Subgroups — #28371950

Order 13952.

452 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812721	C₂	1	1	yes	maximal, characteristic
2	#20295792	C₂	1	2	yes	maximal, complemented
3	#20295149	C₂	1	4	yes	maximal, complemented
4	#20295144	C₂	1	4	yes	maximal, complemented
5	C₂×#16812635	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295044	C₂	1	2	yes	maximal, complemented
7	#405×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13182029	C₂²	1	2	yes	—
9	#16812721	C₄	1	2	yes	complemented
10	#16812721	C₄	1	2	yes	complemented
11	#16812721	C₂²	1	2	yes	—
12	#16812721	C₂²	1	2	yes	—
13	C₂×Dic₄₃₆	C₄	1	1	yes	characteristic, complemented
14	C₂×#13182018	C₄	1	2	yes	complemented
15	C₂×D₈₇₂	C₄	1	1	yes	characteristic, complemented
16	#16812707	—	2	4	no	—
17	#16812705	C₂²	1	8	yes	complemented
18	C₂×#13182013	C₂²	1	2	yes	—
19	M₁₆×D₁₀₉	—	2	4	no	—
20	#16812635	C₂²	1	2	yes	complemented
21	#16812635	C₂²	1	2	yes	complemented
22	#16812627	C₂²	1	8	yes	complemented
23	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
24	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
25	#117×C₁₀₉	C₂²	1	1	yes	characteristic
26	#97×C₁₀₉	C₂²	1	2	yes	complemented
27	C₄×C₈×C₁₀₉	C₂²	1	2	yes	complemented
28	#13182029	—	2	2	no	—
29	#13182029	—	2	2	no	—
30	#13182029	C₂×C₄	1	4	yes	—
31	#13182029	C₂³	1	4	yes	—
32	C₂×Dic₂₁₈	C₂×C₄	1	2	yes	—
33	C₂×D₄₃₆	C₂×C₄	1	2	yes	—
34	C₂×C₂²⋊D₁₀₉	—	2	2	no	—
35	C₂×C₄×D₁₀₉	—	2	2	no	—
36	Dic₄₃₆	C₂×C₄	1	2	yes	—
37	Dic₄₃₆	C₂×C₄	1	2	yes	—
38	#13182018	C₂×C₄	1	4	yes	—
39	#13182018	C₂×C₄	1	4	yes	—
40	D₈₇₂	C₂×C₄	1	2	yes	—
41	D₈₇₂	C₂×C₄	1	2	yes	—
42	#13182013	—	2	2	no	—
43	#13182013	C₂³	1	4	yes	—
44	#13182012	—	2	8	no	—
45	C₂×#11724784	—	2	2	no	—
46	C₈×D₁₀₉	—	2	8	no	—
47	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
48	M₁₆×C₁₀₉	D₄	1	2	yes	—
49	M₁₆×C₁₀₉	D₄	1	2	yes	—
50	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
51	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
52	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
53	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
54	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
55	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
56	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
57	C₄⋊C₄×C₁₀₉	—	2	1	no	—
58	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
59	C₄²×C₁₀₉	C₂³	1	2	yes	—
60	C₂³×C₁₀₉	#41	1	1	yes	characteristic
61	C₂²×D₁₀₉	—	2	2	no	—
62	C₂×C₄×C₁₀₉	—	2	2	no	—
63	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
67	C₂×C₄×C₁₀₉	#41	1	1	yes	characteristic
68	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
69	C₂²⋊D₁₀₉	—	2	4	no	—
70	C₂²⋊D₁₀₉	—	2	4	no	—
71	C₂×Dic₁₀₉	—	2	2	no	—
72	D₄₃₆	—	2	2	no	—
73	D₄₃₆	C₂²×C₄	1	4	yes	—
74	C₄×D₁₀₉	—	2	4	no	—
75	C₄×D₁₀₉	—	2	4	no	—
76	Dic₂₁₈	—	2	2	no	—
77	Dic₂₁₈	C₂²×C₄	1	4	yes	—
78	C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
79	C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
80	C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
81	C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
82	C₈×C₁₀₉	C₂×D₄	1	4	yes	—
83	#11724784	—	2	4	no	—
84	C₂²×C₁₀₉	—	4	1	no	—
85	C₂²×C₁₀₉	C₄×D₄	1	1	yes	characteristic
86	C₂²×C₁₀₉	C₄×D₄	1	1	yes	characteristic
87	C₂²×C₁₀₉	C₂×#41	1	1	yes	characteristic
88	C₂×D₁₀₉	—	2	4	no	—
89	C₂×D₁₀₉	—	8	2	no	—
90	C₄×C₁₀₉	—	4	2	no	—
91	C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
92	C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
93	C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
94	C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
95	Dic₁₀₉	—	2	4	no	—
96	C₂×C₁₀₉	—	2	1	no	—
97	C₂×C₁₀₉	—	2	1	no	—
98	C₂×C₁₀₉	—	2	1	no	—
99	C₂×C₁₀₉	C₂×C₄×D₄	1	1	yes	characteristic
100	D₁₀₉	—	4	4	no	—
101	#2957	—	109	1	no	maximal, Sylow
102	C₁₀₉	#2957	1	1	yes	characteristic, complemented, Sylow
103	C₂×#135	—	109	1	no	—
104	#568	—	109	2	no	—
105	#444	—	109	4	no	—
106	#443	—	109	4	no	—
107	C₂×#108	—	109	1	no	—
108	#421	—	109	2	no	—
109	#405	D₁₀₉	1	1	yes	characteristic, complemented
110	C₂×#41	—	109	2	no	—
111	#135	—	109	2	no	—
112	#135	—	109	2	no	—
113	#135	—	109	2	no	—
114	#135	—	109	2	no	—
115	C₂×Q₁₆	—	109	1	no	—
116	C₂×SD₁₆	—	109	2	no	—
117	C₂×D₈	—	109	1	no	—
118	#131	—	109	8	no	—
119	#131	—	218	4	no	—
120	C₂×M₁₆	—	109	2	no	—
121	C₂×M₁₆	—	218	4	no	—
122	C₂×M₁₆	C₂×D₁₀₉	1	1	yes	characteristic
123	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
124	#117	C₂×D₁₀₉	1	1	yes	characteristic
125	#108	—	109	2	no	—
126	#108	—	109	2	no	—
127	#104	—	109	8	no	—
128	#97	C₂×D₁₀₉	1	2	yes	complemented
129	C₄×C₈	C₂×D₁₀₉	1	2	yes	complemented
130	#41	—	109	4	no	—
131	#41	—	109	4	no	—
132	#41	—	218	2	no	—
133	#41	—	218	2	no	—
134	C₂×Q₈	—	109	2	no	—
135	C₂×D₄	—	109	2	no	—
136	C₂×D₄	—	218	2	no	—
137	C₂²×C₄	—	218	2	no	—
138	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
139	Q₁₆	—	109	2	no	—
140	Q₁₆	—	109	2	no	—
141	SD₁₆	—	109	4	no	—
142	SD₁₆	—	109	4	no	—
143	D₈	—	109	2	no	—
144	D₈	—	109	2	no	—
145	M₁₆	—	109	4	no	—
146	M₁₆	—	218	2	no	—
147	M₁₆	—	218	8	no	—
148	M₁₆	D₄₃₆	1	2	yes	—
149	M₁₆	D₄₃₆	1	2	yes	—
150	C₂×C₈	—	218	2	no	—
151	C₂×C₈	—	218	8	no	—
152	C₂×C₈	C₄×D₁₀₉	1	1	yes	characteristic
153	C₂×C₈	C₄×D₁₀₉	1	1	yes	characteristic
154	C₂×C₈	C₄×D₁₀₉	1	1	yes	characteristic
155	C₂×C₈	C₄×D₁₀₉	1	1	yes	characteristic
156	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
157	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
158	C₂×C₈	C₂²×D₁₀₉	1	2	yes	—
159	C₄⋊C₄	—	2	1	no	—
160	C₂²⋊C₄	—	2	1	no	—
161	C₄²	C₂²×D₁₀₉	1	2	yes	—
162	C₂³	—	218	2	no	—
163	C₂³	#13182029	1	1	yes	characteristic
164	Q₈	—	109	4	no	—
165	Q₈	—	218	2	no	—
166	D₄	—	109	4	no	—
167	D₄	—	218	4	no	—
168	D₄	—	218	4	no	—
169	D₄	—	218	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	218	2	no	—
172	C₂×C₄	—	218	4	no	—
173	C₂×C₄	—	218	4	no	—
174	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
175	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
176	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
177	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
178	C₂×C₄	#13182029	1	1	yes	characteristic
179	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
180	C₈	—	218	4	no	—
181	C₈	C₂×C₄×D₁₀₉	1	1	yes	characteristic
182	C₈	C₂×C₄×D₁₀₉	1	1	yes	characteristic
183	C₈	C₂×C₄×D₁₀₉	1	1	yes	characteristic
184	C₈	C₂×C₄×D₁₀₉	1	1	yes	characteristic
185	C₈	C₂×D₄₃₆	1	4	yes	—
186	C₂²	—	4	1	no	—
187	C₂²	—	872	2	no	—
188	C₂²	—	218	4	no	—
189	C₂²	C₄×D₄₃₆	1	1	yes	characteristic
190	C₂²	C₄×D₄₃₆	1	1	yes	characteristic
191	C₂²	C₂×#13182029	1	1	yes	characteristic
192	C₄	—	4	2	no	—
193	C₄	—	218	4	no	—
194	C₄	C₄×D₄₃₆	1	1	yes	characteristic
195	C₄	C₄×D₄₃₆	1	1	yes	characteristic
196	C₄	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
197	C₄	C₂²×D₄₃₆	1	1	yes	characteristic
198	C₂	—	2	1	no	—
199	C₂	—	2	1	no	—
200	C₂	—	2	1	no	—
201	C₂	—	436	4	no	—
202	C₂	C₂×C₄×D₄₃₆	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×D₄₃₆
commutator	C₄×C₁₀₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#405×C₁₀₉	C₂
socle	C₂×C₁₀₉	C₂×C₄×D₄
prime radical	C₂×#16812721	C₂
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	#405	D₁₀₉
hypocenter	C₁₀₉	#2957
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #405

Complements

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

N	H ≅ G/N	type
2. #20295792	C₂	semidirect
3. #20295149	C₂	semidirect
4. #20295144	C₂	semidirect
5. C₂×#16812635	C₂	semidirect
6. #20295044	C₂	semidirect
7. #405×C₁₀₉	C₂	semidirect
9. #16812721	C₄	semidirect
10. #16812721	C₄	semidirect
13. C₂×Dic₄₃₆	C₄	semidirect
14. C₂×#13182018	C₄	semidirect
15. C₂×D₈₇₂	C₄	semidirect
17. #16812705	C₂²	semidirect
20. #16812635	C₂²	semidirect
21. #16812635	C₂²	semidirect
22. #16812627	C₂²	semidirect
26. #97×C₁₀₉	C₂²	semidirect
27. C₄×C₈×C₁₀₉	C₂²	semidirect
102. C₁₀₉	#2957	semidirect
109. #405	D₁₀₉	semidirect
128. #97	C₂×D₁₀₉	semidirect
129. C₄×C₈	C₂×D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16812721	2
#20295792	2
#20295149	2
#20295144	2
C₂×#16812635	2
#20295044	2
#405×C₁₀₉	2
#2957	109