Subgroups — #28371970

Order 13952.

348 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295579	C₂	1	1	yes	maximal, characteristic
2	#20295156	C₂	1	4	yes	maximal, complemented
3	#20295154	C₂	1	4	yes	maximal, complemented
4	#20295054	C₂	1	1	yes	maximal, characteristic
5	#20295053	C₂	1	1	yes	maximal, characteristic
6	#20295049	C₂	1	1	yes	maximal, characteristic
7	#20294981	C₂	1	1	yes	maximal, characteristic
8	C₂×#16812609	C₂	1	1	yes	maximal, characteristic
9	#405×C₁₀₉	C₂	1	1	yes	maximal, characteristic
10	C₂×#13182013	C₂²	1	1	yes	characteristic
11	C₂²×#11724784	C₂²	1	1	yes	characteristic
12	#16812698	C₄	1	2	yes	—
13	#16812690	C₄	1	2	yes	—
14	#16812676	C₄	1	4	yes	—
15	C₄×Dic₂₁₈	C₂²	1	4	yes	—
16	#16812658	C₂²	1	1	yes	characteristic
17	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
18	C₄⋊#11724784	C₂²	1	4	yes	—
19	#16812628	C₂²	1	4	yes	—
20	#16812628	C₂²	1	4	yes	—
21	#16812609	C₂²	1	2	yes	—
22	#16812609	C₂²	1	2	yes	—
23	#16812609	C₂²	1	2	yes	—
24	C₈×Dic₁₀₉	C₂²	1	2	yes	—
25	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
26	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
27	#117×C₁₀₉	C₂²	1	1	yes	characteristic
28	#97×C₁₀₉	C₂²	1	2	yes	—
29	C₄×C₈×C₁₀₉	C₂²	1	2	yes	—
30	C₂×Dic₂₁₈	C₂×C₄	1	4	yes	—
31	C₂²×Dic₁₀₉	C₂×C₄	1	2	yes	—
32	#13182013	—	2	2	no	—
33	C₂×#11724784	—	2	2	no	—
34	C₂×#11724784	C₂³	1	2	yes	—
35	C₂×#11724784	C₂³	1	2	yes	—
36	C₄⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
37	C₄⋊Dic₁₀₉	C₂×C₄	1	4	yes	—
38	#13182008	C₂×C₄	1	4	yes	—
39	#13182008	C₂×C₄	1	4	yes	—
40	C₂²⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
41	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
42	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
43	C₄×Dic₁₀₉	C₂³	1	2	yes	—
44	C₄×Dic₁₀₉	C₂³	1	2	yes	—
45	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
46	M₁₆×C₁₀₉	—	2	2	no	—
47	C₂×C₈×C₁₀₉	Q₈	1	4	yes	complemented
48	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
49	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
50	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
51	C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
52	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
53	C₄²×C₁₀₉	C₂³	1	2	yes	—
54	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
56	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
58	C₂×C₄×C₁₀₉	#41	1	2	yes	—
59	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
60	C₂×Dic₁₀₉	—	4	2	no	—
61	C₂×Dic₁₀₉	C₂²×C₄	1	2	yes	—
62	C₂×Dic₁₀₉	C₂²×C₄	1	2	yes	—
63	C₂×Dic₁₀₉	C₂²×C₄	1	4	yes	—
64	Dic₂₁₈	—	4	4	no	—
65	C₈×C₁₀₉	—	2	2	no	—
66	C₈×C₁₀₉	C₂×Q₈	1	2	yes	—
67	C₈×C₁₀₉	C₂×Q₈	1	2	yes	—
68	#11724784	—	2	2	no	—
69	#11724784	—	2	2	no	—
70	C₂²×C₁₀₉	—	2	2	no	—
71	C₂²×C₁₀₉	C₄×Q₈	1	2	yes	—
72	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
73	C₄×C₁₀₉	—	2	4	no	—
74	C₄×C₁₀₉	C₄×Q₈	1	2	yes	—
75	C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
76	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
77	Dic₁₀₉	—	2	2	no	—
78	Dic₁₀₉	—	2	4	no	—
79	Dic₁₀₉	—	2	2	no	—
80	C₂×C₁₀₉	—	2	2	no	—
81	C₂×C₁₀₉	C₂×C₄×Q₈	1	1	yes	characteristic
82	C₂×C₁₀₉	#568	1	2	yes	—
83	#2961	—	109	1	no	maximal, Sylow
84	C₁₀₉	#2961	1	1	yes	characteristic, complemented, Sylow
85	#533	—	109	1	no	—
86	#446	—	109	4	no	—
87	#446	—	109	4	no	—
88	#424	—	109	1	no	—
89	#424	—	109	1	no	—
90	#423	—	109	1	no	—
91	#405	—	109	1	no	—
92	#405	D₁₀₉	1	1	yes	characteristic
93	C₂×#97	—	109	1	no	—
94	C₂×M₁₆	—	109	1	no	—
95	C₂×M₁₆	C₂×D₁₀₉	1	1	yes	characteristic
96	C₂²×C₈	—	109	1	no	—
97	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
98	C₄⋊Q₈	—	109	2	no	—
99	#125	—	109	2	no	—
100	C₂²⋊Q₈	—	109	4	no	—
101	C₄×Q₈	—	109	4	no	—
102	#117	—	109	1	no	—
103	#117	C₂×D₁₀₉	1	1	yes	characteristic
104	C₂×C₄²	—	109	1	no	—
105	C₄⋊C₈	—	109	4	no	—
106	C₄⋊C₈	—	109	4	no	—
107	C₄⋊C₈	—	109	4	no	—
108	#97	—	109	2	no	—
109	#97	—	109	2	no	—
110	#97	—	109	2	no	—
111	#97	C₂×D₁₀₉	1	2	yes	—
112	C₄×C₈	—	109	2	no	—
113	C₄×C₈	C₂×D₁₀₉	1	2	yes	—
114	C₂×Q₈	—	109	4	no	—
115	C₂²×C₄	—	109	2	no	—
116	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
117	M₁₆	—	2	2	no	—
118	M₁₆	—	218	2	no	—
119	C₂×C₈	—	109	2	no	—
120	C₂×C₈	—	109	2	no	—
121	C₂×C₈	—	218	2	no	—
122	C₂×C₈	Dic₂₁₈	1	4	yes	complemented
123	C₂×C₈	C₂²×D₁₀₉	1	2	yes	—
124	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
125	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
126	C₄⋊C₄	—	109	4	no	—
127	C₄⋊C₄	—	109	4	no	—
128	C₄⋊C₄	—	109	2	no	—
129	C₄⋊C₄	—	109	4	no	—
130	C₄⋊C₄	C₄×D₁₀₉	1	2	yes	—
131	C₂²⋊C₄	—	109	2	no	—
132	C₂²⋊C₄	C₄×D₁₀₉	1	2	yes	—
133	C₄²	—	109	1	no	—
134	C₄²	—	109	1	no	—
135	C₄²	—	109	2	no	—
136	C₄²	—	109	2	no	—
137	C₄²	C₂²×D₁₀₉	1	2	yes	—
138	C₂³	C₂×C₄×D₁₀₉	1	1	yes	characteristic
139	Q₈	—	436	4	no	—
140	C₂×C₄	—	109	2	no	—
141	C₂×C₄	—	109	2	no	—
142	C₂×C₄	—	109	4	no	—
143	C₂×C₄	—	436	2	no	—
144	C₂×C₄	C₂×C₄×D₁₀₉	1	4	yes	—
145	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
146	C₂×C₄	C₂×Dic₂₁₈	1	2	yes	—
147	C₂×C₄	#13182029	1	2	yes	—
148	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
149	C₈	—	2	2	no	—
150	C₈	—	218	2	no	—
151	C₈	—	218	2	no	—
152	C₈	C₂×Dic₂₁₈	1	2	yes	—
153	C₈	C₂×Dic₂₁₈	1	2	yes	—
154	C₂²	—	2	2	no	—
155	C₂²	C₄×Dic₂₁₈	1	2	yes	—
156	C₂²	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
157	C₄	—	2	4	no	—
158	C₄	—	218	2	no	—
159	C₄	—	218	2	no	—
160	C₄	—	218	4	no	—
161	C₄	C₄×Dic₂₁₈	1	2	yes	—
162	C₄	C₂²×Dic₂₁₈	1	1	yes	characteristic
163	C₄	C₂×#13182029	1	1	yes	characteristic
164	C₂	—	2	2	no	—
165	C₂	C₂×C₄×Dic₂₁₈	1	1	yes	characteristic
166	C₂	#20295791	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₀₉
commutator, socle	C₂²×C₁₀₉	C₂³×C₄
Fitting	#405×C₁₀₉	C₂
prime radical	C₂³×C₁₀₉	C₂²×C₄
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	#405	D₁₀₉
hypocenter	C₁₀₉	#2961
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ #405

Complements

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

N	H ≅ G/N	type
2. #20295156	C₂	semidirect
3. #20295154	C₂	semidirect
47. C₂×C₈×C₁₀₉	Q₈	semidirect
84. C₁₀₉	#2961	semidirect
122. C₂×C₈	Dic₂₁₈	semidirect

Maximal subgroups

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

subgroup	index
#20295579	2
#20295156	2
#20295154	2
#20295054	2
#20295053	2
#20295049	2
#20294981	2
C₂×#16812609	2
#405×C₁₀₉	2
#2961	109