Subgroups — #28371760

Order 13952.

502 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812707	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×M₁₆×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295538	C₂	1	1	yes	maximal, characteristic
4	#20295105	C₂	1	2	yes	maximal, complemented
5	#20295099	C₂	1	2	yes	maximal, complemented
6	#20295099	C₂	1	2	yes	maximal, complemented
7	C₈×D₄₃₆	C₂	1	2	yes	maximal, complemented
8	#20295053	C₂	1	1	yes	maximal, characteristic, complemented
9	#20294992	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#16812611	C₂	1	1	yes	maximal, characteristic, complemented
11	#405×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13182029	C₂²	1	1	yes	characteristic
13	C₂²×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
14	#16812707	—	2	4	no	—
15	C₂×#13182013	C₂²	1	1	yes	characteristic
16	C₂×#13182012	C₂²	1	2	yes	—
17	C₂×#13182012	C₂²	1	2	yes	complemented
18	M₁₆×D₁₀₉	—	2	4	no	—
19	C₂²×#11724784	C₂²	1	1	yes	characteristic
20	C₂×C₈×D₁₀₉	C₂²	1	2	yes	—
21	C₂×C₈×D₁₀₉	C₂²	1	2	yes	complemented
22	#16812683	C₄	1	2	yes	—
23	#16812679	C₄	1	1	yes	characteristic
24	#16812679	C₄	1	1	yes	characteristic
25	#16812674	C₄	1	1	yes	characteristic
26	#16812674	C₄	1	1	yes	characteristic
27	#16812669	C₄	1	2	yes	—
28	C₄×D₄₃₆	C₂²	1	2	yes	—
29	C₄×D₄₃₆	C₂²	1	2	yes	—
30	C₄⋊#11724784	C₂²	1	2	yes	complemented
31	C₄⋊#11724784	C₂²	1	2	yes	complemented
32	#16812611	C₂²	1	1	yes	characteristic
33	#16812611	C₂²	1	1	yes	characteristic
34	#16812611	C₂²	1	1	yes	characteristic
35	#16812611	C₂²	1	1	yes	characteristic
36	#16812611	C₂²	1	2	yes	complemented
37	#16812611	C₂²	1	2	yes	complemented
38	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
39	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
40	#117×C₁₀₉	C₂²	1	1	yes	characteristic
41	#97×C₁₀₉	C₂²	1	2	yes	complemented
42	C₄×C₈×C₁₀₉	C₂²	1	2	yes	complemented
43	C₂³×D₁₀₉	C₂×C₄	1	1	yes	characteristic
44	#13182029	—	2	2	no	—
45	#13182029	—	2	2	no	—
46	C₂×Dic₂₁₈	C₂×C₄	1	1	yes	characteristic
47	C₂×D₄₃₆	C₂×C₄	1	2	yes	—
48	C₂×D₄₃₆	C₂×C₄	1	1	yes	characteristic
49	C₂×D₄₃₆	C₂×C₄	1	2	yes	—
50	C₂×C₂²⋊D₁₀₉	C₂×C₄	1	2	yes	—
51	C₂²×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₁₀₉	—	2	2	no	—
53	C₂×C₄×D₁₀₉	—	2	2	no	—
54	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
57	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
58	C₂×C₄×D₁₀₉	C₂³	1	2	yes	—
59	#13182013	—	2	2	no	—
60	#13182012	—	2	4	no	—
61	#13182012	—	2	4	no	—
62	C₂×#11724784	—	2	2	no	—
63	C₂×#11724784	C₂³	1	1	yes	characteristic
64	C₂×#11724784	C₂³	1	1	yes	characteristic
65	C₂×#11724784	C₂³	1	2	yes	—
66	C₈×D₁₀₉	—	2	4	no	—
67	C₈×D₁₀₉	—	2	4	no	—
68	C₄⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
69	C₄⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
70	#13182006	C₂×C₄	1	2	yes	—
71	#13182006	C₂×C₄	1	2	yes	—
72	#13182006	C₂×C₄	1	2	yes	—
73	#13182006	C₂×C₄	1	2	yes	—
74	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
75	M₁₆×C₁₀₉	D₄	1	4	yes	complemented
76	C₂×C₈×C₁₀₉	—	2	2	no	—
77	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
78	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
79	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
80	C₄⋊C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
81	C₄⋊C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
82	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
83	C₄²×C₁₀₉	C₂³	1	2	yes	—
84	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
85	C₂²×D₁₀₉	—	2	2	no	—
86	C₂²×D₁₀₉	—	2	2	no	—
87	C₂²×D₁₀₉	—	4	1	no	—
88	C₂²×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
89	C₂²×D₁₀₉	C₂²×C₄	1	2	yes	—
90	C₂²×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
92	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
93	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
95	C₂×C₄×C₁₀₉	#41	1	2	yes	—
96	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
97	C₂²⋊D₁₀₉	—	2	4	no	—
98	C₂×Dic₁₀₉	—	2	2	no	—
99	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
100	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
101	C₂×Dic₁₀₉	C₂²×C₄	1	2	yes	—
102	D₄₃₆	—	2	4	no	—
103	D₄₃₆	—	2	2	no	—
104	D₄₃₆	—	4	2	no	—
105	C₄×D₁₀₉	—	2	2	no	—
106	C₄×D₁₀₉	—	2	2	no	—
107	C₄×D₁₀₉	—	2	1	no	—
108	C₄×D₁₀₉	—	2	2	no	—
109	C₄×D₁₀₉	—	2	1	no	—
110	C₄×D₁₀₉	—	2	1	no	—
111	C₄×D₁₀₉	—	2	1	no	—
112	C₄×D₁₀₉	—	2	2	no	—
113	Dic₂₁₈	—	2	2	no	—
114	C₈×C₁₀₉	—	2	1	no	—
115	C₈×C₁₀₉	—	2	1	no	—
116	C₈×C₁₀₉	C₂×D₄	1	4	yes	—
117	#11724784	—	2	1	no	—
118	#11724784	—	2	2	no	—
119	#11724784	—	2	1	no	—
120	C₂²×C₁₀₉	—	2	1	no	—
121	C₂²×C₁₀₉	—	2	1	no	—
122	C₂²×C₁₀₉	C₄×D₄	1	2	yes	—
123	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
124	C₂×D₁₀₉	—	2	1	no	—
125	C₂×D₁₀₉	—	2	1	no	—
126	C₂×D₁₀₉	—	2	1	no	—
127	C₂×D₁₀₉	—	2	2	no	—
128	C₂×D₁₀₉	—	2	2	no	—
129	C₂×D₁₀₉	—	2	1	no	—
130	C₂×D₁₀₉	—	2	1	no	—
131	C₂×D₁₀₉	—	2	2	no	—
132	C₂×D₁₀₉	—	4	4	no	—
133	C₂×D₁₀₉	#131	1	2	yes	complemented
134	C₄×C₁₀₉	—	2	2	no	—
135	C₄×C₁₀₉	—	2	2	no	—
136	C₄×C₁₀₉	C₄×D₄	1	2	yes	—
137	C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
138	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
139	Dic₁₀₉	—	2	2	no	—
140	Dic₁₀₉	—	2	1	no	—
141	Dic₁₀₉	—	2	1	no	—
142	C₂×C₁₀₉	—	2	2	no	—
143	C₂×C₁₀₉	C₂×C₄×D₄	1	1	yes	characteristic
144	C₂×C₁₀₉	C₂×#131	1	1	yes	characteristic
145	C₂×C₁₀₉	#568	1	1	yes	characteristic
146	D₁₀₉	—	2	2	no	—
147	D₁₀₉	—	4	2	no	—
148	D₁₀₉	—	4	1	no	—
149	#2933	—	109	1	no	maximal, Sylow
150	C₁₀₉	#2933	1	1	yes	characteristic, complemented, Sylow
151	C₂×#131	—	109	1	no	—
152	C₂²×M₁₆	—	109	1	no	—
153	#525	—	109	1	no	—
154	#436	—	109	2	no	—
155	#435	—	109	2	no	—
156	#435	—	109	2	no	—
157	C₈×D₄	—	109	2	no	—
158	#424	—	109	1	no	—
159	#408	—	109	1	no	—
160	C₂×C₂²⋊C₈	—	109	1	no	—
161	#405	D₁₀₉	1	1	yes	characteristic, complemented
162	C₂×#41	—	109	1	no	—
163	C₂³×C₄	—	109	1	no	—
164	#131	—	218	4	no	—
165	C₂×M₁₆	—	109	1	no	—
166	C₂×M₁₆	—	109	2	no	—
167	C₂×M₁₆	—	109	2	no	—
168	C₂×M₁₆	—	218	4	no	—
169	C₂×M₁₆	C₂×D₁₀₉	1	1	yes	characteristic
170	C₂²×C₈	—	109	1	no	—
171	C₂²×C₈	—	109	2	no	—
172	C₂²×C₈	—	109	2	no	—
173	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
174	#123	—	109	2	no	—
175	C₂²⋊Q₈	—	109	1	no	—
176	C₂²⋊Q₈	—	109	1	no	—
177	#121	—	109	1	no	—
178	#121	—	109	1	no	—
179	#120	—	109	2	no	—
180	C₄×D₄	—	109	2	no	—
181	C₄×D₄	—	109	2	no	—
182	#117	C₂×D₁₀₉	1	1	yes	characteristic
183	C₄⋊C₈	—	109	2	no	—
184	C₄⋊C₈	—	109	2	no	—
185	C₂²⋊C₈	—	109	1	no	—
186	C₂²⋊C₈	—	109	1	no	—
187	C₂²⋊C₈	—	109	1	no	—
188	C₂²⋊C₈	—	109	1	no	—
189	C₂²⋊C₈	—	109	2	no	—
190	C₂²⋊C₈	—	109	2	no	—
191	#97	C₂×D₁₀₉	1	2	yes	complemented
192	C₄×C₈	C₂×D₁₀₉	1	2	yes	complemented
193	C₂⁴	—	109	1	no	—
194	#41	—	218	2	no	—
195	#41	—	218	2	no	—
196	C₂×Q₈	—	109	1	no	—
197	C₂×D₄	—	109	2	no	—
198	C₂×D₄	—	109	2	no	—
199	C₂×D₄	—	109	2	no	—
200	C₂×D₄	—	109	1	no	—
201	C₂²×C₄	—	109	1	no	—
202	C₂²×C₄	—	109	1	no	—
203	C₂²×C₄	—	109	1	no	—
204	C₂²×C₄	—	109	1	no	—
205	C₂²×C₄	—	109	2	no	—
206	C₂²×C₄	—	109	1	no	—
207	C₂²×C₄	—	218	2	no	—
208	C₂²×C₄	—	218	2	no	—
209	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
210	M₁₆	—	218	2	no	—
211	M₁₆	—	218	4	no	—
212	M₁₆	—	218	4	no	—
213	M₁₆	D₄₃₆	1	4	yes	complemented
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	109	1	no	—
216	C₂×C₈	—	109	1	no	—
217	C₂×C₈	—	109	2	no	—
218	C₂×C₈	—	218	2	no	—
219	C₂×C₈	—	218	4	no	—
220	C₂×C₈	—	218	4	no	—
221	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
222	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
223	C₂×C₈	C₂²×D₁₀₉	1	2	yes	—
224	C₄⋊C₄	—	109	2	no	—
225	C₄⋊C₄	—	109	2	no	—
226	C₄⋊C₄	C₄×D₁₀₉	1	1	yes	characteristic
227	C₄⋊C₄	C₄×D₁₀₉	1	1	yes	characteristic
228	C₂²⋊C₄	—	109	2	no	—
229	C₂²⋊C₄	—	109	2	no	—
230	C₂²⋊C₄	—	109	2	no	—
231	C₂²⋊C₄	—	109	2	no	—
232	C₂²⋊C₄	C₄×D₁₀₉	1	2	yes	—
233	C₄²	C₂²×D₁₀₉	1	2	yes	—
234	C₂³	—	109	1	no	—
235	C₂³	—	109	1	no	—
236	C₂³	—	109	2	no	—
237	C₂³	—	436	1	no	—
238	C₂³	—	218	2	no	—
239	C₂³	—	218	2	no	—
240	C₂³	C₂×C₄×D₁₀₉	1	1	yes	characteristic
241	Q₈	—	218	2	no	—
242	D₄	—	436	2	no	—
243	D₄	—	218	4	no	—
244	D₄	—	218	4	no	—
245	D₄	—	218	2	no	—
246	C₂×C₄	—	109	1	no	—
247	C₂×C₄	—	109	1	no	—
248	C₂×C₄	—	109	2	no	—
249	C₂×C₄	—	218	2	no	—
250	C₂×C₄	—	218	2	no	—
251	C₂×C₄	—	218	2	no	—
252	C₂×C₄	—	218	1	no	—
253	C₂×C₄	—	218	2	no	—
254	C₂×C₄	—	218	1	no	—
255	C₂×C₄	—	218	1	no	—
256	C₂×C₄	—	218	1	no	—
257	C₂×C₄	—	218	2	no	—
258	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
259	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
260	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
261	C₂×C₄	C₂×D₄₃₆	1	2	yes	—
262	C₂×C₄	#13182029	1	2	yes	—
263	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
264	C₈	—	2	1	no	—
265	C₈	—	2	1	no	—
266	C₈	—	218	1	no	—
267	C₈	—	218	1	no	—
268	C₈	—	218	2	no	—
269	C₈	C₂×D₄₃₆	1	4	yes	—
270	C₂²	—	2	1	no	—
271	C₂²	—	2	1	no	—
272	C₂²	—	109	2	no	—
273	C₂²	—	436	4	no	—
274	C₂²	—	218	1	no	—
275	C₂²	—	218	2	no	—
276	C₂²	—	218	2	no	—
277	C₂²	—	218	1	no	—
278	C₂²	—	218	1	no	—
279	C₂²	—	218	2	no	—
280	C₂²	—	218	1	no	—
281	C₂²	—	218	1	no	—
282	C₂²	C₄×D₄₃₆	1	2	yes	—
283	C₂²	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	2	2	no	—
286	C₄	—	218	1	no	—
287	C₄	—	218	1	no	—
288	C₄	—	218	2	no	—
289	C₄	C₄×D₄₃₆	1	2	yes	—
290	C₄	C₂²×D₄₃₆	1	1	yes	characteristic
291	C₄	C₂×#13182029	1	1	yes	characteristic
292	C₂	—	2	2	no	—
293	C₂	—	436	1	no	—
294	C₂	—	436	2	no	—
295	C₂	—	218	2	no	—
296	C₂	C₂×C₄×D₄₃₆	1	1	yes	characteristic
297	C₂	#131×D₁₀₉	1	1	yes	characteristic
298	C₂	#20295791	1	1	yes	characteristic

	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	#2933
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×#16812707	C₂	semidirect
2. C₂×M₁₆×D₁₀₉	C₂	semidirect
4. #20295105	C₂	semidirect
5. #20295099	C₂	semidirect
6. #20295099	C₂	semidirect
7. C₈×D₄₃₆	C₂	semidirect
8. #20295053	C₂	semidirect
9. #20294992	C₂	semidirect
10. C₂×#16812611	C₂	semidirect
11. #405×C₁₀₉	C₂	semidirect
17. C₂×#13182012	C₂²	semidirect
21. C₂×C₈×D₁₀₉	C₂²	semidirect
30. C₄⋊#11724784	C₂²	semidirect
31. C₄⋊#11724784	C₂²	semidirect
36. #16812611	C₂²	semidirect
37. #16812611	C₂²	semidirect
41. #97×C₁₀₉	C₂²	semidirect
42. C₄×C₈×C₁₀₉	C₂²	semidirect
75. M₁₆×C₁₀₉	D₄	semidirect
133. C₂×D₁₀₉	#131	semidirect
150. C₁₀₉	#2933	semidirect
161. #405	D₁₀₉	semidirect
191. #97	C₂×D₁₀₉	semidirect
192. C₄×C₈	C₂×D₁₀₉	semidirect
213. M₁₆	D₄₃₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16812707	2
C₂×M₁₆×D₁₀₉	2
#20295538	2
#20295105	2
#20295099	2
C₈×D₄₃₆	2
#20295053	2
#20294992	2
C₂×#16812611	2
#405×C₁₀₉	2
#2933	109