Subgroups — #72262703

Order 4096.

At least 19520 conjugacy classes of subgroups; only the 449 normal subgroups are shown.

The subgroup lattice has 250 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72262550	C₂	1	1	yes	maximal, characteristic
2	#72262540	C₄	1	1	yes	characteristic, complemented
3	#72262540	C₄	1	1	yes	characteristic, complemented
4	#72262474	C₂	1	8	yes	maximal, complemented
5	#72238449	C₂	1	1	yes	maximal, characteristic, complemented
6	#72238383	C₂	1	1	yes	maximal, characteristic, complemented
7	#72237985	C₂	1	1	yes	maximal, characteristic, complemented
8	#72237864	C₂	1	1	yes	maximal, characteristic, complemented
9	#72237816	C₂²	1	4	yes	complemented
10	#72237581	C₂	1	1	yes	maximal, characteristic, complemented
11	#72237576	C₂²	1	4	yes	complemented
12	#72237479	C₂²	1	4	yes	complemented
13	#72237471	C₂	1	1	yes	maximal, characteristic, complemented
14	#72237427	C₂²	1	4	yes	complemented
15	#72234625	C₂²	1	1	yes	characteristic
16	#72234625	C₂²	1	1	yes	characteristic
17	#72234607	C₄	1	4	yes	complemented
18	#72234607	C₂²	1	4	yes	—
19	#72234515	C₂²	1	4	yes	complemented
20	C₂×#10586472	C₄	1	1	yes	characteristic, complemented
21	#72226029	C₂²	1	4	yes	complemented
22	#72225870	C₂²	1	1	yes	characteristic, complemented
23	#72225870	C₂²	1	1	yes	characteristic, complemented
24	C₂×#188035	C₄	1	1	yes	characteristic, complemented
25	#72225665	C₂²	1	1	yes	characteristic
26	#72219311	C₂²	1	1	yes	characteristic, complemented
27	#72219310	C₂²	1	1	yes	characteristic, complemented
28	#10586472	C₂×C₄	1	4	yes	—
29	C₂×#62817	C₂×C₄	1	1	yes	characteristic
30	C₂×#62817	C₂×C₄	1	1	yes	characteristic
31	C₂³×D₄²	C₂×C₄	1	1	yes	characteristic
32	C₂×#34980	D₄	1	2	yes	—
33	C₂×#34980	C₂³	1	1	yes	characteristic
34	#6499870	C₂×C₄	1	2	yes	—
35	#6499870	C₂×C₄	1	2	yes	—
36	#6499870	D₄	1	2	yes	—
37	#6499870	D₄	1	2	yes	—
38	#6499870	C₂³	1	2	yes	—
39	#6499870	C₂³	1	2	yes	—
40	#6499868	D₄	1	2	yes	—
41	#6499867	D₄	1	2	yes	—
42	#6451874	D₄	1	2	yes	—
43	#6451874	D₄	1	2	yes	—
44	C₂×#21746	C₂×C₄	1	1	yes	characteristic
45	C₂×#21746	C₂×C₄	1	1	yes	characteristic
46	C₂×#21746	D₄	1	2	yes	—
47	#4420586	C₂×C₄	1	2	yes	—
48	#4420586	C₂³	1	2	yes	—
49	#663621	D₄	1	2	yes	—
50	#647656	D₄	1	2	yes	—
51	#617710	D₄	1	1	yes	characteristic
52	#617710	D₄	1	1	yes	characteristic
53	#616960	D₄	1	1	yes	characteristic
54	#616960	D₄	1	1	yes	characteristic
55	#556752	C₂×C₄	1	1	yes	characteristic
56	#188035	C₂×C₄	1	4	yes	—
57	#187949	C₂×C₄	1	4	yes	—
58	#187949	C₂×C₄	1	4	yes	—
59	#134037	C₂³	1	2	yes	complemented
60	#134036	C₂³	1	2	yes	complemented
61	#115633	C₂³	1	2	yes	complemented
62	#115631	C₂³	1	2	yes	complemented
63	C₂⁵×D₄	C₂×D₄	1	1	yes	characteristic
64	#62817	C₂²⋊C₄	1	2	yes	—
65	#62817	C₂²⋊C₄	1	2	yes	—
66	#62817	C₂²×C₄	1	2	yes	—
67	#62817	C₂²×C₄	1	2	yes	—
68	C₂×#3490	C₂²⋊C₄	1	2	yes	—
69	C₂²×D₄²	C₂²⋊C₄	1	2	yes	—
70	C₂²×D₄²	C₂²×C₄	1	2	yes	—
71	C₂²×#534	C₂×D₄	1	1	yes	characteristic
72	C₂²×#534	C₂×D₄	1	1	yes	characteristic
73	C₂³×#127	C₂²⋊C₄	1	1	yes	characteristic
74	C₂³×#120	C₂²⋊C₄	1	2	yes	—
75	C₂³×#120	C₂²⋊C₄	1	1	yes	characteristic
76	C₂³×#120	C₂²×C₄	1	1	yes	characteristic
77	C₂×#2849	C₂×D₄	1	1	yes	characteristic
78	C₂×#2849	C₂×D₄	1	1	yes	characteristic
79	C₂×#2846	C₂×D₄	1	1	yes	characteristic
80	C₂×#2846	C₂×D₄	1	1	yes	characteristic
81	#34980	C₂²⋊C₄	1	4	yes	—
82	#34980	C₂×D₄	1	4	yes	—
83	C₂×#2613	C₂²×C₄	1	1	yes	characteristic
84	C₂×#2613	C₂×D₄	1	1	yes	characteristic
85	C₂×#2613	C₂×D₄	1	1	yes	characteristic
86	C₂×#2613	C₂⁴	1	1	yes	characteristic
87	C₂×#2308	C₂×D₄	1	1	yes	characteristic
88	C₂×#2308	C₂×D₄	1	1	yes	characteristic
89	C₂²×#379	C₂²⋊C₄	1	2	yes	—
90	C₂²×#379	C₂×D₄	1	1	yes	characteristic
91	#23985	C₂²⋊C₄	1	4	yes	—
92	#23985	C₂×D₄	1	4	yes	—
93	#23984	C₂²⋊C₄	1	4	yes	—
94	#23984	C₂×D₄	1	4	yes	—
95	#21746	C₂²⋊C₄	1	4	yes	—
96	#21746	C₂²⋊C₄	1	2	yes	—
97	#21746	C₂²⋊C₄	1	2	yes	—
98	#21746	C₂²×C₄	1	2	yes	—
99	#21746	C₂²×C₄	1	2	yes	—
100	#21746	C₂×D₄	1	4	yes	—
101	C₂×#1819	C₂²⋊C₄	1	2	yes	—
102	#15948	C₂²⋊C₄	1	2	yes	—
103	#15948	C₂×D₄	1	2	yes	—
104	#15947	C₂²⋊C₄	1	2	yes	—
105	#15947	C₂×D₄	1	2	yes	—
106	#15261	C₂²⋊C₄	1	2	yes	—
107	#15261	C₂×D₄	1	2	yes	—
108	#15258	C₂²⋊C₄	1	2	yes	—
109	#15258	C₂×D₄	1	2	yes	—
110	#14657	C₂²×C₄	1	2	yes	—
111	#14335	C₂²⋊C₄	1	1	yes	characteristic
112	#14335	C₂²⋊C₄	1	1	yes	characteristic
113	#14134	C₂²⋊C₄	1	2	yes	—
114	C₂⁷	#99	1	1	yes	characteristic
115	C₂⁷	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂⁷	#120	1	1	yes	characteristic
117	C₂⁴×D₄	#99	1	2	yes	—
118	C₂⁴×D₄	#99	1	1	yes	characteristic
119	C₂⁴×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂⁴×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂⁴×D₄	#120	1	1	yes	characteristic
122	C₂⁴×D₄	#120	1	1	yes	characteristic
123	C₂⁴×D₄	C₂²×D₄	1	1	yes	characteristic
124	C₂⁵×C₄	#120	1	1	yes	characteristic
125	#3490	#99	1	4	yes	—
126	#3490	C₂×C₂²⋊C₄	1	4	yes	—
127	C₂×#543	#99	1	2	yes	—
128	C₂×#534	#99	1	2	yes	—
129	C₂²×#127	#99	1	2	yes	—
130	C₂²×#127	C₂×C₂²⋊C₄	1	2	yes	—
131	C₂²×#123	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂²×#123	#120	1	1	yes	characteristic
133	C₂²×#123	#120	1	1	yes	characteristic
134	C₂²×#123	C₂²×D₄	1	1	yes	characteristic
135	C₂²×#120	#99	1	4	yes	—
136	C₂²×#120	#99	1	2	yes	—
137	C₂²×#120	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₂²×#120	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₂²×#120	C₂×C₂²⋊C₄	1	1	yes	characteristic
140	C₂²×#120	C₂×C₂²⋊C₄	1	1	yes	characteristic
141	C₂²×#120	C₂×C₂²⋊C₄	1	4	yes	—
142	C₂²×#120	C₂×C₂²⋊C₄	1	2	yes	—
143	C₂²×#120	#120	1	1	yes	characteristic
144	C₂²×#120	#120	1	1	yes	characteristic
145	C₂²×#120	C₂³×C₄	1	1	yes	characteristic
146	C₂²×#120	C₂²×D₄	1	1	yes	characteristic
147	C₂³×C₂²⋊C₄	#99	1	2	yes	—
148	C₂³×C₂²⋊C₄	#99	1	1	yes	characteristic
149	C₂³×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
150	C₂³×C₂²⋊C₄	#120	1	1	yes	characteristic
151	C₂³×C₂²⋊C₄	#120	1	1	yes	characteristic
152	C₂³×C₄²	#99	1	1	yes	characteristic
153	C₂×#381	C₂×C₂²⋊C₄	1	1	yes	characteristic
154	C₂×#381	#120	1	1	yes	characteristic
155	C₂×#381	#120	1	1	yes	characteristic
156	C₂×#381	C₂²×D₄	1	1	yes	characteristic
157	C₂×#379	C₂×C₂²⋊C₄	1	4	yes	—
158	C₂×#379	C₂×C₂²⋊C₄	1	1	yes	characteristic
159	C₂×#379	C₂×C₂²⋊C₄	1	1	yes	characteristic
160	C₂×#379	#120	1	1	yes	characteristic
161	C₂×#379	#120	1	1	yes	characteristic
162	C₂×#379	#120	1	1	yes	characteristic
163	C₂×#379	#120	1	1	yes	characteristic
164	C₂×#379	C₂²×D₄	1	1	yes	characteristic
165	C₂×#379	C₂²×D₄	1	1	yes	characteristic
166	#1819	C₂×C₂²⋊C₄	1	4	yes	—
167	#1438	C₂×C₂²⋊C₄	1	2	yes	—
168	#1438	C₂×C₂²⋊C₄	1	2	yes	—
169	C₂⁶	AΣL(1,𝔽₁₆)	1	2	yes	complemented
170	C₂⁶	#379	1	1	yes	characteristic
171	C₂⁶	#379	1	1	yes	characteristic
172	C₂⁶	#379	1	1	yes	characteristic
173	C₂⁶	C₂×#99	1	2	yes	—
174	C₂⁶	C₂×#99	1	2	yes	—
175	C₂⁶	C₂×#99	1	1	yes	characteristic
176	C₂⁶	C₂×#99	1	2	yes	—
177	C₂⁶	C₂²×C₂²⋊C₄	1	1	yes	characteristic
178	C₂⁶	C₂×#120	1	1	yes	characteristic
179	C₂³×D₄	AΣL(1,𝔽₁₆)	1	4	yes	complemented
180	C₂³×D₄	#379	1	1	yes	characteristic
181	C₂³×D₄	C₂×#99	1	1	yes	characteristic
182	C₂³×D₄	C₂×#99	1	2	yes	—
183	C₂³×D₄	C₂×#99	1	4	yes	—
184	C₂³×D₄	C₂×#99	1	1	yes	characteristic
185	C₂³×D₄	C₂²×C₂²⋊C₄	1	1	yes	characteristic
186	C₂⁴×C₄	AΣL(1,𝔽₁₆)	1	2	yes	complemented
187	C₂⁴×C₄	#379	1	1	yes	characteristic
188	C₂⁴×C₄	C₂×#99	1	1	yes	characteristic
189	C₂⁴×C₄	C₂×#120	1	1	yes	characteristic
190	#543	C₂×#99	1	4	yes	—
191	#534	C₂×#99	1	4	yes	—
192	C₂²×C₂²⋊C₄	AΣL(1,𝔽₁₆)	1	4	yes	complemented
193	C₂²×C₂²⋊C₄	AΣL(1,𝔽₁₆)	1	4	yes	complemented
194	C₂²×C₂²⋊C₄	#379	1	1	yes	characteristic
195	C₂²×C₂²⋊C₄	#379	1	1	yes	characteristic
196	C₂²×C₂²⋊C₄	#379	1	1	yes	characteristic
197	C₂²×C₂²⋊C₄	C₂×#99	1	2	yes	—
198	C₂²×C₂²⋊C₄	C₂×#99	1	2	yes	—
199	C₂²×C₂²⋊C₄	C₂×#99	1	4	yes	—
200	C₂²×C₂²⋊C₄	C₂×#99	1	2	yes	—
201	C₂²×C₂²⋊C₄	C₂²×C₂²⋊C₄	1	1	yes	characteristic
202	C₂²×C₂²⋊C₄	C₂×#120	1	1	yes	characteristic
203	C₂²×C₂²⋊C₄	C₂×#120	1	1	yes	characteristic
204	C₂²×C₄²	C₂×#99	1	2	yes	—
205	C₂⁵	#1781	1	1	yes	characteristic
206	C₂⁵	#1781	1	1	yes	characteristic
207	C₂⁵	#1781	1	2	yes	—
208	C₂⁵	#1781	1	1	yes	characteristic
209	C₂⁵	#1781	1	1	yes	characteristic
210	C₂⁵	#1781	1	1	yes	characteristic
211	C₂⁵	#1781	1	1	yes	characteristic
212	C₂⁵	#1781	1	1	yes	characteristic
213	C₂⁵	C₂×AΣL(1,𝔽₁₆)	1	2	yes	—
214	C₂⁵	C₂×AΣL(1,𝔽₁₆)	1	2	yes	—
215	C₂⁵	C₂×AΣL(1,𝔽₁₆)	1	2	yes	—
216	C₂⁵	C₂×AΣL(1,𝔽₁₆)	1	2	yes	—
217	C₂⁵	C₂×AΣL(1,𝔽₁₆)	1	4	yes	—
218	C₂⁵	C₂×#379	1	1	yes	characteristic
219	C₂⁵	C₂²×#99	1	2	yes	—
220	C₂⁵	C₂²×#99	1	1	yes	characteristic
221	C₂³×C₄	#1781	1	1	yes	characteristic
222	C₂³×C₄	C₂×AΣL(1,𝔽₁₆)	1	4	yes	—
223	C₂³×C₄	C₂²×#99	1	1	yes	characteristic
224	C₂×C₂²⋊C₄	#1781	1	2	yes	—
225	C₂×C₂²⋊C₄	C₂×AΣL(1,𝔽₁₆)	1	8	yes	—
226	C₂×C₂²⋊C₄	C₂²×#99	1	2	yes	—
227	C₂⁴	#8541	1	1	yes	characteristic
228	C₂⁴	#8541	1	1	yes	characteristic
229	C₂⁴	#8541	1	1	yes	characteristic
230	C₂⁴	#8541	1	1	yes	characteristic
231	C₂⁴	#12687	1	2	yes	—
232	C₂⁴	#12687	1	2	yes	—
233	C₂⁴	#12687	1	2	yes	—
234	C₂⁴	#12687	1	2	yes	—
235	C₂⁴	C₂×#1781	1	1	yes	characteristic
236	C₂⁴	C₂×#1781	1	1	yes	characteristic
237	C₂⁴	C₂×#1781	1	1	yes	characteristic
238	C₂⁴	C₂²×AΣL(1,𝔽₁₆)	1	2	yes	—
239	C₂⁴	C₂²×AΣL(1,𝔽₁₆)	1	2	yes	—
240	C₂³	#133822	1	1	yes	characteristic
241	C₂³	#133822	1	2	yes	—
242	C₂³	#133822	1	1	yes	characteristic
243	C₂³	C₂×#8541	1	1	yes	characteristic
244	C₂³	C₂×#12687	1	2	yes	—
245	C₂²	C₂×#133822	1	1	yes	characteristic
246	C₂²	#72240419	1	1	yes	characteristic
247	C₂²	#72240419	1	1	yes	characteristic
248	C₂	#72240424	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72240424
commutator	C₂²×#120	C₂³×C₄
Frattini, cosocle	C₂×#2613	C₂⁴
prime radical	#72262550	C₂
LCS	G ▷ C₂²×#120 ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂⁶ ◁ C₂×#34980 ◁ G
LECS	G ▷ C₂×#2613 ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #72262540	C₄	semidirect
3. #72262540	C₄	semidirect
4. #72262474	C₂	semidirect
5. #72238449	C₂	semidirect
6. #72238383	C₂	semidirect
7. #72237985	C₂	semidirect
8. #72237864	C₂	semidirect
9. #72237816	C₂²	semidirect
10. #72237581	C₂	semidirect
11. #72237576	C₂²	semidirect
12. #72237479	C₂²	semidirect
13. #72237471	C₂	semidirect
14. #72237427	C₂²	semidirect
17. #72234607	C₄	semidirect
19. #72234515	C₂²	semidirect
20. C₂×#10586472	C₄	semidirect
21. #72226029	C₂²	semidirect
22. #72225870	C₂²	semidirect
23. #72225870	C₂²	semidirect
24. C₂×#188035	C₄	semidirect
26. #72219311	C₂²	semidirect
27. #72219310	C₂²	semidirect
59. #134037	C₂³	semidirect
60. #134036	C₂³	semidirect
61. #115633	C₂³	semidirect
62. #115631	C₂³	semidirect
169. C₂⁶	AΣL(1,𝔽₁₆)	semidirect
179. C₂³×D₄	AΣL(1,𝔽₁₆)	semidirect
186. C₂⁴×C₄	AΣL(1,𝔽₁₆)	semidirect
192. C₂²×C₂²⋊C₄	AΣL(1,𝔽₁₆)	semidirect
193. C₂²×C₂²⋊C₄	AΣL(1,𝔽₁₆)	semidirect

Maximal subgroups

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

subgroup	index
#72262550	2
#72262474	2
#72238449	2
#72238383	2
#72237985	2
#72237864	2
#72237581	2
#72237471	2