Subgroups — #30624225

Order 15616.

710 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16840820	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×#11758187	C₂	1	1	yes	maximal, characteristic, complemented
3	#21462199	C₂	1	4	yes	maximal, complemented
4	D₄×#11758187	C₂	1	4	yes	maximal, complemented
5	#21462154	C₂	1	1	yes	maximal, characteristic, complemented
6	#21462049	C₂	1	1	yes	maximal, characteristic, complemented
7	#21462046	C₂	1	1	yes	maximal, characteristic, complemented
8	#21462028	C₂	1	1	yes	maximal, characteristic, complemented
9	#21454514	C₂	1	1	yes	maximal, characteristic
10	#16840820	—	2	4	no	—
11	C₂×#14519851	C₂²	1	1	yes	characteristic
12	C₂×#14519851	C₂²	1	1	yes	characteristic
13	C₂×#14519851	C₂²	1	2	yes	complemented
14	C₂²×#11758187	—	2	4	no	—
15	C₂²×#11758187	C₂²	1	1	yes	characteristic
16	C₂²×#11758187	C₂²	1	2	yes	—
17	C₂²×#11758187	C₂²	1	2	yes	—
18	C₂²×#11758187	C₂²	1	1	yes	characteristic
19	C₂²×#11758187	C₂²	1	2	yes	complemented
20	C₄⋊#11758187	C₂²	1	4	yes	complemented
21	C₂²⋊#11758187	C₂²	1	4	yes	—
22	C₂²⋊#11758187	C₂²	1	4	yes	complemented
23	#16840650	C₂²	1	2	yes	complemented
24	C₄×#11758187	C₂²	1	2	yes	complemented
25	C₂×#14519784	C₂²	1	1	yes	characteristic
26	C₂³×#92418	C₂²	1	1	yes	characteristic
27	#16839834	C₄	1	4	yes	—
28	D₄×#92418	C₂²	1	4	yes	—
29	#16839789	C₄	1	1	yes	characteristic
30	#16839729	C₄	1	1	yes	characteristic
31	#16839726	C₄	1	1	yes	characteristic
32	#16839718	C₂²	1	1	yes	characteristic
33	C₆₁×#525	C₄	1	1	yes	characteristic
34	#14519851	—	2	2	no	—
35	#14519851	—	2	2	no	—
36	#14519851	—	2	4	no	—
37	C₂×#11758187	—	2	2	no	—
38	C₂×#11758187	—	2	4	no	—
39	C₂×#11758187	—	2	4	no	—
40	C₂×#11758187	—	2	4	no	—
41	C₂×#11758187	D₄	1	4	yes	complemented
42	C₂×#11758187	C₂³	1	2	yes	—
43	C₂×#11758187	C₂³	1	2	yes	—
44	C₂×#11758187	C₂³	1	2	yes	—
45	C₂×#11758187	C₂³	1	1	yes	characteristic
46	C₂×#11758187	C₂³	1	1	yes	characteristic
47	#14519784	—	2	2	no	—
48	#14519784	—	2	2	no	—
49	C₂×#11758194	C₂×C₄	1	1	yes	characteristic
50	C₂×#11758194	C₂×C₄	1	1	yes	characteristic
51	C₂×#11758194	C₂×C₄	1	2	yes	—
52	C₂²×#92418	—	2	2	no	—
53	C₂²×#92418	—	2	2	no	—
54	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
55	C₂²×#92418	C₂×C₄	1	2	yes	—
56	C₂²×#92418	C₂³	1	2	yes	—
57	C₂²×#92418	C₂³	1	2	yes	—
58	C₂²×#92418	C₂³	1	1	yes	characteristic
59	C₄⋊#92418	C₂×C₄	1	4	yes	—
60	C₂²⋊#92418	C₂×C₄	1	4	yes	—
61	C₂²⋊#92418	C₂×C₄	1	4	yes	—
62	#14519688	C₂×C₄	1	2	yes	—
63	C₄×#92418	C₂³	1	2	yes	—
64	C₂×#41×C₆₁	C₂×C₄	1	1	yes	characteristic
65	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
66	#123×C₆₁	C₈	1	2	yes	—
67	C₂²⋊Q₈×C₆₁	C₈	1	2	yes	—
68	#121×C₆₁	C₈	1	2	yes	—
69	#120×C₆₁	C₈	1	2	yes	—
70	C₄×D₄×C₆₁	C₂×C₄	1	4	yes	—
71	#117×C₆₁	C₂×C₄	1	1	yes	characteristic
72	C₂⁴×C₆₁	C₂×C₈	1	1	yes	characteristic
73	#41×C₆₁	—	2	2	no	—
74	#41×C₆₁	—	2	2	no	—
75	C₂×Q₈×C₆₁	C₂×C₈	1	1	yes	characteristic
76	C₂×D₄×C₆₁	C₂×C₈	1	4	yes	—
77	C₂×D₄×C₆₁	C₂×C₈	1	2	yes	—
78	C₂×D₄×C₆₁	C₂×C₈	1	1	yes	characteristic
79	C₂²×C₄×C₆₁	—	2	2	no	—
80	C₂²×C₄×C₆₁	—	2	2	no	—
81	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
82	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
83	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
86	C₄⋊C₄×C₆₁	C₂×C₈	1	2	yes	—
87	C₄⋊C₄×C₆₁	C₂×C₈	1	4	yes	—
88	C₂²⋊C₄×C₆₁	C₂×C₈	1	2	yes	—
89	C₂²⋊C₄×C₆₁	C₂×C₈	1	4	yes	—
90	C₂²⋊C₄×C₆₁	C₂×C₈	1	4	yes	—
91	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
92	#11758194	—	2	2	no	—
93	#11758194	—	2	2	no	—
94	#11758194	—	2	4	no	—
95	C₂×#92418	—	2	2	no	—
96	C₂×#92418	—	2	4	no	—
97	C₂×#92418	—	2	2	no	—
98	C₂×#92418	—	2	2	no	—
99	C₂×#92418	—	2	2	no	—
100	C₂×#92418	—	2	2	no	—
101	C₂×#92418	C₂²×C₄	1	2	yes	—
102	C₂×#92418	C₂²×C₄	1	2	yes	—
103	C₂×#92418	C₂²×C₄	1	2	yes	—
104	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
105	C₂×#92418	C₂×D₄	1	2	yes	—
106	C₂×#92418	#41	1	2	yes	—
107	C₂×#92418	C₂⁴	1	1	yes	characteristic
108	#11758187	—	2	2	no	—
109	#11758187	—	2	2	no	—
110	#11758187	—	2	2	no	—
111	#11758187	C₂×D₄	1	2	yes	—
112	#11758187	C₂×D₄	1	2	yes	—
113	C₂³×C₆₁	—	2	2	no	—
114	C₂³×C₆₁	—	2	4	no	—
115	C₂³×C₆₁	C₂²×C₈	1	2	yes	—
116	C₂³×C₆₁	C₂²×C₈	1	1	yes	characteristic
117	C₂³×C₆₁	C₂²×C₈	1	2	yes	—
118	Q₈×C₆₁	—	2	2	no	—
119	D₄×C₆₁	—	2	4	no	—
120	D₄×C₆₁	—	2	8	no	—
121	D₄×C₆₁	—	2	2	no	—
122	C₂×C₄×C₆₁	—	2	2	no	—
123	C₂×C₄×C₆₁	—	2	2	no	—
124	C₂×C₄×C₆₁	—	2	2	no	—
125	C₂×C₄×C₆₁	—	2	2	no	—
126	C₂×C₄×C₆₁	—	2	2	no	—
127	C₂×C₄×C₆₁	—	2	4	no	—
128	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
129	C₂×C₄×C₆₁	C₂²×C₈	1	4	yes	—
130	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
131	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
132	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
133	C₂×C₄×C₆₁	#131	1	2	yes	—
134	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
135	#92418	—	2	2	no	—
136	#92418	—	2	2	no	—
137	#92418	—	2	2	no	—
138	#92418	C₄×D₄	1	2	yes	—
139	#92418	C₂²×D₄	1	1	yes	characteristic
140	#92418	C₂×#41	1	1	yes	characteristic
141	#33432	—	61	1	no	maximal, Sylow
142	C₂²×C₆₁	—	2	8	no	—
143	C₂²×C₆₁	—	2	2	no	—
144	C₂²×C₆₁	—	2	2	no	—
145	C₂²×C₆₁	—	2	2	no	—
146	C₂²×C₆₁	—	2	4	no	—
147	C₂²×C₆₁	—	2	2	no	—
148	C₂²×C₆₁	C₈×D₄	1	2	yes	—
149	C₂²×C₆₁	#504	1	4	yes	complemented
150	C₂²×C₆₁	C₂³×C₈	1	1	yes	characteristic
151	C₄×C₆₁	—	2	4	no	—
152	C₄×C₆₁	—	2	2	no	—
153	C₄×C₆₁	—	2	2	no	—
154	C₄×C₆₁	C₈×D₄	1	2	yes	—
155	C₄×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
156	C₄×C₆₁	C₂×#131	1	1	yes	characteristic
157	C₂×#504	—	61	1	no	—
158	C₂³×C₁₆	—	61	1	no	—
159	#2929	—	61	1	no	—
160	#2168	—	61	4	no	—
161	D₄×C₁₆	—	61	4	no	—
162	#2151	—	61	1	no	—
163	#2113	—	61	1	no	—
164	#2112	—	61	1	no	—
165	#2108	—	61	1	no	—
166	C₂×C₆₁	—	2	2	no	—
167	C₂×C₆₁	—	2	4	no	—
168	C₂×C₆₁	—	4	2	no	—
169	C₂×C₆₁	C₂×C₈×D₄	1	1	yes	characteristic
170	C₂×C₆₁	C₂×#504	1	2	yes	—
171	C₂×#131	—	61	1	no	—
172	C₂³×C₈	—	61	1	no	—
173	#525	#6923	1	1	yes	characteristic
174	#504	—	122	4	no	—
175	C₂×M₃₂	—	61	1	no	—
176	C₂×M₃₂	—	61	1	no	—
177	C₂×M₃₂	—	61	2	no	—
178	C₂²×C₁₆	—	61	1	no	—
179	C₂²×C₁₆	—	61	1	no	—
180	C₂²×C₁₆	—	61	2	no	—
181	C₂²×C₁₆	—	61	2	no	—
182	C₂²×C₁₆	—	61	2	no	—
183	C₂²×C₁₆	—	122	4	no	—
184	#435	—	61	4	no	—
185	C₈×D₄	—	61	4	no	—
186	#424	—	61	1	no	—
187	#408	—	61	1	no	—
188	#407	—	61	1	no	—
189	#405	—	61	1	no	—
190	C₄⋊C₁₆	—	61	4	no	—
191	C₂²⋊C₁₆	—	61	4	no	—
192	C₂²⋊C₁₆	—	61	4	no	—
193	#346	—	61	2	no	—
194	C₄×C₁₆	—	61	2	no	—
195	C₆₁	#33432	1	1	yes	characteristic, complemented, Sylow
196	C₂×#41	C₂×#6923	1	1	yes	characteristic
197	C₂³×C₄	C₂×#6923	1	1	yes	characteristic
198	#131	—	122	2	no	—
199	#131	—	122	2	no	—
200	C₂×M₁₆	—	61	1	no	—
201	C₂×M₁₆	—	61	2	no	—
202	C₂×M₁₆	—	61	1	no	—
203	C₂²×C₈	—	61	1	no	—
204	C₂²×C₈	—	61	2	no	—
205	C₂²×C₈	—	61	1	no	—
206	C₂²×C₈	—	61	2	no	—
207	C₂²×C₈	—	61	2	no	—
208	C₂²×C₈	—	122	2	no	—
209	C₂²×C₈	—	122	2	no	—
210	#123	#92420	1	2	yes	—
211	C₂²⋊Q₈	#92420	1	2	yes	—
212	#121	#92420	1	2	yes	—
213	#120	#92420	1	2	yes	—
214	C₄×D₄	C₂×#6923	1	4	yes	—
215	#117	C₂×#6923	1	1	yes	characteristic
216	M₃₂	—	122	4	no	—
217	M₃₂	—	122	2	no	—
218	M₃₂	—	122	2	no	—
219	C₂×C₁₆	—	61	4	no	—
220	C₂×C₁₆	—	61	1	no	—
221	C₂×C₁₆	—	61	1	no	—
222	C₂×C₁₆	—	61	2	no	—
223	C₂×C₁₆	—	61	2	no	—
224	C₂×C₁₆	—	61	2	no	—
225	C₂×C₁₆	—	122	4	no	—
226	C₂×C₁₆	—	122	4	no	—
227	C₂×C₁₆	—	122	4	no	—
228	C₂×C₁₆	—	122	2	no	—
229	C₄⋊C₈	—	61	4	no	—
230	C₂²⋊C₈	—	61	4	no	—
231	C₂²⋊C₈	—	61	4	no	—
232	#97	—	61	2	no	—
233	C₄×C₈	—	61	2	no	—
234	C₂⁴	C₂×#92420	1	1	yes	characteristic
235	#41	—	2	2	no	—
236	#41	—	2	2	no	—
237	C₂×Q₈	C₂×#92420	1	1	yes	characteristic
238	C₂×D₄	C₂×#92420	1	4	yes	—
239	C₂×D₄	C₂×#92420	1	2	yes	—
240	C₂×D₄	C₂×#92420	1	1	yes	characteristic
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	C₂×#92420	1	2	yes	—
244	C₂²×C₄	C₂×#92420	1	1	yes	characteristic
245	C₂²×C₄	C₂²×#6923	1	2	yes	—
246	C₂²×C₄	C₂²×#6923	1	2	yes	—
247	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
248	M₁₆	—	122	4	no	—
249	M₁₆	—	122	2	no	—
250	M₁₆	—	122	2	no	—
251	C₂×C₈	—	61	2	no	—
252	C₂×C₈	—	61	2	no	—
253	C₂×C₈	—	61	1	no	—
254	C₂×C₈	—	61	1	no	—
255	C₂×C₈	—	61	2	no	—
256	C₂×C₈	—	61	2	no	—
257	C₂×C₈	—	61	2	no	—
258	C₂×C₈	—	122	2	no	—
259	C₂×C₈	—	122	2	no	—
260	C₂×C₈	—	122	4	no	—
261	C₂×C₈	—	122	2	no	—
262	C₂×C₈	—	122	2	no	—
263	C₂×C₈	—	122	2	no	—
264	C₄⋊C₄	C₂×#92420	1	2	yes	—
265	C₄⋊C₄	C₂×#92420	1	4	yes	—
266	C₂²⋊C₄	C₂×#92420	1	2	yes	—
267	C₂²⋊C₄	C₂×#92420	1	4	yes	—
268	C₂²⋊C₄	C₂×#92420	1	4	yes	—
269	C₄²	C₂²×#6923	1	2	yes	—
270	C₁₆	—	61	2	no	—
271	C₁₆	—	61	2	no	—
272	C₁₆	—	122	2	no	—
273	C₁₆	—	122	2	no	—
274	C₁₆	—	122	2	no	—
275	C₂³	—	2	2	no	—
276	C₂³	—	2	4	no	—
277	C₂³	C₂²×#92420	1	2	yes	—
278	C₂³	C₂²×#92420	1	1	yes	characteristic
279	C₂³	C₂²×#92420	1	2	yes	—
280	Q₈	—	2	2	no	—
281	D₄	—	2	4	no	—
282	D₄	—	2	8	no	—
283	D₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	D₄×#6923	1	2	yes	—
291	C₂×C₄	C₂²×#92420	1	4	yes	—
292	C₂×C₄	C₂²×#92420	1	2	yes	—
293	C₂×C₄	C₂²×#92420	1	2	yes	—
294	C₂×C₄	C₂²×#92420	1	1	yes	characteristic
295	C₂×C₄	#14519864	1	2	yes	—
296	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
297	C₈	—	61	2	no	—
298	C₈	—	61	1	no	—
299	C₈	—	61	1	no	—
300	C₈	—	122	2	no	—
301	C₈	—	122	2	no	—
302	C₈	—	122	2	no	—
303	C₂²	—	2	8	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	4	no	—
309	C₂²	D₄×#92420	1	2	yes	—
310	C₂²	#16840820	1	4	yes	complemented
311	C₂²	C₂³×#92420	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	—	2	2	no	—
314	C₄	—	2	4	no	—
315	C₄	D₄×#92420	1	2	yes	—
316	C₄	C₂×D₄×#6923	1	1	yes	characteristic
317	C₄	C₂×#14519864	1	1	yes	characteristic
318	C₂	—	2	2	no	—
319	C₂	—	2	4	no	—
320	C₂	—	4	2	no	—
321	C₂	C₂×D₄×#92420	1	1	yes	characteristic
322	C₂	C₂×#16840820	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#6923
commutator, socle	C₂²×C₆₁	C₂³×C₈
Fitting, prime radical	C₆₁×#525	C₄
cosocle	C₂×#92418	C₂⁴
hypercenter	#525	#6923
hypocenter	C₆₁	#33432
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ #525

Complements

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

N	H ≅ G/N	type
1. C₂×#16840820	C₂	semidirect
2. C₂³×#11758187	C₂	semidirect
3. #21462199	C₂	semidirect
4. D₄×#11758187	C₂	semidirect
5. #21462154	C₂	semidirect
6. #21462049	C₂	semidirect
7. #21462046	C₂	semidirect
8. #21462028	C₂	semidirect
13. C₂×#14519851	C₂²	semidirect
19. C₂²×#11758187	C₂²	semidirect
20. C₄⋊#11758187	C₂²	semidirect
22. C₂²⋊#11758187	C₂²	semidirect
23. #16840650	C₂²	semidirect
24. C₄×#11758187	C₂²	semidirect
41. C₂×#11758187	D₄	semidirect
149. C₂²×C₆₁	#504	semidirect
195. C₆₁	#33432	semidirect
310. C₂²	#16840820	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16840820	2
C₂³×#11758187	2
#21462199	2
D₄×#11758187	2
#21462154	2
#21462049	2
#21462046	2
#21462028	2
#21454514	2
#33432	61