Subgroups — #72341940

Order 6144.

At least 10390 conjugacy classes of subgroups; only the 215 normal subgroups are shown.

The subgroup lattice has 164 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72341928	C₂	1	1	yes	maximal, characteristic, complemented
2	#72341923	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#72236904	C₂	1	1	yes	maximal, characteristic
4	#72341911	C₂	1	1	yes	maximal, characteristic, complemented
5	#72341905	C₂	1	1	yes	maximal, characteristic, complemented
6	#72341898	C₂	1	1	yes	maximal, characteristic, complemented
7	#72341894	C₂	1	1	yes	maximal, characteristic, complemented
8	#72341891	C₂	1	1	yes	maximal, characteristic, complemented
9	#72341890	C₂²	1	1	yes	characteristic, complemented
10	#72341887	C₂²	1	1	yes	characteristic, complemented
11	#72341881	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#11711587	C₂²	1	1	yes	characteristic
13	#72341872	C₂²	1	1	yes	characteristic, complemented
14	#72341862	C₂	1	1	yes	maximal, characteristic, complemented
15	#72341859	C₂²	1	1	yes	characteristic, complemented
16	#72341854	C₂²	1	1	yes	characteristic, complemented
17	#72341847	C₂	1	1	yes	maximal, characteristic, complemented
18	#72341845	C₂²	1	1	yes	characteristic, complemented
19	#72341843	C₂²	1	1	yes	characteristic, complemented
20	#72341840	C₂²	1	1	yes	characteristic, complemented
21	#72341828	C₂	1	1	yes	maximal, characteristic, complemented
22	#72341821	C₂²	1	1	yes	characteristic, complemented
23	#72341742	C₂	1	1	yes	maximal, characteristic, complemented
24	C₂×#11711638	C₂²	1	1	yes	characteristic
25	#72341731	C₂²	1	1	yes	characteristic, complemented
26	#72341716	C₂²	1	1	yes	characteristic, complemented
27	#72341715	C₂²	1	1	yes	characteristic, complemented
28	#72341685	C₂²	1	1	yes	characteristic, complemented
29	C₂×#11711643	C₂²	1	1	yes	characteristic
30	C₂×#32894⋊C₃	C₂²	1	1	yes	characteristic
31	#72341364	C₂	1	1	yes	maximal, characteristic, complemented
32	#72341361	C₂²	1	1	yes	characteristic, complemented
33	#72341357	C₂²	1	1	yes	characteristic, complemented
34	#72341341	C₂²	1	1	yes	characteristic, complemented
35	#72341289	C₂	1	1	yes	maximal, characteristic, complemented
36	#72341287	C₂²	1	1	yes	characteristic, complemented
37	#72341283	C₂²	1	1	yes	characteristic
38	#72341280	C₂²	1	1	yes	characteristic
39	#72341274	C₂²	1	1	yes	characteristic, complemented
40	#72341273	C₂²	1	1	yes	characteristic, complemented
41	#72341264	C₂²	1	1	yes	characteristic, complemented
42	C₂×#11715708	C₂²	1	1	yes	characteristic
43	#72281044	D₃	1	1	yes	characteristic, complemented
44	#72253586	C₂²	1	1	yes	characteristic
45	C₂×#11715687	C₂²	1	1	yes	characteristic
46	#72236904	C₄	1	8	yes	complemented
47	C₂×#11714174	C₂²	1	1	yes	characteristic
48	#72227857	C₂²	1	1	yes	characteristic, complemented
49	#72227330	C₂²	1	1	yes	characteristic, complemented
50	#72227325	C₂²	1	1	yes	characteristic, complemented
51	#72227309	C₂²	1	1	yes	characteristic, complemented
52	C₂×#11715839	C₂²	1	1	yes	characteristic
53	#11715839	C₂×C₄	1	4	yes	complemented
54	C₂×#87521	C₂³	1	1	yes	characteristic
55	C₂×#87519	C₂³	1	1	yes	characteristic
56	C₂×#87522	C₂³	1	1	yes	characteristic
57	#11715708	C₂×C₄	1	4	yes	complemented
58	C₂×#85567	C₂³	1	1	yes	characteristic
59	#11715665	C₂³	1	1	yes	characteristic
60	#11715663	C₂³	1	1	yes	characteristic
61	C₂×#85566	C₂³	1	1	yes	characteristic
62	#11715659	C₂³	1	1	yes	characteristic
63	#11714174	C₂×C₄	1	4	yes	complemented
64	#11711643	C₂×C₄	1	4	yes	complemented
65	#11711638	C₂×C₄	1	4	yes	complemented
66	#11711587	C₂×C₄	1	4	yes	complemented
67	#11711580	D₄	1	1	yes	characteristic, complemented
68	#11711580	D₄	1	1	yes	characteristic, complemented
69	#11711578	D₄	1	1	yes	characteristic, complemented
70	#11711578	D₄	1	1	yes	characteristic, complemented
71	C₂×#87547	C₂³	1	1	yes	characteristic
72	C₂×#85666	C₂³	1	1	yes	characteristic
73	#11710604	C₂³	1	1	yes	characteristic
74	C₂×#3027⋊C₃	C₂³	1	1	yes	characteristic
75	#32894⋊C₃	C₂×C₄	1	4	yes	complemented
76	C₂×#1791⋊C₃	C₂³	1	1	yes	characteristic
77	#20610⋊C₃	C₂³	1	1	yes	characteristic
78	#20608⋊C₃	C₂³	1	1	yes	characteristic
79	#10574063	C₂×D₃	1	1	yes	characteristic, complemented
80	C₂×#32894	C₂×D₃	1	1	yes	characteristic
81	#6372734	C₂×D₃	1	1	yes	characteristic, complemented
82	#6370735	C₂×D₃	1	1	yes	characteristic, complemented
83	#392717	C₂×D₃	1	1	yes	characteristic, complemented
84	#392611	C₂×D₃	1	1	yes	characteristic, complemented
85	#354788	C₂×D₃	1	1	yes	characteristic, complemented
86	#87547	C₂²×C₄	1	2	yes	complemented
87	C₂×#5950	C₂⁴	1	1	yes	characteristic
88	#87522	C₂²×C₄	1	2	yes	complemented
89	#87521	C₂²×C₄	1	2	yes	complemented
90	#87519	C₂²×C₄	1	2	yes	—
91	#87519	#41	1	2	yes	—
92	C₂×#5934	C₂×D₄	1	1	yes	characteristic
93	C₂×#5932	C₂×D₄	1	1	yes	characteristic
94	#85567	C₂²×C₄	1	2	yes	complemented
95	#85565	C₂×D₄	1	1	yes	characteristic
96	#85565	C₂×D₄	1	1	yes	characteristic
97	#3027⋊C₃	C₂²×C₄	1	2	yes	complemented
98	#1791⋊C₃	C₂²×C₄	1	2	yes	complemented
99	#1790⋊C₃	C₂×D₄	1	1	yes	characteristic
100	#1790⋊C₃	C₂×D₄	1	1	yes	characteristic
101	C₂×H(7,𝔽₂)	C₂²×D₃	1	1	yes	characteristic
102	C₂×#3573	C₂²×D₃	1	1	yes	characteristic
103	#62654	C₂²×D₃	1	1	yes	characteristic
104	C₂×#3027	C₂²×D₃	1	1	yes	characteristic
105	C₂²×#442	S₄	1	1	yes	characteristic, complemented
106	#32894	C₄×D₃	1	4	yes	complemented
107	C₂×#1791	C₂²×D₃	1	1	yes	characteristic
108	#20610	C₂²×D₃	1	1	yes	characteristic
109	#20608	C₂²×D₃	1	1	yes	characteristic
110	#5950	C₂³×C₄	1	1	yes	characteristic
111	#5950	C₂×#41	1	1	yes	characteristic
112	C₂×#997	C₂²×D₄	1	1	yes	characteristic
113	#5934	C₄×D₄	1	2	yes	—
114	#5932	C₄×D₄	1	2	yes	—
115	H(7,𝔽₂)	C₂×C₄×D₃	1	2	yes	complemented
116	C₂×#585	C₂³×D₃	1	1	yes	characteristic
117	C₂³×D₈	C₂×S₄	1	1	yes	characteristic
118	C₂²×C₄×D₄	C₂×S₄	1	1	yes	characteristic
119	C₂²×C₄×D₄	C₂×S₄	1	1	yes	characteristic, complemented
120	#3027	C₂×C₄×D₃	1	2	yes	complemented
121	C₂²×#106	C₂×S₄	1	1	yes	characteristic, complemented
122	C₂²×#102	C₂×S₄	1	1	yes	characteristic
123	C₂²×#102	C₂×S₄	1	1	yes	characteristic, complemented
124	C₂²×#97	C₂×S₄	1	1	yes	characteristic, complemented
125	#1791	C₂×C₄×D₃	1	2	yes	complemented
126	#1790	D₃×D₄	1	1	yes	characteristic
127	#1790	D₃×D₄	1	1	yes	characteristic
128	#997	C₂×C₄×D₄	1	1	yes	characteristic
129	#585	C₂²×C₄×D₃	1	1	yes	characteristic
130	#585	D₃×#41	1	1	yes	characteristic
131	C₂×H(5,𝔽₂)	C₂×D₃×D₄	1	1	yes	characteristic
132	C₂³×D₄	C₂²×S₄	1	1	yes	characteristic
133	C₂³×D₄	C₂²×S₄	1	1	yes	characteristic
134	C₂²×D₈	C₄×S₄	1	4	yes	complemented
135	C₂³×C₈	C₂²×S₄	1	1	yes	characteristic
136	C₂³×C₈	C₂²×S₄	1	1	yes	characteristic
137	C₂²×C₄⋊C₄	C₂²×S₄	1	1	yes	characteristic
138	C₂²×C₄⋊C₄	C₂²×S₄	1	1	yes	characteristic
139	C₂²×C₄²	C₂²×S₄	1	1	yes	characteristic
140	H(5,𝔽₂)	C₄×D₃×D₄	1	1	yes	characteristic
141	C₂²×D₄	C₂×C₄×S₄	1	2	yes	—
142	C₂²×D₄	C₂×C₄×S₄	1	2	yes	—
143	C₂³×C₄	D₄×S₄	1	1	yes	characteristic
144	C₂³×C₄	D₄×S₄	1	1	yes	characteristic
145	C₂³×C₄	C₂³×S₄	1	1	yes	characteristic
146	C₂²×C₈	C₂×C₄×S₄	1	2	yes	—
147	#102	#1020⋊C₂	1	1	yes	characteristic
148	C₂⁴	C₂×D₄×S₄	1	1	yes	characteristic
149	C₂×D₄	C₂×#1020⋊C₂	1	1	yes	characteristic
150	C₂²×C₄	C₂²×C₄×S₄	1	1	yes	characteristic
151	C₂²×C₄	#41×S₄	1	1	yes	characteristic
152	C₂×C₈	C₂×#1020⋊C₂	1	1	yes	characteristic
153	C₄⋊C₄	C₂×#1020⋊C₂	1	1	yes	characteristic
154	C₂³	C₄×D₄×S₄	1	1	yes	characteristic
155	D₄	C₄×#1020⋊C₂	1	2	yes	—
156	C₂×C₄	C₂²×#1020⋊C₂	1	1	yes	characteristic
157	C₂²	D₄×#1020⋊C₂	1	1	yes	characteristic
158	C₄	#72220624	1	1	yes	characteristic
159	C₄	C₂×C₄×#1020⋊C₂	1	1	yes	characteristic
160	C₂	#72240981	1	1	yes	characteristic
161	C₂	#72240982	1	1	yes	characteristic
162	C₂	#72341561	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	D₄×#1020⋊C₂
commutator	#5950	C₂³×C₄
Frattini	C₂³×C₄	C₂³×S₄
Fitting	#72281044	D₃
prime radical	C₂×#72236904	C₂
cosocle	C₂×#5950	C₂⁴
hypercenter	#102	#1020⋊C₂
hypocenter	#997	C₂×C₄×D₄
LCS	G ▷ #5950 ▷ #997
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #102

Complements

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

N	H ≅ G/N	type
1. #72341928	C₂	semidirect
2. #72341923	C₂	semidirect
4. #72341911	C₂	semidirect
5. #72341905	C₂	semidirect
6. #72341898	C₂	semidirect
7. #72341894	C₂	semidirect
8. #72341891	C₂	semidirect
9. #72341890	C₂²	semidirect
10. #72341887	C₂²	semidirect
11. #72341881	C₂	semidirect
13. #72341872	C₂²	semidirect
14. #72341862	C₂	semidirect
15. #72341859	C₂²	semidirect
16. #72341854	C₂²	semidirect
17. #72341847	C₂	semidirect
18. #72341845	C₂²	semidirect
19. #72341843	C₂²	semidirect
20. #72341840	C₂²	semidirect
21. #72341828	C₂	semidirect
22. #72341821	C₂²	semidirect
23. #72341742	C₂	semidirect
25. #72341731	C₂²	semidirect
26. #72341716	C₂²	semidirect
27. #72341715	C₂²	semidirect
28. #72341685	C₂²	semidirect
31. #72341364	C₂	semidirect
32. #72341361	C₂²	semidirect
33. #72341357	C₂²	semidirect
34. #72341341	C₂²	semidirect
35. #72341289	C₂	semidirect
36. #72341287	C₂²	semidirect
39. #72341274	C₂²	semidirect
40. #72341273	C₂²	semidirect
41. #72341264	C₂²	semidirect
43. #72281044	D₃	semidirect
46. #72236904	C₄	semidirect
48. #72227857	C₂²	semidirect
49. #72227330	C₂²	semidirect
50. #72227325	C₂²	semidirect
51. #72227309	C₂²	semidirect
53. #11715839	C₂×C₄	semidirect
57. #11715708	C₂×C₄	semidirect
63. #11714174	C₂×C₄	semidirect
64. #11711643	C₂×C₄	semidirect
65. #11711638	C₂×C₄	semidirect
66. #11711587	C₂×C₄	semidirect
67. #11711580	D₄	semidirect
68. #11711580	D₄	semidirect
69. #11711578	D₄	semidirect
70. #11711578	D₄	semidirect
75. #32894⋊C₃	C₂×C₄	semidirect
79. #10574063	C₂×D₃	semidirect
81. #6372734	C₂×D₃	semidirect
82. #6370735	C₂×D₃	semidirect
83. #392717	C₂×D₃	semidirect
84. #392611	C₂×D₃	semidirect
85. #354788	C₂×D₃	semidirect
86. #87547	C₂²×C₄	semidirect
88. #87522	C₂²×C₄	semidirect
89. #87521	C₂²×C₄	semidirect
94. #85567	C₂²×C₄	semidirect
97. #3027⋊C₃	C₂²×C₄	semidirect
98. #1791⋊C₃	C₂²×C₄	semidirect
105. C₂²×#442	S₄	semidirect
106. #32894	C₄×D₃	semidirect
115. H(7,𝔽₂)	C₂×C₄×D₃	semidirect
119. C₂²×C₄×D₄	C₂×S₄	semidirect
120. #3027	C₂×C₄×D₃	semidirect
121. C₂²×#106	C₂×S₄	semidirect
123. C₂²×#102	C₂×S₄	semidirect
124. C₂²×#97	C₂×S₄	semidirect
125. #1791	C₂×C₄×D₃	semidirect
134. C₂²×D₈	C₄×S₄	semidirect

Maximal subgroups

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

subgroup	index
#72341928	2
#72341923	2
C₂×#72236904	2
#72341911	2
#72341905	2
#72341898	2
#72341894	2
#72341891	2
#72341881	2
#72341862	2
#72341847	2
#72341828	2
#72341742	2
#72341364	2
#72341289	2
#72312418	3
S₄×#442	4