Subgroups — #72334283

Order 2048.

At least 9377 conjugacy classes of subgroups; only the 444 normal subgroups are shown.

The subgroup lattice has 212 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72317049	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#97555	C₂	1	1	yes	maximal, characteristic, complemented
3	#72235059	C₂	1	4	yes	maximal, complemented
4	C₂×#97550	C₂	1	1	yes	maximal, characteristic, complemented
5	#72234682	C₂	1	1	yes	maximal, characteristic, complemented
6	#72221174	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#417744	C₂	1	1	yes	maximal, characteristic, complemented
8	#72218776	C₂	1	4	yes	maximal, complemented
9	C₂×#449475	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#2846	C₂²	1	1	yes	characteristic
11	C₂²×#2846	C₂²	1	2	yes	—
12	C₂²×#2846	C₂²	1	2	yes	complemented
13	C₂×#35390	C₂²	1	1	yes	characteristic
14	C₂×#35346	C₂²	1	1	yes	characteristic
15	#5955303	C₂²	1	4	yes	—
16	C₂×#14679	C₂²	1	2	yes	complemented
17	#515326	C₂²	1	1	yes	characteristic
18	C₂×#10965	C₂²	1	1	yes	characteristic
19	C₂×#10965	C₂²	1	2	yes	—
20	C₂×#10965	C₂²	1	4	yes	—
21	C₂×#10965	C₂²	1	1	yes	characteristic
22	C₂×#10965	C₂²	1	1	yes	characteristic
23	C₂×#10965	C₂²	1	2	yes	complemented
24	C₂×#10965	C₂²	1	4	yes	complemented
25	#123513	C₂²	1	2	yes	complemented
26	#123513	C₂²	1	4	yes	complemented
27	C₂⁸	C₂³	1	1	yes	characteristic
28	C₂³×H(5,𝔽₂)	D₄	1	1	yes	characteristic, complemented
29	C₂²×#534	D₄	1	2	yes	complemented
30	C₂²×#534	D₄	1	1	yes	characteristic, complemented
31	C₂²×#534	D₄	1	4	yes	complemented
32	C₂³×#120	C₂³	1	2	yes	—
33	C₂³×#120	C₂³	1	2	yes	—
34	C₂³×#120	C₂³	1	1	yes	characteristic
35	C₂×#2846	D₄	1	4	yes	complemented
36	C₂×#2315	D₄	1	1	yes	characteristic, complemented
37	C₂×#2308	D₄	1	1	yes	characteristic, complemented
38	C₂×#2308	D₄	1	2	yes	complemented
39	C₂×#2308	D₄	1	2	yes	complemented
40	C₂×#2308	D₄	1	4	yes	complemented
41	C₂×#2308	D₄	1	4	yes	complemented
42	C₂²×#379	C₂³	1	2	yes	—
43	C₂²×#379	C₂³	1	2	yes	—
44	C₂²×#379	C₂³	1	1	yes	characteristic
45	C₂²×#379	C₂³	1	2	yes	—
46	C₂²×#379	C₂³	1	2	yes	—
47	C₂×#2273	D₄	1	1	yes	characteristic, complemented
48	C₂×#2273	D₄	1	1	yes	characteristic, complemented
49	#13794	D₄	1	4	yes	complemented
50	#10965	D₄	1	4	yes	complemented
51	#10965	D₄	1	4	yes	complemented
52	C₂⁷	C₂×D₄	1	2	yes	—
53	C₂⁷	C₂⁴	1	1	yes	characteristic
54	C₂⁴×D₄	C₂×D₄	1	2	yes	—
55	C₂⁴×D₄	C₂×D₄	1	4	yes	—
56	C₂⁴×D₄	C₂×D₄	1	4	yes	—
57	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
58	C₂⁴×D₄	C₂×D₄	1	2	yes	—
59	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
60	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
61	C₂²×#120	C₂×D₄	1	2	yes	—
62	C₂²×#120	C₂×D₄	1	2	yes	—
63	C₂²×#120	C₂×D₄	1	2	yes	—
64	C₂²×#120	#41	1	2	yes	—
65	C₂³×C₂²⋊C₄	C₂×D₄	1	2	yes	—
66	C₂³×C₂²⋊C₄	C₂×D₄	1	4	yes	—
67	C₂³×C₂²⋊C₄	C₂×D₄	1	4	yes	—
68	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
69	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
70	C₂³×C₂²⋊C₄	C₂×D₄	1	2	yes	—
71	C₂³×C₂²⋊C₄	C₂×D₄	1	2	yes	—
72	C₂³×C₂²⋊C₄	C₂×D₄	1	4	yes	—
73	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
74	C₂³×C₂²⋊C₄	C₂×D₄	1	2	yes	—
75	C₂³×C₂²⋊C₄	C₂×D₄	1	4	yes	—
76	C₂×#379	C₂×D₄	1	4	yes	—
77	C₂×#379	C₂×D₄	1	4	yes	—
78	C₂×#379	C₂×D₄	1	2	yes	—
79	C₂⁶	#120	1	2	yes	—
80	C₂⁶	#120	1	1	yes	characteristic
81	C₂⁶	#120	1	2	yes	—
82	C₂⁶	C₂²×D₄	1	1	yes	characteristic
83	C₂⁶	C₂²×D₄	1	1	yes	characteristic
84	C₂⁶	C₂²×D₄	1	1	yes	characteristic
85	C₂⁶	C₂²×D₄	1	4	yes	—
86	C₂⁶	C₂²×D₄	1	1	yes	characteristic
87	C₂⁶	C₂²×D₄	1	1	yes	characteristic
88	C₂⁶	C₂²×D₄	1	1	yes	characteristic
89	C₂⁶	C₂×#41	1	1	yes	characteristic
90	C₂⁶	H(5,𝔽₂)	1	4	yes	—
91	C₂³×D₄	#120	1	2	yes	—
92	C₂³×D₄	#120	1	2	yes	—
93	C₂³×D₄	#121	1	2	yes	—
94	C₂³×D₄	#121	1	2	yes	—
95	C₂³×D₄	#121	1	2	yes	—
96	C₂³×D₄	#127	1	2	yes	—
97	C₂⁴×C₄	#120	1	2	yes	—
98	C₂⁴×C₄	#120	1	4	yes	—
99	C₂⁴×C₄	#120	1	1	yes	characteristic
100	C₂⁴×C₄	#120	1	1	yes	characteristic
101	C₂⁴×C₄	#120	1	2	yes	—
102	C₂⁴×C₄	#120	1	4	yes	—
103	C₂⁴×C₄	#120	1	1	yes	characteristic
104	C₂⁴×C₄	#120	1	4	yes	—
105	C₂⁴×C₄	#120	1	2	yes	—
106	C₂⁴×C₄	#120	1	2	yes	—
107	C₂²×C₂²⋊C₄	#120	1	2	yes	—
108	C₂²×C₂²⋊C₄	#120	1	2	yes	—
109	C₂²×C₂²⋊C₄	#121	1	2	yes	—
110	C₂²×C₂²⋊C₄	#121	1	2	yes	—
111	C₂²×C₂²⋊C₄	#121	1	2	yes	—
112	C₂²×C₂²⋊C₄	#127	1	2	yes	—
113	C₂⁵	#392	1	2	yes	—
114	C₂⁵	B(4,𝔽₂)	1	4	yes	—
115	C₂⁵	C₂×#120	1	1	yes	characteristic
116	C₂⁵	C₂×#120	1	2	yes	—
117	C₂⁵	C₂×#120	1	2	yes	—
118	C₂⁵	C₂×#120	1	2	yes	—
119	C₂⁵	C₂×#120	1	1	yes	characteristic
120	C₂⁵	C₂×#120	1	1	yes	characteristic
121	C₂⁵	C₂×#121	1	1	yes	characteristic
122	C₂⁵	C₂×#121	1	1	yes	characteristic
123	C₂⁵	C₂×#121	1	1	yes	characteristic
124	C₂⁵	C₂×#127	1	1	yes	characteristic
125	C₂⁵	#534	1	2	yes	—
126	C₂⁵	#534	1	2	yes	—
127	C₂⁵	#535	1	2	yes	—
128	C₂⁵	D₄²	1	4	yes	—
129	C₂⁵	D₄²	1	4	yes	—
130	C₂⁵	D₄²	1	4	yes	—
131	C₂⁵	#546	1	4	yes	—
132	C₂³×C₄	#392	1	2	yes	—
133	C₂³×C₄	#392	1	2	yes	—
134	C₂³×C₄	#392	1	2	yes	—
135	C₂³×C₄	B(4,𝔽₂)	1	4	yes	—
136	C₂³×C₄	B(4,𝔽₂)	1	2	yes	—
137	C₂³×C₄	B(4,𝔽₂)	1	2	yes	—
138	C₂³×C₄	B(4,𝔽₂)	1	8	yes	complemented
139	C₂³×C₄	B(4,𝔽₂)	1	4	yes	complemented
140	C₂⁴	C₂×#392	1	1	yes	characteristic
141	C₂⁴	#2403	1	2	yes	—
142	C₂⁴	#2403	1	2	yes	—
143	C₂⁴	#2408	1	2	yes	—
144	C₂⁴	#2410	1	2	yes	—
145	C₂⁴	#2410	1	2	yes	—
146	C₂⁴	#2433	1	2	yes	—
147	C₂⁴	#2613	1	1	yes	characteristic
148	C₂⁴	#2846	1	1	yes	characteristic
149	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
150	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
151	C₂⁴	C₂×B(4,𝔽₂)	1	4	yes	—
152	C₂⁴	C₂×B(4,𝔽₂)	1	1	yes	characteristic
153	C₂⁴	C₂×B(4,𝔽₂)	1	1	yes	characteristic
154	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
155	C₂⁴	#3025	1	2	yes	—
156	C₂⁴	#3026	1	2	yes	—
157	C₂⁴	#3026	1	4	yes	—
158	C₂⁴	#3026	1	2	yes	—
159	C₂⁴	#3026	1	1	yes	characteristic
160	C₂⁴	#3026	1	1	yes	characteristic
161	C₂⁴	#3026	1	2	yes	—
162	C₂⁴	#3027	1	2	yes	—
163	C₂²×C₄	#2021	1	4	yes	complemented
164	C₂³	#14715	1	1	yes	characteristic
165	C₂³	C₂×#2021	1	1	yes	characteristic
166	C₂³	#23444	1	1	yes	characteristic
167	C₂³	#23900	1	2	yes	—
168	C₂³	#23900	1	2	yes	—
169	C₂³	#23903	1	2	yes	—
170	C₂³	#23903	1	2	yes	—
171	C₂³	#24260	1	2	yes	—
172	C₂³	#24262	1	2	yes	—
173	C₂³	#24404	1	4	yes	—
174	C₂³	#24404	1	2	yes	—
175	C₂³	#24404	1	2	yes	—
176	C₂³	#25771	1	2	yes	—
177	C₂³	#25773	1	2	yes	—
178	C₂³	#29634	1	1	yes	characteristic
179	C₂³	#29634	1	1	yes	characteristic
180	C₂³	#32898	1	2	yes	—
181	C₂³	#32898	1	2	yes	—
182	C₂³	#32898	1	4	yes	—
183	C₂³	#32898	1	1	yes	characteristic
184	C₂³	#32898	1	2	yes	—
185	C₂³	#32898	1	1	yes	characteristic
186	C₂³	#32899	1	2	yes	—
187	C₂²	#347042	1	4	yes	—
188	C₂²	#347042	1	2	yes	—
189	C₂²	#347042	1	1	yes	characteristic
190	C₂²	#347042	1	1	yes	characteristic
191	C₂²	#347043	1	4	yes	—
192	C₂²	#347043	1	4	yes	—
193	C₂²	#347043	1	2	yes	—
194	C₂²	#347045	1	1	yes	characteristic
195	C₂²	#347047	1	4	yes	—
196	C₂²	#347047	1	1	yes	characteristic
197	C₂²	#347047	1	2	yes	—
198	C₂²	#347047	1	2	yes	—
199	C₂²	#393929	1	1	yes	characteristic
200	C₂²	#393934	1	2	yes	—
201	C₂²	#394066	1	2	yes	—
202	C₂²	#394066	1	2	yes	—
203	C₂	#72219236	1	4	yes	—
204	C₂	#72219240	1	1	yes	characteristic
205	C₂	#72225663	1	1	yes	characteristic
206	C₂	#72266447	1	1	yes	characteristic
207	C₂	#72266457	1	2	yes	—
208	C₂	#72266458	1	2	yes	—
209	C₂	#72334274	1	2	yes	—
210	C₂	#72334281	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴	#2846
commutator, Frattini, cosocle	C₂⁷	C₂⁴
LCS	G ▷ C₂⁷ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂⁷ ◁ G
LECS	G ▷ C₂⁷ ▷ C₂³ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72317049	C₂	semidirect
2. C₂×#97555	C₂	semidirect
3. #72235059	C₂	semidirect
4. C₂×#97550	C₂	semidirect
5. #72234682	C₂	semidirect
6. #72221174	C₂	semidirect
7. C₂×#417744	C₂	semidirect
8. #72218776	C₂	semidirect
9. C₂×#449475	C₂	semidirect
12. C₂²×#2846	C₂²	semidirect
16. C₂×#14679	C₂²	semidirect
23. C₂×#10965	C₂²	semidirect
24. C₂×#10965	C₂²	semidirect
25. #123513	C₂²	semidirect
26. #123513	C₂²	semidirect
28. C₂³×H(5,𝔽₂)	D₄	semidirect
29. C₂²×#534	D₄	semidirect
30. C₂²×#534	D₄	semidirect
31. C₂²×#534	D₄	semidirect
35. C₂×#2846	D₄	semidirect
36. C₂×#2315	D₄	semidirect
37. C₂×#2308	D₄	semidirect
38. C₂×#2308	D₄	semidirect
39. C₂×#2308	D₄	semidirect
40. C₂×#2308	D₄	semidirect
41. C₂×#2308	D₄	semidirect
47. C₂×#2273	D₄	semidirect
48. C₂×#2273	D₄	semidirect
49. #13794	D₄	semidirect
50. #10965	D₄	semidirect
51. #10965	D₄	semidirect
138. C₂³×C₄	B(4,𝔽₂)	semidirect
139. C₂³×C₄	B(4,𝔽₂)	semidirect
163. C₂²×C₄	#2021	semidirect

Maximal subgroups

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

subgroup	index
#72317049	2
C₂×#97555	2
#72235059	2
C₂×#97550	2
#72234682	2
#72221174	2
C₂×#417744	2
#72218776	2
C₂×#449475	2