Finite Group Databasebeta

Subgroups — #14558342

Order 2368.

258 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11924584	C₂	1	1	yes	maximal, characteristic, complemented
2	#11924582	C₂	1	1	yes	maximal, characteristic, complemented
3	#11924568	C₂	1	2	yes	maximal, complemented
4	#11924564	C₂	1	1	yes	maximal, characteristic, complemented
5	#11924563	C₂	1	2	yes	maximal, complemented
6	#11924559	C₂	1	2	yes	maximal, complemented
7	#11924556	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10596941	C₂	1	1	yes	maximal, characteristic, complemented
9	SD₁₆×D₃₇	C₂	1	2	yes	maximal, complemented
10	#11924544	C₂	1	1	yes	maximal, characteristic, complemented
11	#137×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	#41×C₃₇	C₂²	1	1	yes	characteristic
13	C₂×Q₈×C₃₇	C₂²	1	1	yes	characteristic, complemented
14	#10596966	—	2	1	no	—
15	#10596966	C₂²	1	2	yes	—
16	#10596966	C₂²	1	2	yes	complemented
17	Q₈×D₃₇	—	2	1	no	—
18	Q₈×D₃₇	C₂²	1	2	yes	complemented
19	D₄×D₃₇	—	2	2	no	—
20	D₄×D₃₇	C₂²	1	2	yes	—
21	#10596962	—	2	2	no	—
22	#10596962	—	2	1	no	—
23	#10596962	C₂²	1	1	yes	characteristic
24	#10596962	C₂²	1	1	yes	characteristic, complemented
25	C₂×D₁₄₈	—	2	1	no	—
26	C₂×D₁₄₈	C₂²	1	1	yes	characteristic
27	Q₁₆×C₃₇	C₂²	1	2	yes	complemented
28	SD₁₆×C₃₇	C₂²	1	2	yes	complemented
29	M₁₆×C₃₇	C₂²	1	1	yes	characteristic, complemented
30	#10596942	C₂²	1	1	yes	characteristic, complemented
31	#10596941	C₂²	1	2	yes	complemented
32	#10596941	C₂²	1	2	yes	complemented
33	#10596941	C₂²	1	1	yes	characteristic, complemented
34	#10596940	C₂²	1	1	yes	characteristic, complemented
35	#10596939	C₂²	1	1	yes	characteristic, complemented
36	C₂×#64436	C₂²	1	1	yes	characteristic, complemented
37	D₂₉₆	C₂²	1	2	yes	complemented
38	#10596930	C₂²	1	2	yes	complemented
39	#10596929	C₂²	1	2	yes	complemented
40	C₈×D₃₇	C₂²	1	2	yes	complemented
41	C₂²×D₃₇	—	2	1	no	—
42	C₂²×D₃₇	—	2	2	no	—
43	Q₈×C₃₇	—	2	1	no	—
44	Q₈×C₃₇	C₂³	1	2	yes	—
45	Q₈×C₃₇	C₂³	1	1	yes	characteristic
46	D₄×C₃₇	—	2	1	no	—
47	D₄×C₃₇	C₂³	1	1	yes	characteristic
48	C₂×C₄×C₃₇	—	2	1	no	—
49	C₂×C₄×C₃₇	—	2	1	no	—
50	C₂×C₄×C₃₇	C₂³	1	1	yes	characteristic
51	C₂²⋊D₃₇	—	2	1	no	—
52	C₂²⋊D₃₇	—	2	2	no	—
53	C₂²⋊D₃₇	D₄	1	2	yes	—
54	D₁₄₈	—	2	2	no	—
55	D₁₄₈	—	2	2	no	—
56	D₁₄₈	—	2	1	no	—
57	D₁₄₈	D₄	1	1	yes	characteristic
58	D₁₄₈	C₂³	1	2	yes	—
59	D₁₄₈	C₂³	1	1	yes	characteristic
60	C₄×D₃₇	—	2	2	no	—
61	C₄×D₃₇	—	2	1	no	—
62	C₄×D₃₇	—	2	2	no	—
63	C₄×D₃₇	—	2	2	no	—
64	C₄×D₃₇	C₂³	1	2	yes	—
65	Dic₇₄	—	2	2	no	—
66	Dic₇₄	D₄	1	1	yes	characteristic
67	Dic₇₄	C₂³	1	1	yes	characteristic
68	C₈×C₃₇	C₂³	1	2	yes	—
69	#64436	C₂³	1	1	yes	characteristic
70	#64436	C₂³	1	1	yes	characteristic
71	C₂²×C₃₇	—	2	1	no	—
72	C₂²×C₃₇	C₂×D₄	1	1	yes	characteristic
73	C₂×D₃₇	—	2	1	no	—
74	C₂×D₃₇	—	2	2	no	—
75	C₂×D₃₇	—	8	2	no	—
76	C₂×D₃₇	—	8	1	no	—
77	C₂×D₃₇	C₂×D₄	1	2	yes	—
78	C₄×C₃₇	—	2	2	no	—
79	C₄×C₃₇	—	2	1	no	—
80	C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
81	C₄×C₃₇	C₂⁴	1	1	yes	characteristic
82	Dic₃₇	—	2	1	no	—
83	Dic₃₇	C₂×D₄	1	2	yes	—
84	C₂×C₃₇	—	2	1	no	—
85	C₂×C₃₇	—	4	1	no	—
86	C₂×C₃₇	C₂²×D₄	1	1	yes	characteristic
87	D₃₇	—	2	2	no	—
88	D₃₇	—	4	1	no	—
89	D₃₇	—	4	2	no	—
90	#577	—	37	1	no	maximal, Sylow
91	C₃₇	#577	1	1	yes	characteristic, complemented, Sylow
92	2₋¹⁺⁴	—	37	1	no	—
93	H(5,𝔽₂)	—	37	1	no	—
94	#137	—	37	2	no	—
95	#137	D₃₇	1	1	yes	characteristic, complemented
96	AGL(1,ℤ/8)	—	37	2	no	—
97	AGL(1,ℤ/8)	—	37	1	no	—
98	#135	—	37	1	no	—
99	#135	—	37	2	no	—
100	C₂×SD₁₆	—	37	1	no	—
101	C₂×SD₁₆	—	37	2	no	—
102	#131	—	37	1	no	—
103	#41	—	37	1	no	—
104	#41	—	37	2	no	—
105	#41	—	37	2	no	—
106	#41	—	37	1	no	—
107	#41	—	74	2	no	—
108	#41	—	74	1	no	—
109	#41	—	74	1	no	—
110	#41	C₂×D₃₇	1	1	yes	characteristic
111	C₂×Q₈	—	37	2	no	—
112	C₂×Q₈	—	74	1	no	—
113	C₂×Q₈	C₂×D₃₇	1	1	yes	characteristic, complemented
114	C₂×D₄	—	37	2	no	—
115	C₂×D₄	—	37	1	no	—
116	C₂×D₄	—	74	1	no	—
117	C₂×D₄	—	74	2	no	—
118	Q₁₆	—	37	1	no	—
119	Q₁₆	C₂×D₃₇	1	2	yes	complemented
120	SD₁₆	—	37	2	no	—
121	SD₁₆	—	37	2	no	—
122	SD₁₆	—	37	1	no	—
123	SD₁₆	—	37	1	no	—
124	SD₁₆	—	37	2	no	—
125	SD₁₆	C₂×D₃₇	1	2	yes	complemented
126	D₈	—	37	1	no	—
127	D₈	—	37	2	no	—
128	M₁₆	—	37	2	no	—
129	M₁₆	C₂×D₃₇	1	1	yes	characteristic, complemented
130	C₂×C₈	—	37	1	no	—
131	C₂×C₈	—	37	2	no	—
132	C₂³	—	74	2	no	—
133	C₂³	—	74	1	no	—
134	Q₈	—	2	1	no	—
135	Q₈	—	37	1	no	—
136	Q₈	—	37	1	no	—
137	Q₈	—	74	2	no	—
138	Q₈	C₂²×D₃₇	1	2	yes	—
139	Q₈	C₂²×D₃₇	1	1	yes	characteristic
140	D₄	—	2	1	no	—
141	D₄	—	37	2	no	—
142	D₄	—	37	1	no	—
143	D₄	—	37	1	no	—
144	D₄	—	37	2	no	—
145	D₄	—	74	2	no	—
146	D₄	—	74	2	no	—
147	D₄	—	74	2	no	—
148	D₄	—	74	1	no	—
149	D₄	—	74	1	no	—
150	D₄	C₂²×D₃₇	1	1	yes	characteristic
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	37	2	no	—
154	C₂×C₄	—	74	2	no	—
155	C₂×C₄	—	74	2	no	—
156	C₂×C₄	—	74	2	no	—
157	C₂×C₄	—	74	1	no	—
158	C₂×C₄	C₂²×D₃₇	1	1	yes	characteristic
159	C₈	—	37	1	no	—
160	C₈	—	37	1	no	—
161	C₈	C₂²×D₃₇	1	2	yes	—
162	C₂²	—	2	1	no	—
163	C₂²	—	37	2	no	—
164	C₂²	—	296	1	no	—
165	C₂²	—	296	2	no	—
166	C₂²	—	74	1	no	—
167	C₂²	—	74	2	no	—
168	C₂²	D₄×D₃₇	1	1	yes	characteristic
169	C₄	—	2	1	no	—
170	C₄	—	2	2	no	—
171	C₄	—	37	2	no	—
172	C₄	—	74	1	no	—
173	C₄	D₄×D₃₇	1	1	yes	characteristic
174	C₄	C₂³×D₃₇	1	1	yes	characteristic
175	C₂	—	2	1	no	—
176	C₂	—	4	1	no	—
177	C₂	—	74	2	no	—
178	C₂	—	148	2	no	—
179	C₂	—	148	1	no	—
180	C₂	C₂×D₄×D₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×D₃₇
commutator, cosocle	C₄×C₃₇	C₂⁴
Frattini	C₄	C₂³×D₃₇
Fitting	#137×C₃₇	C₂
socle	C₂×C₃₇	C₂²×D₄
hypercenter	#137	D₃₇
hypocenter	C₃₇	#577
LCS	G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #137

Complements

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

N	H ≅ G/N	type
1. #11924584	C₂	semidirect
2. #11924582	C₂	semidirect
3. #11924568	C₂	semidirect
4. #11924564	C₂	semidirect
5. #11924563	C₂	semidirect
6. #11924559	C₂	semidirect
7. #11924556	C₂	semidirect
8. C₂×#10596941	C₂	semidirect
9. SD₁₆×D₃₇	C₂	semidirect
10. #11924544	C₂	semidirect
11. #137×C₃₇	C₂	semidirect
13. C₂×Q₈×C₃₇	C₂²	semidirect
16. #10596966	C₂²	semidirect
18. Q₈×D₃₇	C₂²	semidirect
24. #10596962	C₂²	semidirect
27. Q₁₆×C₃₇	C₂²	semidirect
28. SD₁₆×C₃₇	C₂²	semidirect
29. M₁₆×C₃₇	C₂²	semidirect
30. #10596942	C₂²	semidirect
31. #10596941	C₂²	semidirect
32. #10596941	C₂²	semidirect
33. #10596941	C₂²	semidirect
34. #10596940	C₂²	semidirect
35. #10596939	C₂²	semidirect
36. C₂×#64436	C₂²	semidirect
37. D₂₉₆	C₂²	semidirect
38. #10596930	C₂²	semidirect
39. #10596929	C₂²	semidirect
40. C₈×D₃₇	C₂²	semidirect
91. C₃₇	#577	semidirect
95. #137	D₃₇	semidirect
113. C₂×Q₈	C₂×D₃₇	semidirect
119. Q₁₆	C₂×D₃₇	semidirect
125. SD₁₆	C₂×D₃₇	semidirect
129. M₁₆	C₂×D₃₇	semidirect

Maximal subgroups

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

subgroup	index
#11924584	2
#11924582	2
#11924568	2
#11924564	2
#11924563	2
#11924559	2
#11924556	2
C₂×#10596941	2
SD₁₆×D₃₇	2
#11924544	2
#137×C₃₇	2
#577	37