Finite Group Databasebeta

Subgroups — #20295041

Order 6976.

154 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 51 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	#41×D₁₀₉	C₂	1	1	yes	maximal, characteristic
2	M₁₆×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#16812656	C₂	1	1	yes	maximal, characteristic, complemented
4	#16812627	C₂	1	1	yes	maximal, characteristic, complemented
5	#16812626	C₂	1	1	yes	maximal, characteristic, complemented
6	#16812625	C₂	1	1	yes	maximal, characteristic, complemented
7	#104×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#13182030	C₄	1	1	yes	characteristic, complemented
9	#13182029	—	2	1	no	—
10	#13182029	C₂²	1	1	yes	characteristic
11	#13182028	—	2	1	no	—
12	#13182028	C₄	1	1	yes	characteristic, complemented
13	Q₈×D₁₀₉	C₄	1	1	yes	characteristic, complemented
14	D₄×D₁₀₉	—	2	1	no	—
15	D₄×D₁₀₉	C₄	1	1	yes	characteristic, complemented
16	C₂×C₄×D₁₀₉	—	2	1	no	—
17	C₂×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
18	#13182013	C₂²	1	1	yes	characteristic, complemented
19	#13182012	—	2	1	no	—
20	C₈×D₁₀₉	—	2	1	no	—
21	#13182008	—	2	1	no	—
22	#13182006	—	2	1	no	—
23	C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic, complemented
24	#41×C₁₀₉	C₂²	1	1	yes	characteristic
25	M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
26	C₄²×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
27	C₂²×D₁₀₉	—	2	1	no	—
28	C₂²×D₁₀₉	D₄	1	1	yes	characteristic
29	Q₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
30	D₄×C₁₀₉	—	2	1	no	—
31	D₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×C₁₀₉	—	2	1	no	—
33	C₂×C₄×C₁₀₉	—	2	1	no	—
34	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
35	C₂²⋊D₁₀₉	—	2	1	no	—
36	C₂²⋊D₁₀₉	—	2	1	no	—
37	C₂²⋊D₁₀₉	—	2	1	no	—
38	C₂×Dic₁₀₉	—	2	1	no	—
39	C₂×Dic₁₀₉	—	2	1	no	—
40	C₂×Dic₁₀₉	D₄	1	1	yes	characteristic
41	D₄₃₆	—	2	1	no	—
42	D₄₃₆	C₂×C₄	1	1	yes	characteristic
43	C₄×D₁₀₉	—	2	1	no	—
44	C₄×D₁₀₉	—	2	1	no	—
45	C₄×D₁₀₉	—	2	1	no	—
46	C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
47	C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
48	C₄×D₁₀₉	D₄	1	1	yes	characteristic
49	C₄×D₁₀₉	D₄	1	1	yes	characteristic
50	Dic₂₁₈	—	2	1	no	—
51	Dic₂₁₈	C₂×C₄	1	1	yes	characteristic
52	C₈×C₁₀₉	—	2	1	no	—
53	#11724784	—	2	1	no	—
54	C₂²×C₁₀₉	—	2	1	no	—
55	C₂²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
56	C₂×D₁₀₉	—	2	1	no	—
57	C₂×D₁₀₉	—	4	1	no	—
58	C₂×D₁₀₉	—	4	2	no	—
59	C₂×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
61	C₄×C₁₀₉	—	2	1	no	—
62	C₄×C₁₀₉	—	4	1	no	—
63	C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
64	C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
65	Dic₁₀₉	—	2	1	no	—
66	Dic₁₀₉	—	4	1	no	—
67	Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
68	Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₁₀₉	—	2	1	no	—
70	C₂×C₁₀₉	—	4	1	no	—
71	C₂×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	D₁₀₉	—	2	1	no	—
73	D₁₀₉	—	2	1	no	—
74	D₁₀₉	—	4	1	no	—
75	C₁₀₉	#421	1	1	yes	characteristic, complemented, Sylow
76	#421	—	109	1	no	maximal, Sylow
77	C₂×#41	—	109	1	no	—
78	C₂×M₁₆	—	109	1	no	—
79	#117	—	109	1	no	—
80	#104	—	109	1	no	—
81	#104	—	109	1	no	—
82	#104	—	109	1	no	—
83	#104	D₁₀₉	1	1	yes	characteristic, complemented
84	#41	—	109	1	no	—
85	#41	—	109	1	no	—
86	#41	—	109	1	no	—
87	#41	—	218	1	no	—
88	#41	—	218	1	no	—
89	#41	C₂×D₁₀₉	1	1	yes	characteristic
90	C₂×Q₈	—	109	1	no	—
91	C₂×D₄	—	109	1	no	—
92	C₂×D₄	—	218	1	no	—
93	C₂²×C₄	—	109	1	no	—
94	C₂²×C₄	—	218	1	no	—
95	M₁₆	—	109	1	no	—
96	M₁₆	—	218	1	no	—
97	M₁₆	C₂×D₁₀₉	1	1	yes	characteristic, complemented
98	C₂×C₈	—	218	1	no	—
99	C₄⋊C₄	—	218	1	no	—
100	C₂²⋊C₄	—	218	1	no	—
101	C₄²	—	109	1	no	—
102	C₄²	C₂×D₁₀₉	1	1	yes	characteristic, complemented
103	C₂³	—	109	1	no	—
104	C₂³	—	218	1	no	—
105	Q₈	—	109	1	no	—
106	Q₈	—	218	1	no	—
107	Q₈	C₄×D₁₀₉	1	1	yes	characteristic
108	D₄	—	2	1	no	—
109	D₄	—	109	1	no	—
110	D₄	—	218	1	no	—
111	D₄	—	218	1	no	—
112	D₄	—	218	1	no	—
113	D₄	—	218	1	no	—
114	D₄	C₄×D₁₀₉	1	1	yes	characteristic
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	109	1	no	—
118	C₂×C₄	—	109	1	no	—
119	C₂×C₄	—	109	1	no	—
120	C₂×C₄	—	109	1	no	—
121	C₂×C₄	—	109	1	no	—
122	C₂×C₄	—	218	1	no	—
123	C₂×C₄	—	218	1	no	—
124	C₂×C₄	—	218	1	no	—
125	C₂×C₄	—	218	1	no	—
126	C₂×C₄	—	218	1	no	—
127	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
128	C₈	—	2	1	no	—
129	C₈	—	218	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	109	1	no	—
132	C₂²	—	109	1	no	—
133	C₂²	—	436	1	no	—
134	C₂²	—	436	2	no	—
135	C₂²	—	218	1	no	—
136	C₂²	D₄×D₁₀₉	1	1	yes	characteristic
137	C₄	—	2	1	no	—
138	C₄	—	4	1	no	—
139	C₄	—	109	1	no	—
140	C₄	—	109	1	no	—
141	C₄	—	436	1	no	—
142	C₄	—	218	1	no	—
143	C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
144	C₄	D₄×D₁₀₉	1	1	yes	characteristic
145	C₂	—	2	1	no	—
146	C₂	—	4	1	no	—
147	C₂	—	436	1	no	—
148	C₂	—	218	1	no	—
149	C₂	—	218	1	no	—
150	C₂	C₂²⋊C₄×D₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×D₁₀₉
commutator	C₄×C₁₀₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₀₉
Fitting	#104×C₁₀₉	C₂
socle	C₂×C₁₀₉	C₂×C₂²⋊C₄
hypercenter	#104	D₁₀₉
hypocenter	C₁₀₉	#421
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #104

Complements

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

N	H ≅ G/N	type
2. M₁₆×D₁₀₉	C₂	semidirect
3. #16812656	C₂	semidirect
4. #16812627	C₂	semidirect
5. #16812626	C₂	semidirect
6. #16812625	C₂	semidirect
7. #104×C₁₀₉	C₂	semidirect
8. #13182030	C₄	semidirect
12. #13182028	C₄	semidirect
13. Q₈×D₁₀₉	C₄	semidirect
15. D₄×D₁₀₉	C₄	semidirect
18. #13182013	C₂²	semidirect
23. C₄×Dic₁₀₉	C₂²	semidirect
25. M₁₆×C₁₀₉	C₂²	semidirect
26. C₄²×C₁₀₉	C₂²	semidirect
75. C₁₀₉	#421	semidirect
83. #104	D₁₀₉	semidirect
97. M₁₆	C₂×D₁₀₉	semidirect
102. C₄²	C₂×D₁₀₉	semidirect