Subgroups — #30598945

Order 15616.

598 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21461673	C₂	1	2	yes	maximal
2	C₂×#16840624	C₂	1	2	yes	maximal, complemented
3	#21460649	C₂	1	2	yes	maximal, complemented
4	C₂²×C₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#92420	C₂²	1	4	yes	—
6	C₂×C₂²⋊#92420	—	2	2	no	—
7	#16840628	—	2	2	no	—
8	#16840624	—	2	4	no	—
9	C₂³×#92418	C₄	1	1	yes	characteristic
10	C₂²×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₆₁	—	2	4	no	—
12	C₂×C₈×Dic₆₁	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₆₁	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
15	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
16	C₂³×C₈×C₆₁	C₄	1	1	yes	characteristic
17	C₂²×#92420	—	2	4	no	—
18	C₄⋊#92420	—	2	4	no	—
19	C₄⋊#92420	—	2	4	no	—
20	C₂²⋊#92420	—	4	4	no	—
21	C₂³×Dic₆₁	D₄	1	1	yes	characteristic
22	C₂³×Dic₆₁	Q₈	1	1	yes	characteristic
23	C₂²×#92418	—	2	2	no	—
24	C₂²×#92418	—	2	2	no	—
25	C₂²×#92418	C₈	1	4	yes	complemented
26	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
27	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×Dic₆₁	—	2	2	no	—
29	C₂×C₄×Dic₆₁	—	2	2	no	—
30	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
32	C₂×C₄×Dic₆₁	Q₈	1	1	yes	characteristic
33	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
34	C₈×Dic₆₁	—	2	2	no	—
35	C₈×Dic₆₁	—	2	2	no	—
36	C₈×Dic₆₁	—	2	2	no	—
37	C₈×Dic₆₁	D₄	1	4	yes	—
38	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₆₁	—	2	2	no	—
40	C₂²×C₈×C₆₁	—	2	2	no	—
41	C₂²×C₈×C₆₁	—	2	2	no	—
42	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
44	C₂⁴×C₆₁	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₆₁	—	2	1	no	—
46	C₂²×C₄×C₆₁	—	2	1	no	—
47	C₂²×C₄×C₆₁	—	2	1	no	—
48	C₂²×C₄×C₆₁	—	2	1	no	—
49	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
51	C₂²×C₄×C₆₁	M₁₆	1	2	yes	—
52	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
53	C₂²×Dic₆₁	—	2	2	no	—
54	C₂²×Dic₆₁	—	2	2	no	—
55	C₂²×Dic₆₁	—	2	2	no	—
56	C₂²×Dic₆₁	—	2	2	no	—
57	C₂²×Dic₆₁	—	2	2	no	—
58	C₂²×Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×Dic₆₁	SD₁₆	1	4	yes	—
61	C₂²×Dic₆₁	C₂×D₄	1	1	yes	characteristic
62	C₂²×Dic₆₁	C₂×Q₈	1	1	yes	characteristic
63	C₂×#92420	—	4	4	no	—
64	C₂×#92420	—	4	4	no	—
65	C₂×#92420	—	4	4	no	—
66	C₂×C₈×C₆₁	—	2	8	no	—
67	C₂×C₈×C₆₁	—	2	1	no	—
68	C₂×C₈×C₆₁	—	2	1	no	—
69	C₂×C₈×C₆₁	—	2	1	no	—
70	C₂×C₈×C₆₁	—	2	1	no	—
71	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
72	C₂×C₈×C₆₁	M₁₆	1	2	yes	—
73	C₄×Dic₆₁	—	2	1	no	—
74	C₄×Dic₆₁	—	2	1	no	—
75	C₄×Dic₆₁	—	2	2	no	—
76	C₄×Dic₆₁	—	2	2	no	—
77	C₄×Dic₆₁	C₂×D₄	1	2	yes	—
78	C₄×Dic₆₁	#41	1	2	yes	—
79	C₂×#92418	—	2	1	no	—
80	C₂×#92418	—	2	1	no	—
81	C₂×#92418	—	2	8	no	—
82	C₂×#92418	C₂²⋊C₄	1	2	yes	—
83	C₂×#92418	C₂×C₈	1	2	yes	—
84	C₂×#92418	C₂×C₈	1	2	yes	—
85	C₂×#92418	M₁₆	1	2	yes	—
86	C₂³×C₆₁	—	2	1	no	—
87	C₂³×C₆₁	—	2	1	no	—
88	C₂³×C₆₁	—	2	1	no	—
89	C₂³×C₆₁	—	2	1	no	—
90	C₂³×C₆₁	C₄⋊C₈	1	2	yes	—
91	C₂³×C₆₁	#106	1	2	yes	—
92	C₂³×C₆₁	#108	1	2	yes	—
93	C₂³×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₆₁	—	2	1	no	—
95	C₂×C₄×C₆₁	—	2	1	no	—
96	C₂×C₄×C₆₁	—	2	2	no	—
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	C₄⋊C₈	1	2	yes	—
101	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
103	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
104	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
105	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
106	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
107	C₂×Dic₆₁	—	2	8	no	—
108	C₂×Dic₆₁	—	2	1	no	—
109	C₂×Dic₆₁	—	2	1	no	—
110	C₂×Dic₆₁	—	2	1	no	—
111	C₂×Dic₆₁	—	2	1	no	—
112	C₂×Dic₆₁	—	2	1	no	—
113	C₂×Dic₆₁	—	2	1	no	—
114	C₂×Dic₆₁	—	2	8	no	—
115	C₂×Dic₆₁	#106	1	2	yes	—
116	C₂×Dic₆₁	#108	1	2	yes	—
117	C₂×Dic₆₁	#121	1	2	yes	—
118	C₂×Dic₆₁	C₂²⋊Q₈	1	2	yes	—
119	C₂×Dic₆₁	C₂×SD₁₆	1	2	yes	—
120	C₂×Dic₆₁	#135	1	2	yes	—
121	#92420	—	8	4	no	—
122	C₈×C₆₁	—	2	2	no	—
123	C₈×C₆₁	—	2	2	no	—
124	#92418	—	2	2	no	—
125	#92418	C₂²⋊C₈	1	4	yes	complemented
126	#8595	—	61	1	no	maximal, Sylow
127	C₂²×C₆₁	—	2	2	no	—
128	C₂²×C₆₁	—	2	2	no	—
129	C₂²×C₆₁	—	2	2	no	—
130	C₂²×C₆₁	—	2	2	no	—
131	C₂²×C₆₁	—	2	2	no	—
132	C₂²×C₆₁	—	2	2	no	—
133	C₂²×C₆₁	#334	1	4	yes	complemented
134	C₂²×C₆₁	#380	1	1	yes	characteristic
135	C₂²×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
136	C₂²×C₆₁	#423	1	1	yes	characteristic
137	C₂²×C₆₁	C₂×#106	1	1	yes	characteristic
138	C₂²×C₆₁	#427	1	1	yes	characteristic
139	C₂²×C₆₁	C₂×#108	1	1	yes	characteristic
140	C₂²×C₆₁	C₂×#108	1	1	yes	characteristic
141	C₄×C₆₁	—	2	2	no	—
142	C₄×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
143	C₄×C₆₁	#407	1	1	yes	characteristic
144	C₄×C₆₁	#431	1	1	yes	characteristic
145	C₄×C₆₁	#432	1	1	yes	characteristic
146	Dic₆₁	—	2	2	no	—
147	Dic₆₁	—	2	2	no	—
148	Dic₆₁	—	2	2	no	—
149	Dic₆₁	#465	1	4	yes	—
150	C₂²×C₄×C₈	—	61	1	no	—
151	#1797	—	61	2	no	—
152	C₂×#334	—	61	2	no	—
153	#1277	—	61	2	no	—
154	C₂×C₆₁	—	2	4	no	—
155	C₂×C₆₁	C₂×#334	1	1	yes	characteristic
156	C₂×C₆₁	#1564	1	1	yes	characteristic
157	C₂×C₆₁	#1566	1	1	yes	characteristic
158	C₂×C₆₁	#1568	1	1	yes	characteristic
159	C₂×C₆₁	#1797	1	1	yes	characteristic
160	C₂×C₆₁	#1807	1	1	yes	characteristic
161	C₂×C₆₁	#1810	1	1	yes	characteristic
162	C₂³×C₈	—	61	1	no	—
163	C₂³×C₈	#6923	1	1	yes	characteristic
164	C₂²×C₄²	—	61	1	no	—
165	C₂×C₄⋊C₈	—	61	4	no	—
166	C₂×C₂²⋊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	1	no	—
171	C₂×C₄×C₈	—	122	4	no	—
172	#336	—	122	2	no	—
173	#334	—	122	4	no	—
174	C₆₁	#8595	1	1	yes	characteristic, complemented, Sylow
175	C₂³×C₄	—	61	1	no	—
176	C₂³×C₄	—	61	1	no	—
177	C₂³×C₄	C₂×#6923	1	1	yes	characteristic
178	C₂²×C₈	—	2	2	no	—
179	C₂²×C₈	—	2	2	no	—
180	C₂²×C₈	—	2	2	no	—
181	C₂²×C₈	—	61	1	no	—
182	C₂²×C₈	—	61	4	no	—
183	C₂²×C₈	—	61	1	no	—
184	C₂²×C₈	—	122	2	no	—
185	C₂²×C₈	—	122	2	no	—
186	C₂²×C₈	—	122	4	no	—
187	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
188	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
189	C₂×C₄²	—	61	1	no	—
190	C₂×C₄²	—	61	1	no	—
191	C₂×C₄²	—	61	1	no	—
192	C₂×C₄²	—	61	1	no	—
193	C₂×C₄²	—	122	2	no	—
194	C₂×C₄²	—	122	2	no	—
195	C₄⋊C₈	—	122	4	no	—
196	C₄⋊C₈	—	122	4	no	—
197	C₂²⋊C₈	—	244	4	no	—
198	C₄×C₈	—	61	4	no	—
199	C₄×C₈	—	122	2	no	—
200	C₄×C₈	—	122	2	no	—
201	C₄×C₈	—	122	2	no	—
202	C₂⁴	C₄⋊#6923	1	1	yes	characteristic
203	C₂²×C₄	—	2	1	no	—
204	C₂²×C₄	—	2	1	no	—
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	—	61	1	no	—
208	C₂²×C₄	—	61	4	no	—
209	C₂²×C₄	—	61	1	no	—
210	C₂²×C₄	—	61	1	no	—
211	C₂²×C₄	—	61	1	no	—
212	C₂²×C₄	—	122	2	no	—
213	C₂²×C₄	—	122	2	no	—
214	C₂²×C₄	—	122	2	no	—
215	C₂²×C₄	—	122	2	no	—
216	C₂²×C₄	—	122	2	no	—
217	C₂²×C₄	D₆₁⋊C₈	1	2	yes	—
218	C₂²×C₄	C₄⋊#6923	1	1	yes	characteristic
219	C₂²×C₄	#11758217	1	2	yes	—
220	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
221	C₂×C₈	—	2	8	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	61	2	no	—
227	C₂×C₈	—	61	2	no	—
228	C₂×C₈	—	61	2	no	—
229	C₂×C₈	—	61	2	no	—
230	C₂×C₈	—	122	8	no	—
231	C₂×C₈	—	122	1	no	—
232	C₂×C₈	—	122	1	no	—
233	C₂×C₈	—	244	4	no	—
234	C₂×C₈	—	244	4	no	—
235	C₂×C₈	—	244	4	no	—
236	C₂×C₈	#11758217	1	2	yes	—
237	C₂×C₈	C₂²⋊#6923	1	2	yes	—
238	C₄²	—	61	2	no	—
239	C₄²	—	61	2	no	—
240	C₄²	—	122	2	no	—
241	C₄²	—	122	2	no	—
242	C₄²	—	122	1	no	—
243	C₄²	—	122	1	no	—
244	C₂³	—	2	1	no	—
245	C₂³	—	2	1	no	—
246	C₂³	—	2	1	no	—
247	C₂³	—	2	1	no	—
248	C₂³	#14519844	1	2	yes	—
249	C₂³	#14519845	1	2	yes	—
250	C₂³	#14519848	1	2	yes	—
251	C₂³	C₂×C₄⋊#6923	1	1	yes	characteristic
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	61	2	no	—
259	C₂×C₄	—	61	2	no	—
260	C₂×C₄	—	61	2	no	—
261	C₂×C₄	—	61	2	no	—
262	C₂×C₄	—	61	2	no	—
263	C₂×C₄	—	61	2	no	—
264	C₂×C₄	—	122	8	no	—
265	C₂×C₄	—	122	8	no	—
266	C₂×C₄	—	122	1	no	—
267	C₂×C₄	—	122	1	no	—
268	C₂×C₄	—	122	1	no	—
269	C₂×C₄	—	122	1	no	—
270	C₂×C₄	—	122	1	no	—
271	C₂×C₄	—	122	1	no	—
272	C₂×C₄	#14519844	1	2	yes	—
273	C₂×C₄	C₂×C₂²⋊#6923	1	1	yes	characteristic
274	C₂×C₄	#14519856	1	1	yes	characteristic
275	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
276	C₂×C₄	C₂×#11758217	1	1	yes	characteristic
277	C₂×C₄	C₂×#11758217	1	1	yes	characteristic
278	C₂×C₄	D₆₁⋊M₁₆	1	1	yes	characteristic
279	C₈	—	2	2	no	—
280	C₈	—	2	2	no	—
281	C₈	—	61	4	no	—
282	C₈	—	122	2	no	—
283	C₈	—	488	4	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	#16840624	1	4	yes	complemented
291	C₂²	#16840704	1	1	yes	characteristic
292	C₂²	C₂×#14519844	1	1	yes	characteristic
293	C₂²	#16840765	1	1	yes	characteristic
294	C₂²	C₂×#14519845	1	1	yes	characteristic
295	C₂²	#16840772	1	1	yes	characteristic
296	C₂²	D₆₁⋊#108	1	1	yes	characteristic
297	C₂²	C₂×#14519848	1	1	yes	characteristic
298	C₄	—	2	2	no	—
299	C₄	—	61	4	no	—
300	C₄	—	122	2	no	—
301	C₄	—	122	2	no	—
302	C₄	—	122	2	no	—
303	C₄	D₆₁⋊(C₂²⋊C₈)	1	1	yes	characteristic
304	C₄	#16840732	1	1	yes	characteristic
305	C₄	#16840779	1	1	yes	characteristic
306	C₄	#16840782	1	1	yes	characteristic
307	C₂	—	2	4	no	—
308	C₂	C₂×#16840624	1	1	yes	characteristic
309	C₂	#21461368	1	1	yes	characteristic
310	C₂	#21461373	1	1	yes	characteristic
311	C₂	#21461381	1	1	yes	characteristic
312	C₂	#21461675	1	1	yes	characteristic
313	C₂	#21461695	1	1	yes	characteristic
314	C₂	#21461699	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊#6923
commutator	C₂×C₄×C₆₁	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#6923
Fitting	C₂³×C₈×C₆₁	C₄
socle	C₂³×C₆₁	C₂×C₄⋊C₄
prime radical	C₂⁴×C₆₁	C₄⋊C₄
cosocle	C₂×C₄×Dic₆₁	C₂³
hypercenter	C₂³×C₈	#6923
hypocenter	C₆₁	#8595
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. C₂×#16840624	C₂	semidirect
3. #21460649	C₂	semidirect
25. C₂²×#92418	C₈	semidirect
125. #92418	C₂²⋊C₈	semidirect
133. C₂²×C₆₁	#334	semidirect
174. C₆₁	#8595	semidirect
290. C₂²	#16840624	semidirect

Maximal subgroups

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

subgroup	index
#21461673	2
C₂×#16840624	2
#21460649	2
C₂²×C₈×Dic₆₁	2
#8595	61