Finite Group Databasebeta

Subgroups — #34061091

Order 19072.

146 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 46 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	#23750669	C₂	1	1	yes	maximal, characteristic, complemented
2	#23750667	C₂	1	1	yes	maximal, characteristic, complemented
3	#23750554	C₂	1	1	yes	maximal, characteristic, complemented
4	#23750540	C₂	1	1	yes	maximal, characteristic, complemented
5	#23750539	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₆×#10597010	C₂	1	1	yes	maximal, characteristic, complemented
7	#23750056	C₂	1	1	yes	maximal, characteristic
8	#18011999	—	2	1	no	—
9	#18011998	—	2	1	no	—
10	D₁₄₉⋊C₁₆	C₂²	1	1	yes	characteristic
11	#18011982	C₂²	1	1	yes	characteristic
12	#18011982	C₂²	1	1	yes	characteristic
13	#18011976	—	2	1	no	—
14	#18011975	—	2	1	no	—
15	C₈×#10597010	C₂²	1	1	yes	characteristic
16	#18011934	C₂²	1	1	yes	characteristic
17	#18011930	C₂²	1	1	yes	characteristic
18	#18011857	C₄	1	1	yes	characteristic, complemented
19	#18011856	C₄	1	1	yes	characteristic, complemented
20	#18011852	C₄	1	1	yes	characteristic, complemented
21	C₁₆×D₁₄₉	C₂²	1	1	yes	characteristic
22	SD₃₂×C₁₄₉	C₄	1	1	yes	characteristic, complemented
23	#14558746	—	2	1	no	—
24	#14558746	—	2	1	no	—
25	#14558746	—	4	1	no	—
26	#14558745	—	4	1	no	—
27	D₁₄₉⋊C₈	—	4	1	no	—
28	D₁₄₉⋊C₈	D₄	1	1	yes	characteristic
29	C₂×#11924753	—	4	1	no	—
30	C₄×#10597010	D₄	1	1	yes	characteristic
31	#14558739	—	2	1	no	—
32	#14558737	—	2	1	no	—
33	#14558735	—	2	1	no	—
34	Dic₅₉₆	C₂×C₄	1	1	yes	characteristic
35	#14558724	—	2	1	no	—
36	#14558723	—	2	1	no	—
37	D₁₁₉₂	C₂×C₄	1	1	yes	characteristic
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	#11924753	—	4	1	no	—
45	#11924753	—	4	1	no	—
46	#11924753	D₈	1	1	yes	characteristic
47	#11924753	D₈	1	1	yes	characteristic
48	Dic₂₉₈	—	2	1	no	—
49	D₅₉₆	—	2	1	no	—
50	D₅₉₆	—	4	1	no	—
51	C₂²⋊D₁₄₉	—	4	1	no	—
52	C₂×Dic₁₄₉	—	4	1	no	—
53	C₄×D₁₄₉	—	4	1	no	—
54	C₄×D₁₄₉	C₂×D₄	1	1	yes	characteristic
55	#11924746	#41	1	1	yes	characteristic
56	Q₈×C₁₄₉	—	2	1	no	—
57	D₄×C₁₄₉	—	2	1	no	—
58	C₈×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₁₄₉	—	4	1	no	—
60	C₂×D₁₄₉	—	4	1	no	—
61	C₂×D₁₄₉	#135	1	1	yes	characteristic
62	#10597010	—	2	2	no	—
63	C₄×C₁₄₉	—	4	1	no	—
64	C₄×C₁₄₉	C₄×D₄	1	1	yes	characteristic
65	Dic₁₄₉	—	4	1	no	—
66	Dic₁₄₉	C₂×D₈	1	1	yes	characteristic
67	C₂×C₁₄₉	—	8	1	no	—
68	C₂×C₁₄₉	C₄×D₈	1	1	yes	characteristic
69	D₁₄₉	—	2	1	no	—
70	D₁₄₉	—	8	1	no	—
71	C₁₄₉	#2178	1	1	yes	characteristic, complemented, Sylow
72	#2178	—	149	1	no	maximal, Sylow
73	#508	—	149	1	no	—
74	#443	—	149	1	no	—
75	#443	—	149	1	no	—
76	#368	—	149	1	no	—
77	#359	—	149	1	no	—
78	#359	—	149	1	no	—
79	C₄×C₁₆	—	149	1	no	—
80	#135	—	149	1	no	—
81	#135	—	149	1	no	—
82	#131	—	298	1	no	—
83	#131	—	298	1	no	—
84	Q₃₂	—	149	1	no	—
85	SD₃₂	—	149	1	no	—
86	SD₃₂	#10597010	1	1	yes	characteristic, complemented
87	D₁₆	—	149	1	no	—
88	C₂×C₁₆	—	149	1	no	—
89	C₂×C₁₆	—	149	1	no	—
90	#108	—	149	1	no	—
91	#108	—	149	1	no	—
92	#104	—	298	1	no	—
93	#104	—	298	1	no	—
94	C₄×C₈	—	149	1	no	—
95	#41	—	298	1	no	—
96	#41	—	298	1	no	—
97	Q₁₆	—	149	1	no	—
98	Q₁₆	C₂×#10597010	1	1	yes	characteristic
99	SD₁₆	—	298	1	no	—
100	SD₁₆	—	298	1	no	—
101	D₈	—	149	1	no	—
102	D₈	C₂×#10597010	1	1	yes	characteristic
103	M₁₆	—	298	1	no	—
104	M₁₆	—	298	1	no	—
105	M₁₆	—	596	1	no	—
106	M₁₆	—	596	1	no	—
107	C₂×C₈	—	596	1	no	—
108	C₂×C₈	—	596	1	no	—
109	C₂×C₈	—	149	1	no	—
110	C₂×C₈	—	149	1	no	—
111	C₄²	—	149	1	no	—
112	C₁₆	—	298	1	no	—
113	C₁₆	—	149	1	no	—
114	C₁₆	C₂×#10597010	1	1	yes	characteristic
115	Q₈	—	2	1	no	—
116	Q₈	—	298	1	no	—
117	D₄	—	2	1	no	—
118	D₄	—	298	1	no	—
119	D₄	—	596	1	no	—
120	D₄	—	596	1	no	—
121	C₂×C₄	—	298	1	no	—
122	C₂×C₄	—	596	1	no	—
123	C₂×C₄	—	596	1	no	—
124	C₂×C₄	—	149	1	no	—
125	C₈	—	596	1	no	—
126	C₈	—	596	1	no	—
127	C₈	—	149	1	no	—
128	C₈	—	149	1	no	—
129	C₈	—	149	1	no	—
130	C₈	C₂²×#10597010	1	1	yes	characteristic
131	C₂²	—	4	1	no	—
132	C₂²	—	596	1	no	—
133	C₂²	—	149	1	no	—
134	C₄	—	4	1	no	—
135	C₄	—	298	2	no	—
136	C₄	—	596	1	no	—
137	C₄	—	149	1	no	—
138	C₄	D₄×#10597010	1	1	yes	characteristic
139	C₂	—	8	1	no	—
140	C₂	—	298	1	no	—
141	C₂	—	1192	1	no	—
142	C₂	D₈×#10597010	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₈×#10597010
commutator	C₈×C₁₄₉	C₂²×C₄
Frattini	C₈	C₂²×#10597010
Fitting	SD₃₂×C₁₄₉	C₄
socle	C₂×C₁₄₉	C₄×D₈
prime radical	#23750056	C₂
cosocle	C₈×D₁₄₉	C₂³
hypercenter	SD₃₂	#10597010
hypocenter	C₁₄₉	#2178
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
1. #23750669	C₂	semidirect
2. #23750667	C₂	semidirect
3. #23750554	C₂	semidirect
4. #23750540	C₂	semidirect
5. #23750539	C₂	semidirect
6. C₁₆×#10597010	C₂	semidirect
18. #18011857	C₄	semidirect
19. #18011856	C₄	semidirect
20. #18011852	C₄	semidirect
22. SD₃₂×C₁₄₉	C₄	semidirect
71. C₁₄₉	#2178	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
#23750669	2
#23750667	2
#23750554	2
#23750540	2
#23750539	2
C₁₆×#10597010	2
#23750056	2
#2178	149