Subgroups — #28371737

Order 13952.

492 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812705	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#13182012	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295538	C₂	1	1	yes	maximal, characteristic
4	#20295105	C₂	1	4	yes	maximal, complemented
5	#20295099	C₂	1	4	yes	maximal, complemented
6	#20295053	C₂	1	1	yes	maximal, characteristic, complemented
7	#20294994	C₂	1	1	yes	maximal, characteristic, complemented
8	#20294989	C₂	1	1	yes	maximal, characteristic, complemented
9	#405×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13182029	C₂²	1	1	yes	characteristic
11	C₂²×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
12	#16812705	—	2	4	no	—
13	C₂×#13182013	C₂²	1	1	yes	characteristic
14	C₂×#13182012	—	2	4	no	—
15	C₂×#13182012	C₂²	1	2	yes	—
16	C₂×#13182012	C₂²	1	2	yes	—
17	C₂×#13182012	C₂²	1	2	yes	complemented
18	C₂²×#11724784	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₁₀₉	C₂²	1	2	yes	complemented
20	#16812683	C₄	1	2	yes	—
21	#16812679	C₄	1	2	yes	—
22	#16812674	C₄	1	2	yes	—
23	#16812669	C₄	1	2	yes	—
24	C₄×D₄₃₆	C₂²	1	4	yes	—
25	C₄⋊#11724784	C₂²	1	4	yes	complemented
26	#16812611	C₂²	1	4	yes	—
27	#16812611	C₂²	1	4	yes	complemented
28	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
29	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
30	#117×C₁₀₉	C₂²	1	1	yes	characteristic
31	#97×C₁₀₉	C₂²	1	2	yes	complemented
32	C₄×C₈×C₁₀₉	C₂²	1	2	yes	complemented
33	C₂³×D₁₀₉	C₂×C₄	1	1	yes	characteristic
34	#13182029	—	2	2	no	—
35	#13182029	—	2	2	no	—
36	C₂×Dic₂₁₈	C₂×C₄	1	1	yes	characteristic
37	C₂×D₄₃₆	C₂×C₄	1	1	yes	characteristic
38	C₂×D₄₃₆	C₂×C₄	1	4	yes	—
39	C₂×C₂²⋊D₁₀₉	C₂×C₄	1	2	yes	—
40	C₂²×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×D₁₀₉	—	2	2	no	—
42	C₂×C₄×D₁₀₉	—	2	2	no	—
43	C₂×C₄×D₁₀₉	C₂×C₄	1	2	yes	—
44	C₂×C₄×D₁₀₉	C₂³	1	2	yes	—
45	C₂×C₄×D₁₀₉	C₂³	1	2	yes	—
46	#13182013	—	2	2	no	—
47	#13182012	—	2	4	no	—
48	#13182012	—	2	4	no	—
49	#13182012	—	2	4	no	—
50	C₂×#11724784	—	2	2	no	—
51	C₂×#11724784	C₂³	1	2	yes	—
52	C₂×#11724784	C₂³	1	2	yes	—
53	C₈×D₁₀₉	—	2	4	no	—
54	C₄⋊Dic₁₀₉	C₂×C₄	1	4	yes	—
55	#13182006	C₂×C₄	1	4	yes	—
56	#13182006	C₂×C₄	1	4	yes	—
57	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
58	M₁₆×C₁₀₉	—	2	2	no	—
59	C₂×C₈×C₁₀₉	D₄	1	4	yes	complemented
60	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
61	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
62	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
63	C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
64	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
65	C₄²×C₁₀₉	C₂³	1	2	yes	—
66	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
67	C₂²×D₁₀₉	—	2	2	no	—
68	C₂²×D₁₀₉	—	4	2	no	—
69	C₂²×D₁₀₉	C₂²×C₄	1	2	yes	—
70	C₂²×D₁₀₉	C₂²×C₄	1	2	yes	—
71	C₂×C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
72	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
74	C₂×C₄×C₁₀₉	#41	1	2	yes	—
75	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
76	C₂²⋊D₁₀₉	—	2	4	no	—
77	C₂×Dic₁₀₉	—	2	2	no	—
78	C₂×Dic₁₀₉	C₂²×C₄	1	2	yes	—
79	C₂×Dic₁₀₉	C₂²×C₄	1	2	yes	—
80	D₄₃₆	—	2	2	no	—
81	D₄₃₆	—	4	4	no	—
82	C₄×D₁₀₉	—	2	4	no	—
83	C₄×D₁₀₉	—	2	2	no	—
84	C₄×D₁₀₉	—	2	2	no	—
85	C₄×D₁₀₉	—	2	2	no	—
86	C₄×D₁₀₉	—	2	2	no	—
87	Dic₂₁₈	—	2	2	no	—
88	C₈×C₁₀₉	—	2	2	no	—
89	C₈×C₁₀₉	C₂×D₄	1	2	yes	—
90	C₈×C₁₀₉	C₂×D₄	1	2	yes	—
91	#11724784	—	2	2	no	—
92	#11724784	—	2	2	no	—
93	C₂²×C₁₀₉	—	2	2	no	—
94	C₂²×C₁₀₉	C₄×D₄	1	2	yes	—
95	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
96	C₂×D₁₀₉	—	2	2	no	—
97	C₂×D₁₀₉	—	2	2	no	—
98	C₂×D₁₀₉	—	2	2	no	—
99	C₂×D₁₀₉	—	2	4	no	—
100	C₂×D₁₀₉	—	2	2	no	—
101	C₂×D₁₀₉	—	4	4	no	—
102	C₄×C₁₀₉	—	2	4	no	—
103	C₄×C₁₀₉	C₄×D₄	1	2	yes	—
104	C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
105	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
106	Dic₁₀₉	—	2	2	no	—
107	Dic₁₀₉	—	2	2	no	—
108	C₂×C₁₀₉	—	2	2	no	—
109	C₂×C₁₀₉	C₂×C₄×D₄	1	1	yes	characteristic
110	C₂×C₁₀₉	#568	1	2	yes	—
111	D₁₀₉	—	4	2	no	—
112	D₁₀₉	—	4	2	no	—
113	#2930	—	109	1	no	maximal, Sylow
114	C₁₀₉	#2930	1	1	yes	characteristic, complemented, Sylow
115	C₂×#131	—	109	1	no	—
116	C₂²×M₁₆	—	109	1	no	—
117	#525	—	109	1	no	—
118	#436	—	109	4	no	—
119	#435	—	109	4	no	—
120	#424	—	109	1	no	—
121	#408	—	109	1	no	—
122	#407	—	109	1	no	—
123	#405	D₁₀₉	1	1	yes	characteristic, complemented
124	C₂×#41	—	109	1	no	—
125	C₂³×C₄	—	109	1	no	—
126	#131	—	218	4	no	—
127	C₂×M₁₆	—	109	1	no	—
128	C₂×M₁₆	—	109	2	no	—
129	C₂×M₁₆	—	109	2	no	—
130	C₂×M₁₆	—	109	2	no	—
131	C₂×M₁₆	—	218	4	no	—
132	C₂×M₁₆	C₂×D₁₀₉	1	1	yes	characteristic
133	C₂²×C₈	—	109	1	no	—
134	C₂²×C₈	—	109	2	no	—
135	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
136	#123	—	109	2	no	—
137	C₂²⋊Q₈	—	109	2	no	—
138	#121	—	109	2	no	—
139	#120	—	109	2	no	—
140	C₄×D₄	—	109	4	no	—
141	#117	C₂×D₁₀₉	1	1	yes	characteristic
142	C₄⋊C₈	—	109	4	no	—
143	C₂²⋊C₈	—	109	4	no	—
144	C₂²⋊C₈	—	109	4	no	—
145	#97	C₂×D₁₀₉	1	2	yes	complemented
146	C₄×C₈	C₂×D₁₀₉	1	2	yes	complemented
147	C₂⁴	—	109	1	no	—
148	#41	—	218	2	no	—
149	#41	—	218	2	no	—
150	C₂×Q₈	—	109	1	no	—
151	C₂×D₄	—	109	2	no	—
152	C₂×D₄	—	109	4	no	—
153	C₂×D₄	—	109	1	no	—
154	C₂²×C₄	—	109	1	no	—
155	C₂²×C₄	—	109	2	no	—
156	C₂²×C₄	—	109	2	no	—
157	C₂²×C₄	—	109	2	no	—
158	C₂²×C₄	—	218	2	no	—
159	C₂²×C₄	—	218	2	no	—
160	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
161	M₁₆	—	2	2	no	—
162	M₁₆	—	218	2	no	—
163	M₁₆	—	218	4	no	—
164	M₁₆	—	218	4	no	—
165	M₁₆	—	218	4	no	—
166	C₂×C₈	—	109	2	no	—
167	C₂×C₈	—	109	2	no	—
168	C₂×C₈	—	218	2	no	—
169	C₂×C₈	—	218	4	no	—
170	C₂×C₈	D₄₃₆	1	4	yes	complemented
171	C₂×C₈	C₂²×D₁₀₉	1	2	yes	—
172	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
173	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
174	C₄⋊C₄	—	109	4	no	—
175	C₄⋊C₄	C₄×D₁₀₉	1	2	yes	—
176	C₂²⋊C₄	—	109	4	no	—
177	C₂²⋊C₄	—	109	4	no	—
178	C₂²⋊C₄	C₄×D₁₀₉	1	2	yes	—
179	C₄²	C₂²×D₁₀₉	1	2	yes	—
180	C₂³	—	109	2	no	—
181	C₂³	—	109	2	no	—
182	C₂³	—	436	2	no	—
183	C₂³	—	218	2	no	—
184	C₂³	C₂×C₄×D₁₀₉	1	1	yes	characteristic
185	Q₈	—	218	2	no	—
186	D₄	—	436	4	no	—
187	D₄	—	218	4	no	—
188	D₄	—	218	2	no	—
189	C₂×C₄	—	109	2	no	—
190	C₂×C₄	—	109	2	no	—
191	C₂×C₄	—	218	2	no	—
192	C₂×C₄	—	218	4	no	—
193	C₂×C₄	—	218	2	no	—
194	C₂×C₄	—	218	2	no	—
195	C₂×C₄	—	218	2	no	—
196	C₂×C₄	—	218	2	no	—
197	C₂×C₄	C₂×C₄×D₁₀₉	1	4	yes	—
198	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
199	C₂×C₄	C₂×D₄₃₆	1	2	yes	—
200	C₂×C₄	#13182029	1	2	yes	—
201	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
202	C₈	—	2	2	no	—
203	C₈	—	218	2	no	—
204	C₈	—	218	2	no	—
205	C₈	C₂×D₄₃₆	1	2	yes	—
206	C₈	C₂×D₄₃₆	1	2	yes	—
207	C₂²	—	2	2	no	—
208	C₂²	—	436	4	no	—
209	C₂²	—	218	2	no	—
210	C₂²	—	218	2	no	—
211	C₂²	—	218	2	no	—
212	C₂²	—	218	4	no	—
213	C₂²	—	218	2	no	—
214	C₂²	C₄×D₄₃₆	1	2	yes	—
215	C₂²	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
216	C₄	—	2	4	no	—
217	C₄	—	218	2	no	—
218	C₄	—	218	2	no	—
219	C₄	C₄×D₄₃₆	1	2	yes	—
220	C₄	C₂²×D₄₃₆	1	1	yes	characteristic
221	C₄	C₂×#13182029	1	1	yes	characteristic
222	C₂	—	2	2	no	—
223	C₂	—	436	2	no	—
224	C₂	—	436	2	no	—
225	C₂	C₂×C₄×D₄₃₆	1	1	yes	characteristic
226	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	#20295538	C₂
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	#405	D₁₀₉
hypocenter	C₁₀₉	#2930
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
1. C₂×#16812705	C₂	semidirect
2. C₂²×#13182012	C₂	semidirect
4. #20295105	C₂	semidirect
5. #20295099	C₂	semidirect
6. #20295053	C₂	semidirect
7. #20294994	C₂	semidirect
8. #20294989	C₂	semidirect
9. #405×C₁₀₉	C₂	semidirect
17. C₂×#13182012	C₂²	semidirect
19. C₂×C₈×D₁₀₉	C₂²	semidirect
25. C₄⋊#11724784	C₂²	semidirect
27. #16812611	C₂²	semidirect
31. #97×C₁₀₉	C₂²	semidirect
32. C₄×C₈×C₁₀₉	C₂²	semidirect
59. C₂×C₈×C₁₀₉	D₄	semidirect
114. C₁₀₉	#2930	semidirect
123. #405	D₁₀₉	semidirect
145. #97	C₂×D₁₀₉	semidirect
146. C₄×C₈	C₂×D₁₀₉	semidirect
170. C₂×C₈	D₄₃₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16812705	2
C₂²×#13182012	2
#20295538	2
#20295105	2
#20295099	2
#20295053	2
#20294994	2
#20294989	2
#405×C₁₀₉	2
#2930	109