Subgroups — #72323238

Order 4096.

At least 10564 conjugacy classes of subgroups; only the 233 normal subgroups are shown.

The subgroup lattice has 171 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72323228	C₂	1	1	yes	maximal, characteristic, complemented
2	#72323217	C₂	1	1	yes	maximal, characteristic, complemented
3	#72313556	C₂	1	1	yes	maximal, characteristic, complemented
4	#72309421	C₂	1	1	yes	maximal, characteristic, complemented
5	#72266163	C₂	1	1	yes	maximal, characteristic, complemented
6	#72265766	C₂²	1	1	yes	characteristic
7	#72237605	C₂	1	4	yes	maximal, complemented
8	#72235847	C₂²	1	1	yes	characteristic
9	#72234789	C₂	1	1	yes	maximal, characteristic, complemented
10	#72234639	C₂	1	1	yes	maximal, characteristic, complemented
11	#72234636	C₂²	1	1	yes	characteristic
12	#72227566	C₂²	1	2	yes	complemented
13	#72227564	C₂²	1	2	yes	complemented
14	#72225870	C₂²	1	1	yes	characteristic
15	#72224136	C₂²	1	1	yes	characteristic
16	#72224136	C₂²	1	1	yes	characteristic
17	#72219311	C₂²	1	1	yes	characteristic
18	#72216504	C₂	1	4	yes	maximal, complemented
19	#72216484	C₂²	1	4	yes	complemented
20	#72216427	C₂²	1	4	yes	complemented
21	#72215653	C₂²	1	2	yes	complemented
22	#72213827	C₂²	1	2	yes	complemented
23	#72211186	C₂²	1	2	yes	complemented
24	#72210578	C₂²	1	2	yes	complemented
25	#72210482	C₂²	1	4	yes	complemented
26	#72209844	C₂²	1	4	yes	—
27	#10586404	D₄	1	1	yes	characteristic
28	#10586404	D₄	1	1	yes	characteristic, complemented
29	C₂×#46996	D₄	1	1	yes	characteristic, complemented
30	C₂×#34980	C₂³	1	1	yes	characteristic
31	#6499870	D₄	1	2	yes	—
32	#6499870	D₄	1	2	yes	complemented
33	#6499867	C₂³	1	2	yes	—
34	#6499867	C₂³	1	2	yes	—
35	#6374128	D₄	1	1	yes	characteristic
36	#6374128	D₄	1	1	yes	characteristic, complemented
37	#5024691	D₄	1	1	yes	characteristic, complemented
38	#5020751	C₂³	1	1	yes	characteristic
39	#4430405	D₄	1	1	yes	characteristic, complemented
40	#648360	C₂³	1	1	yes	characteristic
41	#576160	D₄	1	1	yes	characteristic, complemented
42	C₂×#11499	D₄	1	2	yes	complemented
43	#392109	D₄	1	2	yes	complemented
44	C₂×#8547	D₄	1	1	yes	characteristic, complemented
45	C₂×#8541	D₄	1	1	yes	characteristic, complemented
46	#134035	D₄	1	2	yes	complemented
47	#134033	C₂³	1	2	yes	—
48	#133823	D₄	1	2	yes	complemented
49	#133822	C₂³	1	2	yes	—
50	#115690	D₄	1	2	yes	complemented
51	#115690	C₂³	1	2	yes	—
52	#115690	C₂³	1	2	yes	—
53	C₂⁵×D₄	C₂×D₄	1	1	yes	characteristic
54	#62817	C₂×D₄	1	2	yes	—
55	#62817	C₂×D₄	1	2	yes	—
56	C₂²×#534	C₂×D₄	1	1	yes	characteristic
57	C₂²×#534	C₂×D₄	1	1	yes	characteristic
58	C₂³×#120	C₂×D₄	1	1	yes	characteristic
59	C₂×#2840	C₂×D₄	1	1	yes	characteristic, complemented
60	C₂×#2849	C₂×D₄	1	1	yes	characteristic
61	C₂×#2849	C₂×D₄	1	1	yes	characteristic
62	C₂×#2846	C₂×D₄	1	1	yes	characteristic
63	C₂×#2846	C₂⁴	1	1	yes	characteristic
64	C₂×#2403	C₂×D₄	1	1	yes	characteristic, complemented
65	C₂×#2613	C₂×D₄	1	1	yes	characteristic
66	C₂×#2613	C₂×D₄	1	1	yes	characteristic
67	C₂×#2613	C₂×D₄	1	1	yes	characteristic
68	C₂×#2613	#41	1	1	yes	characteristic
69	C₂×#2308	C₂×D₄	1	1	yes	characteristic
70	C₂×#2308	#41	1	1	yes	characteristic
71	C₂²×#379	C₂×D₄	1	1	yes	characteristic
72	#21746	C₂×D₄	1	2	yes	—
73	#21746	C₂×D₄	1	2	yes	—
74	#14679	C₂×D₄	1	4	yes	complemented
75	#13827	C₂×D₄	1	1	yes	characteristic, complemented
76	C₂²⋊C₄²	C₂×D₄	1	1	yes	characteristic, complemented
77	#11499	C₂×D₄	1	4	yes	complemented
78	#8547	C₂×D₄	1	2	yes	—
79	#8541	C₂×D₄	1	2	yes	—
80	C₂⁷	#127	1	1	yes	characteristic
81	C₂⁷	C₂²×D₄	1	1	yes	characteristic
82	C₂⁴×D₄	#120	1	1	yes	characteristic
83	C₂⁴×D₄	#120	1	1	yes	characteristic
84	C₂⁴×D₄	#120	1	1	yes	characteristic
85	C₂⁴×D₄	#121	1	1	yes	characteristic
86	C₂⁴×D₄	#121	1	1	yes	characteristic
87	C₂⁵×C₄	#127	1	1	yes	characteristic
88	C₂²×#123	#120	1	1	yes	characteristic
89	C₂²×#123	#120	1	1	yes	characteristic
90	C₂²×#123	#120	1	1	yes	characteristic
91	C₂²×#123	#121	1	1	yes	characteristic
92	C₂²×#120	#120	1	1	yes	characteristic
93	C₂²×#120	#120	1	1	yes	characteristic
94	C₂²×#120	#121	1	1	yes	characteristic
95	C₂²×#120	#121	1	1	yes	characteristic
96	C₂²×#120	#127	1	1	yes	characteristic
97	C₂²×#120	C₂²×D₄	1	1	yes	characteristic
98	C₂²×#120	C₂²×D₄	1	1	yes	characteristic
99	C₂²×#120	C₂²×D₄	1	1	yes	characteristic
100	C₂³×C₂²⋊C₄	#120	1	1	yes	characteristic
101	C₂³×C₂²⋊C₄	#121	1	1	yes	characteristic
102	C₂³×C₂²⋊C₄	#121	1	1	yes	characteristic
103	C₂×#381	#120	1	1	yes	characteristic
104	C₂×#381	#121	1	1	yes	characteristic
105	C₂×#381	#121	1	1	yes	characteristic
106	C₂×#381	#121	1	1	yes	characteristic
107	C₂×#379	#120	1	1	yes	characteristic
108	C₂×#379	#120	1	1	yes	characteristic
109	C₂×#379	#121	1	1	yes	characteristic
110	C₂×#379	#121	1	1	yes	characteristic
111	C₂×#379	#127	1	1	yes	characteristic
112	C₂×#379	C₂²×D₄	1	1	yes	characteristic
113	C₂×#379	C₂²×D₄	1	1	yes	characteristic
114	C₂×#379	C₂×#41	1	1	yes	characteristic
115	C₂⁶	#392	1	1	yes	characteristic
116	C₂⁶	#392	1	1	yes	characteristic
117	C₂⁶	B(4,𝔽₂)	1	1	yes	characteristic
118	C₂⁶	C₂×#120	1	1	yes	characteristic
119	C₂⁶	C₂×#121	1	1	yes	characteristic
120	C₂⁶	C₂×#127	1	1	yes	characteristic
121	C₂³×D₄	#392	1	1	yes	characteristic
122	C₂³×D₄	B(4,𝔽₂)	1	1	yes	characteristic
123	C₂³×D₄	B(4,𝔽₂)	1	2	yes	—
124	C₂³×D₄	B(4,𝔽₂)	1	2	yes	—
125	C₂³×D₄	B(4,𝔽₂)	1	1	yes	characteristic
126	C₂³×D₄	C₂×#120	1	1	yes	characteristic
127	C₂⁴×C₄	#392	1	1	yes	characteristic
128	C₂⁴×C₄	#392	1	1	yes	characteristic
129	C₂⁴×C₄	B(4,𝔽₂)	1	1	yes	characteristic
130	C₂²×C₂²⋊C₄	#392	1	1	yes	characteristic
131	C₂²×C₂²⋊C₄	#392	1	1	yes	characteristic
132	C₂²×C₂²⋊C₄	#392	1	1	yes	characteristic
133	C₂²×C₂²⋊C₄	B(4,𝔽₂)	1	2	yes	—
134	C₂²×C₂²⋊C₄	B(4,𝔽₂)	1	2	yes	—
135	C₂²×C₂²⋊C₄	C₂×#120	1	1	yes	characteristic
136	C₂²×C₂²⋊C₄	C₂×#121	1	1	yes	characteristic
137	C₂²×C₂²⋊C₄	C₂×#121	1	1	yes	characteristic
138	C₂⁵	#2021	1	1	yes	characteristic
139	C₂⁵	#2021	1	1	yes	characteristic
140	C₂⁵	#2021	1	1	yes	characteristic
141	C₂⁵	#2021	1	1	yes	characteristic
142	C₂⁵	#2021	1	1	yes	characteristic
143	C₂⁵	#2021	1	1	yes	characteristic
144	C₂⁵	#2021	1	1	yes	characteristic
145	C₂⁵	C₂×#392	1	1	yes	characteristic
146	C₂⁵	C₂×B(4,𝔽₂)	1	2	yes	—
147	C₂⁵	C₂×B(4,𝔽₂)	1	1	yes	characteristic
148	C₂³×C₄	#2021	1	2	yes	—
149	C₂³×C₄	#2021	1	1	yes	characteristic
150	C₂³×C₄	C₂×B(4,𝔽₂)	1	1	yes	characteristic
151	C₂×C₂²⋊C₄	#2021	1	2	yes	—
152	C₂×C₂²⋊C₄	C₂×B(4,𝔽₂)	1	2	yes	—
153	C₂⁴	#10336	1	1	yes	characteristic
154	C₂⁴	#10336	1	1	yes	characteristic
155	C₂⁴	#10336	1	1	yes	characteristic
156	C₂⁴	#10336	1	1	yes	characteristic
157	C₂⁴	C₂×#2021	1	1	yes	characteristic
158	C₂⁴	C₂×#2021	1	1	yes	characteristic
159	C₂⁴	C₂×#2021	1	1	yes	characteristic
160	C₂⁴	#33551	1	4	yes	—
161	C₂³	#144364	1	1	yes	characteristic
162	C₂³	#144364	1	1	yes	characteristic
163	C₂³	#144378	1	2	yes	—
164	C₂³	C₂×#10336	1	1	yes	characteristic
165	C₂³	#498927	1	2	yes	—
166	C₂²	C₂×#144378	1	1	yes	characteristic
167	C₂²	C₂×#144364	1	1	yes	characteristic
168	C₂²	#72218215	1	1	yes	characteristic
169	C₂	#72266301	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72266301
commutator, Frattini, cosocle	C₂×#2846	C₂⁴
LCS	G ▷ C₂×#2846 ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂⁶ ◁ C₂×#34980 ◁ G
LECS	G ▷ C₂×#2846 ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72323228	C₂	semidirect
2. #72323217	C₂	semidirect
3. #72313556	C₂	semidirect
4. #72309421	C₂	semidirect
5. #72266163	C₂	semidirect
7. #72237605	C₂	semidirect
9. #72234789	C₂	semidirect
10. #72234639	C₂	semidirect
12. #72227566	C₂²	semidirect
13. #72227564	C₂²	semidirect
18. #72216504	C₂	semidirect
19. #72216484	C₂²	semidirect
20. #72216427	C₂²	semidirect
21. #72215653	C₂²	semidirect
22. #72213827	C₂²	semidirect
23. #72211186	C₂²	semidirect
24. #72210578	C₂²	semidirect
25. #72210482	C₂²	semidirect
28. #10586404	D₄	semidirect
29. C₂×#46996	D₄	semidirect
32. #6499870	D₄	semidirect
36. #6374128	D₄	semidirect
37. #5024691	D₄	semidirect
39. #4430405	D₄	semidirect
41. #576160	D₄	semidirect
42. C₂×#11499	D₄	semidirect
43. #392109	D₄	semidirect
44. C₂×#8547	D₄	semidirect
45. C₂×#8541	D₄	semidirect
46. #134035	D₄	semidirect
48. #133823	D₄	semidirect
50. #115690	D₄	semidirect
59. C₂×#2840	C₂×D₄	semidirect
64. C₂×#2403	C₂×D₄	semidirect
74. #14679	C₂×D₄	semidirect
75. #13827	C₂×D₄	semidirect
76. C₂²⋊C₄²	C₂×D₄	semidirect
77. #11499	C₂×D₄	semidirect

Maximal subgroups

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

subgroup	index
#72323228	2
#72323217	2
#72313556	2
#72309421	2
#72266163	2
#72237605	2
#72234789	2
#72234639	2
#72216504	2