Subgroups — #28371757

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	#20295537	C₂	1	1	yes	maximal, characteristic
3	#20295102	C₂	1	2	yes	maximal, complemented
4	#20295097	C₂	1	2	yes	maximal, complemented
5	#20295095	C₂	1	2	yes	maximal, complemented
6	C₈×C₂²⋊D₁₀₉	C₂	1	2	yes	maximal, complemented
7	#20295054	C₂	1	1	yes	maximal, characteristic, complemented
8	#20294994	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₂²⋊#11724784	C₂	1	1	yes	maximal, characteristic, complemented
10	#20294984	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×M₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13182029	C₂²	1	1	yes	characteristic
13	#16812707	—	2	4	no	—
14	C₂×#13182012	C₂²	1	2	yes	complemented
15	C₂²×#11724784	C₂²	1	1	yes	characteristic
16	C₂²×#11724784	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₁₀₉	C₂²	1	2	yes	complemented
18	#16812682	C₄	1	2	yes	—
19	#16812678	C₄	1	1	yes	characteristic
20	#16812678	C₄	1	1	yes	characteristic
21	#16812673	C₄	1	1	yes	characteristic
22	#16812673	C₄	1	1	yes	characteristic
23	#16812668	C₄	1	2	yes	—
24	C₄×C₂²⋊D₁₀₉	C₂²	1	2	yes	—
25	C₄×C₂²⋊D₁₀₉	C₂²	1	2	yes	—
26	#16812658	C₂²	1	1	yes	characteristic
27	#16812628	C₂²	1	2	yes	complemented
28	#16812628	C₂²	1	2	yes	complemented
29	C₂²⋊#11724784	C₂²	1	1	yes	characteristic
30	C₂²⋊#11724784	C₂²	1	1	yes	characteristic
31	C₂²⋊#11724784	C₂²	1	1	yes	characteristic
32	C₂²⋊#11724784	C₂²	1	1	yes	characteristic
33	#16812611	C₂²	1	2	yes	complemented
34	#16812611	C₂²	1	2	yes	complemented
35	#16812609	C₂²	1	2	yes	complemented
36	C₈×Dic₁₀₉	C₂²	1	2	yes	complemented
37	C₂³×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
38	C₂×M₁₆×C₁₀₉	—	2	4	no	—
39	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
40	C₂×M₁₆×C₁₀₉	C₂²	1	2	yes	—
41	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
42	C₂²×C₈×C₁₀₉	C₂²	1	2	yes	—
43	#13182029	—	2	2	no	—
44	#13182029	—	2	2	no	—
45	C₂×Dic₂₁₈	C₂×C₄	1	1	yes	characteristic
46	C₂×D₄₃₆	C₂×C₄	1	1	yes	characteristic
47	C₂×C₂²⋊D₁₀₉	C₂×C₄	1	2	yes	—
48	C₂×C₂²⋊D₁₀₉	C₂×C₄	1	2	yes	—
49	C₂×C₂²⋊D₁₀₉	C₂×C₄	1	2	yes	—
50	C₂×C₄×D₁₀₉	C₂³	1	2	yes	—
51	#13182012	—	2	4	no	—
52	C₂×#11724784	—	2	2	no	—
53	C₂×#11724784	—	2	2	no	—
54	C₂×#11724784	C₂³	1	1	yes	characteristic
55	C₂×#11724784	C₂³	1	1	yes	characteristic
56	C₂×#11724784	C₂³	1	1	yes	characteristic
57	C₂×#11724784	C₂³	1	1	yes	characteristic
58	C₈×D₁₀₉	—	2	4	no	—
59	C₄⋊Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
60	C₄⋊Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
61	#13182008	C₂×C₄	1	2	yes	—
62	#13182008	C₂×C₄	1	2	yes	—
63	C₂²⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
64	C₂²⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
65	C₂²⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
66	#13182006	C₂×C₄	1	2	yes	—
67	#13182006	C₂×C₄	1	2	yes	—
68	C₄×Dic₁₀₉	C₂³	1	2	yes	—
69	C₂⁴×C₁₀₉	C₂×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₀₉	—	2	2	no	—
71	C₂²×C₄×C₁₀₉	—	2	2	no	—
72	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
76	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
77	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
78	M₁₆×C₁₀₉	—	2	2	no	—
79	M₁₆×C₁₀₉	—	2	4	no	—
80	M₁₆×C₁₀₉	D₄	1	4	yes	complemented
81	C₂×C₈×C₁₀₉	—	2	4	no	—
82	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
83	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
84	C₂³×C₁₀₉	—	2	2	no	—
85	C₂³×C₁₀₉	—	2	2	no	—
86	C₂³×C₁₀₉	—	4	1	no	—
87	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
88	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
89	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
90	C₂²×D₁₀₉	C₂²×C₄	1	2	yes	—
91	C₂×C₄×C₁₀₉	—	2	2	no	—
92	C₂×C₄×C₁₀₉	—	2	2	no	—
93	C₂×C₄×C₁₀₉	—	2	2	no	—
94	C₂×C₄×C₁₀₉	—	2	1	no	—
95	C₂×C₄×C₁₀₉	—	2	1	no	—
96	C₂×C₄×C₁₀₉	—	2	1	no	—
97	C₂×C₄×C₁₀₉	—	2	1	no	—
98	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
102	C₂×C₄×C₁₀₉	#41	1	2	yes	—
103	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
104	C₂²⋊D₁₀₉	—	2	4	no	—
105	C₂²⋊D₁₀₉	—	2	4	no	—
106	C₂²⋊D₁₀₉	—	4	2	no	—
107	C₂×Dic₁₀₉	C₂²×C₄	1	2	yes	—
108	C₂×Dic₁₀₉	C₂²×C₄	1	2	yes	—
109	C₂×Dic₁₀₉	C₂²×C₄	1	2	yes	—
110	D₄₃₆	—	2	2	no	—
111	C₄×D₁₀₉	—	2	2	no	—
112	C₄×D₁₀₉	—	2	2	no	—
113	Dic₂₁₈	—	2	2	no	—
114	C₈×C₁₀₉	—	2	2	no	—
115	C₈×C₁₀₉	C₂×D₄	1	4	yes	—
116	#11724784	—	2	1	no	—
117	#11724784	—	2	1	no	—
118	#11724784	—	2	1	no	—
119	#11724784	—	2	1	no	—
120	C₂²×C₁₀₉	—	2	1	no	—
121	C₂²×C₁₀₉	—	2	1	no	—
122	C₂²×C₁₀₉	—	2	1	no	—
123	C₂²×C₁₀₉	—	2	1	no	—
124	C₂²×C₁₀₉	—	2	1	no	—
125	C₂²×C₁₀₉	—	2	1	no	—
126	C₂²×C₁₀₉	—	2	1	no	—
127	C₂²×C₁₀₉	—	4	4	no	—
128	C₂²×C₁₀₉	C₄×D₄	1	2	yes	—
129	C₂²×C₁₀₉	#131	1	2	yes	complemented
130	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
131	C₂×D₁₀₉	—	2	2	no	—
132	C₂×D₁₀₉	—	2	2	no	—
133	C₂×D₁₀₉	—	2	2	no	—
134	C₄×C₁₀₉	—	2	1	no	—
135	C₄×C₁₀₉	—	2	1	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	2	no	—
141	Dic₁₀₉	—	2	2	no	—
142	C₂×C₁₀₉	—	2	2	no	—
143	C₂×C₁₀₉	—	2	2	no	—
144	C₂×C₁₀₉	—	4	1	no	—
145	C₂×C₁₀₉	C₂×C₄×D₄	1	1	yes	characteristic
146	C₂×C₁₀₉	C₂×#131	1	1	yes	characteristic
147	C₂×C₁₀₉	#568	1	1	yes	characteristic
148	D₁₀₉	—	4	2	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	D₁₀₉	1	1	yes	characteristic, complemented
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	—	109	1	no	—
162	C₂×#41	—	109	1	no	—
163	C₂³×C₄	C₂×D₁₀₉	1	1	yes	characteristic
164	#131	—	218	4	no	—
165	C₂×M₁₆	—	2	4	no	—
166	C₂×M₁₆	—	109	2	no	—
167	C₂×M₁₆	C₂×D₁₀₉	1	1	yes	characteristic
168	C₂×M₁₆	C₂×D₁₀₉	1	2	yes	—
169	C₂×M₁₆	C₂×D₁₀₉	1	1	yes	characteristic
170	C₂²×C₈	—	109	1	no	—
171	C₂²×C₈	—	109	1	no	—
172	C₂²×C₈	—	109	2	no	—
173	C₂²×C₈	C₂×D₁₀₉	1	2	yes	—
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	—	109	1	no	—
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	—	109	2	no	—
192	C₄×C₈	—	109	2	no	—
193	C₂⁴	C₄×D₁₀₉	1	1	yes	characteristic
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	—	2	2	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	109	2	no	—
204	C₂²×C₄	C₄×D₁₀₉	1	1	yes	characteristic
205	C₂²×C₄	C₄×D₁₀₉	1	1	yes	characteristic
206	C₂²×C₄	C₄×D₁₀₉	1	1	yes	characteristic
207	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
208	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
209	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
210	M₁₆	—	2	2	no	—
211	M₁₆	—	2	4	no	—
212	M₁₆	—	218	4	no	—
213	M₁₆	C₂²⋊D₁₀₉	1	4	yes	complemented
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	109	1	no	—
216	C₂×C₈	—	109	1	no	—
217	C₂×C₈	—	109	1	no	—
218	C₂×C₈	—	109	1	no	—
219	C₂×C₈	—	218	2	no	—
220	C₂×C₈	—	218	2	no	—
221	C₂×C₈	—	218	4	no	—
222	C₂×C₈	C₂²×D₁₀₉	1	2	yes	—
223	C₂×C₈	C₂²×D₁₀₉	1	2	yes	—
224	C₄⋊C₄	—	109	2	no	—
225	C₄⋊C₄	—	109	2	no	—
226	C₄⋊C₄	—	109	1	no	—
227	C₄⋊C₄	—	109	1	no	—
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	—	109	2	no	—
233	C₄²	—	109	2	no	—
234	C₂³	—	2	2	no	—
235	C₂³	—	2	2	no	—
236	C₂³	—	4	1	no	—
237	C₂³	—	109	2	no	—
238	C₂³	C₂×C₄×D₁₀₉	1	1	yes	characteristic
239	C₂³	C₂×C₄×D₁₀₉	1	1	yes	characteristic
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	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	109	2	no	—
254	C₂×C₄	—	109	2	no	—
255	C₂×C₄	—	109	2	no	—
256	C₂×C₄	—	218	2	no	—
257	C₂×C₄	—	218	2	no	—
258	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
259	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
260	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
261	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	2	yes	—
262	C₂×C₄	#13182029	1	2	yes	—
263	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
264	C₈	—	2	2	no	—
265	C₈	—	218	1	no	—
266	C₈	—	218	1	no	—
267	C₈	—	218	1	no	—
268	C₈	—	218	1	no	—
269	C₈	C₂×C₂²⋊D₁₀₉	1	4	yes	—
270	C₂²	—	2	1	no	—
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	—	4	4	no	—
278	C₂²	—	218	2	no	—
279	C₂²	—	218	2	no	—
280	C₂²	—	218	2	no	—
281	C₂²	C₄×C₂²⋊D₁₀₉	1	2	yes	—
282	C₂²	#16812707	1	2	yes	complemented
283	C₂²	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	218	2	no	—
287	C₄	—	218	2	no	—
288	C₄	—	218	2	no	—
289	C₄	C₄×C₂²⋊D₁₀₉	1	2	yes	—
290	C₄	C₂²×C₂²⋊D₁₀₉	1	1	yes	characteristic
291	C₄	C₂×#13182029	1	1	yes	characteristic
292	C₂	—	2	2	no	—
293	C₂	—	2	2	no	—
294	C₂	—	4	1	no	—
295	C₂	—	436	2	no	—
296	C₂	C₂×C₄×C₂²⋊D₁₀₉	1	1	yes	characteristic
297	C₂	C₂×#16812707	1	1	yes	characteristic
298	C₂	#20295792	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₀₉
commutator, socle	C₂²×C₁₀₉	C₂³×C₄
Fitting	C₂²×M₁₆×C₁₀₉	C₂
prime radical	#20295537	C₂
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	C₂²×M₁₆	D₁₀₉
hypocenter	C₁₀₉	#2933
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#16812707	C₂	semidirect
3. #20295102	C₂	semidirect
4. #20295097	C₂	semidirect
5. #20295095	C₂	semidirect
6. C₈×C₂²⋊D₁₀₉	C₂	semidirect
7. #20295054	C₂	semidirect
8. #20294994	C₂	semidirect
9. C₂×C₂²⋊#11724784	C₂	semidirect
10. #20294984	C₂	semidirect
11. C₂²×M₁₆×C₁₀₉	C₂	semidirect
14. C₂×#13182012	C₂²	semidirect
17. C₂×C₈×D₁₀₉	C₂²	semidirect
27. #16812628	C₂²	semidirect
28. #16812628	C₂²	semidirect
33. #16812611	C₂²	semidirect
34. #16812611	C₂²	semidirect
35. #16812609	C₂²	semidirect
36. C₈×Dic₁₀₉	C₂²	semidirect
80. M₁₆×C₁₀₉	D₄	semidirect
129. C₂²×C₁₀₉	#131	semidirect
150. C₁₀₉	#2933	semidirect
152. C₂²×M₁₆	D₁₀₉	semidirect
213. M₁₆	C₂²⋊D₁₀₉	semidirect
282. C₂²	#16812707	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
#20295537	2
#20295102	2
#20295097	2
#20295095	2
C₈×C₂²⋊D₁₀₉	2
#20295054	2
#20294994	2
C₂×C₂²⋊#11724784	2
#20294984	2
C₂²×M₁₆×C₁₀₉	2
#2933	109