Finite Group Databasebeta

Subgroups — #34061097

Order 19072.

142 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 142 conjugacy classes of subgroups — too many to draw. Only its 44 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	#23750672	C₂	1	1	yes	maximal, characteristic
2	#23750665	C₂	1	1	yes	maximal, characteristic, complemented
3	#23750549	C₂	1	1	yes	maximal, characteristic, complemented
4	#23750544	C₂	1	1	yes	maximal, characteristic, complemented
5	#23750541	C₂	1	1	yes	maximal, characteristic
6	#23750516	C₂	1	1	yes	maximal, characteristic, complemented
7	#23750064	C₂	1	1	yes	maximal, characteristic
8	#18011998	—	2	1	no	—
9	Q₈×#10597010	—	2	1	no	—
10	#18011986	C₂²	1	1	yes	characteristic
11	#18011981	C₂²	1	1	yes	characteristic
12	#18011979	C₂²	1	1	yes	characteristic
13	#18011975	—	2	1	no	—
14	#18011974	—	2	1	no	—
15	#18011965	C₂²	1	1	yes	characteristic
16	#18011930	C₂²	1	1	yes	characteristic
17	Q₁₆×D₁₄₉	C₂²	1	1	yes	characteristic
18	Dic₁₁₉₂	C₄	1	1	yes	characteristic, complemented
19	#18011857	C₄	1	1	yes	characteristic, complemented
20	#18011855	C₄	1	1	yes	characteristic, complemented
21	#18011850	C₂²	1	1	yes	characteristic
22	SD₃₂×C₁₄₉	C₄	1	1	yes	characteristic, complemented
23	#14558746	—	2	1	no	—
24	#14558745	—	4	1	no	—
25	D₁₄₉⋊C₈	D₄	1	1	yes	characteristic
26	C₂×#11924753	—	4	1	no	—
27	C₄⋊#10597010	—	2	1	no	—
28	C₄⋊#10597010	—	4	1	no	—
29	C₄×#10597010	—	4	1	no	—
30	C₄×#10597010	D₄	1	1	yes	characteristic
31	#14558739	—	2	1	no	—
32	#14558735	—	2	1	no	—
33	Q₈×D₁₄₉	—	2	1	no	—
34	Dic₅₉₆	C₂×C₄	1	1	yes	characteristic
35	Dic₅₉₆	C₂×C₄	1	1	yes	characteristic
36	#14558726	—	2	1	no	—
37	#14558724	—	2	1	no	—
38	C₈×D₁₄₉	C₂³	1	1	yes	characteristic
39	#14558711	C₂×C₄	1	1	yes	characteristic
40	Q₁₆×C₁₄₉	C₂×C₄	1	1	yes	characteristic
41	D₈×C₁₄₉	C₂×C₄	1	1	yes	characteristic
42	C₁₆×C₁₄₉	C₂×C₄	1	1	yes	characteristic
43	C₂×#10597010	—	2	1	no	—
44	C₂×#10597010	—	4	1	no	—
45	#11924753	—	2	1	no	—
46	#11924753	—	4	1	no	—
47	Dic₂₉₈	—	2	1	no	—
48	Dic₂₉₈	—	2	1	no	—
49	Dic₂₉₈	—	4	1	no	—
50	C₂²⋊D₁₄₉	—	4	1	no	—
51	C₂×Dic₁₄₉	—	4	1	no	—
52	C₄×D₁₄₉	—	4	1	no	—
53	C₄×D₁₄₉	C₂×D₄	1	1	yes	characteristic
54	#11924746	#41	1	1	yes	characteristic
55	Q₈×C₁₄₉	—	2	1	no	—
56	D₄×C₁₄₉	—	2	1	no	—
57	C₈×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₁₄₉	—	4	1	no	—
59	C₂×D₁₄₉	AGL(1,ℤ/8)	1	1	yes	characteristic
60	#10597010	—	4	1	no	—
61	#10597010	—	8	1	no	—
62	C₄×C₁₄₉	—	4	1	no	—
63	C₄×C₁₄₉	C₄×D₄	1	1	yes	characteristic
64	Dic₁₄₉	—	4	1	no	—
65	Dic₁₄₉	—	4	1	no	—
66	Dic₁₄₉	AGL(1,ℤ/8)	1	1	yes	characteristic
67	C₂×C₁₄₉	—	8	1	no	—
68	C₂×C₁₄₉	#442	1	1	yes	characteristic
69	D₁₄₉	—	2	1	no	—
70	C₁₄₉	#2180	1	1	yes	characteristic, complemented, Sylow
71	#2180	—	149	1	no	maximal, Sylow
72	#510	—	149	1	no	—
73	#444	—	149	1	no	—
74	#441	—	149	1	no	—
75	#365	—	149	1	no	—
76	#362	—	149	1	no	—
77	#360	—	149	1	no	—
78	#347	—	149	1	no	—
79	#135	—	149	1	no	—
80	C₂×Q₁₆	—	149	1	no	—
81	#131	—	298	1	no	—
82	C₄×Q₈	—	298	1	no	—
83	Q₃₂	—	149	1	no	—
84	Q₃₂	—	149	1	no	—
85	SD₃₂	—	149	1	no	—
86	SD₃₂	#10597010	1	1	yes	characteristic, complemented
87	M₃₂	—	149	1	no	—
88	M₃₂	—	149	1	no	—
89	#108	—	149	1	no	—
90	#106	—	149	1	no	—
91	#104	—	298	1	no	—
92	#103	—	298	1	no	—
93	#97	—	149	1	no	—
94	#41	—	298	1	no	—
95	C₂×Q₈	—	298	1	no	—
96	Q₁₆	—	298	1	no	—
97	Q₁₆	—	149	1	no	—
98	Q₁₆	—	149	1	no	—
99	Q₁₆	C₂×#10597010	1	1	yes	characteristic
100	SD₁₆	—	298	1	no	—
101	D₈	C₂×#10597010	1	1	yes	characteristic
102	M₁₆	—	298	1	no	—
103	M₁₆	—	596	1	no	—
104	C₂×C₈	—	596	1	no	—
105	C₂×C₈	—	149	1	no	—
106	C₂×C₈	—	149	1	no	—
107	C₄⋊C₄	—	298	1	no	—
108	C₄⋊C₄	—	596	1	no	—
109	C₄²	—	596	1	no	—
110	C₄²	—	149	1	no	—
111	C₁₆	—	298	1	no	—
112	C₁₆	—	149	1	no	—
113	C₁₆	C₂×#10597010	1	1	yes	characteristic
114	Q₈	—	2	1	no	—
115	Q₈	—	298	1	no	—
116	Q₈	—	298	1	no	—
117	Q₈	—	596	1	no	—
118	D₄	—	2	1	no	—
119	D₄	—	596	1	no	—
120	C₂×C₄	—	298	1	no	—
121	C₂×C₄	—	596	1	no	—
122	C₂×C₄	—	596	1	no	—
123	C₂×C₄	—	596	1	no	—
124	C₂×C₄	—	149	1	no	—
125	C₈	—	298	1	no	—
126	C₈	—	596	1	no	—
127	C₈	—	149	1	no	—
128	C₈	C₂²×#10597010	1	1	yes	characteristic
129	C₂²	—	4	1	no	—
130	C₂²	—	149	1	no	—
131	C₄	—	4	1	no	—
132	C₄	—	596	1	no	—
133	C₄	—	596	1	no	—
134	C₄	—	596	1	no	—
135	C₄	—	149	1	no	—
136	C₄	—	1192	1	no	—
137	C₄	D₄×#10597010	1	1	yes	characteristic
138	C₂	—	8	1	no	—
139	C₂	—	298	1	no	—
140	C₂	#23750666	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#23750666
commutator	C₈×C₁₄₉	C₂²×C₄
Frattini	C₈	C₂²×#10597010
Fitting	SD₃₂×C₁₄₉	C₄
socle	C₂×C₁₄₉	#442
prime radical	#18011930	C₂²
cosocle	C₈×D₁₄₉	C₂³
hypercenter	SD₃₂	#10597010
hypocenter	C₁₄₉	#2180
LCS	G ▷ C₈×C₁₄₉ ▷ C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ SD₃₂

Complements

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

N	H ≅ G/N	type
2. #23750665	C₂	semidirect
3. #23750549	C₂	semidirect
4. #23750544	C₂	semidirect
6. #23750516	C₂	semidirect
18. Dic₁₁₉₂	C₄	semidirect
19. #18011857	C₄	semidirect
20. #18011855	C₄	semidirect
22. SD₃₂×C₁₄₉	C₄	semidirect
70. C₁₄₉	#2180	semidirect
86. SD₃₂	#10597010	semidirect

Maximal subgroups

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

subgroup	index
#23750672	2
#23750665	2
#23750549	2
#23750544	2
#23750541	2
#23750516	2
#23750064	2
#2180	149