Subgroups — #28371461

Order 13952.

556 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812707	C₂	1	2	yes	maximal
2	C₄×#13182029	C₂	1	1	yes	maximal, characteristic
3	#20294984	C₂	1	2	yes	maximal, complemented
4	#20294980	C₂	1	4	yes	maximal, complemented
5	#20294978	C₂	1	2	yes	maximal, complemented
6	C₂×#16812610	C₂	1	1	yes	maximal, characteristic, complemented
7	#97×D₁₀₉	C₂	1	2	yes	maximal, complemented
8	C₄×M₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13182029	C₂²	1	1	yes	characteristic
10	#16812707	C₄	1	16	yes	complemented
11	C₂×#13182012	C₂²	1	4	yes	—
12	C₂²×#11724784	C₂²	1	2	yes	—
13	C₂×C₈×D₁₀₉	C₂²	1	4	yes	—
14	C₄×Dic₂₁₈	C₄	1	2	yes	—
15	C₄×D₄₃₆	C₄	1	2	yes	—
16	C₄×C₂²⋊D₁₀₉	C₄	1	4	yes	—
17	#16812658	C₂²	1	1	yes	characteristic
18	#16812656	C₂²	1	2	yes	—
19	C₄²×D₁₀₉	C₂²	1	2	yes	—
20	#16812610	C₂²	1	1	yes	characteristic
21	#16812610	C₂²	1	1	yes	characteristic
22	#16812610	C₂²	1	2	yes	—
23	#16812609	C₂²	1	4	yes	—
24	C₈×Dic₁₀₉	C₂²	1	4	yes	—
25	C₂×M₁₆×C₁₀₉	C₂²	1	2	yes	—
26	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
27	#97×C₁₀₉	C₂²	1	2	yes	—
28	C₄×C₈×C₁₀₉	C₂²	1	2	yes	—
29	#13182029	C₂×C₄	1	8	yes	—
30	C₂×Dic₂₁₈	C₂×C₄	1	1	yes	characteristic
31	C₂×D₄₃₆	C₂×C₄	1	1	yes	characteristic
32	C₂×C₂²⋊D₁₀₉	C₂×C₄	1	2	yes	—
33	C₂×C₄×D₁₀₉	C₂×C₄	1	4	yes	—
34	C₂×C₄×D₁₀₉	C₂³	1	2	yes	—
35	#13182012	C₂×C₄	1	16	yes	complemented
36	C₂×#11724784	C₂×C₄	1	8	yes	complemented
37	C₂×#11724784	C₂³	1	2	yes	—
38	C₂×#11724784	C₂³	1	2	yes	—
39	C₈×D₁₀₉	C₂×C₄	1	16	yes	complemented
40	C₄⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
41	#13182008	C₂×C₄	1	4	yes	—
42	C₂²⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
43	#13182006	C₂×C₄	1	4	yes	—
44	C₄×Dic₁₀₉	C₂×C₄	1	4	yes	—
45	C₄×Dic₁₀₉	C₂³	1	2	yes	—
46	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
47	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
48	M₁₆×C₁₀₉	C₂×C₄	1	8	yes	complemented
49	C₂×C₈×C₁₀₉	C₂³	1	4	yes	—
50	C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
51	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
52	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
53	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
54	C₂²×D₁₀₉	C₂²×C₄	1	2	yes	—
55	C₂×C₄×C₁₀₉	—	4	2	no	—
56	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
58	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
60	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
61	C₂²⋊D₁₀₉	C₄²	1	8	yes	—
62	C₂×Dic₁₀₉	C₂²×C₄	1	4	yes	—
63	C₂×Dic₁₀₉	C₂²×C₄	1	2	yes	—
64	D₄₃₆	C₄²	1	4	yes	—
65	C₄×D₁₀₉	—	4	4	no	—
66	C₄×D₁₀₉	C₂²×C₄	1	8	yes	—
67	Dic₂₁₈	C₄²	1	4	yes	—
68	C₈×C₁₀₉	C₂²×C₄	1	8	yes	—
69	#11724784	C₂²×C₄	1	4	yes	—
70	#11724784	C₂²×C₄	1	4	yes	—
71	C₂²×C₁₀₉	—	2	2	no	—
72	C₂²×C₁₀₉	C₂×C₄²	1	2	yes	—
73	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
74	C₂×D₁₀₉	—	2	4	no	—
75	C₂×D₁₀₉	C₂×C₄²	1	4	yes	—
76	C₄×C₁₀₉	—	2	2	no	—
77	C₄×C₁₀₉	—	2	2	no	—
78	C₄×C₁₀₉	C₂×C₄²	1	2	yes	—
79	C₄×C₁₀₉	C₂³×C₄	1	2	yes	—
80	Dic₁₀₉	—	2	4	no	—
81	Dic₁₀₉	C₂×C₄²	1	4	yes	—
82	C₂×C₁₀₉	—	2	2	no	—
83	C₂×C₁₀₉	C₂²×C₄²	1	1	yes	characteristic
84	C₂×C₁₀₉	#568	1	2	yes	—
85	D₁₀₉	—	2	4	no	—
86	#2875	—	109	1	no	maximal, Sylow
87	C₁₀₉	#2875	1	1	yes	characteristic, complemented, Sylow
88	C₂×#131	—	109	2	no	—
89	C₄×#41	—	109	1	no	—
90	#405	—	109	2	no	—
91	#405	—	109	4	no	—
92	C₄×M₁₆	—	109	2	no	—
93	C₄×M₁₆	D₁₀₉	1	1	yes	characteristic, complemented
94	C₂×#97	—	109	1	no	—
95	C₂×#97	—	109	2	no	—
96	C₂×#41	—	109	1	no	—
97	#131	—	109	16	no	—
98	C₂×M₁₆	—	109	4	no	—
99	C₂×M₁₆	C₂×D₁₀₉	1	2	yes	—
100	C₂²×C₈	—	109	2	no	—
101	C₂²×C₈	—	109	4	no	—
102	C₄×Q₈	—	109	2	no	—
103	C₄×D₄	—	109	4	no	—
104	C₄×D₄	—	109	2	no	—
105	#117	—	109	1	no	—
106	#117	—	109	2	no	—
107	C₂×C₄²	—	109	2	no	—
108	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
109	#97	—	109	4	no	—
110	#97	—	109	1	no	—
111	#97	—	109	1	no	—
112	#97	—	109	2	no	—
113	#97	C₂×D₁₀₉	1	2	yes	—
114	C₄×C₈	—	109	4	no	—
115	C₄×C₈	C₂×D₁₀₉	1	2	yes	—
116	#41	—	109	8	no	—
117	C₂×Q₈	—	109	1	no	—
118	C₂×D₄	—	109	2	no	—
119	C₂×D₄	—	109	1	no	—
120	C₂²×C₄	—	109	2	no	—
121	C₂²×C₄	—	109	4	no	—
122	C₂²×C₄	C₄×D₁₀₉	1	2	yes	—
123	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
124	M₁₆	—	109	16	no	—
125	M₁₆	C₄×D₁₀₉	1	8	yes	complemented
126	C₂×C₈	—	109	8	no	—
127	C₂×C₈	—	109	2	no	—
128	C₂×C₈	—	109	2	no	—
129	C₂×C₈	—	109	16	no	—
130	C₂×C₈	C₂²×D₁₀₉	1	4	yes	—
131	C₄⋊C₄	—	109	4	no	—
132	C₄⋊C₄	—	109	2	no	—
133	C₂²⋊C₄	—	109	2	no	—
134	C₂²⋊C₄	—	109	4	no	—
135	C₄²	—	109	4	no	—
136	C₄²	—	109	2	no	—
137	C₄²	C₄×D₁₀₉	1	2	yes	—
138	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
139	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
140	C₂³	—	109	2	no	—
141	C₂³	C₂×C₄×D₁₀₉	1	1	yes	characteristic
142	Q₈	—	109	4	no	—
143	D₄	—	109	8	no	—
144	D₄	—	109	4	no	—
145	C₂×C₄	—	4	2	no	—
146	C₂×C₄	—	109	2	no	—
147	C₂×C₄	—	109	8	no	—
148	C₂×C₄	—	109	4	no	—
149	C₂×C₄	—	436	4	no	—
150	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
151	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
152	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄×D₁₀₉	1	4	yes	—
154	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
155	C₈	—	109	4	no	—
156	C₈	—	109	4	no	—
157	C₈	C₂×C₄×D₁₀₉	1	8	yes	—
158	C₂²	—	2	2	no	—
159	C₂²	—	109	4	no	—
160	C₂²	—	218	4	no	—
161	C₂²	C₄²×D₁₀₉	1	2	yes	—
162	C₂²	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
163	C₄	—	2	2	no	—
164	C₄	—	2	2	no	—
165	C₄	—	109	4	no	—
166	C₄	—	218	4	no	—
167	C₄	C₄²×D₁₀₉	1	2	yes	—
168	C₄	C₂²×C₄×D₁₀₉	1	2	yes	—
169	C₂	—	2	2	no	—
170	C₂	—	218	4	no	—
171	C₂	C₂×C₄²×D₁₀₉	1	1	yes	characteristic
172	C₂	#20295792	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₀₉
commutator	C₂×C₁₀₉	C₂²×C₄²
Fitting	C₄×M₁₆×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂³×C₄
prime radical	C₂×#13182029	C₂²
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	C₄×M₁₆	D₁₀₉
hypocenter	C₁₀₉	#2875
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. #20294984	C₂	semidirect
4. #20294980	C₂	semidirect
5. #20294978	C₂	semidirect
6. C₂×#16812610	C₂	semidirect
7. #97×D₁₀₉	C₂	semidirect
8. C₄×M₁₆×C₁₀₉	C₂	semidirect
10. #16812707	C₄	semidirect
35. #13182012	C₂×C₄	semidirect
36. C₂×#11724784	C₂×C₄	semidirect
39. C₈×D₁₀₉	C₂×C₄	semidirect
48. M₁₆×C₁₀₉	C₂×C₄	semidirect
87. C₁₀₉	#2875	semidirect
93. C₄×M₁₆	D₁₀₉	semidirect
125. M₁₆	C₄×D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16812707	2
C₄×#13182029	2
#20294984	2
#20294980	2
#20294978	2
C₂×#16812610	2
#97×D₁₀₉	2
C₄×M₁₆×C₁₀₉	2
#2875	109