329	C₄	—	61	1	no	—
330	C₄	—	61	2	no	—
331	C₄	—	122	8	no	—
332	C₄	C₂×C₈×#6923	1	1	yes	characteristic
333	C₄	C₂×C₈×#6923	1	2	yes	—
334	C₄	C₂²×C₄×#6923	1	1	yes	characteristic
335	C₂	—	2	2	no	—
336	C₂	—	61	2	no	—
337	C₂	—	61	2	no	—
338	C₂	—	122	2	no	—
339	C₂	#21462027	1	2	yes	direct product factor
340	C₂	C₂²×C₈×#6923	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆	C₂×#6923
commutator	C₂×C₆₁	C₂²×C₄×C₈
Frattini	C₈	C₂³×#6923
Fitting	C₂²×C₁₆×C₆₁	C₄
socle	C₂²×C₆₁	C₂×C₄×C₈
prime radical	C₂³×D₆₁	C₂×C₈
cosocle	C₈×D₆₁	C₂⁴
hypercenter	C₂²×C₁₆	#6923
hypocenter	C₆₁	C₂×#2108
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
3. #21462027	C₂	direct (N × H)
4. C₂×#16840649	C₂	semidirect
5. C₂×C₁₆×#6923	C₂	semidirect
9. C₂×#14519852	C₄	semidirect
10. D₆₁⋊M₃₂	C₄	semidirect
12. C₂×#14519851	C₄	semidirect
18. #16840649	C₂²	semidirect
19. C₁₆×#6923	C₂²	semidirect
21. C₂²×#11758185	C₄	semidirect
22. C₂×C₁₆×D₆₁	C₄	semidirect
23. C₂×C₁₆×D₆₁	C₄	semidirect
24. C₂×C₁₆×D₆₁	C₄	semidirect
28. C₂²×C₁₆×C₆₁	C₄	semidirect
39. #14519852	C₂×C₄	semidirect
40. #14519851	C₂×C₄	semidirect
59. C₂×#11758185	C₂×C₄	semidirect
62. C₁₆×D₆₁	C₂×C₄	semidirect
63. C₁₆×D₆₁	C₂×C₄	semidirect
69. C₂×C₁₆×C₆₁	C₂×C₄	semidirect
172. C₂×C₆₁	#2108	semidirect
175. D₆₁	#2108	semidirect
192. C₂²×C₁₆	#6923	semidirect
199. C₆₁	C₂×#2108	semidirect
237. C₂×C₁₆	C₂×#6923	semidirect
339. C₂	#21462027	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×D₆₁⋊M₃₂	2
C₂×#16840725	2
#21462027	2
C₂×#16840649	2
C₂×C₁₆×#6923	2
C₂²×C₁₆×D₆₁	2
C₂×#2108	61

Subgroups — C2×#21462027 #30623816

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21462027 #30623816