Subgroups — #72340539

Order 8192.

Only the 172 normal subgroups are shown; non-normal subgroups are omitted.

The subgroup lattice has 153 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72340535	C₂	1	1	yes	maximal, characteristic, complemented
2	#72340532	C₂	1	1	yes	maximal, characteristic, complemented
3	#72340529	C₂	1	1	yes	maximal, characteristic, complemented
4	#72340527	C₂	1	1	yes	maximal, characteristic, complemented
5	#72340523	C₂	1	1	yes	maximal, characteristic, complemented
6	#72340522	C₂²	1	1	yes	characteristic, complemented
7	#72340518	C₂²	1	1	yes	characteristic, complemented
8	#72337700	C₂	1	1	yes	maximal, characteristic, complemented
9	#72337699	C₂	1	1	yes	maximal, characteristic, complemented
10	#72337698	C₂²	1	1	yes	characteristic, complemented
11	#72337697	C₂	1	1	yes	maximal, characteristic, complemented
12	#72337696	C₂²	1	1	yes	characteristic, complemented
13	#72337695	C₂²	1	1	yes	characteristic, complemented
14	#72337693	C₂	1	1	yes	maximal, characteristic, complemented
15	#72337692	C₂²	1	1	yes	characteristic, complemented
16	#72337686	C₂²	1	1	yes	characteristic, complemented
17	#72337685	C₂²	1	1	yes	characteristic
18	#72337684	D₄	1	1	yes	characteristic, complemented
19	#72337683	C₂³	1	1	yes	characteristic
20	#72337679	D₄	1	1	yes	characteristic, complemented
21	#72267058	C₂	1	1	yes	maximal, characteristic, complemented
22	#72267057	C₂	1	1	yes	maximal, characteristic, complemented
23	#72252412	C₂²	1	1	yes	characteristic, complemented
24	#72252408	C₂²	1	1	yes	characteristic, complemented
25	#72252408	C₂²	1	1	yes	characteristic, complemented
26	#72241341	C₂	1	1	yes	maximal, characteristic, complemented
27	#72241330	C₂	1	1	yes	maximal, characteristic, complemented
28	#72241323	C₂²	1	1	yes	characteristic, complemented
29	#72241319	C₂²	1	1	yes	characteristic, complemented
30	#72241317	C₂²	1	1	yes	characteristic
31	#72241314	C₂	1	1	yes	maximal, characteristic, complemented
32	#72241312	C₂²	1	1	yes	characteristic, complemented
33	#72241312	C₂²	1	1	yes	characteristic, complemented
34	#72241306	C₂²	1	1	yes	characteristic
35	#72241085	C₂²	1	1	yes	characteristic
36	#72241085	C₂²	1	1	yes	characteristic
37	#72241085	C₂²	1	1	yes	characteristic, complemented
38	#72237019	C₂	1	1	yes	maximal, characteristic, complemented
39	#72237011	C₂²	1	1	yes	characteristic, complemented
40	#72236999	C₂²	1	1	yes	characteristic, complemented
41	#72236999	C₂²	1	1	yes	characteristic, complemented
42	#72236986	C₂²	1	1	yes	characteristic
43	#72236979	D₄	1	1	yes	characteristic, complemented
44	#72236975	D₄	1	1	yes	characteristic, complemented
45	#72236975	D₄	1	1	yes	characteristic, complemented
46	#72236936	C₂²	1	1	yes	characteristic, complemented
47	#72236936	C₂²	1	1	yes	characteristic, complemented
48	#72236935	C₂²	1	1	yes	characteristic, complemented
49	#72236925	D₄	1	2	yes	complemented
50	#72236189	D₄	1	1	yes	characteristic, complemented
51	#72227588	C₂³	1	1	yes	characteristic
52	#72219530	C₂²	1	1	yes	characteristic, complemented
53	#72219526	C₂²	1	1	yes	characteristic, complemented
54	#72219524	C₂²	1	1	yes	characteristic
55	#72219518	C₂²	1	1	yes	characteristic
56	#72213834	C₂²	1	1	yes	characteristic, complemented
57	#72213758	C₂³	1	1	yes	characteristic
58	#72213758	C₂³	1	1	yes	characteristic
59	#72213565	C₂²	1	1	yes	characteristic
60	#72213565	C₂²	1	1	yes	characteristic
61	#72213523	D₄	1	2	yes	complemented
62	#72213513	D₄	1	2	yes	complemented
63	#72213500	C₂³	1	1	yes	characteristic
64	#72213462	C₂³	1	1	yes	characteristic
65	#72213462	C₂³	1	1	yes	characteristic
66	#72213462	C₂³	1	1	yes	characteristic
67	#72213423	C₂³	1	1	yes	characteristic
68	#72213269	D₄	1	1	yes	characteristic, complemented
69	#72213260	D₄	1	1	yes	characteristic, complemented
70	#72210591	C₂²	1	1	yes	characteristic
71	#72210549	D₄	1	1	yes	characteristic, complemented
72	#72210539	D₄	1	1	yes	characteristic, complemented
73	#72210529	C₂³	1	1	yes	characteristic
74	#72210529	C₂³	1	1	yes	characteristic
75	#72210524	C₂³	1	1	yes	characteristic
76	#72210498	D₄	1	1	yes	characteristic, complemented
77	#72210483	D₄	1	1	yes	characteristic, complemented
78	#72210439	C₂³	1	1	yes	characteristic
79	#72210439	C₂³	1	1	yes	characteristic
80	#72210439	C₂³	1	1	yes	characteristic
81	#72210025	D₄	1	2	yes	complemented
82	H(9,𝔽₂)	C₂×D₄	1	1	yes	characteristic
83	#8934366	C₂×D₄	1	1	yes	characteristic
84	#8934366	C₂×D₄	1	1	yes	characteristic
85	#8934366	C₂×D₄	1	1	yes	characteristic, complemented
86	#8933848	C₂×D₄	1	1	yes	characteristic, complemented
87	#7672751	C₂×D₄	1	1	yes	characteristic
88	#6628901	C₂⁴	1	1	yes	characteristic
89	#6628896	C₂×D₄	1	2	yes	—
90	#6437630	C₂×D₄	1	1	yes	characteristic
91	#449492	C₂×D₄	1	1	yes	characteristic
92	#449492	C₂×D₄	1	1	yes	characteristic
93	#417807	C₂×D₄	1	1	yes	characteristic
94	#417807	C₂×D₄	1	1	yes	characteristic
95	#392136	C₂×D₄	1	1	yes	characteristic
96	#392135	C₂×D₄	1	2	yes	—
97	#374009	C₂×D₄	1	1	yes	characteristic
98	#356056	C₂×D₄	1	1	yes	characteristic
99	#213861	C₂×D₄	1	1	yes	characteristic, complemented
100	#213860	C₂×D₄	1	1	yes	characteristic
101	#213860	C₂×D₄	1	1	yes	characteristic
102	#213860	C₂×D₄	1	1	yes	characteristic, complemented
103	#213854	C₂×D₄	1	2	yes	—
104	#133835	C₂×D₄	1	1	yes	characteristic, complemented
105	#133835	C₂×D₄	1	1	yes	characteristic, complemented
106	#128062	C₂×D₄	1	2	yes	—
107	#97460	C₂×D₄	1	1	yes	characteristic, complemented
108	#97460	C₂×D₄	1	1	yes	characteristic, complemented
109	C₂×H(7,𝔽₂)	C₂²×D₄	1	1	yes	characteristic
110	C₂×#2867	#120	1	1	yes	characteristic
111	#35527	#120	1	1	yes	characteristic
112	#35527	C₂²×D₄	1	1	yes	characteristic
113	#35527	C₂²×D₄	1	1	yes	characteristic
114	#35527	C₂²×D₄	1	1	yes	characteristic
115	#35346	#120	1	2	yes	—
116	#20586	C₂²×D₄	1	1	yes	characteristic
117	#20586	H(5,𝔽₂)	1	1	yes	characteristic
118	#20586	H(5,𝔽₂)	1	1	yes	characteristic
119	#20585	#120	1	2	yes	—
120	#10981	#120	1	1	yes	characteristic
121	#10981	#120	1	1	yes	characteristic
122	C₂²×H(5,𝔽₂)	B(4,𝔽₂)	1	1	yes	characteristic
123	C₂²×H(5,𝔽₂)	C₂×#120	1	1	yes	characteristic
124	C₂²×H(5,𝔽₂)	D₄²	1	1	yes	characteristic
125	C₂²×H(5,𝔽₂)	D₄²	1	1	yes	characteristic
126	C₂²×#127	B(4,𝔽₂)	1	1	yes	characteristic
127	C₂²×#127	C₂×#120	1	1	yes	characteristic
128	C₂²×#120	B(4,𝔽₂)	1	2	yes	—
129	#2309	B(4,𝔽₂)	1	1	yes	characteristic
130	#2309	#534	1	1	yes	characteristic
131	#2309	D₄²	1	1	yes	characteristic
132	#2309	D₄²	1	1	yes	characteristic
133	#2308	B(4,𝔽₂)	1	2	yes	—
134	C₂×C₄³	B(4,𝔽₂)	1	1	yes	characteristic
135	C₂⁶	#2196	1	2	yes	—
136	C₂³×D₄	#2403	1	1	yes	characteristic
137	C₂³×D₄	C₂×B(4,𝔽₂)	1	2	yes	—
138	C₂³×D₄	C₂×B(4,𝔽₂)	1	1	yes	characteristic
139	C₂²×C₂²⋊C₄	#2196	1	2	yes	—
140	C₂²×C₄²	C₂×B(4,𝔽₂)	1	1	yes	characteristic
141	C₂⁵	#23895	1	1	yes	characteristic
142	C₂⁵	#23900	1	1	yes	characteristic
143	C₂⁵	C₂×#2196	1	2	yes	—
144	C₂³×C₄	#23900	1	1	yes	characteristic
145	C₂³×C₄	#33551	1	2	yes	—
146	C₂⁴	#245225	1	1	yes	characteristic
147	C₂⁴	#499787	1	2	yes	—
148	C₂³	B(5,𝔽₂)	1	2	yes	—
149	C₂³	#72210587	1	1	yes	characteristic
150	C₂²	#72216737	1	1	yes	characteristic
151	C₂	#72216741	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72216741
commutator, Frattini, cosocle	#6628901	C₂⁴
LCS	G ▷ #6628901 ▷ C₂³×D₄ ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂⁴ ◁ C₂³×D₄ ◁ #6628901 ◁ G
LECS	G ▷ #6628901 ▷ C₂³×D₄ ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72340535	C₂	semidirect
2. #72340532	C₂	semidirect
3. #72340529	C₂	semidirect
4. #72340527	C₂	semidirect
5. #72340523	C₂	semidirect
6. #72340522	C₂²	semidirect
7. #72340518	C₂²	semidirect
8. #72337700	C₂	semidirect
9. #72337699	C₂	semidirect
10. #72337698	C₂²	semidirect
11. #72337697	C₂	semidirect
12. #72337696	C₂²	semidirect
13. #72337695	C₂²	semidirect
14. #72337693	C₂	semidirect
15. #72337692	C₂²	semidirect
16. #72337686	C₂²	semidirect
18. #72337684	D₄	semidirect
20. #72337679	D₄	semidirect
21. #72267058	C₂	semidirect
22. #72267057	C₂	semidirect
23. #72252412	C₂²	semidirect
24. #72252408	C₂²	semidirect
25. #72252408	C₂²	semidirect
26. #72241341	C₂	semidirect
27. #72241330	C₂	semidirect
28. #72241323	C₂²	semidirect
29. #72241319	C₂²	semidirect
31. #72241314	C₂	semidirect
32. #72241312	C₂²	semidirect
33. #72241312	C₂²	semidirect
37. #72241085	C₂²	semidirect
38. #72237019	C₂	semidirect
39. #72237011	C₂²	semidirect
40. #72236999	C₂²	semidirect
41. #72236999	C₂²	semidirect
43. #72236979	D₄	semidirect
44. #72236975	D₄	semidirect
45. #72236975	D₄	semidirect
46. #72236936	C₂²	semidirect
47. #72236936	C₂²	semidirect
48. #72236935	C₂²	semidirect
49. #72236925	D₄	semidirect
50. #72236189	D₄	semidirect
52. #72219530	C₂²	semidirect
53. #72219526	C₂²	semidirect
56. #72213834	C₂²	semidirect
61. #72213523	D₄	semidirect
62. #72213513	D₄	semidirect
68. #72213269	D₄	semidirect
69. #72213260	D₄	semidirect
71. #72210549	D₄	semidirect
72. #72210539	D₄	semidirect
76. #72210498	D₄	semidirect
77. #72210483	D₄	semidirect
81. #72210025	D₄	semidirect
85. #8934366	C₂×D₄	semidirect
86. #8933848	C₂×D₄	semidirect
99. #213861	C₂×D₄	semidirect
102. #213860	C₂×D₄	semidirect
104. #133835	C₂×D₄	semidirect
105. #133835	C₂×D₄	semidirect
107. #97460	C₂×D₄	semidirect
108. #97460	C₂×D₄	semidirect

Maximal subgroups

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

subgroup	index
#72340535	2
#72340532	2
#72340529	2
#72340527	2
#72340523	2
#72337700	2
#72337699	2
#72337697	2
#72337693	2
#72267058	2
#72267057	2
#72241341	2
#72241330	2
#72241314	2
#72237019	2