Subgroups — #30624348

Order 15616.

506 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21462236	C₂	1	2	yes	maximal, complemented
2	#21462235	C₂	1	1	yes	maximal, characteristic, complemented
3	#21462234	C₂	1	1	yes	maximal, characteristic, complemented
4	#21462233	C₂	1	1	yes	maximal, characteristic, complemented
5	#21462233	C₂	1	2	yes	maximal, complemented
6	Q₈×#11758187	C₂	1	1	yes	maximal, characteristic, complemented
7	#21462155	C₂	1	2	yes	maximal
8	#21462152	C₂	1	1	yes	maximal, characteristic
9	#21462029	C₂	1	2	yes	maximal
10	C₄×D₆₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
11	#21454750	C₂	1	1	yes	maximal, characteristic
12	C₂×#14519852	C₂²	1	2	yes	—
13	C₂×D₆₁⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₂×D₆₁⋊C₁₆	C₂²	1	1	yes	characteristic
15	#16840683	C₂²	1	2	yes	—
16	#16840683	C₂²	1	2	yes	—
17	#16840683	C₂²	1	2	yes	—
18	C₄⋊#11758187	C₂²	1	1	yes	characteristic
19	C₄⋊#11758187	C₂²	1	1	yes	characteristic
20	C₄⋊#11758187	C₂²	1	2	yes	—
21	C₄⋊#11758187	C₂²	1	2	yes	—
22	#16840651	C₂²	1	2	yes	—
23	#16840650	C₂²	1	2	yes	—
24	#16840648	C₂²	1	1	yes	characteristic
25	#16840648	C₂²	1	1	yes	characteristic
26	#16840648	C₂²	1	2	yes	—
27	C₄×#11758187	C₂²	1	1	yes	characteristic
28	C₄×#11758187	C₂²	1	1	yes	characteristic
29	C₄×#11758187	C₂²	1	2	yes	—
30	#16840316	C₄	1	1	yes	characteristic
31	#16839892	C₄	1	1	yes	characteristic
32	#16839890	C₄	1	2	yes	—
33	Q₈×#92418	C₂²	1	1	yes	characteristic
34	#16839886	C₂²	1	1	yes	characteristic
35	#16839886	C₂²	1	2	yes	—
36	#16839787	C₄	1	2	yes	—
37	#16839784	C₄	1	1	yes	characteristic
38	#16839719	C₂²	1	2	yes	—
39	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
40	C₆₁×#446	C₄	1	1	yes	characteristic
41	#14519852	—	2	4	no	—
42	D₆₁⋊C₁₆	—	2	2	no	—
43	D₆₁⋊C₁₆	Q₈	1	4	yes	complemented
44	C₂×#11758187	C₂³	1	1	yes	characteristic
45	C₂×#11758187	C₂³	1	1	yes	characteristic
46	C₂×#11758187	C₂³	1	2	yes	—
47	C₂×#11758187	C₂³	1	2	yes	—
48	C₂×#11758187	C₂³	1	1	yes	characteristic
49	C₂×#11758187	C₂³	1	1	yes	characteristic
50	C₂×#11758189	C₂×C₄	1	2	yes	—
51	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
52	C₂×C₈×D₆₁	C₂³	1	1	yes	characteristic
53	#14519776	C₈	1	1	yes	characteristic
54	#14519775	C₈	1	1	yes	characteristic
55	#14519767	C₈	1	2	yes	—
56	#14519757	C₈	1	1	yes	characteristic
57	#14519755	C₈	1	2	yes	—
58	#14519753	C₈	1	1	yes	characteristic
59	C₄×Dic₁₂₂	C₂×C₄	1	2	yes	—
60	C₄×Dic₁₂₂	C₂×C₄	1	1	yes	characteristic
61	#14519734	C₂×C₄	1	2	yes	—
62	C₄²×D₆₁	C₂×C₄	1	1	yes	characteristic
63	C₄⋊#92418	C₂×C₄	1	1	yes	characteristic
64	C₄⋊#92418	C₂×C₄	1	2	yes	—
65	#14519706	C₂×C₄	1	2	yes	—
66	#14519706	C₂×C₄	1	2	yes	—
67	#14519706	C₂×C₄	1	2	yes	—
68	#14519687	C₂×C₄	1	2	yes	—
69	C₄×#92418	C₂³	1	1	yes	characteristic
70	C₄×#92418	C₂³	1	2	yes	—
71	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
72	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
73	C₈×Dic₆₁	C₂³	1	2	yes	—
74	C₄×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
75	C₄⋊C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
76	C₄⋊C₈×C₆₁	C₂×C₄	1	2	yes	—
77	#97×C₆₁	C₂×C₄	1	2	yes	—
78	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
79	C₂×Q₈×C₆₁	C₂×C₈	1	1	yes	characteristic
80	C₂×C₄×D₆₁	C₂×C₈	1	1	yes	characteristic
81	C₂×C₄×D₆₁	C₂×C₈	1	1	yes	characteristic
82	C₂×C₄×D₆₁	C₂²×C₄	1	1	yes	characteristic
83	C₂×Dic₁₂₂	C₂×C₈	1	2	yes	—
84	C₂×Dic₁₂₂	C₂×C₈	1	1	yes	characteristic
85	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
88	C₄⋊C₄×C₆₁	C₂×C₈	1	2	yes	—
89	C₄⋊C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
90	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
91	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
92	#11758198	C₂×C₈	1	2	yes	—
93	#11758198	C₂×C₈	1	2	yes	—
94	C₄⋊Dic₆₁	C₂×C₈	1	1	yes	characteristic
95	C₄⋊Dic₆₁	C₂×C₈	1	2	yes	—
96	#11758196	C₂×C₈	1	2	yes	—
97	#11758196	C₂×C₈	1	2	yes	—
98	#11758196	C₂×C₈	1	2	yes	—
99	#11758196	C₂×C₈	1	2	yes	—
100	#11758196	C₂×C₈	1	2	yes	—
101	C₄×Dic₆₁	C₂×C₈	1	1	yes	characteristic
102	C₄×Dic₆₁	C₂×C₈	1	1	yes	characteristic
103	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
104	C₄×Dic₆₁	C₂²×C₄	1	2	yes	—
105	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
106	C₂×#92418	C₂²×C₄	1	2	yes	—
107	C₂×#92418	C₂⁴	1	1	yes	characteristic
108	#11758189	—	2	4	no	—
109	C₈×D₆₁	—	2	2	no	—
110	C₈×D₆₁	C₂×Q₈	1	2	yes	—
111	C₈×D₆₁	#41	1	2	yes	—
112	#11758187	—	2	1	no	—
113	#11758187	—	2	1	no	—
114	#11758187	—	2	2	no	—
115	#11758187	—	2	2	no	—
116	#11758187	C₂×Q₈	1	2	yes	—
117	#11758187	C₂×Q₈	1	2	yes	—
118	C₂²×D₆₁	C₂²×C₈	1	1	yes	characteristic
119	Q₈×C₆₁	—	2	2	no	—
120	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
122	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
123	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
124	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
125	C₂×Dic₆₁	C₂²×C₈	1	1	yes	characteristic
126	C₂×Dic₆₁	C₂²×C₈	1	2	yes	—
127	C₂×Dic₆₁	C₂²×C₈	1	2	yes	—
128	C₂×Dic₆₁	C₂²×C₈	1	1	yes	characteristic
129	C₂×Dic₆₁	C₂²×C₈	1	1	yes	characteristic
130	C₄×D₆₁	—	2	2	no	—
131	C₄×D₆₁	—	2	2	no	—
132	C₄×D₆₁	C₄×Q₈	1	2	yes	—
133	C₄×D₆₁	#131	1	2	yes	—
134	Dic₁₂₂	—	2	2	no	—
135	Dic₁₂₂	—	2	4	no	—
136	C₈×C₆₁	—	2	1	no	—
137	C₈×C₆₁	—	2	2	no	—
138	C₈×C₆₁	C₄×Q₈	1	2	yes	—
139	#92418	—	2	1	no	—
140	#92418	—	2	2	no	—
141	#92418	C₂²×Q₈	1	1	yes	characteristic
142	#92418	C₂×#41	1	1	yes	characteristic
143	#33474	—	61	1	no	maximal, Sylow
144	C₂²×C₆₁	C₂³×C₈	1	1	yes	characteristic
145	C₂×D₆₁	—	2	1	no	—
146	C₂×D₆₁	—	2	1	no	—
147	C₂×D₆₁	C₈×Q₈	1	2	yes	—
148	C₄×C₆₁	—	2	2	no	—
149	C₄×C₆₁	—	2	1	no	—
150	C₄×C₆₁	—	2	2	no	—
151	C₄×C₆₁	#504	1	2	yes	—
152	C₄×C₆₁	C₂×C₄×Q₈	1	1	yes	characteristic
153	C₄×C₆₁	C₂×#131	1	1	yes	characteristic
154	Dic₆₁	—	2	2	no	—
155	Dic₆₁	—	2	2	no	—
156	Dic₆₁	—	2	1	no	—
157	Dic₆₁	C₈×Q₈	1	2	yes	—
158	Dic₆₁	#504	1	2	yes	—
159	#2960	—	61	1	no	—
160	#2183	—	61	1	no	—
161	#2183	—	61	1	no	—
162	#2183	—	61	2	no	—
163	Q₈×C₁₆	—	61	1	no	—
164	Q₈×C₁₆	—	61	1	no	—
165	Q₈×C₁₆	—	61	2	no	—
166	#2151	—	61	2	no	—
167	#2150	—	61	1	no	—
168	#2108	—	61	2	no	—
169	C₂×C₄×C₁₆	—	61	1	no	—
170	C₂×C₆₁	C₂×C₈×Q₈	1	1	yes	characteristic
171	C₂×C₆₁	C₂×#504	1	1	yes	characteristic
172	C₂×C₆₁	C₂×#504	1	1	yes	characteristic
173	D₆₁	—	2	2	no	—
174	#533	—	61	1	no	—
175	C₂×M₃₂	—	61	2	no	—
176	C₂²×C₁₆	—	61	1	no	—
177	C₂²×C₁₆	—	61	1	no	—
178	#446	—	61	2	no	—
179	#446	—	61	1	no	—
180	#446	#6923	1	1	yes	characteristic
181	C₈×Q₈	—	61	1	no	—
182	C₈×Q₈	—	61	2	no	—
183	C₈×Q₈	—	61	1	no	—
184	#424	—	61	2	no	—
185	#423	—	61	1	no	—
186	#405	—	61	2	no	—
187	C₂×C₄×C₈	—	61	1	no	—
188	C₄⋊C₁₆	—	61	1	no	—
189	C₄⋊C₁₆	—	61	1	no	—
190	C₄⋊C₁₆	—	61	2	no	—
191	C₄⋊C₁₆	—	61	2	no	—
192	C₄⋊C₁₆	—	61	2	no	—
193	C₄⋊C₁₆	—	61	2	no	—
194	C₄⋊C₁₆	—	61	2	no	—
195	#346	—	61	2	no	—
196	#346	—	61	2	no	—
197	C₄×C₁₆	—	61	1	no	—
198	C₄×C₁₆	—	61	1	no	—
199	C₄×C₁₆	—	61	1	no	—
200	C₄×C₁₆	—	61	1	no	—
201	C₄×C₁₆	—	61	2	no	—
202	C₄×C₁₆	—	61	2	no	—
203	C₆₁	#33474	1	1	yes	characteristic, complemented, Sylow
204	C₂×M₁₆	—	61	2	no	—
205	C₂²×C₈	—	61	1	no	—
206	C₂²×C₈	—	61	1	no	—
207	C₄⋊Q₈	—	61	1	no	—
208	C₄⋊Q₈	—	61	1	no	—
209	#125	—	61	2	no	—
210	C₂²⋊Q₈	—	61	1	no	—
211	C₂²⋊Q₈	—	61	2	no	—
212	C₂²⋊Q₈	—	61	1	no	—
213	C₄×Q₈	—	61	2	no	—
214	C₄×Q₈	—	61	1	no	—
215	C₄×Q₈	C₂×#6923	1	1	yes	characteristic
216	#117	—	61	2	no	—
217	C₂×C₄²	—	61	1	no	—
218	M₃₂	—	122	4	no	—
219	C₂×C₁₆	—	61	1	no	—
220	C₂×C₁₆	—	61	1	no	—
221	C₂×C₁₆	—	61	4	no	—
222	C₂×C₁₆	—	61	1	no	—
223	C₂×C₁₆	—	61	1	no	—
224	C₂×C₁₆	—	61	2	no	—
225	C₂×C₁₆	—	61	2	no	—
226	C₂×C₁₆	—	122	2	no	—
227	C₄⋊C₈	—	61	2	no	—
228	C₄⋊C₈	—	61	2	no	—
229	C₄⋊C₈	—	61	2	no	—
230	C₄⋊C₈	—	61	1	no	—
231	C₄⋊C₈	—	61	2	no	—
232	C₄⋊C₈	C₂×#6923	1	1	yes	characteristic
233	C₄⋊C₈	C₂×#6923	1	2	yes	—
234	#97	—	61	2	no	—
235	#97	C₂×#6923	1	2	yes	—
236	C₄×C₈	—	61	1	no	—
237	C₄×C₈	—	61	2	no	—
238	C₄×C₈	—	61	1	no	—
239	C₄×C₈	—	61	1	no	—
240	C₄×C₈	—	61	2	no	—
241	C₄×C₈	C₂×#6923	1	1	yes	characteristic
242	C₂×Q₈	—	61	2	no	—
243	C₂×Q₈	—	61	1	no	—
244	C₂×Q₈	D₆₁⋊C₈	1	1	yes	characteristic
245	C₂²×C₄	—	61	1	no	—
246	C₂²×C₄	—	61	1	no	—
247	C₂²×C₄	—	61	1	no	—
248	M₁₆	—	122	4	no	—
249	C₂×C₈	—	61	1	no	—
250	C₂×C₈	—	61	2	no	—
251	C₂×C₈	—	61	2	no	—
252	C₂×C₈	—	61	1	no	—
253	C₂×C₈	—	61	2	no	—
254	C₂×C₈	—	122	2	no	—
255	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
256	C₂×C₈	C₂²×#6923	1	2	yes	—
257	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
258	C₄⋊C₄	—	61	2	no	—
259	C₄⋊C₄	—	61	2	no	—
260	C₄⋊C₄	—	61	2	no	—
261	C₄⋊C₄	—	61	2	no	—
262	C₄⋊C₄	—	61	2	no	—
263	C₄⋊C₄	—	61	2	no	—
264	C₄⋊C₄	—	61	1	no	—
265	C₄⋊C₄	D₆₁⋊C₈	1	2	yes	—
266	C₄⋊C₄	D₆₁⋊C₈	1	1	yes	characteristic
267	C₂²⋊C₄	—	61	2	no	—
268	C₂²⋊C₄	—	61	2	no	—
269	C₄²	—	61	1	no	—
270	C₄²	—	61	1	no	—
271	C₄²	—	61	1	no	—
272	C₄²	—	61	2	no	—
273	C₄²	C₂²×#6923	1	1	yes	characteristic
274	C₄²	C₂²×#6923	1	2	yes	—
275	C₁₆	—	61	2	no	—
276	C₁₆	—	61	2	no	—
277	C₁₆	—	122	1	no	—
278	C₁₆	—	122	1	no	—
279	C₁₆	—	122	2	no	—
280	C₁₆	—	122	2	no	—
281	C₂³	—	61	1	no	—
282	Q₈	—	2	2	no	—
283	Q₈	—	122	4	no	—
284	Q₈	—	122	2	no	—
285	C₂×C₄	—	61	1	no	—
286	C₂×C₄	—	61	1	no	—
287	C₂×C₄	—	61	2	no	—
288	C₂×C₄	—	61	2	no	—
289	C₂×C₄	—	61	1	no	—
290	C₂×C₄	—	61	2	no	—
291	C₂×C₄	—	61	2	no	—
292	C₂×C₄	—	122	2	no	—
293	C₂×C₄	—	122	2	no	—
294	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
295	C₂×C₄	C₂×D₆₁⋊C₈	1	2	yes	—
296	C₂×C₄	C₂×D₆₁⋊C₈	1	2	yes	—
297	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
298	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
299	C₈	—	2	1	no	—
300	C₈	—	2	2	no	—
301	C₈	—	61	1	no	—
302	C₈	—	61	1	no	—
303	C₈	—	122	1	no	—
304	C₈	—	122	2	no	—
305	C₈	Q₈×#6923	1	2	yes	—
306	C₂²	—	61	2	no	—
307	C₂²	—	122	1	no	—
308	C₂²	—	122	1	no	—
309	C₂²	C₂²×D₆₁⋊C₈	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	2	no	—
312	C₄	—	2	2	no	—
313	C₄	—	61	2	no	—
314	C₄	—	61	2	no	—
315	C₄	—	122	2	no	—
316	C₄	—	122	1	no	—
317	C₄	—	122	2	no	—
318	C₄	#16840821	1	2	yes	—
319	C₄	C₂×Q₈×#6923	1	1	yes	characteristic
320	C₄	D₆₁⋊#131	1	1	yes	characteristic
321	C₂	—	122	2	no	—
322	C₂	Q₈×D₆₁⋊C₈	1	1	yes	characteristic
323	C₂	D₆₁⋊#504	1	1	yes	characteristic
324	C₂	C₂×#16840821	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#6923
commutator, socle	C₂²×C₆₁	C₂³×C₈
Fitting	C₆₁×#446	C₄
prime radical	C₂²×D₆₁	C₂²×C₈
cosocle	C₂×#92418	C₂⁴
hypercenter	#446	#6923
hypocenter	C₆₁	#33474
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
1. #21462236	C₂	semidirect
2. #21462235	C₂	semidirect
3. #21462234	C₂	semidirect
4. #21462233	C₂	semidirect
5. #21462233	C₂	semidirect
6. Q₈×#11758187	C₂	semidirect
43. D₆₁⋊C₁₆	Q₈	semidirect
203. C₆₁	#33474	semidirect

Maximal subgroups

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

subgroup	index
#21462236	2
#21462235	2
#21462234	2
#21462233	2
Q₈×#11758187	2
#21462155	2
#21462152	2
#21462029	2
C₄×D₆₁⋊C₁₆	2
#21454750	2
#33474	61