Subgroups — #30624178

Order 15616.

590 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21462184	C₂	1	4	yes	maximal, complemented
2	#21462180	C₂	1	2	yes	maximal, complemented
3	#21462176	C₂	1	2	yes	maximal, complemented
4	#21462151	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄⋊#11758187	C₂	1	1	yes	maximal, characteristic, complemented
6	#21462049	C₂	1	2	yes	maximal
7	#21462028	C₂	1	2	yes	maximal, complemented
8	C₄×#14519784	C₂	1	1	yes	maximal, characteristic
9	C₂×#14519851	C₂²	1	2	yes	—
10	C₂²×#11758187	C₂²	1	2	yes	—
11	C₄⋊#11758187	C₂²	1	1	yes	characteristic
12	C₄⋊#11758187	C₂²	1	1	yes	characteristic
13	C₄⋊#11758187	C₂²	1	2	yes	—
14	C₄⋊#11758187	C₂²	1	2	yes	—
15	C₄⋊#11758187	C₂²	1	2	yes	—
16	C₂²⋊#11758187	C₂²	1	4	yes	—
17	C₂²⋊#11758187	C₂²	1	4	yes	—
18	#16840650	C₂²	1	4	yes	—
19	C₄×#11758187	C₂²	1	4	yes	—
20	C₂×#14519784	C₄	1	1	yes	characteristic
21	C₂×#14519784	C₂²	1	1	yes	characteristic
22	#16839718	C₄	1	2	yes	—
23	#16839718	C₄	1	1	yes	characteristic
24	#16839718	C₂²	1	1	yes	characteristic
25	#16839718	C₂²	1	2	yes	—
26	C₄×#11758194	C₄	1	2	yes	—
27	C₄×#11758194	C₄	1	1	yes	characteristic
28	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
29	C₂×C₄×#92418	C₂²	1	2	yes	—
30	C₄×#41×C₆₁	C₄	1	1	yes	characteristic
31	#14519851	—	2	4	no	—
32	C₂×#11758187	—	2	4	no	—
33	C₂×#11758187	C₂³	1	2	yes	—
34	C₂×#11758187	C₂³	1	2	yes	—
35	C₂×#11758187	C₂³	1	4	yes	—
36	#14519784	—	2	4	no	—
37	#14519784	—	2	2	no	—
38	#14519784	—	2	2	no	—
39	C₂×#11758194	C₂×C₄	1	1	yes	characteristic
40	C₂×#11758194	C₂×C₄	1	2	yes	—
41	C₂×#11758194	C₂×C₄	1	2	yes	—
42	C₂×#11758194	C₂×C₄	1	1	yes	characteristic
43	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
44	C₂²×#92418	C₂×C₄	1	2	yes	—
45	C₂²×#92418	C₂³	1	2	yes	—
46	C₂²×#92418	C₂³	1	1	yes	characteristic
47	#14519688	C₂×C₄	1	2	yes	—
48	#14519688	C₂×C₄	1	2	yes	—
49	#14519688	C₂×C₄	1	2	yes	—
50	C₄×#92418	C₂×C₄	1	2	yes	—
51	C₄×#92418	C₂×C₄	1	1	yes	characteristic
52	C₄×#92418	C₂×C₄	1	2	yes	—
53	C₄×#92418	C₂×C₄	1	1	yes	characteristic
54	C₄×#92418	C₂³	1	1	yes	characteristic
55	C₄×#92418	C₂³	1	2	yes	—
56	C₄×#92418	C₂³	1	1	yes	characteristic
57	C₂×#41×C₆₁	C₂×C₄	1	1	yes	characteristic
58	C₄×Q₈×C₆₁	C₈	1	1	yes	characteristic
59	C₄×Q₈×C₆₁	C₈	1	1	yes	characteristic
60	C₄×D₄×C₆₁	C₈	1	2	yes	—
61	C₄×D₄×C₆₁	C₈	1	1	yes	characteristic
62	C₄×D₄×C₆₁	C₈	1	2	yes	—
63	C₄×D₄×C₆₁	C₈	1	1	yes	characteristic
64	#117×C₆₁	C₂×C₄	1	2	yes	—
65	#117×C₆₁	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
67	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
68	#41×C₆₁	—	2	2	no	—
69	#41×C₆₁	—	2	2	no	—
70	C₂×Q₈×C₆₁	C₂×C₈	1	1	yes	characteristic
71	C₂×D₄×C₆₁	C₂×C₈	1	2	yes	—
72	C₂×D₄×C₆₁	C₂×C₈	1	1	yes	characteristic
73	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
74	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
75	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
76	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
77	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
79	C₄⋊C₄×C₆₁	C₂×C₈	1	2	yes	—
80	C₄⋊C₄×C₆₁	C₂×C₈	1	2	yes	—
81	C₄⋊C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
82	C₄⋊C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
83	C₂²⋊C₄×C₆₁	C₂×C₈	1	2	yes	—
84	C₂²⋊C₄×C₆₁	C₂×C₈	1	2	yes	—
85	C₂²⋊C₄×C₆₁	C₂×C₈	1	2	yes	—
86	C₄²×C₆₁	C₂×C₈	1	2	yes	—
87	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
88	C₄²×C₆₁	C₂×C₈	1	2	yes	—
89	C₄²×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	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
93	#11758194	—	2	2	no	—
94	#11758194	—	2	4	no	—
95	#11758194	—	2	4	no	—
96	#11758194	M₁₆	1	4	yes	—
97	C₂×#92418	—	2	2	no	—
98	C₂×#92418	—	2	4	no	—
99	C₂×#92418	—	2	2	no	—
100	C₂×#92418	—	2	2	no	—
101	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
102	C₂×#92418	C₂²×C₄	1	2	yes	—
103	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
104	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
105	C₂×#92418	C₂²×C₄	1	2	yes	—
106	C₂×#92418	#41	1	2	yes	—
107	C₂×#92418	#41	1	2	yes	—
108	C₂×#92418	C₂⁴	1	1	yes	characteristic
109	#11758187	—	2	2	no	—
110	#11758187	—	2	2	no	—
111	#11758187	—	2	4	no	—
112	C₂³×C₆₁	C₂²×C₈	1	2	yes	—
113	C₂³×C₆₁	C₂²×C₈	1	1	yes	characteristic
114	Q₈×C₆₁	—	2	2	no	—
115	D₄×C₆₁	—	2	4	no	—
116	D₄×C₆₁	—	2	2	no	—
117	C₂×C₄×C₆₁	—	2	2	no	—
118	C₂×C₄×C₆₁	—	2	2	no	—
119	C₂×C₄×C₆₁	—	2	2	no	—
120	C₂×C₄×C₆₁	—	2	4	no	—
121	C₂×C₄×C₆₁	—	4	1	no	—
122	C₂×C₄×C₆₁	—	4	2	no	—
123	C₂×C₄×C₆₁	#117	1	2	yes	—
124	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
125	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
126	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
127	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
128	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
129	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
130	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
131	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
132	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
133	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
134	#92418	—	2	2	no	—
135	#92418	—	2	1	no	—
136	#92418	—	2	1	no	—
137	#92418	#117	1	2	yes	—
138	#92418	C₂×M₁₆	1	4	yes	—
139	#92418	C₂×#41	1	1	yes	characteristic
140	#92418	C₂×#41	1	1	yes	characteristic
141	#33423	—	61	1	no	maximal, Sylow
142	C₂²×C₆₁	—	2	1	no	—
143	C₂²×C₆₁	—	2	1	no	—
144	C₂²×C₆₁	—	2	2	no	—
145	C₂²×C₆₁	—	2	2	no	—
146	C₂²×C₆₁	—	2	2	no	—
147	C₂²×C₆₁	#431	1	2	yes	—
148	C₂²×C₆₁	C₂³×C₈	1	1	yes	characteristic
149	C₄×C₆₁	—	2	1	no	—
150	C₄×C₆₁	—	2	2	no	—
151	C₄×C₆₁	—	2	1	no	—
152	C₄×C₆₁	—	2	1	no	—
153	C₄×C₆₁	—	2	2	no	—
154	C₄×C₆₁	—	2	2	no	—
155	C₄×C₆₁	#431	1	2	yes	—
156	C₄×C₆₁	#504	1	2	yes	—
157	C₄×C₆₁	C₂×#117	1	1	yes	characteristic
158	C₄×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
159	C₄×#131	—	61	1	no	—
160	#2164	—	61	4	no	—
161	#2163	—	61	2	no	—
162	#2162	—	61	2	no	—
163	#2150	—	61	1	no	—
164	C₂×C₄⋊C₁₆	—	61	1	no	—
165	#2113	—	61	2	no	—
166	#2108	—	61	2	no	—
167	C₂×C₆₁	—	2	2	no	—
168	C₂×C₆₁	—	4	2	no	—
169	C₂×C₆₁	C₂×#431	1	1	yes	characteristic
170	C₂×C₆₁	C₂×#504	1	1	yes	characteristic
171	C₂×C₆₁	#3407	1	1	yes	characteristic
172	C₂×#131	—	61	1	no	—
173	C₂×#131	—	61	1	no	—
174	C₄×#41	#6923	1	1	yes	characteristic
175	C₂×M₃₂	—	61	2	no	—
176	C₂²×C₁₆	—	61	2	no	—
177	#405	—	61	2	no	—
178	#405	—	61	1	no	—
179	#405	—	61	1	no	—
180	#405	—	61	2	no	—
181	C₄×M₁₆	—	61	1	no	—
182	C₄×M₁₆	—	61	2	no	—
183	C₂×C₄×C₈	—	61	1	no	—
184	C₂×C₄×C₈	—	61	2	no	—
185	C₄⋊C₁₆	—	61	1	no	—
186	C₄⋊C₁₆	—	61	1	no	—
187	C₄⋊C₁₆	—	61	2	no	—
188	C₄⋊C₁₆	—	61	2	no	—
189	C₄⋊C₁₆	—	61	2	no	—
190	C₂²⋊C₁₆	—	61	4	no	—
191	C₂²⋊C₁₆	—	61	4	no	—
192	#346	—	61	4	no	—
193	C₄×C₁₆	—	61	4	no	—
194	C₆₁	#33423	1	1	yes	characteristic, complemented, Sylow
195	C₂×#41	C₂×#6923	1	1	yes	characteristic
196	#131	—	122	2	no	—
197	#131	—	122	2	no	—
198	#131	—	122	4	no	—
199	C₂×M₁₆	—	61	1	no	—
200	C₂×M₁₆	—	61	2	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	2	no	—
206	C₂²×C₈	—	61	1	no	—
207	C₄×Q₈	#92420	1	1	yes	characteristic
208	C₄×Q₈	#92420	1	1	yes	characteristic
209	C₄×D₄	#92420	1	2	yes	—
210	C₄×D₄	#92420	1	1	yes	characteristic
211	C₄×D₄	#92420	1	2	yes	—
212	C₄×D₄	#92420	1	1	yes	characteristic
213	#117	C₂×#6923	1	2	yes	—
214	#117	C₂×#6923	1	1	yes	characteristic
215	C₂×C₄²	C₂×#6923	1	2	yes	—
216	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
217	M₃₂	—	122	4	no	—
218	C₂×C₁₆	—	61	2	no	—
219	C₂×C₁₆	—	61	4	no	—
220	C₂×C₁₆	—	61	2	no	—
221	C₂×C₁₆	—	122	4	no	—
222	#97	—	61	2	no	—
223	#97	—	61	2	no	—
224	#97	—	61	2	no	—
225	C₄×C₈	—	61	2	no	—
226	C₄×C₈	—	61	1	no	—
227	C₄×C₈	—	61	1	no	—
228	C₄×C₈	—	61	1	no	—
229	C₄×C₈	—	61	2	no	—
230	C₄×C₈	—	61	1	no	—
231	C₄×C₈	—	61	2	no	—
232	#41	—	2	2	no	—
233	#41	—	2	2	no	—
234	C₂×Q₈	C₂×#92420	1	1	yes	characteristic
235	C₂×D₄	C₂×#92420	1	2	yes	—
236	C₂×D₄	C₂×#92420	1	1	yes	characteristic
237	C₂²×C₄	C₂×#92420	1	2	yes	—
238	C₂²×C₄	C₂×#92420	1	1	yes	characteristic
239	C₂²×C₄	C₂×#92420	1	2	yes	—
240	C₂²×C₄	C₂×#92420	1	1	yes	characteristic
241	C₂²×C₄	C₂²×#6923	1	2	yes	—
242	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
243	M₁₆	—	61	4	no	—
244	M₁₆	—	122	4	no	—
245	M₁₆	—	122	4	no	—
246	M₁₆	—	122	2	no	—
247	C₂×C₈	—	61	2	no	—
248	C₂×C₈	—	61	2	no	—
249	C₂×C₈	—	61	1	no	—
250	C₂×C₈	—	61	1	no	—
251	C₂×C₈	—	61	2	no	—
252	C₂×C₈	—	61	1	no	—
253	C₂×C₈	—	61	2	no	—
254	C₂×C₈	—	61	1	no	—
255	C₂×C₈	—	122	2	no	—
256	C₂×C₈	—	122	2	no	—
257	C₂×C₈	—	122	4	no	—
258	C₂×C₈	—	122	2	no	—
259	C₄⋊C₄	C₂×#92420	1	2	yes	—
260	C₄⋊C₄	C₂×#92420	1	2	yes	—
261	C₄⋊C₄	C₂×#92420	1	1	yes	characteristic
262	C₄⋊C₄	C₂×#92420	1	1	yes	characteristic
263	C₂²⋊C₄	C₂×#92420	1	2	yes	—
264	C₂²⋊C₄	C₂×#92420	1	2	yes	—
265	C₂²⋊C₄	C₂×#92420	1	2	yes	—
266	C₄²	C₂×#92420	1	2	yes	—
267	C₄²	C₂×#92420	1	1	yes	characteristic
268	C₄²	C₂×#92420	1	2	yes	—
269	C₄²	C₂×#92420	1	1	yes	characteristic
270	C₄²	C₂²×#6923	1	1	yes	characteristic
271	C₄²	C₂²×#6923	1	2	yes	—
272	C₄²	C₂²×#6923	1	1	yes	characteristic
273	C₁₆	—	122	2	no	—
274	C₁₆	—	122	2	no	—
275	C₁₆	—	122	4	no	—
276	C₂³	C₂²×#92420	1	2	yes	—
277	C₂³	C₂²×#92420	1	1	yes	characteristic
278	Q₈	—	2	2	no	—
279	D₄	—	2	4	no	—
280	D₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	4	1	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	#14519856	1	2	yes	—
288	C₂×C₄	C₂²×#92420	1	1	yes	characteristic
289	C₂×C₄	C₂²×#92420	1	2	yes	—
290	C₂×C₄	C₂²×#92420	1	2	yes	—
291	C₂×C₄	C₂²×#92420	1	1	yes	characteristic
292	C₂×C₄	C₂²×#92420	1	2	yes	—
293	C₂×C₄	C₂²×#92420	1	1	yes	characteristic
294	C₂×C₄	C₂²×#92420	1	1	yes	characteristic
295	C₂×C₄	C₂²×#92420	1	1	yes	characteristic
296	C₂×C₄	D₆₁⋊M₁₆	1	2	yes	—
297	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
298	C₈	—	61	2	no	—
299	C₈	—	61	1	no	—
300	C₈	—	61	1	no	—
301	C₈	—	61	4	no	—
302	C₈	—	122	2	no	—
303	C₈	—	122	1	no	—
304	C₈	—	122	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	2	2	no	—
310	C₂²	#16840778	1	2	yes	—
311	C₂²	C₂³×#92420	1	1	yes	characteristic
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	2	no	—
315	C₄	—	2	2	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	2	no	—
318	C₄	#16840778	1	2	yes	—
319	C₄	#16840820	1	2	yes	—
320	C₄	C₂×#14519856	1	1	yes	characteristic
321	C₄	C₂×D₆₁⋊M₁₆	1	1	yes	characteristic
322	C₂	—	2	2	no	—
323	C₂	—	4	2	no	—
324	C₂	C₂×#16840778	1	1	yes	characteristic
325	C₂	C₂×#16840820	1	1	yes	characteristic
326	C₂	#21462786	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#6923
commutator, socle	C₂²×C₆₁	C₂³×C₈
Fitting	C₄×#41×C₆₁	C₄
prime radical	C₂×#41×C₆₁	C₂×C₄
cosocle	C₂×#92418	C₂⁴
hypercenter	C₄×#41	#6923
hypocenter	C₆₁	#33423
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. #21462184	C₂	semidirect
2. #21462180	C₂	semidirect
3. #21462176	C₂	semidirect
4. #21462151	C₂	semidirect
5. C₂×C₄⋊#11758187	C₂	semidirect
7. #21462028	C₂	semidirect
194. C₆₁	#33423	semidirect

Maximal subgroups

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

subgroup	index
#21462184	2
#21462180	2
#21462176	2
#21462151	2
C₂×C₄⋊#11758187	2
#21462049	2
#21462028	2
C₄×#14519784	2
#33423	61