Subgroups — #72310383

Order 4096.

At least 10871 conjugacy classes of subgroups; only the 465 normal subgroups are shown.

The subgroup lattice has 319 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72310380	C₂	1	1	yes	maximal, characteristic, complemented
2	#72310339	C₂	1	2	yes	maximal, complemented
3	#72310332	C₂	1	1	yes	maximal, characteristic, complemented
4	#72310299	C₂	1	1	yes	maximal, characteristic, complemented
5	#72309591	C₂	1	1	yes	maximal, characteristic, complemented
6	#72309452	C₂	1	2	yes	maximal, complemented
7	#72309379	C₂	1	1	yes	maximal, characteristic, complemented
8	#72266875	C₂	1	1	yes	maximal, characteristic, complemented
9	#72266073	C₂²	1	1	yes	characteristic
10	#72266028	C₂²	1	1	yes	characteristic
11	#72264784	C₂	1	2	yes	maximal, complemented
12	#72262357	C₂	1	2	yes	maximal, complemented
13	#72251218	C₂	1	1	yes	maximal, characteristic
14	#72251171	C₄	1	1	yes	characteristic, complemented
15	#72238613	C₄	1	2	yes	complemented
16	#72238612	C₄	1	1	yes	characteristic, complemented
17	#72237900	C₄	1	2	yes	complemented
18	#72237899	C₄	1	1	yes	characteristic, complemented
19	#72237829	C₄	1	1	yes	characteristic, complemented
20	#72237398	C₂²	1	1	yes	characteristic
21	#72234663	C₂²	1	1	yes	characteristic
22	#72225667	C₂²	1	1	yes	characteristic
23	#72225619	C₂²	1	1	yes	characteristic
24	#72225617	C₂²	1	2	yes	—
25	#72219418	C₂²	1	1	yes	characteristic
26	#72219418	C₂²	1	1	yes	characteristic
27	#72219418	C₂²	1	2	yes	—
28	#72219301	C₂²	1	2	yes	—
29	#72219248	C₂²	1	2	yes	—
30	#72219237	C₂²	1	2	yes	—
31	#72219229	C₂²	1	1	yes	characteristic
32	#72219228	C₂²	1	1	yes	characteristic
33	C₂²×#8534	C₂²	1	1	yes	characteristic
34	#72218498	C₂²	1	2	yes	—
35	#72218381	C₂²	1	2	yes	—
36	#72218381	C₂²	1	2	yes	—
37	#72218180	C₂²	1	2	yes	—
38	#72218057	C₂²	1	2	yes	—
39	#72218057	C₂²	1	2	yes	—
40	#72213862	C₂²	1	2	yes	—
41	C₂²×#3484	C₂³	1	1	yes	characteristic
42	#7630939	C₂³	1	2	yes	—
43	#7626202	D₄	1	1	yes	characteristic
44	#7626202	D₄	1	1	yes	characteristic
45	C₂×#32873	C₂×C₄	1	1	yes	characteristic, complemented
46	C₂×#23895	C₂×C₄	1	1	yes	characteristic, complemented
47	C₂×#23895	D₄	1	2	yes	—
48	C₂×#23895	D₄	1	2	yes	—
49	C₂×#23895	D₄	1	2	yes	—
50	C₂×#23895	D₄	1	2	yes	—
51	#6437621	C₂×C₄	1	2	yes	—
52	#6437621	C₂×C₄	1	2	yes	—
53	#6374251	C₂×C₄	1	1	yes	characteristic
54	#6374251	C₂×C₄	1	1	yes	characteristic
55	C₂×#21776	C₂×C₄	1	2	yes	complemented
56	#6374128	C₂×C₄	1	1	yes	characteristic
57	#6374128	C₂×C₄	1	1	yes	characteristic
58	C₂×#21746	D₄	1	1	yes	characteristic
59	C₂×#21746	D₄	1	1	yes	characteristic
60	#6371007	D₄	1	1	yes	characteristic
61	#6371007	D₄	1	1	yes	characteristic
62	#6370693	D₄	1	1	yes	characteristic
63	#6370693	D₄	1	1	yes	characteristic
64	C₂²×#1781	C₂³	1	1	yes	characteristic
65	C₂²×#1781	C₂³	1	1	yes	characteristic
66	C₂²×#1781	C₂³	1	1	yes	characteristic
67	C₂²×#1781	C₂³	1	1	yes	characteristic
68	#517411	C₂³	1	1	yes	characteristic
69	C₂²×#1438	C₂³	1	1	yes	characteristic
70	#392109	C₂³	1	2	yes	—
71	#392109	C₂³	1	2	yes	—
72	#392109	C₂³	1	2	yes	—
73	#392030	C₂×C₄	1	1	yes	characteristic, complemented
74	#392030	C₂×C₄	1	2	yes	complemented
75	#382544	C₂×C₄	1	2	yes	complemented
76	#382434	C₂×C₄	1	2	yes	complemented
77	#373637	C₂×C₄	1	1	yes	characteristic, complemented
78	#359549	C₂×C₄	1	2	yes	complemented
79	#359549	D₄	1	2	yes	—
80	#359549	D₄	1	2	yes	—
81	#359549	D₄	1	2	yes	—
82	#359549	D₄	1	2	yes	—
83	#359442	C₂×C₄	1	2	yes	—
84	#359442	C₂×C₄	1	2	yes	—
85	#355096	C₂×C₄	1	2	yes	complemented
86	C₂³×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
87	C₂⁵×D₄	C₂⁴	1	1	yes	characteristic
88	C₂×#3484	C₂²×C₄	1	2	yes	—
89	C₂×#3484	C₂×D₄	1	2	yes	—
90	C₂×#3484	C₂×D₄	1	2	yes	—
91	C₂×#3484	C₂×D₄	1	2	yes	—
92	C₂×#3484	C₂×D₄	1	2	yes	—
93	C₂×#3484	C₂×D₄	1	2	yes	—
94	C₂×#3484	C₂×D₄	1	2	yes	—
95	C₂²×#534	C₂²×C₄	1	1	yes	characteristic
96	C₂²×#534	C₂²×C₄	1	1	yes	characteristic
97	C₂³×#120	C₂×D₄	1	1	yes	characteristic
98	C₂³×#120	C₂×D₄	1	1	yes	characteristic
99	C₂²×#518	C₂×D₄	1	1	yes	characteristic
100	C₂×#3026	C₂²⋊C₄	1	2	yes	—
101	C₂×#3026	C₂²⋊C₄	1	2	yes	—
102	C₂×#3026	C₂²⋊C₄	1	2	yes	—
103	C₂×#3026	C₂²⋊C₄	1	2	yes	—
104	#34176	C₂²⋊C₄	1	1	yes	characteristic
105	#34176	C₂²⋊C₄	1	1	yes	characteristic
106	#34176	C₂²⋊C₄	1	1	yes	characteristic
107	#34176	C₂²⋊C₄	1	1	yes	characteristic
108	#34171	C₂²⋊C₄	1	1	yes	characteristic
109	#34171	C₂²⋊C₄	1	1	yes	characteristic
110	#34171	C₂²⋊C₄	1	1	yes	characteristic
111	#34171	C₂²⋊C₄	1	1	yes	characteristic
112	#34171	C₂²×C₄	1	2	yes	—
113	#34171	C₂×D₄	1	2	yes	—
114	#34171	C₂×D₄	1	2	yes	—
115	C₂²×#379	C₂×D₄	1	1	yes	characteristic
116	C₂²×#379	C₂×D₄	1	1	yes	characteristic
117	C₂²×#379	C₂×D₄	1	1	yes	characteristic
118	C₂²×#379	C₂×D₄	1	1	yes	characteristic
119	C₂×#1913	C₂²⋊C₄	1	2	yes	—
120	C₂×#1913	C₂²⋊C₄	1	2	yes	—
121	C₂×#1913	C₂²⋊C₄	1	2	yes	—
122	C₂×#1913	C₂²⋊C₄	1	2	yes	—
123	C₂×#1913	C₂²⋊C₄	1	2	yes	—
124	C₂×#1913	C₂²⋊C₄	1	2	yes	—
125	C₂×#1913	C₂²⋊C₄	1	2	yes	—
126	C₂×#1913	C₂²⋊C₄	1	2	yes	—
127	#20960	C₂²⋊C₄	1	2	yes	—
128	#20960	C₂²⋊C₄	1	2	yes	—
129	#20960	C₂²⋊C₄	1	2	yes	—
130	#20960	C₂²⋊C₄	1	2	yes	—
131	C₂×#1819	C₂²⋊C₄	1	1	yes	characteristic
132	C₂×#1819	C₂²⋊C₄	1	1	yes	characteristic
133	C₂×#1819	C₂²⋊C₄	1	1	yes	characteristic
134	C₂×#1819	C₂²⋊C₄	1	1	yes	characteristic
135	C₂×#1781	C₂²⋊C₄	1	1	yes	characteristic
136	C₂×#1781	C₂²⋊C₄	1	1	yes	characteristic
137	C₂×#1781	C₂²⋊C₄	1	1	yes	characteristic
138	C₂×#1781	C₂²⋊C₄	1	1	yes	characteristic
139	C₂×#1781	C₂²×C₄	1	1	yes	characteristic
140	C₂×#1781	C₂²×C₄	1	1	yes	characteristic
141	C₂×#1781	C₂²×C₄	1	1	yes	characteristic
142	C₂×#1781	C₂²×C₄	1	1	yes	characteristic
143	C₂×#1781	C₂²×C₄	1	2	yes	—
144	C₂×#1781	C₂²×C₄	1	2	yes	—
145	C₂×#1781	C₂×D₄	1	1	yes	characteristic
146	C₂×#1781	C₂×D₄	1	1	yes	characteristic
147	C₂×#1781	C₂×D₄	1	1	yes	characteristic
148	C₂×#1781	C₂×D₄	1	1	yes	characteristic
149	C₂×#1781	C₂×D₄	1	2	yes	—
150	C₂×#1781	C₂×D₄	1	2	yes	—
151	C₂×#1781	C₂×D₄	1	2	yes	—
152	C₂×#1781	C₂×D₄	1	2	yes	—
153	C₂×#1781	#41	1	4	yes	complemented
154	C₂×#1781	#41	1	4	yes	complemented
155	C₂⁷	C₂²×D₄	1	1	yes	characteristic
156	C₂⁷	H(5,𝔽₂)	1	1	yes	characteristic
157	C₂²×H(5,𝔽₂)	C₂×C₂²⋊C₄	1	2	yes	—
158	C₂²×H(5,𝔽₂)	#120	1	2	yes	—
159	C₂²×H(5,𝔽₂)	#120	1	2	yes	—
160	C₂⁴×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
161	C₂⁴×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
162	C₂⁴×D₄	C₂³×C₄	1	1	yes	characteristic
163	C₂⁴×D₄	C₂²×D₄	1	1	yes	characteristic
164	C₂⁴×D₄	C₂²×D₄	1	2	yes	—
165	C₂⁴×D₄	C₂²×D₄	1	2	yes	—
166	C₂⁴×D₄	C₂×#41	1	2	yes	—
167	C₂⁴×D₄	C₂×#41	1	2	yes	—
168	C₂⁴×D₄	H(5,𝔽₂)	1	1	yes	characteristic
169	C₂⁴×D₄	2₋¹⁺⁴	1	1	yes	characteristic
170	C₂⁵×C₄	2₋¹⁺⁴	1	1	yes	characteristic
171	C₂×#534	C₂×C₂²⋊C₄	1	2	yes	—
172	C₂×#534	C₂×C₂²⋊C₄	1	2	yes	—
173	C₂×#534	C₂×C₂²⋊C₄	1	2	yes	—
174	C₂×#534	C₂×C₂²⋊C₄	1	2	yes	—
175	C₂²×#120	#120	1	1	yes	characteristic
176	C₂²×#120	#120	1	1	yes	characteristic
177	C₂²×#120	#120	1	2	yes	—
178	C₂²×#120	#120	1	2	yes	—
179	C₂²×#120	#120	1	1	yes	characteristic
180	C₂²×#120	#120	1	1	yes	characteristic
181	C₂×#518	C₂×C₂²⋊C₄	1	2	yes	—
182	C₂×#518	C₂×C₂²⋊C₄	1	2	yes	—
183	C₂×#518	C₂×C₂²⋊C₄	1	2	yes	—
184	C₂×#518	#120	1	2	yes	—
185	C₂×#518	#120	1	2	yes	—
186	C₂³×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
187	C₂³×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
188	C₂²×#99	C₂×C₂²⋊C₄	1	2	yes	—
189	C₂²×#99	C₂×C₂²⋊C₄	1	2	yes	—
190	C₂²×#99	C₂×C₂²⋊C₄	1	2	yes	—
191	C₂²×#99	C₂×C₂²⋊C₄	1	2	yes	—
192	#2309	#99	1	2	yes	complemented
193	#2308	#99	1	2	yes	complemented
194	#2280	#99	1	2	yes	complemented
195	C₂×#379	#99	1	2	yes	complemented
196	C₂×#379	C₂×C₂²⋊C₄	1	1	yes	characteristic
197	C₂×#379	C₂×C₂²⋊C₄	1	1	yes	characteristic
198	C₂×#379	C₂×C₂²⋊C₄	1	1	yes	characteristic
199	C₂×#379	C₂×C₂²⋊C₄	1	1	yes	characteristic
200	C₂×#379	C₂×C₂²⋊C₄	1	1	yes	characteristic
201	C₂×#379	C₂×C₂²⋊C₄	1	1	yes	characteristic
202	C₂×#379	C₂×C₂²⋊C₄	1	1	yes	characteristic
203	C₂×#379	C₂×C₂²⋊C₄	1	1	yes	characteristic
204	C₂⁶	C₂×#99	1	1	yes	characteristic
205	C₂⁶	#410	1	1	yes	characteristic
206	C₂⁶	C₂²×C₂²⋊C₄	1	1	yes	characteristic
207	C₂⁶	#520	1	1	yes	characteristic
208	C₂⁶	C₂×#120	1	1	yes	characteristic
209	C₂⁶	C₂×#120	1	1	yes	characteristic
210	C₂⁶	#532	1	1	yes	characteristic
211	C₂⁶	#532	1	1	yes	characteristic
212	C₂⁶	#535	1	1	yes	characteristic
213	C₂⁶	#537	1	1	yes	characteristic
214	C₂⁶	#537	1	1	yes	characteristic
215	C₂⁶	#546	1	2	yes	—
216	C₂⁶	#546	1	2	yes	—
217	C₂³×D₄	#379	1	1	yes	characteristic
218	C₂³×D₄	#379	1	1	yes	characteristic
219	C₂³×D₄	#379	1	2	yes	—
220	C₂³×D₄	C₂×#99	1	1	yes	characteristic
221	C₂³×D₄	C₂×#99	1	1	yes	characteristic
222	C₂³×D₄	#410	1	2	yes	—
223	C₂³×D₄	C₂²×C₂²⋊C₄	1	2	yes	—
224	C₂³×D₄	C₄×#41	1	2	yes	—
225	C₂³×D₄	C₂×#120	1	2	yes	—
226	C₂³×D₄	#525	1	2	yes	—
227	C₂³×D₄	#546	1	2	yes	—
228	C₂³×D₄	#546	1	2	yes	—
229	C₂³×D₄	#547	1	2	yes	—
230	C₂³×D₄	#547	1	2	yes	—
231	C₂⁴×C₄	C₂×#99	1	1	yes	characteristic
232	C₂⁴×C₄	#410	1	1	yes	characteristic
233	C₂⁴×C₄	#520	1	1	yes	characteristic
234	C₂⁴×C₄	#536	1	1	yes	characteristic
235	C₂⁴×C₄	#547	1	2	yes	—
236	C₂⁴×C₄	#547	1	2	yes	—
237	C₂×#120	B(4,𝔽₂)	1	2	yes	—
238	C₂×#120	B(4,𝔽₂)	1	2	yes	—
239	C₂×#120	#458	1	2	yes	—
240	C₂×#120	#458	1	2	yes	—
241	C₂²×C₂²⋊C₄	#379	1	2	yes	—
242	C₂²×C₂²⋊C₄	#379	1	1	yes	characteristic
243	C₂²×C₂²⋊C₄	#379	1	1	yes	characteristic
244	#379	C₂×#99	1	2	yes	—
245	#379	C₂×#99	1	2	yes	—
246	#379	C₂×#99	1	2	yes	—
247	#379	C₂×#99	1	2	yes	—
248	C₂⁵	#1781	1	1	yes	characteristic
249	C₂⁵	#1781	1	1	yes	characteristic
250	C₂⁵	#1782	1	2	yes	—
251	C₂⁵	C₂×#379	1	1	yes	characteristic
252	C₂⁵	#2297	1	1	yes	characteristic
253	C₂⁵	#2311	1	1	yes	characteristic
254	C₂⁵	#2339	1	2	yes	—
255	C₂⁵	#2404	1	2	yes	—
256	C₂⁵	#2430	1	1	yes	characteristic
257	C₂⁵	#2430	1	1	yes	characteristic
258	C₂⁵	#2430	1	1	yes	characteristic
259	C₂⁵	#2430	1	1	yes	characteristic
260	C₂⁵	C₂²×#99	1	1	yes	characteristic
261	C₂⁵	C₂²×#99	1	1	yes	characteristic
262	C₂⁵	C₂×#410	1	1	yes	characteristic
263	C₂⁵	C₂×#410	1	1	yes	characteristic
264	C₂⁵	C₂×B(4,𝔽₂)	1	1	yes	characteristic
265	C₂⁵	C₂×B(4,𝔽₂)	1	1	yes	characteristic
266	C₂⁵	C₂×#458	1	1	yes	characteristic
267	C₂⁵	C₂×#458	1	1	yes	characteristic
268	C₂⁵	#3026	1	1	yes	characteristic
269	C₂⁵	#3026	1	1	yes	characteristic
270	C₂⁵	#3026	1	1	yes	characteristic
271	C₂⁵	#3026	1	1	yes	characteristic
272	C₂²×D₄	#1790	1	2	yes	—
273	C₂²×D₄	#1790	1	2	yes	—
274	C₂³×C₄	#1781	1	1	yes	characteristic
275	C₂³×C₄	#1781	1	1	yes	characteristic
276	C₂³×C₄	#1782	1	2	yes	—
277	C₂³×C₄	#2339	1	2	yes	—
278	C₂³×C₄	#2406	1	2	yes	—
279	C₂×C₂²⋊C₄	#1790	1	2	yes	—
280	C₂×C₂²⋊C₄	#1790	1	2	yes	—
281	C₂⁴	#8534	1	2	yes	—
282	C₂⁴	#13956	1	1	yes	characteristic
283	C₂⁴	C₂×#1781	1	1	yes	characteristic
284	C₂⁴	C₂×#1782	1	1	yes	characteristic
285	C₂⁴	C₂×#1790	1	1	yes	characteristic
286	C₂⁴	C₂×#1790	1	1	yes	characteristic
287	C₂⁴	#20609	1	1	yes	characteristic
288	C₂⁴	#20609	1	1	yes	characteristic
289	C₂⁴	#21420	1	1	yes	characteristic
290	C₂⁴	#21420	1	1	yes	characteristic
291	C₂⁴	#21874	1	1	yes	characteristic
292	C₂⁴	#21874	1	1	yes	characteristic
293	C₂⁴	#23996	1	4	yes	—
294	C₂⁴	#25563	1	1	yes	characteristic
295	C₂⁴	#25563	1	1	yes	characteristic
296	C₂⁴	#25563	1	1	yes	characteristic
297	C₂⁴	#25563	1	1	yes	characteristic
298	C₂⁴	#25565	1	1	yes	characteristic
299	C₂⁴	#25565	1	1	yes	characteristic
300	C₂⁴	#25565	1	1	yes	characteristic
301	C₂⁴	#25565	1	1	yes	characteristic
302	C₂²×C₄	#8534	1	2	yes	—
303	C₂³	C₂×#8534	1	1	yes	characteristic
304	C₂³	#187929	1	1	yes	characteristic
305	C₂³	#187929	1	1	yes	characteristic
306	C₂³	#187944	1	1	yes	characteristic
307	C₂³	#363858	1	1	yes	characteristic
308	C₂³	#363895	1	1	yes	characteristic
309	C₂³	#363895	1	1	yes	characteristic
310	C₂³	#363953	1	2	yes	—
311	C₂³	#363969	1	2	yes	—
312	C₂²	#72233199	1	1	yes	characteristic
313	C₂²	#72234957	1	2	yes	—
314	C₂²	#72265239	1	2	yes	—
315	C₂²	#72265667	1	1	yes	characteristic
316	C₂²	#72265949	1	1	yes	characteristic
317	C₂	#72266152	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72266152
commutator	C₂⁴×D₄	C₂³×C₄
Frattini, cosocle	C₂⁵×D₄	C₂⁴
prime radical	#72251218	C₂
LCS	G ▷ C₂⁴×D₄ ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ C₂⁵×D₄ ◁ G
LECS	G ▷ C₂⁵×D₄ ▷ C₂⁴ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72310380	C₂	semidirect
2. #72310339	C₂	semidirect
3. #72310332	C₂	semidirect
4. #72310299	C₂	semidirect
5. #72309591	C₂	semidirect
6. #72309452	C₂	semidirect
7. #72309379	C₂	semidirect
8. #72266875	C₂	semidirect
11. #72264784	C₂	semidirect
12. #72262357	C₂	semidirect
14. #72251171	C₄	semidirect
15. #72238613	C₄	semidirect
16. #72238612	C₄	semidirect
17. #72237900	C₄	semidirect
18. #72237899	C₄	semidirect
19. #72237829	C₄	semidirect
45. C₂×#32873	C₂×C₄	semidirect
46. C₂×#23895	C₂×C₄	semidirect
55. C₂×#21776	C₂×C₄	semidirect
73. #392030	C₂×C₄	semidirect
74. #392030	C₂×C₄	semidirect
75. #382544	C₂×C₄	semidirect
76. #382434	C₂×C₄	semidirect
77. #373637	C₂×C₄	semidirect
78. #359549	C₂×C₄	semidirect
85. #355096	C₂×C₄	semidirect
153. C₂×#1781	#41	semidirect
154. C₂×#1781	#41	semidirect
192. #2309	#99	semidirect
193. #2308	#99	semidirect
194. #2280	#99	semidirect
195. C₂×#379	#99	semidirect

Maximal subgroups

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

subgroup	index
#72310380	2
#72310339	2
#72310332	2
#72310299	2
#72309591	2
#72309452	2
#72309379	2
#72266875	2
#72264784	2
#72262357	2
#72251218	2