Subgroups — #72337615

Order 8192.

At least 28449 conjugacy classes of subgroups; only the 208 normal subgroups are shown.

The subgroup lattice has 198 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72334286	C₂	1	1	yes	maximal, characteristic
2	#72316216	C₂	1	1	yes	maximal, characteristic, complemented
3	#72314621	C₂	1	1	yes	maximal, characteristic, complemented
4	#72302102	C₂	1	1	yes	maximal, characteristic, complemented
5	#72265845	C₂	1	1	yes	maximal, characteristic
6	#72265825	C₄	1	1	yes	characteristic, complemented
7	#72265785	C₂²	1	1	yes	characteristic
8	#72265763	C₄	1	1	yes	characteristic, complemented
9	#72265678	C₄	1	1	yes	characteristic, complemented
10	#72238363	C₂	1	1	yes	maximal, characteristic, complemented
11	#72237630	C₂²	1	1	yes	characteristic
12	#72237613	D₄	1	2	yes	—
13	#72234688	C₂²	1	1	yes	characteristic
14	#72234688	C₂²	1	1	yes	characteristic
15	#72225776	D₄	1	1	yes	characteristic
16	#72225776	D₄	1	1	yes	characteristic
17	#72225775	C₂	1	1	yes	maximal, characteristic
18	#72219397	C₂²	1	1	yes	characteristic
19	#72219311	D₄	1	1	yes	characteristic
20	#72219311	D₄	1	1	yes	characteristic
21	#72219311	D₄	1	1	yes	characteristic
22	#72219310	D₄	1	1	yes	characteristic
23	#72219309	C₂²	1	1	yes	characteristic
24	#72219301	D₄	1	1	yes	characteristic
25	#72219301	D₄	1	1	yes	characteristic
26	#72219301	D₄	1	1	yes	characteristic
27	#72219299	D₄	1	1	yes	characteristic
28	C₂×#72217546	C₂²	1	1	yes	characteristic
29	C₂×#374022	Q₈	1	1	yes	characteristic
30	C₂×#359428	C₂×C₄	1	1	yes	characteristic
31	C₂×#359428	D₄	1	1	yes	characteristic
32	C₂×#7670203	C₂³	1	1	yes	characteristic
33	#72218836	C₄	1	1	yes	characteristic, complemented
34	#72218518	C₂×C₄	1	1	yes	characteristic
35	#72218518	C₂×C₄	1	1	yes	characteristic
36	#72218518	C₂×C₄	1	1	yes	characteristic
37	#72218291	C₂×C₄	1	1	yes	characteristic
38	C₂³×#2846	C₂×C₄	1	1	yes	characteristic
39	C₂³×#2846	D₄	1	1	yes	characteristic
40	C₂×#6342371	Q₈	1	1	yes	characteristic
41	C₂⁹	C₄⋊C₄	1	1	yes	characteristic, complemented
42	C₂²×#3484	C₂×D₄	1	1	yes	characteristic
43	C₂²×#3484	C₂×D₄	1	1	yes	characteristic
44	C₂⁴×#120	C₂²×C₄	1	1	yes	characteristic
45	C₂⁴×#120	C₂×D₄	1	1	yes	characteristic
46	C₂²×#2846	C₂×Q₈	1	1	yes	characteristic
47	C₂×#35390	#41	1	1	yes	characteristic
48	C₂×#35348	C₂×D₄	1	1	yes	characteristic
49	C₂×#35348	#41	1	1	yes	characteristic
50	C₂×#35346	C₂×D₄	1	1	yes	characteristic
51	C₂×#34981	#41	1	1	yes	characteristic
52	C₂×#34981	#41	1	1	yes	characteristic
53	C₂×#34980	C₂×D₄	1	1	yes	characteristic
54	C₂×#34980	C₂×D₄	1	1	yes	characteristic
55	C₂×#34198	#41	1	1	yes	characteristic
56	C₂×#34198	#41	1	1	yes	characteristic
57	C₂³×#379	C₄⋊C₄	1	1	yes	characteristic, complemented
58	C₂³×#379	C₄⋊C₄	1	1	yes	characteristic, complemented
59	C₂×#14679	C₄⋊C₄	1	1	yes	characteristic, complemented
60	#392109	C₂²⋊C₄	1	1	yes	characteristic
61	#392109	C₂²⋊C₄	1	1	yes	characteristic
62	#392109	C₂²⋊C₄	1	1	yes	characteristic
63	#392109	C₂²⋊C₄	1	1	yes	characteristic
64	C₂⁸	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂³×H(5,𝔽₂)	#120	1	1	yes	characteristic
66	C₂⁵×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂⁵×D₄	#120	1	1	yes	characteristic
68	C₂²×#534	#121	1	1	yes	characteristic
69	C₂²×#534	#121	1	1	yes	characteristic
70	C₂²×#534	#124	1	1	yes	characteristic
71	C₂²×#534	#124	1	1	yes	characteristic
72	C₂²×#534	#127	1	1	yes	characteristic
73	C₂³×#120	C₄×D₄	1	1	yes	characteristic
74	C₂³×#120	C₄×D₄	1	1	yes	characteristic
75	C₂³×#120	C₄×D₄	1	1	yes	characteristic
76	C₂³×#120	C₄×D₄	1	1	yes	characteristic
77	C₂³×#120	#120	1	1	yes	characteristic
78	C₂³×#120	#121	1	1	yes	characteristic
79	C₂³×#120	#121	1	1	yes	characteristic
80	C₂³×#120	C₂²⋊Q₈	1	1	yes	characteristic
81	C₂³×#120	C₂²⋊Q₈	1	1	yes	characteristic
82	C₂³×#120	C₂²⋊Q₈	1	1	yes	characteristic
83	C₂³×#120	#123	1	1	yes	characteristic
84	C₂²×#518	#121	1	1	yes	characteristic
85	C₂²×#518	#123	1	1	yes	characteristic
86	C₂⁴×C₂²⋊C₄	#117	1	1	yes	characteristic
87	C₂⁴×C₂²⋊C₄	#123	1	1	yes	characteristic
88	C₂×#2315	#123	1	1	yes	characteristic
89	C₂×#2315	#126	1	1	yes	characteristic
90	C₂×#2308	#121	1	1	yes	characteristic
91	C₂×#2308	#121	1	1	yes	characteristic
92	C₂×#2308	#121	1	1	yes	characteristic
93	C₂×#2308	#123	1	1	yes	characteristic
94	C₂×#2308	#124	1	1	yes	characteristic
95	C₂²×#379	C₂²⋊Q₈	1	1	yes	characteristic
96	C₂²×#379	C₂²⋊Q₈	1	1	yes	characteristic
97	C₂²×#379	C₂²⋊Q₈	1	1	yes	characteristic
98	C₂×#2273	#126	1	1	yes	characteristic
99	C₂×#1781	#99	1	2	yes	complemented
100	#14679	#99	1	2	yes	complemented
101	C₂⁷	#380	1	1	yes	characteristic
102	C₂⁷	#385	1	1	yes	characteristic
103	C₂⁷	#385	1	1	yes	characteristic
104	C₂⁷	C₂×#99	1	1	yes	characteristic
105	C₂⁷	B(4,𝔽₂)	1	1	yes	characteristic
106	C₂²×H(5,𝔽₂)	B(4,𝔽₂)	1	1	yes	characteristic
107	C₂²×H(5,𝔽₂)	B(4,𝔽₂)	1	1	yes	characteristic
108	C₂²×H(5,𝔽₂)	B(4,𝔽₂)	1	1	yes	characteristic
109	C₂²×H(5,𝔽₂)	B(4,𝔽₂)	1	1	yes	characteristic
110	C₂⁴×D₄	#386	1	1	yes	characteristic
111	C₂⁴×D₄	#390	1	1	yes	characteristic
112	C₂⁴×D₄	#392	1	1	yes	characteristic
113	C₂⁴×D₄	#393	1	1	yes	characteristic
114	C₂⁴×D₄	#394	1	1	yes	characteristic
115	C₂⁴×D₄	C₂×#99	1	1	yes	characteristic
116	C₂⁴×D₄	#410	1	1	yes	characteristic
117	C₂⁴×D₄	B(4,𝔽₂)	1	1	yes	characteristic
118	C₂⁴×D₄	#458	1	1	yes	characteristic
119	C₂⁵×C₄	#410	1	1	yes	characteristic
120	C₂⁵×C₄	#458	1	1	yes	characteristic
121	C₂²×#120	B(4,𝔽₂)	1	2	yes	—
122	C₂²×#120	#458	1	2	yes	—
123	C₂³×C₂²⋊C₄	#388	1	1	yes	characteristic
124	C₂³×C₂²⋊C₄	#388	1	1	yes	characteristic
125	C₂³×C₂²⋊C₄	#394	1	1	yes	characteristic
126	C₂³×C₂²⋊C₄	#396	1	1	yes	characteristic
127	C₂³×C₂²⋊C₄	#396	1	1	yes	characteristic
128	C₂³×C₂²⋊C₄	#397	1	1	yes	characteristic
129	C₂³×C₂²⋊C₄	#399	1	1	yes	characteristic
130	C₂⁶	#1439	1	1	yes	characteristic
131	C₂⁶	#1799	1	1	yes	characteristic
132	C₂⁶	#1799	1	1	yes	characteristic
133	C₂⁶	#1867	1	1	yes	characteristic
134	C₂⁶	#1867	1	1	yes	characteristic
135	C₂⁶	#1867	1	2	yes	—
136	C₂⁶	#1898	1	1	yes	characteristic
137	C₂⁶	#1964	1	1	yes	characteristic
138	C₂⁶	#2021	1	1	yes	characteristic
139	C₂⁶	#2032	1	1	yes	characteristic
140	C₂⁶	#2044	1	1	yes	characteristic
141	C₂³×D₄	#1889	1	1	yes	characteristic
142	C₂³×D₄	#1889	1	1	yes	characteristic
143	C₂³×D₄	#1913	1	1	yes	characteristic
144	C₂³×D₄	#1913	1	1	yes	characteristic
145	C₂³×D₄	#2021	1	1	yes	characteristic
146	C₂³×D₄	#2021	1	1	yes	characteristic
147	C₂³×D₄	#2021	1	1	yes	characteristic
148	C₂³×D₄	#2022	1	1	yes	characteristic
149	C₂³×D₄	#2029	1	1	yes	characteristic
150	C₂³×D₄	#2029	1	1	yes	characteristic
151	C₂³×D₄	#2032	1	1	yes	characteristic
152	C₂³×D₄	#2032	1	1	yes	characteristic
153	C₂⁴×C₄	#1899	1	1	yes	characteristic
154	C₂⁴×C₄	#1964	1	1	yes	characteristic
155	C₂⁴×C₄	#2022	1	1	yes	characteristic
156	C₂⁴×C₄	#2032	1	1	yes	characteristic
157	C₂⁴×C₄	#2044	1	1	yes	characteristic
158	C₂²×C₂²⋊C₄	#2044	1	2	yes	—
159	C₂⁵	#9495	1	1	yes	characteristic
160	C₂⁵	#9497	1	1	yes	characteristic
161	C₂⁵	#10336	1	1	yes	characteristic
162	C₂⁵	#10337	1	1	yes	characteristic
163	C₂⁵	#10390	1	1	yes	characteristic
164	C₂⁵	#10392	1	1	yes	characteristic
165	C₂⁵	#11043	1	1	yes	characteristic
166	C₂⁵	#11046	1	1	yes	characteristic
167	C₂⁵	#11069	1	1	yes	characteristic
168	C₂⁵	#11087	1	1	yes	characteristic
169	C₂⁵	#11087	1	1	yes	characteristic
170	C₂⁵	#11089	1	1	yes	characteristic
171	C₂⁵	#11089	1	1	yes	characteristic
172	C₂³×C₄	#9496	1	1	yes	characteristic
173	C₂³×C₄	#9498	1	1	yes	characteristic
174	C₂³×C₄	#10337	1	1	yes	characteristic
175	C₂³×C₄	#10338	1	1	yes	characteristic
176	C₂³×C₄	#10392	1	1	yes	characteristic
177	C₂³×C₄	#10392	1	1	yes	characteristic
178	C₂⁴	#115723	1	1	yes	characteristic
179	C₂⁴	#115995	1	1	yes	characteristic
180	C₂⁴	#115995	1	1	yes	characteristic
181	C₂⁴	#116022	1	1	yes	characteristic
182	C₂⁴	#116022	1	1	yes	characteristic
183	C₂⁴	#116351	1	1	yes	characteristic
184	C₂⁴	#116356	1	1	yes	characteristic
185	C₂⁴	#139750	1	2	yes	—
186	C₂⁴	#144378	1	2	yes	—
187	C₂³	#72246301	1	1	yes	characteristic
188	C₂³	#72255838	1	1	yes	characteristic
189	C₂³	#72255867	1	1	yes	characteristic
190	C₂³	#72256005	1	1	yes	characteristic
191	C₂³	#72264957	1	1	yes	characteristic
192	C₂³	#72265906	1	2	yes	—
193	C₂²	#72265705	1	1	yes	characteristic
194	C₂²	#72265792	1	1	yes	characteristic
195	C₂²	#72265907	1	1	yes	characteristic
196	C₂	#72266683	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72266683
commutator	C₂⁴×#120	C₂²×C₄
Frattini, cosocle	C₂×#7670203	C₂³
prime radical	C₂×#72217546	C₂²
LCS	G ▷ C₂⁴×#120 ▷ C₂⁶ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂⁶ ◁ C₂×#7670203 ◁ G
LECS	G ▷ C₂×#7670203 ▷ C₂⁶ ▷ C₂³ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #72316216	C₂	semidirect
3. #72314621	C₂	semidirect
4. #72302102	C₂	semidirect
6. #72265825	C₄	semidirect
8. #72265763	C₄	semidirect
9. #72265678	C₄	semidirect
10. #72238363	C₂	semidirect
33. #72218836	C₄	semidirect
41. C₂⁹	C₄⋊C₄	semidirect
57. C₂³×#379	C₄⋊C₄	semidirect
58. C₂³×#379	C₄⋊C₄	semidirect
59. C₂×#14679	C₄⋊C₄	semidirect
99. C₂×#1781	#99	semidirect
100. #14679	#99	semidirect

Maximal subgroups

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

subgroup	index
#72334286	2
#72316216	2
#72314621	2
#72302102	2
#72265845	2
#72238363	2
#72225775	2