Subgroups — #72334275

Order 2048.

At least 13899 conjugacy classes of subgroups; only the 440 normal subgroups are shown.

The subgroup lattice has 263 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72317046	C₂	1	2	yes	maximal, complemented
2	C₂×#97551	C₂	1	1	yes	maximal, characteristic, complemented
3	#72235059	C₂	1	2	yes	maximal, complemented
4	#72235058	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#97550	C₂	1	1	yes	maximal, characteristic, complemented
6	#72234618	C₂	1	1	yes	maximal, characteristic, complemented
7	#72221168	C₂	1	2	yes	maximal, complemented
8	C₂×#417493	C₂	1	1	yes	maximal, characteristic, complemented
9	#72218789	C₂	1	1	yes	maximal, characteristic, complemented
10	#72218776	C₂	1	2	yes	maximal, complemented
11	C₂×#449475	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#2846	C₂²	1	1	yes	characteristic
13	C₂²×#2846	C₂²	1	2	yes	—
14	C₂²×#2846	C₂²	1	1	yes	characteristic, complemented
15	C₂×#35348	C₂²	1	1	yes	characteristic
16	C₂×#35346	C₂²	1	1	yes	characteristic
17	#5955309	C₂²	1	2	yes	—
18	#5955303	C₂²	1	2	yes	—
19	C₂×#14679	C₂²	1	1	yes	characteristic, complemented
20	C₂×#14679	C₂²	1	2	yes	complemented
21	#515285	C₂²	1	1	yes	characteristic
22	C₂×#11196	C₂²	1	1	yes	characteristic, complemented
23	C₂×#11196	C₂²	1	2	yes	complemented
24	C₂×#10965	C₂²	1	1	yes	characteristic
25	C₂×#10965	C₂²	1	2	yes	—
26	C₂×#10965	C₂²	1	2	yes	—
27	C₂×#10965	C₂²	1	2	yes	—
28	C₂×#10965	C₂²	1	1	yes	characteristic
29	C₂×#10965	C₂²	1	1	yes	characteristic
30	C₂×#10965	C₂²	1	1	yes	characteristic, complemented
31	C₂×#10965	C₂²	1	2	yes	complemented
32	#123513	C₂²	1	2	yes	complemented
33	#123513	C₂²	1	2	yes	complemented
34	#123513	C₂²	1	2	yes	complemented
35	C₂⁸	C₂³	1	1	yes	characteristic
36	C₂³×H(5,𝔽₂)	D₄	1	1	yes	characteristic, complemented
37	C₂²×#534	D₄	1	1	yes	characteristic, complemented
38	C₂²×#534	D₄	1	2	yes	complemented
39	C₂³×#120	C₂³	1	2	yes	—
40	C₂³×#120	C₂³	1	1	yes	characteristic
41	C₂³×#120	C₂³	1	1	yes	characteristic
42	C₂²×#518	D₄	1	1	yes	characteristic, complemented
43	C₂²×#518	D₄	1	1	yes	characteristic, complemented
44	C₂×#2846	D₄	1	4	yes	complemented
45	C₂×#2315	D₄	1	1	yes	characteristic, complemented
46	C₂×#2315	D₄	1	2	yes	complemented
47	C₂×#2308	D₄	1	1	yes	characteristic, complemented
48	C₂×#2308	D₄	1	1	yes	characteristic, complemented
49	C₂×#2308	D₄	1	2	yes	complemented
50	C₂×#2308	D₄	1	1	yes	characteristic, complemented
51	C₂×#2308	D₄	1	2	yes	complemented
52	C₂×#2308	D₄	1	2	yes	complemented
53	C₂×#2308	D₄	1	2	yes	complemented
54	C₂×#2308	D₄	1	2	yes	complemented
55	C₂²×#379	C₂³	1	2	yes	—
56	C₂²×#379	C₂³	1	1	yes	characteristic
57	C₂²×#379	C₂³	1	1	yes	characteristic
58	C₂²×#379	C₂³	1	2	yes	—
59	C₂²×#379	C₂³	1	1	yes	characteristic
60	C₂²×#379	C₂³	1	1	yes	characteristic
61	C₂²×#379	C₂³	1	2	yes	—
62	#13794	D₄	1	2	yes	complemented
63	#13794	D₄	1	2	yes	complemented
64	#13794	D₄	1	2	yes	complemented
65	#10965	D₄	1	4	yes	complemented
66	#10965	D₄	1	4	yes	complemented
67	C₂⁷	C₂×D₄	1	2	yes	—
68	C₂⁷	C₂⁴	1	1	yes	characteristic
69	C₂⁴×D₄	C₂×D₄	1	2	yes	—
70	C₂⁴×D₄	C₂×D₄	1	2	yes	—
71	C₂⁴×D₄	C₂×D₄	1	2	yes	—
72	C₂⁴×D₄	C₂×D₄	1	2	yes	—
73	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
74	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
75	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
76	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
77	C₂²×#120	C₂×D₄	1	2	yes	—
78	C₂²×#120	C₂×D₄	1	2	yes	—
79	C₂²×#120	C₂×D₄	1	2	yes	—
80	C₂²×#120	#41	1	2	yes	—
81	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
82	C₂³×C₂²⋊C₄	C₂×D₄	1	2	yes	—
83	C₂³×C₂²⋊C₄	C₂×D₄	1	2	yes	—
84	C₂³×C₂²⋊C₄	C₂×D₄	1	2	yes	—
85	C₂³×C₂²⋊C₄	C₂×D₄	1	2	yes	—
86	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
87	C₂³×C₂²⋊C₄	C₂×D₄	1	2	yes	—
88	C₂³×C₂²⋊C₄	C₂×D₄	1	2	yes	—
89	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
90	C₂³×C₂²⋊C₄	C₂×D₄	1	2	yes	—
91	C₂³×C₂²⋊C₄	C₂×D₄	1	2	yes	—
92	C₂³×C₂²⋊C₄	C₂×D₄	1	2	yes	—
93	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
94	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
95	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
96	C₂³×C₂²⋊C₄	C₂×D₄	1	2	yes	—
97	C₂³×C₂²⋊C₄	C₂×D₄	1	2	yes	—
98	C₂³×C₂²⋊C₄	C₂×D₄	1	2	yes	—
99	C₂×#379	C₂×D₄	1	4	yes	—
100	C₂×#379	C₂×D₄	1	2	yes	—
101	C₂×#379	C₂×D₄	1	2	yes	—
102	C₂×#379	C₂×D₄	1	2	yes	—
103	C₂⁶	#120	1	1	yes	characteristic
104	C₂⁶	#120	1	2	yes	—
105	C₂⁶	#120	1	1	yes	characteristic
106	C₂⁶	C₂²×D₄	1	1	yes	characteristic
107	C₂⁶	C₂²×D₄	1	1	yes	characteristic
108	C₂⁶	C₂²×D₄	1	1	yes	characteristic
109	C₂⁶	C₂²×D₄	1	2	yes	—
110	C₂⁶	C₂²×D₄	1	2	yes	—
111	C₂⁶	C₂²×D₄	1	1	yes	characteristic
112	C₂⁶	C₂²×D₄	1	1	yes	characteristic
113	C₂⁶	C₂²×D₄	1	1	yes	characteristic
114	C₂⁶	C₂×#41	1	1	yes	characteristic
115	C₂⁶	H(5,𝔽₂)	1	2	yes	—
116	C₂⁶	H(5,𝔽₂)	1	2	yes	—
117	C₂³×D₄	#120	1	2	yes	—
118	C₂³×D₄	#120	1	2	yes	—
119	C₂³×D₄	#121	1	2	yes	—
120	C₂³×D₄	#121	1	2	yes	—
121	C₂³×D₄	#121	1	2	yes	—
122	C₂³×D₄	#127	1	2	yes	—
123	C₂⁴×C₄	#120	1	1	yes	characteristic
124	C₂⁴×C₄	#120	1	2	yes	—
125	C₂⁴×C₄	#120	1	2	yes	—
126	C₂⁴×C₄	#120	1	1	yes	characteristic
127	C₂⁴×C₄	#120	1	1	yes	characteristic
128	C₂⁴×C₄	#120	1	1	yes	characteristic
129	C₂⁴×C₄	#120	1	2	yes	—
130	C₂⁴×C₄	#120	1	2	yes	—
131	C₂⁴×C₄	#120	1	2	yes	—
132	C₂⁴×C₄	#120	1	2	yes	—
133	C₂⁴×C₄	#120	1	2	yes	—
134	C₂⁴×C₄	#120	1	1	yes	characteristic
135	C₂⁴×C₄	#120	1	1	yes	characteristic
136	C₂⁴×C₄	#120	1	2	yes	—
137	C₂⁴×C₄	#120	1	2	yes	—
138	C₂²×C₂²⋊C₄	#120	1	2	yes	—
139	C₂²×C₂²⋊C₄	#120	1	2	yes	—
140	C₂²×C₂²⋊C₄	#121	1	2	yes	—
141	C₂²×C₂²⋊C₄	#121	1	2	yes	—
142	C₂²×C₂²⋊C₄	#121	1	2	yes	—
143	C₂²×C₂²⋊C₄	#127	1	2	yes	—
144	C₂⁵	#392	1	2	yes	—
145	C₂⁵	B(4,𝔽₂)	1	4	yes	—
146	C₂⁵	B(4,𝔽₂)	1	2	yes	—
147	C₂⁵	B(4,𝔽₂)	1	2	yes	—
148	C₂⁵	C₂×#120	1	1	yes	characteristic
149	C₂⁵	C₂×#120	1	2	yes	—
150	C₂⁵	C₂×#120	1	1	yes	characteristic
151	C₂⁵	C₂×#120	1	1	yes	characteristic
152	C₂⁵	C₂×#120	1	2	yes	—
153	C₂⁵	C₂×#120	1	1	yes	characteristic
154	C₂⁵	C₂×#120	1	1	yes	characteristic
155	C₂⁵	C₂×#121	1	1	yes	characteristic
156	C₂⁵	C₂×#121	1	1	yes	characteristic
157	C₂⁵	C₂×#121	1	1	yes	characteristic
158	C₂⁵	C₂×#127	1	1	yes	characteristic
159	C₂⁵	#534	1	1	yes	characteristic
160	C₂⁵	#534	1	1	yes	characteristic
161	C₂⁵	#534	1	2	yes	—
162	C₂⁵	#535	1	2	yes	—
163	C₂⁵	D₄²	1	2	yes	—
164	C₂⁵	D₄²	1	2	yes	—
165	C₂⁵	D₄²	1	2	yes	—
166	C₂⁵	D₄²	1	2	yes	—
167	C₂⁵	D₄²	1	2	yes	—
168	C₂⁵	D₄²	1	2	yes	—
169	C₂⁵	#546	1	2	yes	—
170	C₂⁵	#546	1	2	yes	—
171	C₂³×C₄	#392	1	2	yes	—
172	C₂³×C₄	#392	1	2	yes	—
173	C₂³×C₄	#392	1	2	yes	—
174	C₂³×C₄	B(4,𝔽₂)	1	4	yes	—
175	C₂³×C₄	B(4,𝔽₂)	1	4	yes	complemented
176	C₂³×C₄	B(4,𝔽₂)	1	4	yes	complemented
177	C₂⁴	#2021	1	4	yes	complemented
178	C₂⁴	C₂×#392	1	1	yes	characteristic
179	C₂⁴	#2403	1	2	yes	—
180	C₂⁴	#2403	1	1	yes	characteristic
181	C₂⁴	#2403	1	1	yes	characteristic
182	C₂⁴	#2408	1	2	yes	—
183	C₂⁴	#2410	1	2	yes	—
184	C₂⁴	#2410	1	1	yes	characteristic
185	C₂⁴	#2410	1	1	yes	characteristic
186	C₂⁴	#2433	1	2	yes	—
187	C₂⁴	#2613	1	1	yes	characteristic
188	C₂⁴	#2846	1	1	yes	characteristic
189	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
190	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
191	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
192	C₂⁴	C₂×B(4,𝔽₂)	1	1	yes	characteristic
193	C₂⁴	C₂×B(4,𝔽₂)	1	1	yes	characteristic
194	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
195	C₂⁴	#3025	1	2	yes	—
196	C₂⁴	#3025	1	2	yes	—
197	C₂⁴	#3026	1	2	yes	—
198	C₂⁴	#3026	1	2	yes	—
199	C₂⁴	#3026	1	2	yes	—
200	C₂⁴	#3026	1	1	yes	characteristic
201	C₂⁴	#3026	1	1	yes	characteristic
202	C₂⁴	#3026	1	2	yes	—
203	C₂⁴	#3027	1	2	yes	—
204	C₂⁴	#3027	1	2	yes	—
205	C₂³	#14715	1	1	yes	characteristic
206	C₂³	C₂×#2021	1	1	yes	characteristic
207	C₂³	#23444	1	1	yes	characteristic
208	C₂³	#23900	1	1	yes	characteristic
209	C₂³	#23900	1	2	yes	—
210	C₂³	#23902	1	1	yes	characteristic
211	C₂³	#23903	1	1	yes	characteristic
212	C₂³	#23903	1	2	yes	—
213	C₂³	#23904	1	1	yes	characteristic
214	C₂³	#24260	1	2	yes	—
215	C₂³	#24262	1	2	yes	—
216	C₂³	#24404	1	2	yes	—
217	C₂³	#24404	1	2	yes	—
218	C₂³	#24404	1	2	yes	—
219	C₂³	#24406	1	2	yes	—
220	C₂³	#25771	1	2	yes	—
221	C₂³	#25773	1	2	yes	—
222	C₂³	#29634	1	1	yes	characteristic
223	C₂³	#29634	1	1	yes	characteristic
224	C₂³	#32898	1	2	yes	—
225	C₂³	#32898	1	2	yes	—
226	C₂³	#32898	1	2	yes	—
227	C₂³	#32898	1	1	yes	characteristic
228	C₂³	#32898	1	1	yes	characteristic
229	C₂³	#32898	1	2	yes	—
230	C₂³	#32899	1	2	yes	—
231	C₂³	#32899	1	2	yes	—
232	C₂²	#347042	1	2	yes	—
233	C₂²	#347042	1	2	yes	—
234	C₂²	#347042	1	1	yes	characteristic
235	C₂²	#347042	1	1	yes	characteristic
236	C₂²	#347042	1	1	yes	characteristic
237	C₂²	#347043	1	2	yes	—
238	C₂²	#347043	1	2	yes	—
239	C₂²	#347043	1	2	yes	—
240	C₂²	#347043	1	2	yes	—
241	C₂²	#347043	1	2	yes	—
242	C₂²	#347043	1	2	yes	—
243	C₂²	#347043	1	2	yes	—
244	C₂²	#347044	1	2	yes	—
245	C₂²	#347044	1	1	yes	characteristic
246	C₂²	#347045	1	2	yes	—
247	C₂²	#347047	1	2	yes	—
248	C₂²	#393929	1	1	yes	characteristic
249	C₂²	#393934	1	2	yes	—
250	C₂²	#394066	1	1	yes	characteristic
251	C₂²	#394066	1	2	yes	—
252	C₂²	#394067	1	1	yes	characteristic
253	C₂	#72219236	1	2	yes	—
254	C₂	#72219238	1	2	yes	—
255	C₂	#72219240	1	1	yes	characteristic
256	C₂	#72225661	1	1	yes	characteristic
257	C₂	#72266443	1	1	yes	characteristic
258	C₂	#72266457	1	2	yes	—
259	C₂	#72266460	1	2	yes	—
260	C₂	#72334273	1	2	yes	—
261	C₂	#72334274	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. #72317046	C₂	semidirect
2. C₂×#97551	C₂	semidirect
3. #72235059	C₂	semidirect
4. #72235058	C₂	semidirect
5. C₂×#97550	C₂	semidirect
6. #72234618	C₂	semidirect
7. #72221168	C₂	semidirect
8. C₂×#417493	C₂	semidirect
9. #72218789	C₂	semidirect
10. #72218776	C₂	semidirect
11. C₂×#449475	C₂	semidirect
14. C₂²×#2846	C₂²	semidirect
19. C₂×#14679	C₂²	semidirect
20. C₂×#14679	C₂²	semidirect
22. C₂×#11196	C₂²	semidirect
23. C₂×#11196	C₂²	semidirect
30. C₂×#10965	C₂²	semidirect
31. C₂×#10965	C₂²	semidirect
32. #123513	C₂²	semidirect
33. #123513	C₂²	semidirect
34. #123513	C₂²	semidirect
36. C₂³×H(5,𝔽₂)	D₄	semidirect
37. C₂²×#534	D₄	semidirect
38. C₂²×#534	D₄	semidirect
42. C₂²×#518	D₄	semidirect
43. C₂²×#518	D₄	semidirect
44. C₂×#2846	D₄	semidirect
45. C₂×#2315	D₄	semidirect
46. C₂×#2315	D₄	semidirect
47. C₂×#2308	D₄	semidirect
48. C₂×#2308	D₄	semidirect
49. C₂×#2308	D₄	semidirect
50. C₂×#2308	D₄	semidirect
51. C₂×#2308	D₄	semidirect
52. C₂×#2308	D₄	semidirect
53. C₂×#2308	D₄	semidirect
54. C₂×#2308	D₄	semidirect
62. #13794	D₄	semidirect
63. #13794	D₄	semidirect
64. #13794	D₄	semidirect
65. #10965	D₄	semidirect
66. #10965	D₄	semidirect
175. C₂³×C₄	B(4,𝔽₂)	semidirect
176. C₂³×C₄	B(4,𝔽₂)	semidirect
177. C₂⁴	#2021	semidirect

Maximal subgroups

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

subgroup	index
#72317046	2
C₂×#97551	2
#72235059	2
#72235058	2
C₂×#97550	2
#72234618	2
#72221168	2
C₂×#417493	2
#72218789	2
#72218776	2
C₂×#449475	2