Subgroups — #72241264

Order 2048.

At least 7353 conjugacy classes of subgroups; only the 472 normal subgroups are shown.

The subgroup lattice has 290 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72241263	C₂	1	2	yes	maximal, complemented
2	#72241262	C₂	1	2	yes	maximal, complemented
3	#72241261	C₂	1	1	yes	maximal, characteristic, complemented
4	#72241260	C₂	1	2	yes	maximal, complemented
5	#72241259	C₂	1	2	yes	maximal, complemented
6	#72241255	C₂	1	1	yes	maximal, characteristic, complemented
7	#72241097	C₂	1	1	yes	maximal, characteristic
8	#72241091	C₂	1	1	yes	maximal, characteristic, complemented
9	#72232651	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#134033	C₂	1	1	yes	maximal, characteristic, complemented
11	#72217818	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#45457	C₂²	1	1	yes	characteristic
13	C₂²×#2846	C₂²	1	1	yes	characteristic
14	#7626202	C₂²	1	1	yes	characteristic
15	C₂×#21746	C₂²	1	1	yes	characteristic
16	#5020752	C₄	1	2	yes	complemented
17	#5020746	C₄	1	2	yes	complemented
18	#2032594	C₄	1	2	yes	complemented
19	#2031999	C₄	1	2	yes	complemented
20	#846058	C₂²	1	1	yes	characteristic
21	#845678	C₂²	1	1	yes	characteristic
22	#617359	C₂²	1	1	yes	characteristic
23	#617162	C₂²	1	1	yes	characteristic
24	#559861	C₂²	1	1	yes	characteristic
25	#482561	C₂²	1	2	yes	complemented
26	#482551	C₂²	1	2	yes	complemented
27	C₂×#11499	C₂²	1	2	yes	—
28	C₂×#11499	C₂²	1	2	yes	—
29	#359756	C₂²	1	2	yes	—
30	#359756	C₂²	1	2	yes	—
31	#201466	C₂²	1	2	yes	complemented
32	#200636	C₂²	1	2	yes	complemented
33	C₂×#8547	C₂²	1	1	yes	characteristic
34	C₂×#8547	C₂²	1	1	yes	characteristic
35	#127217	C₂²	1	2	yes	complemented
36	#127215	C₂²	1	2	yes	complemented
37	#106380	C₂²	1	2	yes	complemented
38	#106326	C₂²	1	2	yes	complemented
39	C₂⁸	D₄	1	1	yes	characteristic
40	C₂³×#120	D₄	1	1	yes	characteristic
41	C₂³×#120	D₄	1	1	yes	characteristic
42	C₂³×#120	C₂³	1	1	yes	characteristic
43	C₂×#2735	C₂×C₄	1	1	yes	characteristic, complemented
44	C₂×#2668	C₂×C₄	1	1	yes	characteristic, complemented
45	C₂×#2842	C₂×C₄	1	1	yes	characteristic, complemented
46	C₂×#2812	C₂×C₄	1	1	yes	characteristic, complemented
47	C₂×#2846	C₂×C₄	1	2	yes	—
48	C₂×#2846	C₂×C₄	1	2	yes	—
49	C₂×#2846	D₄	1	2	yes	—
50	C₂×#2846	D₄	1	2	yes	—
51	C₂×#2539	C₂³	1	1	yes	characteristic
52	C₂×#2482	D₄	1	1	yes	characteristic
53	C₂×#2482	C₂³	1	1	yes	characteristic
54	C₂×#2433	D₄	1	1	yes	characteristic
55	C₂×#2617	D₄	1	1	yes	characteristic
56	#34176	C₂³	1	2	yes	—
57	C₂×#2309	D₄	1	1	yes	characteristic
58	C₂²×#379	D₄	1	1	yes	characteristic
59	C₂²×#379	C₂³	1	1	yes	characteristic
60	C₂²×#379	C₂³	1	1	yes	characteristic
61	C₂×#1819	C₂³	1	2	yes	—
62	C₂×#1462	C₂³	1	2	yes	—
63	C₂×#1462	C₂³	1	2	yes	—
64	#15480	C₂×C₄	1	2	yes	complemented
65	#15284	C₂×C₄	1	2	yes	complemented
66	#15284	C₂×C₄	1	2	yes	complemented
67	#15283	C₂×C₄	1	2	yes	—
68	#15283	C₂×C₄	1	2	yes	—
69	#15283	C₂×C₄	1	2	yes	—
70	#15283	D₄	1	2	yes	—
71	#15283	D₄	1	2	yes	—
72	#15283	D₄	1	2	yes	—
73	#15267	C₂×C₄	1	2	yes	complemented
74	#15258	C₂×C₄	1	2	yes	—
75	#15258	D₄	1	2	yes	—
76	#14365	C₂×C₄	1	2	yes	complemented
77	#14335	D₄	1	1	yes	characteristic
78	#13903	C₂³	1	1	yes	characteristic
79	#13902	D₄	1	1	yes	characteristic
80	#13868	D₄	1	1	yes	characteristic
81	#13868	C₂³	1	1	yes	characteristic
82	#13827	C₂×C₄	1	2	yes	complemented
83	#13775	D₄	1	1	yes	characteristic
84	#10935	D₄	1	8	yes	complemented
85	C₂⁷	C₂²⋊C₄	1	2	yes	—
86	C₂⁷	C₂²⋊C₄	1	2	yes	—
87	C₂⁷	C₂×D₄	1	1	yes	characteristic
88	C₂⁷	C₂×D₄	1	1	yes	characteristic
89	C₂⁷	C₂×D₄	1	1	yes	characteristic
90	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
91	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
92	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
93	C₂²×#127	C₂²⋊C₄	1	2	yes	complemented
94	C₂²×#121	C₂×D₄	1	1	yes	characteristic
95	C₂²×#120	C₂²⋊C₄	1	2	yes	—
96	C₂²×#120	C₂²⋊C₄	1	2	yes	—
97	C₂²×#120	C₂²⋊C₄	1	2	yes	—
98	C₂²×#120	C₂²⋊C₄	1	2	yes	—
99	C₂²×#120	C₂²×C₄	1	2	yes	—
100	C₂²×#120	C₂×D₄	1	2	yes	—
101	C₂²×#120	C₂×D₄	1	2	yes	—
102	C₂²×#120	C₂×D₄	1	2	yes	—
103	C₂²×#117	C₂×D₄	1	1	yes	characteristic
104	C₂²×#117	C₂×D₄	1	1	yes	characteristic
105	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
106	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
107	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
108	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
109	C₂³×C₂²⋊C₄	C₂⁴	1	1	yes	characteristic
110	C₂×#399	C₂²⋊C₄	1	2	yes	complemented
111	C₂×#399	C₂²×C₄	1	1	yes	characteristic
112	C₂×#399	C₂²×C₄	1	1	yes	characteristic
113	C₂×#399	C₂×D₄	1	1	yes	characteristic
114	C₂×#399	C₂×D₄	1	1	yes	characteristic
115	C₂×#397	C₂²⋊C₄	1	2	yes	complemented
116	C₂×#397	C₂²⋊C₄	1	2	yes	complemented
117	C₂×#397	C₂²×C₄	1	1	yes	characteristic
118	C₂×#397	C₂×D₄	1	1	yes	characteristic
119	C₂×#392	C₂²⋊C₄	1	2	yes	complemented
120	C₂×#392	C₂²×C₄	1	1	yes	characteristic
121	C₂×#392	C₂×D₄	1	1	yes	characteristic
122	C₂×#390	C₂²⋊C₄	1	2	yes	complemented
123	C₂×#390	C₂²⋊C₄	1	2	yes	complemented
124	C₂×#390	C₂²⋊C₄	1	2	yes	complemented
125	C₂×#381	C₂×D₄	1	1	yes	characteristic
126	C₂×#381	C₂×D₄	1	1	yes	characteristic
127	C₂×#380	C₂×D₄	1	1	yes	characteristic
128	C₂×#379	C₂²⋊C₄	1	2	yes	—
129	C₂×#379	C₂²⋊C₄	1	2	yes	—
130	C₂×#379	C₂²×C₄	1	2	yes	—
131	C₂×#379	C₂²×C₄	1	1	yes	characteristic
132	C₂×#379	C₂²×C₄	1	1	yes	characteristic
133	C₂×#379	C₂×D₄	1	2	yes	—
134	C₂×#379	C₂×D₄	1	1	yes	characteristic
135	C₂×#379	C₂×D₄	1	1	yes	characteristic
136	C₂×#379	C₂×D₄	1	4	yes	—
137	C₂×#379	#41	1	4	yes	—
138	#2273	C₂×D₄	1	4	yes	—
139	C₂×C₄×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
140	C₂×C₄×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
141	C₂×#342	C₂×D₄	1	4	yes	—
142	#1438	C₂²⋊C₄	1	2	yes	complemented
143	#1438	C₂²⋊C₄	1	2	yes	complemented
144	#1438	C₂²⋊C₄	1	2	yes	complemented
145	#1438	C₂²⋊C₄	1	2	yes	complemented
146	#1437	C₂²⋊C₄	1	2	yes	complemented
147	#1437	C₂²⋊C₄	1	2	yes	complemented
148	#1437	C₂²⋊C₄	1	2	yes	complemented
149	#1437	C₂²⋊C₄	1	2	yes	complemented
150	#1437	C₂²×C₄	1	2	yes	—
151	#1437	C₂²×C₄	1	2	yes	—
152	#1437	C₂×D₄	1	2	yes	—
153	#1437	C₂×D₄	1	2	yes	—
154	C₂⁶	C₂×C₂²⋊C₄	1	2	yes	—
155	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
156	C₂⁶	C₂×C₂²⋊C₄	1	2	yes	—
157	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
158	C₂⁶	#120	1	1	yes	characteristic
159	C₂⁶	#120	1	2	yes	—
160	C₂⁶	#120	1	1	yes	characteristic
161	C₂⁶	#120	1	1	yes	characteristic
162	C₂⁶	#120	1	2	yes	—
163	C₂⁶	#123	1	2	yes	—
164	C₂⁶	C₂²×D₄	1	1	yes	characteristic
165	C₂³×D₄	C₂×C₂²⋊C₄	1	2	yes	—
166	C₂³×D₄	C₂×C₂²⋊C₄	1	2	yes	—
167	C₂³×D₄	C₂×C₂²⋊C₄	1	2	yes	—
168	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic, complemented
169	C₂³×D₄	#120	1	2	yes	—
170	C₂³×D₄	#120	1	2	yes	—
171	C₂³×D₄	#120	1	2	yes	—
172	C₂⁴×C₄	#120	1	1	yes	characteristic
173	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
174	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
175	C₂²×C₄⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic, complemented
176	C₂²×C₄⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic, complemented
177	C₂²×C₄⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic, complemented
178	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
179	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
180	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
181	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
182	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
183	C₂²×C₂²⋊C₄	#120	1	2	yes	—
184	C₂²×C₂²⋊C₄	#120	1	2	yes	—
185	C₂²×C₂²⋊C₄	#120	1	2	yes	—
186	C₂²×C₂²⋊C₄	#123	1	2	yes	—
187	C₂²×C₂²⋊C₄	C₂³×C₄	1	1	yes	characteristic
188	C₂²×C₂²⋊C₄	C₂²×D₄	1	2	yes	—
189	C₂²×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
190	C₂²×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
191	C₂²×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
192	C₂²×C₂²⋊C₄	C₂×#41	1	2	yes	—
193	C₂²×C₂²⋊C₄	H(5,𝔽₂)	1	2	yes	—
194	C₂²×C₂²⋊C₄	H(5,𝔽₂)	1	2	yes	—
195	C₂²×C₄²	C₂×C₂²⋊C₄	1	2	yes	complemented
196	C₂²×C₄²	C₂×C₂²⋊C₄	1	2	yes	complemented
197	#379	C₄×D₄	1	4	yes	—
198	#379	C₄×D₄	1	4	yes	—
199	#379	#121	1	4	yes	—
200	#379	#121	1	4	yes	—
201	C₂×#95	C₂×C₂²⋊C₄	1	1	yes	characteristic, complemented
202	C₂×#95	C₂×C₂²⋊C₄	1	1	yes	characteristic, complemented
203	C₂×#95	C₂×C₂²⋊C₄	1	1	yes	characteristic, complemented
204	C₂×#95	C₂×C₂²⋊C₄	1	1	yes	characteristic, complemented
205	C₂×#95	C₂×C₂²⋊C₄	1	2	yes	complemented
206	C₂×#95	C₂×C₂²⋊C₄	1	2	yes	complemented
207	C₂⁵	#379	1	2	yes	—
208	C₂⁵	#379	1	2	yes	—
209	C₂⁵	#379	1	2	yes	—
210	C₂⁵	#386	1	2	yes	—
211	C₂⁵	#386	1	2	yes	—
212	C₂⁵	B(4,𝔽₂)	1	1	yes	characteristic
213	C₂⁵	B(4,𝔽₂)	1	1	yes	characteristic
214	C₂⁵	B(4,𝔽₂)	1	1	yes	characteristic
215	C₂⁵	B(4,𝔽₂)	1	1	yes	characteristic
216	C₂⁵	#458	1	2	yes	—
217	C₂⁵	#458	1	2	yes	—
218	C₂⁵	C₂²×C₂²⋊C₄	1	1	yes	characteristic
219	C₂⁵	C₂×#120	1	1	yes	characteristic
220	C₂⁵	C₂×#120	1	1	yes	characteristic
221	C₂⁵	C₂×#120	1	1	yes	characteristic
222	C₂⁵	C₂×#123	1	1	yes	characteristic
223	C₂⁵	#534	1	1	yes	characteristic
224	C₂⁵	#534	1	1	yes	characteristic
225	C₂³×C₄	#379	1	2	yes	—
226	C₂³×C₄	C₂²×C₂²⋊C₄	1	1	yes	characteristic
227	C₂³×C₄	C₂²×C₂²⋊C₄	1	1	yes	characteristic
228	C₂³×C₄	C₂×#120	1	1	yes	characteristic
229	C₂³×C₄	C₂×#120	1	1	yes	characteristic
230	C₂³×C₄	D₄²	1	2	yes	—
231	C₂³×C₄	D₄²	1	2	yes	—
232	C₂³×C₄	#546	1	2	yes	—
233	C₂³×C₄	#546	1	2	yes	—
234	C₂×C₂²⋊C₄	#386	1	2	yes	—
235	C₂×C₂²⋊C₄	#386	1	2	yes	—
236	C₂×C₂²⋊C₄	C₂×C₄×D₄	1	2	yes	—
237	C₂×C₂²⋊C₄	#518	1	2	yes	—
238	C₂×C₂²⋊C₄	C₂×#121	1	2	yes	—
239	C₂×C₂²⋊C₄	#534	1	2	yes	—
240	C₂×C₂²⋊C₄	D₄²	1	2	yes	—
241	C₂×C₂²⋊C₄	D₄²	1	2	yes	—
242	C₂×C₂²⋊C₄	#546	1	2	yes	—
243	C₂×C₂²⋊C₄	#546	1	2	yes	—
244	C₂⁴	#1790	1	2	yes	—
245	C₂⁴	#1790	1	2	yes	—
246	C₂⁴	#1790	1	2	yes	—
247	C₂⁴	#1790	1	2	yes	—
248	C₂⁴	#1911	1	4	yes	—
249	C₂⁴	#2200	1	4	yes	—
250	C₂⁴	C₂×#379	1	1	yes	characteristic
251	C₂⁴	C₂×#386	1	1	yes	characteristic
252	C₂⁴	#2308	1	1	yes	characteristic
253	C₂⁴	#2403	1	1	yes	characteristic
254	C₂⁴	#2403	1	1	yes	characteristic
255	C₂⁴	#2405	1	1	yes	characteristic
256	C₂⁴	#2405	1	1	yes	characteristic
257	C₂⁴	C₂×B(4,𝔽₂)	1	1	yes	characteristic
258	C₂⁴	C₂×B(4,𝔽₂)	1	1	yes	characteristic
259	C₂⁴	C₂×#458	1	1	yes	characteristic
260	C₂⁴	C₂×#458	1	1	yes	characteristic
261	C₂²×C₄	D₄×C₂²⋊C₄	1	2	yes	—
262	C₂²×C₄	D₄×C₂²⋊C₄	1	2	yes	—
263	C₂²×C₄	#2410	1	2	yes	—
264	C₂²×C₄	#2410	1	2	yes	—
265	C₂³	#12753	1	4	yes	—
266	C₂³	#12762	1	4	yes	—
267	C₂³	#13867	1	1	yes	characteristic
268	C₂³	C₂×#1790	1	1	yes	characteristic
269	C₂³	C₂×#1790	1	1	yes	characteristic
270	C₂³	C₂×#1911	1	2	yes	—
271	C₂³	#23897	1	2	yes	—
272	C₂³	#23900	1	1	yes	characteristic
273	C₂³	#23901	1	1	yes	characteristic
274	C₂³	#24056	1	1	yes	characteristic
275	C₂³	#24056	1	1	yes	characteristic
276	C₂³	C₂×#2200	1	2	yes	—
277	C₂³	#33551	1	2	yes	—
278	C₂²	#359509	1	1	yes	characteristic
279	C₂²	#359552	1	1	yes	characteristic
280	C₂²	#359558	1	1	yes	characteristic
281	C₂²	C₂×#12753	1	2	yes	—
282	C₂²	C₂×#12762	1	1	yes	characteristic
283	C₂²	#493582	1	1	yes	characteristic
284	C₂²	#499797	1	2	yes	—
285	C₂²	#500238	1	2	yes	—
286	C₂	#72218088	1	1	yes	characteristic
287	C₂	#72219034	1	1	yes	characteristic
288	C₂	#72241251	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#359552
commutator	C₂²×C₂²⋊C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₂²⋊C₄	C₂⁴
prime radical	#72241097	C₂
LCS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂³×#120 ◁ G
LECS	G ▷ C₂³×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72241263	C₂	semidirect
2. #72241262	C₂	semidirect
3. #72241261	C₂	semidirect
4. #72241260	C₂	semidirect
5. #72241259	C₂	semidirect
6. #72241255	C₂	semidirect
8. #72241091	C₂	semidirect
9. #72232651	C₂	semidirect
10. C₂×#134033	C₂	semidirect
11. #72217818	C₂	semidirect
16. #5020752	C₄	semidirect
17. #5020746	C₄	semidirect
18. #2032594	C₄	semidirect
19. #2031999	C₄	semidirect
25. #482561	C₂²	semidirect
26. #482551	C₂²	semidirect
31. #201466	C₂²	semidirect
32. #200636	C₂²	semidirect
35. #127217	C₂²	semidirect
36. #127215	C₂²	semidirect
37. #106380	C₂²	semidirect
38. #106326	C₂²	semidirect
43. C₂×#2735	C₂×C₄	semidirect
44. C₂×#2668	C₂×C₄	semidirect
45. C₂×#2842	C₂×C₄	semidirect
46. C₂×#2812	C₂×C₄	semidirect
64. #15480	C₂×C₄	semidirect
65. #15284	C₂×C₄	semidirect
66. #15284	C₂×C₄	semidirect
73. #15267	C₂×C₄	semidirect
76. #14365	C₂×C₄	semidirect
82. #13827	C₂×C₄	semidirect
84. #10935	D₄	semidirect
93. C₂²×#127	C₂²⋊C₄	semidirect
110. C₂×#399	C₂²⋊C₄	semidirect
115. C₂×#397	C₂²⋊C₄	semidirect
116. C₂×#397	C₂²⋊C₄	semidirect
119. C₂×#392	C₂²⋊C₄	semidirect
122. C₂×#390	C₂²⋊C₄	semidirect
123. C₂×#390	C₂²⋊C₄	semidirect
124. C₂×#390	C₂²⋊C₄	semidirect
142. #1438	C₂²⋊C₄	semidirect
143. #1438	C₂²⋊C₄	semidirect
144. #1438	C₂²⋊C₄	semidirect
145. #1438	C₂²⋊C₄	semidirect
146. #1437	C₂²⋊C₄	semidirect
147. #1437	C₂²⋊C₄	semidirect
148. #1437	C₂²⋊C₄	semidirect
149. #1437	C₂²⋊C₄	semidirect
168. C₂³×D₄	C₂×C₂²⋊C₄	semidirect
175. C₂²×C₄⋊C₄	C₂×C₂²⋊C₄	semidirect
176. C₂²×C₄⋊C₄	C₂×C₂²⋊C₄	semidirect
177. C₂²×C₄⋊C₄	C₂×C₂²⋊C₄	semidirect
195. C₂²×C₄²	C₂×C₂²⋊C₄	semidirect
196. C₂²×C₄²	C₂×C₂²⋊C₄	semidirect
201. C₂×#95	C₂×C₂²⋊C₄	semidirect
202. C₂×#95	C₂×C₂²⋊C₄	semidirect
203. C₂×#95	C₂×C₂²⋊C₄	semidirect
204. C₂×#95	C₂×C₂²⋊C₄	semidirect
205. C₂×#95	C₂×C₂²⋊C₄	semidirect
206. C₂×#95	C₂×C₂²⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
#72241263	2
#72241262	2
#72241261	2
#72241260	2
#72241259	2
#72241255	2
#72241097	2
#72241091	2
#72232651	2
C₂×#134033	2
#72217818	2