Subgroups — #28373559

Order 13952.

340 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295656	C₂	1	1	yes	maximal, characteristic
2	#20295653	C₂	1	1	yes	maximal, characteristic
3	#20295435	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295434	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295433	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295427	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295425	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295423	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295422	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295067	C₂	1	1	yes	maximal, characteristic
11	#20295065	C₂	1	1	yes	maximal, characteristic
12	#20295063	C₂	1	1	yes	maximal, characteristic
13	#106×D₁₀₉	C₂	1	1	yes	maximal, characteristic
14	#20294983	C₂	1	1	yes	maximal, characteristic
15	#501×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13182012	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
18	#16812699	C₂²	1	1	yes	characteristic
19	#16812698	—	2	1	no	—
20	#16812698	C₂²	1	1	yes	characteristic
21	#16812698	C₂²	1	1	yes	characteristic
22	#16812697	—	2	1	no	—
23	#16812691	C₂²	1	1	yes	characteristic
24	#16812690	C₂²	1	1	yes	characteristic
25	#16812690	C₂²	1	1	yes	characteristic
26	#16812689	—	2	1	no	—
27	#16812689	—	2	1	no	—
28	#16812679	—	2	1	no	—
29	#16812677	—	2	1	no	—
30	#16812677	—	2	1	no	—
31	#16812675	—	2	1	no	—
32	#16812657	C₂²	1	1	yes	characteristic
33	#16812657	C₂²	1	1	yes	characteristic
34	#16812656	C₂²	1	1	yes	characteristic
35	C₄⋊C₄×D₁₀₉	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₁₀₉	C₂²	1	1	yes	characteristic
37	#16812633	C₂²	1	1	yes	characteristic
38	#16812633	C₂²	1	1	yes	characteristic
39	#16812632	C₂²	1	1	yes	characteristic
40	#16812632	C₂²	1	1	yes	characteristic
41	#16812631	C₂²	1	1	yes	characteristic
42	#16812631	C₂²	1	1	yes	characteristic
43	#16812630	C₂²	1	1	yes	characteristic
44	#16812630	C₂²	1	1	yes	characteristic
45	#16812630	C₂²	1	1	yes	characteristic
46	#16812630	C₂²	1	1	yes	characteristic
47	#16812630	C₂²	1	1	yes	characteristic
48	#16812630	C₂²	1	1	yes	characteristic
49	#16812609	C₂²	1	1	yes	characteristic
50	C₄×#11724784	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
52	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
53	#125×C₁₀₉	C₂²	1	1	yes	characteristic
54	#107×C₁₀₉	C₂²	1	1	yes	characteristic
55	#107×C₁₀₉	C₂²	1	1	yes	characteristic
56	#106×C₁₀₉	C₂²	1	1	yes	characteristic
57	#106×C₁₀₉	C₂²	1	1	yes	characteristic
58	#97×C₁₀₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₁₈	—	2	1	no	—
60	C₂×Dic₂₁₈	—	2	1	no	—
61	C₂×Dic₂₁₈	—	2	1	no	—
62	C₂×C₄×D₁₀₉	—	2	1	no	—
63	C₂×C₄×D₁₀₉	—	2	1	no	—
64	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
65	#13182012	Q₈	1	4	yes	complemented
66	C₂×#11724784	C₂³	1	1	yes	characteristic
67	C₂×#11724784	C₂³	1	1	yes	characteristic
68	C₈×D₁₀₉	Q₈	1	4	yes	complemented
69	C₄⋊Dic₁₀₉	—	2	1	no	—
70	C₄⋊Dic₁₀₉	—	2	1	no	—
71	C₄⋊Dic₁₀₉	—	2	1	no	—
72	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
76	#13182008	—	2	1	no	—
77	#13182008	—	2	1	no	—
78	#13182008	—	2	1	no	—
79	#13182008	—	2	1	no	—
80	#13182008	—	2	1	no	—
81	#13182008	—	2	1	no	—
82	#13182008	—	2	1	no	—
83	#13182008	—	2	1	no	—
84	#13182008	D₄	1	1	yes	characteristic
85	#13182008	D₄	1	1	yes	characteristic
86	#13182006	—	2	1	no	—
87	#13182006	—	2	1	no	—
88	#13182006	D₄	1	1	yes	characteristic
89	#13182006	D₄	1	1	yes	characteristic
90	C₄×Dic₁₀₉	—	2	1	no	—
91	C₄×Dic₁₀₉	—	2	1	no	—
92	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
93	C₂×Q₈×C₁₀₉	—	2	1	no	—
94	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
95	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
96	C₄⋊C₄×C₁₀₉	—	2	1	no	—
97	C₄⋊C₄×C₁₀₉	—	2	1	no	—
98	C₄⋊C₄×C₁₀₉	—	2	1	no	—
99	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
100	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
103	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
104	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
105	Q₈×C₁₀₉	—	8	1	no	—
106	C₂×C₄×C₁₀₉	—	2	1	no	—
107	C₂×C₄×C₁₀₉	—	2	1	no	—
108	C₂×C₄×C₁₀₉	—	2	1	no	—
109	C₂×C₄×C₁₀₉	—	2	1	no	—
110	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
113	C₂×Dic₁₀₉	—	2	1	no	—
114	C₂×Dic₁₀₉	—	2	1	no	—
115	C₂×Dic₁₀₉	—	2	1	no	—
116	C₂×Dic₁₀₉	—	2	1	no	—
117	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
118	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
119	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
120	C₄×D₁₀₉	—	8	1	no	—
121	C₄×D₁₀₉	—	8	1	no	—
122	C₄×D₁₀₉	C₂×Q₈	1	2	yes	—
123	C₄×D₁₀₉	C₂×Q₈	1	2	yes	—
124	Dic₂₁₈	—	8	1	no	—
125	Dic₂₁₈	—	8	1	no	—
126	Dic₂₁₈	—	8	1	no	—
127	C₈×C₁₀₉	C₂×Q₈	1	2	yes	—
128	C₈×C₁₀₉	C₂×Q₈	1	2	yes	—
129	#11724784	C₂×Q₈	1	2	yes	—
130	#11724784	C₂×Q₈	1	2	yes	—
131	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
132	C₂×D₁₀₉	—	2	1	no	—
133	C₂×D₁₀₉	—	2	1	no	—
134	C₂×D₁₀₉	C₄⋊Q₈	1	2	yes	—
135	C₄×C₁₀₉	—	2	1	no	—
136	C₄×C₁₀₉	—	2	1	no	—
137	C₄×C₁₀₉	—	4	1	no	—
138	C₄×C₁₀₉	—	4	1	no	—
139	C₄×C₁₀₉	—	4	1	no	—
140	C₄×C₁₀₉	—	4	1	no	—
141	C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
142	C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
143	Dic₁₀₉	—	2	1	no	—
144	Dic₁₀₉	—	2	1	no	—
145	Dic₁₀₉	—	4	1	no	—
146	Dic₁₀₉	—	4	1	no	—
147	Dic₁₀₉	—	4	1	no	—
148	Dic₁₀₉	—	4	1	no	—
149	Dic₁₀₉	C₄⋊Q₈	1	2	yes	—
150	C₂×C₁₀₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
151	C₂×C₁₀₉	#577	1	1	yes	characteristic
152	C₂×C₁₀₉	#577	1	1	yes	characteristic
153	D₁₀₉	—	2	2	no	—
154	#3163	—	109	1	no	maximal, Sylow
155	C₁₀₉	#3163	1	1	yes	characteristic, complemented, Sylow
156	#544	—	109	1	no	—
157	#544	—	109	1	no	—
158	#501	—	109	1	no	—
159	#501	—	109	1	no	—
160	#501	—	109	1	no	—
161	#501	D₁₀₉	1	1	yes	characteristic, complemented
162	#499	—	109	1	no	—
163	#499	—	109	1	no	—
164	#498	—	109	1	no	—
165	#498	—	109	1	no	—
166	#428	—	109	1	no	—
167	#428	—	109	1	no	—
168	#427	—	109	1	no	—
169	C₂×#106	—	109	1	no	—
170	#405	—	109	1	no	—
171	C₂×M₁₆	—	109	1	no	—
172	C₂²×C₈	—	109	1	no	—
173	C₄⋊Q₈	—	109	1	no	—
174	C₄⋊Q₈	—	109	1	no	—
175	C₄⋊Q₈	—	109	1	no	—
176	C₄⋊Q₈	—	218	1	no	—
177	C₄⋊Q₈	—	218	1	no	—
178	C₄⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic
179	#125	—	109	1	no	—
180	#125	—	109	1	no	—
181	#125	—	109	1	no	—
182	#125	—	218	1	no	—
183	#125	—	218	1	no	—
184	#125	C₂×D₁₀₉	1	1	yes	characteristic
185	C₂²⋊Q₈	—	218	1	no	—
186	C₂²⋊Q₈	—	218	1	no	—
187	C₂²⋊Q₈	—	218	1	no	—
188	C₂²⋊Q₈	—	218	1	no	—
189	#117	—	109	1	no	—
190	#117	—	109	1	no	—
191	#117	—	109	1	no	—
192	C₂×C₄⋊C₄	—	109	1	no	—
193	C₂×C₄⋊C₄	—	109	1	no	—
194	#107	—	109	1	no	—
195	#107	—	109	1	no	—
196	#107	—	109	1	no	—
197	#107	—	109	1	no	—
198	#107	C₂×D₁₀₉	1	1	yes	characteristic
199	#107	C₂×D₁₀₉	1	1	yes	characteristic
200	#106	—	109	1	no	—
201	#106	—	109	1	no	—
202	#106	—	109	1	no	—
203	#106	—	109	1	no	—
204	#106	—	109	1	no	—
205	#106	—	109	1	no	—
206	#106	—	109	1	no	—
207	#106	—	109	1	no	—
208	#106	C₂×D₁₀₉	1	1	yes	characteristic
209	#106	C₂×D₁₀₉	1	1	yes	characteristic
210	#97	—	109	1	no	—
211	#97	C₂×D₁₀₉	1	1	yes	characteristic
212	C₄×C₈	—	109	1	no	—
213	C₄×C₈	—	109	1	no	—
214	C₂×Q₈	—	2	1	no	—
215	C₂×Q₈	—	218	1	no	—
216	C₂×Q₈	—	218	1	no	—
217	C₂×Q₈	—	218	1	no	—
218	C₂²×C₄	—	109	1	no	—
219	C₂²×C₄	—	218	1	no	—
220	C₂²×C₄	—	218	1	no	—
221	M₁₆	—	109	4	no	—
222	C₂×C₈	—	109	1	no	—
223	C₂×C₈	—	109	1	no	—
224	C₂×C₈	—	109	4	no	—
225	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
226	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
227	C₄⋊C₄	—	2	1	no	—
228	C₄⋊C₄	—	2	1	no	—
229	C₄⋊C₄	—	2	1	no	—
230	C₄⋊C₄	—	109	1	no	—
231	C₄⋊C₄	—	109	1	no	—
232	C₄⋊C₄	—	109	1	no	—
233	C₄⋊C₄	—	109	1	no	—
234	C₄⋊C₄	—	109	1	no	—
235	C₄⋊C₄	—	109	1	no	—
236	C₄⋊C₄	—	218	1	no	—
237	C₄⋊C₄	—	218	1	no	—
238	C₄⋊C₄	—	218	1	no	—
239	C₄⋊C₄	—	218	1	no	—
240	C₄⋊C₄	—	218	1	no	—
241	C₄⋊C₄	—	218	1	no	—
242	C₄⋊C₄	—	218	1	no	—
243	C₄⋊C₄	—	218	1	no	—
244	C₄⋊C₄	—	218	1	no	—
245	C₄⋊C₄	—	218	1	no	—
246	C₄⋊C₄	—	218	1	no	—
247	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
248	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
249	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
250	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
251	C₂²⋊C₄	—	109	1	no	—
252	C₂²⋊C₄	—	109	1	no	—
253	C₂²⋊C₄	—	218	1	no	—
254	C₂²⋊C₄	—	218	1	no	—
255	C₄²	—	109	1	no	—
256	C₄²	—	218	1	no	—
257	C₄²	—	218	1	no	—
258	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
259	C₂³	—	109	1	no	—
260	Q₈	—	8	1	no	—
261	Q₈	—	872	1	no	—
262	Q₈	—	872	1	no	—
263	Q₈	—	872	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	872	1	no	—
269	C₂×C₄	—	872	1	no	—
270	C₂×C₄	—	109	1	no	—
271	C₂×C₄	—	109	2	no	—
272	C₂×C₄	—	109	2	no	—
273	C₂×C₄	—	109	1	no	—
274	C₂×C₄	—	109	1	no	—
275	C₂×C₄	—	218	1	no	—
276	C₂×C₄	—	218	1	no	—
277	C₂×C₄	—	218	1	no	—
278	C₂×C₄	—	218	1	no	—
279	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
280	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
281	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
282	C₈	—	109	2	no	—
283	C₈	—	109	2	no	—
284	C₈	Q₈×D₁₀₉	1	2	yes	—
285	C₈	Q₈×D₁₀₉	1	2	yes	—
286	C₂²	—	109	2	no	—
287	C₂²	—	218	1	no	—
288	C₂²	—	218	1	no	—
289	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	4	1	no	—
293	C₄	—	4	1	no	—
294	C₄	—	4	1	no	—
295	C₄	—	4	1	no	—
296	C₄	—	109	2	no	—
297	C₄	—	436	1	no	—
298	C₄	—	436	1	no	—
299	C₄	—	436	1	no	—
300	C₄	—	436	1	no	—
301	C₄	—	218	1	no	—
302	C₄	—	218	1	no	—
303	C₄	C₂×Q₈×D₁₀₉	1	1	yes	characteristic
304	C₄	C₂×Q₈×D₁₀₉	1	1	yes	characteristic
305	C₂	—	218	2	no	—
306	C₂	C₄⋊Q₈×D₁₀₉	1	1	yes	characteristic
307	C₂	#20295835	1	1	yes	characteristic
308	C₂	#20295836	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#501×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂²×D₁₀₉	C₂×D₄
hypercenter	#501	D₁₀₉
hypocenter	C₁₀₉	#3163
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #501

Complements

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

N	H ≅ G/N	type
3. #20295435	C₂	semidirect
4. #20295434	C₂	semidirect
5. #20295433	C₂	semidirect
6. #20295427	C₂	semidirect
7. #20295425	C₂	semidirect
8. #20295423	C₂	semidirect
9. #20295422	C₂	semidirect
15. #501×C₁₀₉	C₂	semidirect
65. #13182012	Q₈	semidirect
68. C₈×D₁₀₉	Q₈	semidirect
155. C₁₀₉	#3163	semidirect
161. #501	D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295656	2
#20295653	2
#20295435	2
#20295434	2
#20295433	2
#20295427	2
#20295425	2
#20295423	2
#20295422	2
#20295067	2
#20295065	2
#20295063	2
#106×D₁₀₉	2
#20294983	2
#501×C₁₀₉	2
#3163	109