Subgroups — #28362742

Order 13952.

404 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13182013	C₂	1	1	yes	maximal, characteristic
2	C₂×#16812658	C₂	1	1	yes	maximal, characteristic
3	#20294786	C₂	1	4	yes	maximal, complemented
4	C₂×#117×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#13182013	—	2	1	no	—
6	C₂×#13182013	—	2	2	no	—
7	C₂×#13182013	C₄	1	4	yes	complemented
8	C₂×#13182013	C₂²	1	2	yes	—
9	C₂²×#11724784	—	2	1	no	—
10	#16812658	—	2	1	no	—
11	#16812658	—	2	1	no	—
12	#16812658	C₄	1	2	yes	complemented
13	#16812658	C₂²	1	2	yes	—
14	C₂×C₄⋊Dic₁₀₉	C₄	1	2	yes	complemented
15	C₂×C₂²⋊Dic₁₀₉	—	2	1	no	—
16	C₂×C₄×Dic₁₀₉	—	2	1	no	—
17	C₂³×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
18	#117×C₁₀₉	—	2	2	no	—
19	#117×C₁₀₉	C₄	1	4	yes	complemented
20	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
21	C₂×C₂²⋊C₄×C₁₀₉	—	2	1	no	—
22	C₂×C₄²×C₁₀₉	C₂²	1	2	yes	—
23	C₂²×Dic₁₀₉	—	2	2	no	—
24	#13182013	—	2	4	no	—
25	#13182013	—	2	2	no	—
26	#13182013	C₂×C₄	1	4	yes	complemented
27	C₂×#11724784	—	2	4	no	—
28	C₂×#11724784	—	2	2	no	—
29	C₄⋊Dic₁₀₉	—	2	2	no	—
30	C₄⋊Dic₁₀₉	C₂×C₄	1	4	yes	—
31	C₂²⋊Dic₁₀₉	—	2	2	no	—
32	C₂²⋊Dic₁₀₉	—	2	2	no	—
33	C₄×Dic₁₀₉	—	2	2	no	—
34	C₄×Dic₁₀₉	—	2	2	no	—
35	C₂⁴×C₁₀₉	D₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₀₉	—	2	2	no	—
37	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
38	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
41	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₀₉	D₄	1	2	yes	—
45	C₂²×C₄×C₁₀₉	Q₈	1	2	yes	—
46	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
47	C₄⋊C₄×C₁₀₉	—	2	4	no	—
48	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
49	C₄²×C₁₀₉	—	2	2	no	—
50	C₄²×C₁₀₉	C₂×C₄	1	4	yes	complemented
51	C₂³×C₁₀₉	—	4	2	no	—
52	C₂³×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
53	C₂³×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
54	C₂³×C₁₀₉	C₂²⋊C₄	1	2	yes	—
55	C₂³×C₁₀₉	C₄⋊C₄	1	2	yes	—
56	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₁₀₉	—	2	4	no	—
58	C₂×C₄×C₁₀₉	—	4	4	no	—
59	C₂×C₄×C₁₀₉	C₄²	1	2	yes	—
60	C₂×C₄×C₁₀₉	C₄²	1	2	yes	—
61	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
64	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
65	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
66	C₂×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
67	C₂×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
68	C₂×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
69	C₂×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
70	C₂×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
71	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
77	C₂×Dic₁₀₉	—	2	4	no	—
78	C₂×Dic₁₀₉	—	4	4	no	—
79	#11724784	—	2	4	no	—
80	C₂²×C₁₀₉	—	2	2	no	—
81	C₂²×C₁₀₉	—	2	2	no	—
82	C₂²×C₁₀₉	—	2	2	no	—
83	C₂²×C₁₀₉	—	4	2	no	—
84	C₂²×C₁₀₉	—	4	2	no	—
85	C₂²×C₁₀₉	#95	1	2	yes	—
86	C₂²×C₁₀₉	#95	1	2	yes	—
87	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₄×C₁₀₉	—	4	4	no	—
91	C₄×C₁₀₉	#95	1	2	yes	—
92	C₄×C₁₀₉	#95	1	2	yes	—
93	C₄×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
94	C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
97	Dic₁₀₉	—	4	4	no	—
98	C₂×C₁₀₉	—	2	1	no	—
99	C₂×C₁₀₉	—	2	2	no	—
100	C₂×C₁₀₉	—	2	2	no	—
101	C₂×C₁₀₉	—	2	1	no	—
102	C₂×C₁₀₉	C₂×#95	1	1	yes	characteristic
103	C₂×C₁₀₉	#421	1	2	yes	—
104	#1733	—	109	1	no	maximal, Sylow
105	C₁₀₉	#1733	1	1	yes	characteristic, complemented, Sylow
106	C₂²×M₁₆	—	109	1	no	—
107	C₂×#117	—	109	1	no	—
108	C₂×#117	D₁₀₉	1	1	yes	characteristic
109	#339	—	109	4	no	—
110	C₂³×C₄	C₂×D₁₀₉	1	1	yes	characteristic
111	C₂×M₁₆	—	109	4	no	—
112	C₂×M₁₆	—	109	2	no	—
113	C₂×M₁₆	—	218	1	no	—
114	C₂×M₁₆	—	218	2	no	—
115	C₂²×C₈	—	218	1	no	—
116	#117	—	2	2	no	—
117	#117	—	109	2	no	—
118	#117	—	109	2	no	—
119	#117	—	218	1	no	—
120	#117	—	218	1	no	—
121	#117	Dic₁₀₉	1	4	yes	complemented
122	C₂×C₄⋊C₄	—	2	1	no	—
123	C₂×C₄⋊C₄	—	109	2	no	—
124	C₂×C₂²⋊C₄	—	2	1	no	—
125	C₂×C₂²⋊C₄	—	218	1	no	—
126	C₂×C₄²	—	218	1	no	—
127	C₂×C₄²	C₂×D₁₀₉	1	2	yes	—
128	C₂⁴	C₂²⋊D₁₀₉	1	1	yes	characteristic
129	C₂²×C₄	—	2	2	no	—
130	C₂²×C₄	—	218	2	no	—
131	C₂²×C₄	Dic₂₁₈	1	2	yes	—
132	C₂²×C₄	C₄×D₁₀₉	1	2	yes	—
133	C₂²×C₄	C₄×D₁₀₉	1	1	yes	characteristic
134	C₂²×C₄	C₄×D₁₀₉	1	1	yes	characteristic
135	C₂²×C₄	D₄₃₆	1	1	yes	characteristic
136	C₂²×C₄	D₄₃₆	1	1	yes	characteristic
137	C₂²×C₄	C₂×Dic₁₀₉	1	2	yes	—
138	C₂²×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
139	C₂²×C₄	C₂²⋊D₁₀₉	1	2	yes	—
140	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
141	M₁₆	—	109	4	no	—
142	M₁₆	—	218	4	no	—
143	M₁₆	—	218	2	no	—
144	C₂×C₈	—	218	4	no	—
145	C₂×C₈	—	218	2	no	—
146	C₄⋊C₄	—	2	4	no	—
147	C₄⋊C₄	—	109	4	no	—
148	C₄⋊C₄	—	218	2	no	—
149	C₂²⋊C₄	—	2	4	no	—
150	C₂²⋊C₄	—	218	2	no	—
151	C₂²⋊C₄	—	218	2	no	—
152	C₄²	—	2	2	no	—
153	C₄²	—	218	2	no	—
154	C₄²	—	218	2	no	—
155	C₄²	C₂×Dic₁₀₉	1	4	yes	complemented
156	C₂³	—	4	2	no	—
157	C₂³	#13182006	1	1	yes	characteristic
158	C₂³	#13182006	1	1	yes	characteristic
159	C₂³	C₂²⋊Dic₁₀₉	1	2	yes	—
160	C₂³	#13182008	1	2	yes	—
161	C₂³	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	4	4	no	—
164	C₂×C₄	—	436	4	no	—
165	C₂×C₄	—	218	4	no	—
166	C₂×C₄	C₄×Dic₁₀₉	1	2	yes	—
167	C₂×C₄	C₄×Dic₁₀₉	1	2	yes	—
168	C₂×C₄	#13182006	1	1	yes	characteristic
169	C₂×C₄	#13182006	1	1	yes	characteristic
170	C₂×C₄	#13182006	1	2	yes	—
171	C₂×C₄	#13182006	1	2	yes	—
172	C₂×C₄	C₂²⋊Dic₁₀₉	1	2	yes	—
173	C₂×C₄	#13182008	1	2	yes	—
174	C₂×C₄	#13182008	1	2	yes	—
175	C₂×C₄	#13182008	1	2	yes	—
176	C₂×C₄	C₄⋊Dic₁₀₉	1	2	yes	—
177	C₂×C₄	C₄⋊Dic₁₀₉	1	2	yes	—
178	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
179	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
180	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
181	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
182	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
183	C₂×C₄	C₂×Dic₂₁₈	1	1	yes	characteristic
184	C₈	—	218	4	no	—
185	C₂²	—	2	2	no	—
186	C₂²	—	2	2	no	—
187	C₂²	—	2	2	no	—
188	C₂²	—	4	2	no	—
189	C₂²	—	4	2	no	—
190	C₂²	#16812606	1	2	yes	—
191	C₂²	#16812606	1	2	yes	—
192	C₂²	C₂×#13182006	1	1	yes	characteristic
193	C₂²	C₂×C₂²⋊Dic₁₀₉	1	1	yes	characteristic
194	C₂²	C₂×#13182008	1	1	yes	characteristic
195	C₄	—	4	4	no	—
196	C₄	—	436	4	no	—
197	C₄	#16812606	1	2	yes	—
198	C₄	#16812606	1	2	yes	—
199	C₄	C₂×C₄×Dic₁₀₉	1	1	yes	characteristic
200	C₄	C₂×#13182006	1	1	yes	characteristic
201	C₄	C₂×#13182008	1	1	yes	characteristic
202	C₄	C₂×C₄⋊Dic₁₀₉	1	1	yes	characteristic
203	C₂	—	2	1	no	—
204	C₂	—	2	2	no	—
205	C₂	—	2	2	no	—
206	C₂	—	2	1	no	—
207	C₂	C₂×#16812606	1	1	yes	characteristic
208	C₂	#20295044	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄₃₆
commutator	C₄×C₁₀₉	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	C₂×#117×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₁₀₉	D₄
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	C₂×#117	D₁₀₉
hypocenter	C₁₀₉	#1733
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
3. #20294786	C₂	semidirect
7. C₂×#13182013	C₄	semidirect
12. #16812658	C₄	semidirect
14. C₂×C₄⋊Dic₁₀₉	C₄	semidirect
19. #117×C₁₀₉	C₄	semidirect
26. #13182013	C₂×C₄	semidirect
50. C₄²×C₁₀₉	C₂×C₄	semidirect
105. C₁₀₉	#1733	semidirect
121. #117	Dic₁₀₉	semidirect
155. C₄²	C₂×Dic₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13182013	2
C₂×#16812658	2
#20294786	2
C₂×#117×C₁₀₉	2
#1733	109