Subgroups — #72241163

Order 2048.

At least 7027 conjugacy classes of subgroups; only the 404 normal subgroups are shown.

The subgroup lattice has 244 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72241149	C₂	1	1	yes	maximal, characteristic
2	#72241063	C₂	1	2	yes	maximal, complemented
3	#72233369	C₂	1	4	yes	maximal, complemented
4	#72232761	C₂	1	2	yes	maximal, complemented
5	#72232651	C₂	1	2	yes	maximal, complemented
6	C₂×#133822	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#115690	C₂	1	1	yes	maximal, characteristic, complemented
8	#72215653	C₂	1	2	yes	maximal, complemented
9	C₂×#37436	C₄	1	1	yes	characteristic, complemented
10	C₂²×#2846	C₂²	1	1	yes	characteristic
11	C₂×#21746	C₂²	1	2	yes	complemented
12	#5020746	C₄	1	2	yes	complemented
13	#5020746	C₂²	1	2	yes	—
14	#657004	C₄	1	1	yes	characteristic, complemented
15	#657004	C₂²	1	1	yes	characteristic
16	#648360	C₄	1	1	yes	characteristic, complemented
17	#648360	C₂²	1	1	yes	characteristic
18	#619953	C₄	1	2	yes	complemented
19	#617359	C₂²	1	2	yes	—
20	#513778	C₄	1	1	yes	characteristic, complemented
21	C₂×#8547	C₂²	1	1	yes	characteristic, complemented
22	C₂×#8541	C₂²	1	1	yes	characteristic, complemented
23	#139830	C₂²	1	4	yes	complemented
24	#139718	C₂²	1	4	yes	complemented
25	#139504	C₂²	1	4	yes	complemented
26	#139484	C₂²	1	4	yes	complemented
27	#133822	C₂²	1	2	yes	—
28	#133822	C₂²	1	2	yes	complemented
29	#115690	C₂²	1	2	yes	—
30	#115690	C₂²	1	2	yes	complemented
31	C₂⁸	D₄	1	1	yes	characteristic
32	C₂³×#120	D₄	1	1	yes	characteristic
33	C₂³×#120	D₄	1	1	yes	characteristic
34	C₂³×#120	C₂³	1	1	yes	characteristic
35	C₂×#2812	C₂×C₄	1	2	yes	complemented
36	C₂×#2846	C₂×C₄	1	1	yes	characteristic
37	C₂×#2846	C₂×C₄	1	1	yes	characteristic
38	C₂×#2846	D₄	1	1	yes	characteristic
39	C₂×#2846	D₄	1	1	yes	characteristic
40	C₂×#2846	D₄	1	1	yes	characteristic
41	C₂×#2846	D₄	1	1	yes	characteristic
42	C₂×#2846	C₂³	1	1	yes	characteristic
43	C₂×#2846	C₂³	1	1	yes	characteristic
44	C₂×#2539	D₄	1	2	yes	—
45	C₂×#2483	C₂×C₄	1	1	yes	characteristic, complemented
46	C₂×#2433	D₄	1	2	yes	—
47	C₂×#2403	C₂×C₄	1	1	yes	characteristic, complemented
48	C₂×#2617	C₂×C₄	1	2	yes	—
49	C₂²×#379	C₂×C₄	1	1	yes	characteristic
50	C₂²×#379	C₂×C₄	1	1	yes	characteristic
51	C₂²×#379	D₄	1	1	yes	characteristic
52	#21746	C₂³	1	4	yes	—
53	#15480	C₂×C₄	1	2	yes	complemented
54	#15480	C₂×C₄	1	2	yes	complemented
55	#15283	C₂×C₄	1	2	yes	—
56	#15283	D₄	1	2	yes	—
57	#15283	D₄	1	2	yes	—
58	#15283	C₂³	1	2	yes	—
59	#15267	C₂×C₄	1	2	yes	complemented
60	#15267	C₂×C₄	1	2	yes	complemented
61	#15258	C₂×C₄	1	2	yes	—
62	#15258	D₄	1	2	yes	—
63	#15258	D₄	1	2	yes	—
64	#15258	C₂³	1	2	yes	—
65	#13903	D₄	1	2	yes	—
66	#13902	D₄	1	2	yes	—
67	#13827	C₂×C₄	1	2	yes	complemented
68	C₂²⋊C₄²	C₂×C₄	1	1	yes	characteristic, complemented
69	C₂²⋊C₄²	C₂×C₄	1	1	yes	characteristic, complemented
70	#13775	C₂×C₄	1	2	yes	—
71	#8547	C₂³	1	2	yes	—
72	#8541	C₂³	1	2	yes	—
73	C₂⁷	C₂²⋊C₄	1	1	yes	characteristic
74	C₂⁷	C₂²⋊C₄	1	1	yes	characteristic
75	C₂⁷	C₂²⋊C₄	1	1	yes	characteristic
76	C₂⁷	C₂²⋊C₄	1	1	yes	characteristic
77	C₂⁷	C₂×D₄	1	1	yes	characteristic
78	C₂⁷	C₂×D₄	1	1	yes	characteristic
79	C₂⁷	C₂×D₄	1	1	yes	characteristic
80	C₂⁴×D₄	C₂×D₄	1	2	yes	—
81	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
82	C₂²×#127	C₂²⋊C₄	1	2	yes	complemented
83	C₂²×#127	C₂×D₄	1	2	yes	—
84	C₂²×#121	C₂²⋊C₄	1	2	yes	complemented
85	C₂²×#120	C₂²⋊C₄	1	1	yes	characteristic
86	C₂²×#120	C₂²⋊C₄	1	1	yes	characteristic
87	C₂²×#120	C₂²⋊C₄	1	1	yes	characteristic
88	C₂²×#120	C₂²⋊C₄	1	1	yes	characteristic
89	C₂²×#120	C₂²×C₄	1	1	yes	characteristic
90	C₂²×#120	C₂×D₄	1	1	yes	characteristic
91	C₂²×#120	C₂×D₄	1	1	yes	characteristic
92	C₂²×#120	C₂×D₄	1	1	yes	characteristic
93	C₂²×#120	C₂×D₄	1	1	yes	characteristic
94	C₂²×#120	C₂×D₄	1	2	yes	—
95	C₂²×#120	C₂×D₄	1	1	yes	characteristic
96	C₂²×#120	C₂×D₄	1	1	yes	characteristic
97	C₂²×#120	C₂×D₄	1	2	yes	—
98	C₂²×#120	C₂⁴	1	1	yes	characteristic
99	C₂³×C₂²⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
100	C₂³×C₂²⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
101	C₂³×C₂²⋊C₄	C₂²×C₄	1	1	yes	characteristic
102	C₂³×C₂²⋊C₄	C₂×D₄	1	2	yes	—
103	C₂×#399	C₂²⋊C₄	1	2	yes	complemented
104	C₂×#399	C₂×D₄	1	2	yes	—
105	C₂×#397	C₂²⋊C₄	1	2	yes	complemented
106	C₂×#397	C₂²×C₄	1	2	yes	—
107	C₂×#392	C₂²⋊C₄	1	2	yes	complemented
108	C₂×#392	C₂²⋊C₄	1	2	yes	complemented
109	C₂×#392	C₂²×C₄	1	2	yes	—
110	C₂×#392	C₂×D₄	1	2	yes	—
111	C₂×#390	C₂²⋊C₄	1	2	yes	complemented
112	C₂×#390	C₂×D₄	1	2	yes	—
113	C₂×#380	C₂²⋊C₄	1	2	yes	complemented
114	C₂×#379	C₂²⋊C₄	1	1	yes	characteristic
115	C₂×#379	C₂²⋊C₄	1	1	yes	characteristic
116	C₂×#379	C₂²⋊C₄	1	1	yes	characteristic
117	C₂×#379	C₂²⋊C₄	1	1	yes	characteristic
118	C₂×#379	C₂²⋊C₄	1	1	yes	characteristic
119	C₂×#379	C₂²⋊C₄	1	1	yes	characteristic
120	C₂×#379	C₂²×C₄	1	1	yes	characteristic
121	C₂×#379	C₂²×C₄	1	1	yes	characteristic
122	C₂×#379	C₂²×C₄	1	1	yes	characteristic
123	C₂×#379	C₂²×C₄	1	1	yes	characteristic
124	C₂×#379	C₂×D₄	1	1	yes	characteristic
125	C₂×#379	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₂²⋊C₄	C₂²⋊C₄	1	2	yes	complemented
127	C₂×C₄×C₂²⋊C₄	C₂²⋊C₄	1	2	yes	complemented
128	#1438	C₂²⋊C₄	1	2	yes	complemented
129	#1438	C₂²⋊C₄	1	2	yes	complemented
130	#1438	C₂×D₄	1	2	yes	—
131	#1438	C₂×D₄	1	2	yes	—
132	#1437	C₂²⋊C₄	1	2	yes	complemented
133	#1437	C₂²⋊C₄	1	2	yes	complemented
134	#1437	C₂²⋊C₄	1	2	yes	complemented
135	#1437	C₂²⋊C₄	1	2	yes	complemented
136	#1437	C₂²×C₄	1	2	yes	—
137	#1437	C₂²×C₄	1	2	yes	—
138	#1437	C₂×D₄	1	2	yes	—
139	#1437	C₂×D₄	1	2	yes	—
140	C₂⁶	#99	1	2	yes	—
141	C₂⁶	#99	1	2	yes	—
142	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
143	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
144	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
145	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
146	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
147	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
148	C₂⁶	#120	1	2	yes	—
149	C₂⁶	#120	1	1	yes	characteristic
150	C₂⁶	#120	1	2	yes	—
151	C₂⁶	C₂²×D₄	1	1	yes	characteristic
152	C₂³×D₄	C₂×C₂²⋊C₄	1	2	yes	—
153	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
154	C₂³×D₄	C₂×C₂²⋊C₄	1	2	yes	complemented
155	C₂³×D₄	#120	1	1	yes	characteristic
156	C₂³×D₄	#120	1	1	yes	characteristic
157	C₂³×D₄	#120	1	2	yes	—
158	C₂³×D₄	#120	1	2	yes	—
159	C₂³×D₄	C₂²×D₄	1	2	yes	—
160	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
161	C₂⁴×C₄	C₂×C₂²⋊C₄	1	2	yes	—
162	C₂⁴×C₄	#120	1	1	yes	characteristic
163	C₂²×C₄⋊C₄	C₂×C₂²⋊C₄	1	2	yes	complemented
164	C₂²×C₂²⋊C₄	#99	1	2	yes	—
165	C₂²×C₂²⋊C₄	#99	1	2	yes	—
166	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
167	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
168	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
169	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
170	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
171	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
172	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
173	C₂²×C₂²⋊C₄	#120	1	2	yes	—
174	C₂²×C₂²⋊C₄	#120	1	2	yes	—
175	C₂²×C₂²⋊C₄	C₂³×C₄	1	1	yes	characteristic
176	C₂²×C₂²⋊C₄	C₂²×D₄	1	2	yes	—
177	C₂²×C₄²	C₂×C₂²⋊C₄	1	2	yes	complemented
178	C₂²×C₄²	C₂×C₂²⋊C₄	1	2	yes	complemented
179	C₂×#95	C₂×C₂²⋊C₄	1	2	yes	complemented
180	C₂×#95	C₂×C₂²⋊C₄	1	2	yes	complemented
181	C₂×#95	C₂×C₂²⋊C₄	1	2	yes	complemented
182	C₂×#95	C₂×C₂²⋊C₄	1	2	yes	complemented
183	C₂⁵	#353	1	8	yes	complemented
184	C₂⁵	#379	1	1	yes	characteristic
185	C₂⁵	#379	1	2	yes	—
186	C₂⁵	#379	1	1	yes	characteristic
187	C₂⁵	#379	1	1	yes	characteristic
188	C₂⁵	C₂×#99	1	1	yes	characteristic
189	C₂⁵	C₂×#99	1	1	yes	characteristic
190	C₂⁵	C₂×#99	1	1	yes	characteristic
191	C₂⁵	C₂×#99	1	1	yes	characteristic
192	C₂⁵	C₂×#99	1	2	yes	—
193	C₂⁵	C₂×#99	1	2	yes	—
194	C₂⁵	B(4,𝔽₂)	1	2	yes	—
195	C₂⁵	B(4,𝔽₂)	1	2	yes	—
196	C₂⁵	#458	1	2	yes	—
197	C₂⁵	#458	1	2	yes	—
198	C₂⁵	C₂²×C₂²⋊C₄	1	1	yes	characteristic
199	C₂⁵	C₂×#120	1	2	yes	—
200	C₂⁵	C₂×#120	1	1	yes	characteristic
201	C₂³×C₄	#379	1	1	yes	characteristic
202	C₂³×C₄	#379	1	2	yes	—
203	C₂³×C₄	C₂²×C₂²⋊C₄	1	2	yes	—
204	C₂³×C₄	C₂×#120	1	1	yes	characteristic
205	C₂×C₂²⋊C₄	C₂×#99	1	2	yes	—
206	C₂×C₂²⋊C₄	C₂×#99	1	2	yes	—
207	C₂⁴	#1781	1	1	yes	characteristic
208	C₂⁴	#1781	1	1	yes	characteristic
209	C₂⁴	#1781	1	1	yes	characteristic
210	C₂⁴	#1781	1	1	yes	characteristic
211	C₂⁴	#1790	1	2	yes	—
212	C₂⁴	#1790	1	2	yes	—
213	C₂⁴	#1790	1	2	yes	—
214	C₂⁴	#1790	1	2	yes	—
215	C₂⁴	C₂×#353	1	4	yes	—
216	C₂⁴	C₂×#353	1	4	yes	—
217	C₂⁴	C₂×#353	1	4	yes	—
218	C₂⁴	C₂×#379	1	1	yes	characteristic
219	C₂⁴	C₂²×#99	1	1	yes	characteristic
220	C₂⁴	C₂²×#99	1	1	yes	characteristic
221	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
222	C₂⁴	C₂×#458	1	2	yes	—
223	C₂³	#8534	1	2	yes	—
224	C₂³	#8534	1	1	yes	characteristic
225	C₂³	#8534	1	1	yes	characteristic
226	C₂³	#12693	1	2	yes	—
227	C₂³	#12693	1	2	yes	—
228	C₂³	#12707	1	2	yes	—
229	C₂³	#12753	1	4	yes	—
230	C₂³	C₂×#1781	1	1	yes	characteristic
231	C₂³	C₂×#1790	1	2	yes	—
232	C₂³	C₂²×#353	1	2	yes	—
233	C₂³	#33551	1	4	yes	—
234	C₂²	#133290	1	2	yes	—
235	C₂²	#133290	1	2	yes	—
236	C₂²	C₂×#8534	1	1	yes	characteristic
237	C₂²	C₂×#12693	1	1	yes	characteristic
238	C₂²	C₂×#12707	1	1	yes	characteristic
239	C₂²	C₂×#12753	1	2	yes	—
240	C₂²	#493582	1	2	yes	—
241	C₂	#72218462	1	2	yes	—
242	C₂	C₂×#133290	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#8534
commutator	C₂²×C₂²⋊C₄	C₂³×C₄
Frattini, cosocle	C₂²×#120	C₂⁴
prime radical	#72241149	C₂
LCS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂³×#120 ◁ G
LECS	G ▷ C₂²×#120 ▷ C₂⁴ ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #72241063	C₂	semidirect
3. #72233369	C₂	semidirect
4. #72232761	C₂	semidirect
5. #72232651	C₂	semidirect
6. C₂×#133822	C₂	semidirect
7. C₂×#115690	C₂	semidirect
8. #72215653	C₂	semidirect
9. C₂×#37436	C₄	semidirect
11. C₂×#21746	C₂²	semidirect
12. #5020746	C₄	semidirect
14. #657004	C₄	semidirect
16. #648360	C₄	semidirect
18. #619953	C₄	semidirect
20. #513778	C₄	semidirect
21. C₂×#8547	C₂²	semidirect
22. C₂×#8541	C₂²	semidirect
23. #139830	C₂²	semidirect
24. #139718	C₂²	semidirect
25. #139504	C₂²	semidirect
26. #139484	C₂²	semidirect
28. #133822	C₂²	semidirect
30. #115690	C₂²	semidirect
35. C₂×#2812	C₂×C₄	semidirect
45. C₂×#2483	C₂×C₄	semidirect
47. C₂×#2403	C₂×C₄	semidirect
53. #15480	C₂×C₄	semidirect
54. #15480	C₂×C₄	semidirect
59. #15267	C₂×C₄	semidirect
60. #15267	C₂×C₄	semidirect
67. #13827	C₂×C₄	semidirect
68. C₂²⋊C₄²	C₂×C₄	semidirect
69. C₂²⋊C₄²	C₂×C₄	semidirect
82. C₂²×#127	C₂²⋊C₄	semidirect
84. C₂²×#121	C₂²⋊C₄	semidirect
103. C₂×#399	C₂²⋊C₄	semidirect
105. C₂×#397	C₂²⋊C₄	semidirect
107. C₂×#392	C₂²⋊C₄	semidirect
108. C₂×#392	C₂²⋊C₄	semidirect
111. C₂×#390	C₂²⋊C₄	semidirect
113. C₂×#380	C₂²⋊C₄	semidirect
126. C₂×C₄×C₂²⋊C₄	C₂²⋊C₄	semidirect
127. C₂×C₄×C₂²⋊C₄	C₂²⋊C₄	semidirect
128. #1438	C₂²⋊C₄	semidirect
129. #1438	C₂²⋊C₄	semidirect
132. #1437	C₂²⋊C₄	semidirect
133. #1437	C₂²⋊C₄	semidirect
134. #1437	C₂²⋊C₄	semidirect
135. #1437	C₂²⋊C₄	semidirect
154. C₂³×D₄	C₂×C₂²⋊C₄	semidirect
163. C₂²×C₄⋊C₄	C₂×C₂²⋊C₄	semidirect
177. C₂²×C₄²	C₂×C₂²⋊C₄	semidirect
178. C₂²×C₄²	C₂×C₂²⋊C₄	semidirect
179. C₂×#95	C₂×C₂²⋊C₄	semidirect
180. C₂×#95	C₂×C₂²⋊C₄	semidirect
181. C₂×#95	C₂×C₂²⋊C₄	semidirect
182. C₂×#95	C₂×C₂²⋊C₄	semidirect
183. C₂⁵	#353	semidirect

Maximal subgroups

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

subgroup	index
#72241149	2
#72241063	2
#72233369	2
#72232761	2
#72232651	2
C₂×#133822	2
C₂×#115690	2
#72215653	2