Subgroups — #28365430

Order 13952.

568 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20294894	C₂	1	2	yes	maximal, complemented
2	#20294892	C₂	1	2	yes	maximal, complemented
3	#20294889	C₂	1	2	yes	maximal
4	#20294886	C₂	1	2	yes	maximal
5	C₄×C₄⋊Dic₁₀₉	C₂	1	2	yes	maximal, complemented
6	C₄×C₂²⋊Dic₁₀₉	C₂	1	2	yes	maximal
7	#20294872	C₂	1	2	yes	maximal, complemented
8	C₂²×C₄²×C₁₀₉	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄⋊Dic₁₀₉	C₂²	1	4	yes	—
10	C₂×C₂²⋊Dic₁₀₉	C₂²	1	4	yes	—
11	C₂×C₄×Dic₁₀₉	C₂²	1	4	yes	—
12	#16812606	C₂²	1	4	yes	—
13	#16812606	C₂²	1	4	yes	—
14	#16812606	C₂²	1	4	yes	—
15	#16812606	C₂²	1	4	yes	—
16	C₂³×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₂³×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
18	C₂³×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₁₀₉	C₄	1	8	yes	complemented
20	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
21	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
22	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
23	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
24	C₂²×Dic₁₀₉	C₂³	1	8	yes	—
25	C₄⋊Dic₁₀₉	—	2	8	no	—
26	C₂²⋊Dic₁₀₉	—	2	8	no	—
27	C₄×Dic₁₀₉	—	2	8	no	—
28	C₂⁴×C₁₀₉	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₁₀₉	—	2	4	no	—
30	C₂²×C₄×C₁₀₉	—	2	4	no	—
31	C₂²×C₄×C₁₀₉	—	2	4	no	—
32	C₂²×C₄×C₁₀₉	C₂×C₄	1	4	yes	—
33	C₂²×C₄×C₁₀₉	C₂×C₄	1	4	yes	—
34	C₂²×C₄×C₁₀₉	C₂×C₄	1	4	yes	—
35	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
41	C₄²×C₁₀₉	C₂×C₄	1	4	yes	—
42	C₄²×C₁₀₉	C₂×C₄	1	4	yes	—
43	C₄²×C₁₀₉	C₂×C₄	1	4	yes	—
44	C₄²×C₁₀₉	C₂×C₄	1	4	yes	—
45	C₂³×C₁₀₉	—	2	2	no	—
46	C₂³×C₁₀₉	—	2	2	no	—
47	C₂³×C₁₀₉	C₂²×C₄	1	2	yes	—
48	C₂³×C₁₀₉	#41	1	4	yes	—
49	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
50	C₂×C₄×C₁₀₉	—	2	8	no	—
51	C₂×C₄×C₁₀₉	—	2	2	no	—
52	C₂×C₄×C₁₀₉	—	2	2	no	—
53	C₂×C₄×C₁₀₉	—	2	2	no	—
54	C₂×C₄×C₁₀₉	—	2	2	no	—
55	C₂×C₄×C₁₀₉	—	2	8	no	—
56	C₂×C₄×C₁₀₉	—	2	8	no	—
57	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
58	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
59	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
60	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
61	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
62	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
63	C₂×C₄×C₁₀₉	#41	1	4	yes	—
64	C₂×C₄×C₁₀₉	#41	1	4	yes	—
65	C₂×Dic₁₀₉	—	2	8	no	—
66	C₂×Dic₁₀₉	—	2	8	no	—
67	C₂×Dic₁₀₉	—	2	8	no	—
68	C₂²×C₁₀₉	—	2	2	no	—
69	C₂²×C₁₀₉	—	2	2	no	—
70	C₂²×C₁₀₉	—	2	8	no	—
71	C₂²×C₁₀₉	#117	1	4	yes	—
72	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
73	C₂²×C₁₀₉	C₂×#41	1	2	yes	—
74	C₂²×C₁₀₉	C₂×#41	1	2	yes	—
75	C₂²×C₁₀₉	C₂×#41	1	2	yes	—
76	C₄×C₁₀₉	—	2	2	no	—
77	C₄×C₁₀₉	—	2	2	no	—
78	C₄×C₁₀₉	—	2	2	no	—
79	C₄×C₁₀₉	—	2	2	no	—
80	C₄×C₁₀₉	#117	1	4	yes	—
81	C₄×C₁₀₉	#117	1	4	yes	—
82	Dic₁₀₉	—	4	8	no	—
83	C₂×C₁₀₉	—	2	4	no	—
84	C₂×C₁₀₉	C₂×#117	1	1	yes	characteristic
85	C₂×C₁₀₉	C₂×#117	1	1	yes	characteristic
86	C₂×C₁₀₉	C₂×#117	1	1	yes	characteristic
87	C₂×C₁₀₉	#529	1	4	yes	—
88	#2283	—	109	1	no	maximal, Sylow
89	C₁₀₉	#2283	1	1	yes	characteristic, complemented, Sylow
90	C₂²×C₄²	D₁₀₉	1	1	yes	characteristic
91	#383	—	109	2	no	—
92	#382	—	109	2	no	—
93	#381	—	109	2	no	—
94	#380	—	109	2	no	—
95	C₄×C₄⋊C₄	—	109	2	no	—
96	C₄×C₂²⋊C₄	—	109	2	no	—
97	#376	—	109	2	no	—
98	C₂³×C₄	C₂×D₁₀₉	1	1	yes	characteristic
99	C₂³×C₄	C₂×D₁₀₉	1	1	yes	characteristic
100	C₂³×C₄	C₂×D₁₀₉	1	1	yes	characteristic
101	C₂×C₄⋊C₄	—	109	4	no	—
102	C₂×C₂²⋊C₄	—	109	4	no	—
103	C₂×C₄²	—	109	4	no	—
104	C₂×C₄²	Dic₁₀₉	1	8	yes	complemented
105	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
106	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
107	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
108	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
109	#95	—	109	4	no	—
110	#95	—	109	4	no	—
111	#95	—	109	4	no	—
112	#95	—	109	4	no	—
113	C₂⁴	C₂²×D₁₀₉	1	1	yes	characteristic
114	C₂²×C₄	—	2	4	no	—
115	C₂²×C₄	—	2	4	no	—
116	C₂²×C₄	—	2	4	no	—
117	C₂²×C₄	—	109	8	no	—
118	C₂²×C₄	C₂×Dic₁₀₉	1	4	yes	—
119	C₂²×C₄	C₂×Dic₁₀₉	1	4	yes	—
120	C₂²×C₄	C₂×Dic₁₀₉	1	4	yes	—
121	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
122	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
123	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
124	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
125	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
126	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
127	C₄⋊C₄	—	218	8	no	—
128	C₂²⋊C₄	—	218	8	no	—
129	C₄²	—	218	8	no	—
130	C₄²	C₂×Dic₁₀₉	1	4	yes	—
131	C₄²	C₂×Dic₁₀₉	1	4	yes	—
132	C₄²	C₂×Dic₁₀₉	1	4	yes	—
133	C₄²	C₂×Dic₁₀₉	1	4	yes	—
134	C₂³	—	2	2	no	—
135	C₂³	—	2	2	no	—
136	C₂³	C₂²×Dic₁₀₉	1	2	yes	—
137	C₂³	#13182029	1	4	yes	—
138	C₂³	C₂³×D₁₀₉	1	1	yes	characteristic
139	C₂×C₄	—	2	8	no	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	8	no	—
145	C₂×C₄	—	2	8	no	—
146	C₂×C₄	—	218	8	no	—
147	C₂×C₄	—	218	8	no	—
148	C₂×C₄	—	218	8	no	—
149	C₂×C₄	C₂²×Dic₁₀₉	1	2	yes	—
150	C₂×C₄	C₂²×Dic₁₀₉	1	2	yes	—
151	C₂×C₄	C₂²×Dic₁₀₉	1	2	yes	—
152	C₂×C₄	C₂²×Dic₁₀₉	1	2	yes	—
153	C₂×C₄	C₂²×Dic₁₀₉	1	2	yes	—
154	C₂×C₄	C₂²×Dic₁₀₉	1	2	yes	—
155	C₂×C₄	#13182029	1	4	yes	—
156	C₂×C₄	#13182029	1	4	yes	—
157	C₂²	—	2	2	no	—
158	C₂²	—	2	2	no	—
159	C₂²	—	2	8	no	—
160	C₂²	#16812658	1	4	yes	—
161	C₂²	C₂³×Dic₁₀₉	1	1	yes	characteristic
162	C₂²	C₂×#13182029	1	2	yes	—
163	C₂²	C₂×#13182029	1	2	yes	—
164	C₂²	C₂×#13182029	1	2	yes	—
165	C₄	—	2	2	no	—
166	C₄	—	2	2	no	—
167	C₄	—	2	2	no	—
168	C₄	—	2	2	no	—
169	C₄	—	436	8	no	—
170	C₄	#16812658	1	4	yes	—
171	C₄	#16812658	1	4	yes	—
172	C₂	—	2	4	no	—
173	C₂	C₂×#16812658	1	1	yes	characteristic
174	C₂	C₂×#16812658	1	1	yes	characteristic
175	C₂	C₂×#16812658	1	1	yes	characteristic
176	C₂	#20295560	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₀₉
commutator	C₂²×C₁₀₉	C₂³×C₄
Frattini	C₂³	C₂³×D₁₀₉
Fitting	C₂²×C₄²×C₁₀₉	C₂
socle, cosocle	C₂³×C₁₀₉	C₂⁴
prime radical	C₂⁴×C₁₀₉	C₂³
hypercenter	C₂²×C₄²	D₁₀₉
hypocenter	C₁₀₉	#2283
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
1. #20294894	C₂	semidirect
2. #20294892	C₂	semidirect
5. C₄×C₄⋊Dic₁₀₉	C₂	semidirect
7. #20294872	C₂	semidirect
19. C₂×C₄²×C₁₀₉	C₄	semidirect
89. C₁₀₉	#2283	semidirect
104. C₂×C₄²	Dic₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20294894	2
#20294892	2
#20294889	2
#20294886	2
C₄×C₄⋊Dic₁₀₉	2
C₄×C₂²⋊Dic₁₀₉	2
#20294872	2
C₂²×C₄²×C₁₀₉	2
#2283	109