Subgroups — #28372944

Order 13952.

420 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295536	C₂	1	1	yes	maximal, characteristic
2	#117×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295368	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295367	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295363	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295358	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295347	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295343	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295338	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295063	C₂	1	1	yes	maximal, characteristic, complemented
11	#20295061	C₂	1	1	yes	maximal, characteristic, complemented
12	#20295019	C₂	1	1	yes	maximal, characteristic, complemented
13	#20295019	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂²⋊C₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
15	#482×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13182028	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
18	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
20	#16812681	—	2	1	no	—
21	#16812679	C₂²	1	1	yes	characteristic
22	#16812678	C₂²	1	1	yes	characteristic
23	#16812670	C₂²	1	1	yes	characteristic
24	#16812667	—	2	1	no	—
25	D₄×Dic₁₀₉	—	2	1	no	—
26	#16812659	—	2	1	no	—
27	#16812658	C₂²	1	1	yes	characteristic
28	#16812657	C₂²	1	1	yes	characteristic
29	#16812657	C₂²	1	1	yes	characteristic
30	#16812657	C₂²	1	1	yes	characteristic
31	#16812656	—	2	1	no	—
32	#16812655	—	2	1	no	—
33	C₄⋊C₄×D₁₀₉	C₂²	1	1	yes	characteristic
34	C₄⋊C₄×D₁₀₉	C₂²	1	1	yes	characteristic
35	C₂²⋊C₄×D₁₀₉	—	2	1	no	—
36	C₄²×D₁₀₉	—	2	1	no	—
37	#16812633	C₂²	1	1	yes	characteristic, complemented
38	#16812632	C₂²	1	1	yes	characteristic, complemented
39	#16812632	C₂²	1	1	yes	characteristic, complemented
40	#16812631	C₂²	1	1	yes	characteristic, complemented
41	#16812630	C₂²	1	1	yes	characteristic, complemented
42	#16812630	C₂²	1	1	yes	characteristic, complemented
43	#16812624	C₂²	1	1	yes	characteristic, complemented
44	#16812624	C₂²	1	1	yes	characteristic, complemented
45	#16812622	C₂²	1	1	yes	characteristic, complemented
46	#16812622	C₂²	1	1	yes	characteristic, complemented
47	#16812620	C₂²	1	1	yes	characteristic, complemented
48	#16812620	C₂²	1	1	yes	characteristic, complemented
49	C₂²⋊#11724784	C₂²	1	1	yes	characteristic
50	#16812611	C₂²	1	1	yes	characteristic
51	#16812611	C₂²	1	1	yes	characteristic
52	#121×C₁₀₉	C₂²	1	1	yes	characteristic
53	#117×C₁₀₉	C₂²	1	1	yes	characteristic
54	#107×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
55	#106×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
56	#102×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
57	#102×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
58	C₂²⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
59	C₂³×D₁₀₉	D₄	1	1	yes	characteristic
60	#13182028	—	2	2	no	—
61	#13182028	—	2	2	no	—
62	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
64	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
65	C₂²×Dic₁₀₉	—	2	1	no	—
66	C₂²×Dic₁₀₉	D₄	1	1	yes	characteristic
67	C₂×C₄×D₁₀₉	—	2	2	no	—
68	C₂×C₄×D₁₀₉	—	2	2	no	—
69	C₂×C₄×D₁₀₉	—	2	1	no	—
70	C₂×C₄×D₁₀₉	—	2	1	no	—
71	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
72	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
73	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
74	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
75	C₂×#11724784	C₂³	1	1	yes	characteristic
76	C₂×#11724784	C₂³	1	1	yes	characteristic
77	C₈×D₁₀₉	—	2	2	no	—
78	C₈×D₁₀₉	—	2	2	no	—
79	C₄⋊Dic₁₀₉	—	2	1	no	—
80	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
81	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
82	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
83	#13182008	—	2	1	no	—
84	#13182008	—	2	1	no	—
85	#13182008	—	2	1	no	—
86	C₂²⋊Dic₁₀₉	—	2	1	no	—
87	C₂²⋊Dic₁₀₉	—	2	1	no	—
88	C₂²⋊Dic₁₀₉	—	2	1	no	—
89	#13182006	—	2	1	no	—
90	#13182006	—	2	1	no	—
91	#13182006	—	2	1	no	—
92	#13182006	—	2	1	no	—
93	C₄×Dic₁₀₉	—	2	1	no	—
94	C₄×Dic₁₀₉	—	2	1	no	—
95	C₄×Dic₁₀₉	—	2	1	no	—
96	C₄×Dic₁₀₉	—	2	1	no	—
97	C₂×D₄×C₁₀₉	—	2	1	no	—
98	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
99	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
100	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
101	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
105	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
106	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
107	C₄²×C₁₀₉	—	2	1	no	—
108	C₂³×C₁₀₉	—	2	1	no	—
109	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
110	C₂²×D₁₀₉	—	2	2	no	—
111	C₂²×D₁₀₉	—	2	2	no	—
112	C₂²×D₁₀₉	—	4	1	no	—
113	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
114	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
115	D₄×C₁₀₉	—	2	1	no	—
116	D₄×C₁₀₉	—	2	1	no	—
117	D₄×C₁₀₉	—	4	2	no	—
118	C₂×C₄×C₁₀₉	—	2	1	no	—
119	C₂×C₄×C₁₀₉	—	2	1	no	—
120	C₂×C₄×C₁₀₉	—	2	1	no	—
121	C₂×C₄×C₁₀₉	—	2	1	no	—
122	C₂×C₄×C₁₀₉	—	2	1	no	—
123	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
125	C₂²⋊D₁₀₉	—	4	2	no	—
126	C₂²⋊D₁₀₉	—	4	2	no	—
127	C₂×Dic₁₀₉	—	2	2	no	—
128	C₂×Dic₁₀₉	—	2	1	no	—
129	C₂×Dic₁₀₉	—	2	1	no	—
130	C₂×Dic₁₀₉	—	2	1	no	—
131	C₂×Dic₁₀₉	—	2	1	no	—
132	C₂×Dic₁₀₉	—	4	2	no	—
133	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
134	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
135	C₄×D₁₀₉	—	2	2	no	—
136	C₄×D₁₀₉	—	2	1	no	—
137	C₄×D₁₀₉	—	2	1	no	—
138	C₄×D₁₀₉	—	2	2	no	—
139	C₄×D₁₀₉	—	4	2	no	—
140	C₄×D₁₀₉	—	4	2	no	—
141	C₄×D₁₀₉	#41	1	2	yes	—
142	C₄×D₁₀₉	#41	1	2	yes	—
143	Dic₂₁₈	—	2	1	no	—
144	Dic₂₁₈	—	2	1	no	—
145	C₈×C₁₀₉	—	2	1	no	—
146	C₈×C₁₀₉	—	2	1	no	—
147	#11724784	—	2	1	no	—
148	#11724784	—	2	1	no	—
149	C₂²×C₁₀₉	—	2	1	no	—
150	C₂²×C₁₀₉	—	2	1	no	—
151	C₂²×C₁₀₉	—	2	1	no	—
152	C₂²×C₁₀₉	—	4	1	no	—
153	C₂²×C₁₀₉	—	4	1	no	—
154	C₂²×C₁₀₉	—	4	1	no	—
155	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
156	C₂×D₁₀₉	—	2	1	no	—
157	C₂×D₁₀₉	—	2	1	no	—
158	C₂×D₁₀₉	—	2	1	no	—
159	C₂×D₁₀₉	—	2	1	no	—
160	C₂×D₁₀₉	—	4	4	no	—
161	C₂×D₁₀₉	#123	1	2	yes	—
162	C₂×D₁₀₉	#135	1	2	yes	—
163	C₄×C₁₀₉	—	2	1	no	—
164	C₄×C₁₀₉	—	4	1	no	—
165	C₄×C₁₀₉	—	4	1	no	—
166	C₄×C₁₀₉	—	4	1	no	—
167	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
168	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
169	Dic₁₀₉	—	2	1	no	—
170	Dic₁₀₉	—	4	1	no	—
171	Dic₁₀₉	—	4	1	no	—
172	Dic₁₀₉	—	4	1	no	—
173	Dic₁₀₉	—	4	1	no	—
174	Dic₁₀₉	#123	1	2	yes	—
175	C₂×C₁₀₉	—	4	1	no	—
176	C₂×C₁₀₉	—	8	1	no	—
177	C₂×C₁₀₉	C₂×#123	1	1	yes	characteristic
178	C₂×C₁₀₉	C₂×#135	1	1	yes	characteristic
179	C₂×C₁₀₉	#575	1	1	yes	characteristic
180	D₁₀₉	—	2	2	no	—
181	D₁₀₉	—	2	2	no	—
182	#3091	—	109	1	no	maximal, Sylow
183	C₁₀₉	#3091	1	1	yes	characteristic, complemented, Sylow
184	#525	—	109	1	no	—
185	C₂×#117	—	109	1	no	—
186	#485	—	109	1	no	—
187	#485	—	109	1	no	—
188	#484	—	109	1	no	—
189	#483	—	109	1	no	—
190	#482	—	109	1	no	—
191	#482	D₁₀₉	1	1	yes	characteristic, complemented
192	#481	—	109	1	no	—
193	#480	—	109	1	no	—
194	#427	—	109	1	no	—
195	#427	—	109	1	no	—
196	#416	—	109	1	no	—
197	#416	—	109	1	no	—
198	C₂×C₂²⋊C₈	—	109	1	no	—
199	C₂×#41	—	109	1	no	—
200	C₂³×C₄	—	109	1	no	—
201	C₂²×C₈	—	109	1	no	—
202	C₂²×C₈	—	109	1	no	—
203	#123	—	218	1	no	—
204	C₂²⋊Q₈	—	109	1	no	—
205	C₂²⋊Q₈	—	109	1	no	—
206	#121	—	109	1	no	—
207	#121	C₂×D₁₀₉	1	1	yes	characteristic
208	#120	—	218	1	no	—
209	C₄×D₄	—	218	1	no	—
210	C₄×D₄	—	218	1	no	—
211	#117	—	109	1	no	—
212	#117	—	109	1	no	—
213	#117	—	109	1	no	—
214	#117	—	109	1	no	—
215	#117	—	218	1	no	—
216	#117	—	218	1	no	—
217	#117	C₂×D₁₀₉	1	1	yes	characteristic
218	C₂×C₄⋊C₄	—	109	1	no	—
219	C₂×C₄⋊C₄	—	109	1	no	—
220	C₂×C₂²⋊C₄	—	218	1	no	—
221	C₂×C₄²	—	218	1	no	—
222	#107	—	109	1	no	—
223	#107	—	109	1	no	—
224	#107	—	109	1	no	—
225	#107	C₂×D₁₀₉	1	1	yes	characteristic, complemented
226	#106	—	109	1	no	—
227	#106	—	109	1	no	—
228	#106	—	109	1	no	—
229	#106	C₂×D₁₀₉	1	1	yes	characteristic, complemented
230	#103	—	109	1	no	—
231	#103	—	109	1	no	—
232	#103	—	109	1	no	—
233	#103	—	109	1	no	—
234	#102	—	109	1	no	—
235	#102	—	109	1	no	—
236	#102	C₂×D₁₀₉	1	1	yes	characteristic, complemented
237	#102	C₂×D₁₀₉	1	1	yes	characteristic, complemented
238	C₂²⋊C₈	—	109	1	no	—
239	C₂²⋊C₈	—	109	1	no	—
240	C₂²⋊C₈	—	109	1	no	—
241	C₂²⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
242	C₂⁴	—	109	1	no	—
243	#41	—	218	2	no	—
244	#41	—	218	2	no	—
245	C₂×Q₈	—	109	1	no	—
246	C₂×D₄	—	2	1	no	—
247	C₂×D₄	—	218	1	no	—
248	C₂×D₄	—	218	1	no	—
249	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
250	C₂²×C₄	—	109	1	no	—
251	C₂²×C₄	—	109	1	no	—
252	C₂²×C₄	—	109	1	no	—
253	C₂²×C₄	—	109	1	no	—
254	C₂²×C₄	—	109	1	no	—
255	C₂²×C₄	—	218	2	no	—
256	C₂²×C₄	—	218	2	no	—
257	C₂²×C₄	—	218	1	no	—
258	C₂²×C₄	—	218	1	no	—
259	C₂²×C₄	—	218	1	no	—
260	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
261	C₂×C₈	—	109	1	no	—
262	C₂×C₈	—	109	1	no	—
263	C₂×C₈	—	218	2	no	—
264	C₂×C₈	—	218	2	no	—
265	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
266	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
267	C₄⋊C₄	—	109	1	no	—
268	C₄⋊C₄	—	109	1	no	—
269	C₄⋊C₄	—	109	1	no	—
270	C₄⋊C₄	—	218	1	no	—
271	C₄⋊C₄	—	218	1	no	—
272	C₄⋊C₄	—	218	1	no	—
273	C₄⋊C₄	—	218	1	no	—
274	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
275	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
276	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
277	C₂²⋊C₄	—	2	1	no	—
278	C₂²⋊C₄	—	2	1	no	—
279	C₂²⋊C₄	—	218	1	no	—
280	C₂²⋊C₄	—	218	1	no	—
281	C₂²⋊C₄	—	218	1	no	—
282	C₂²⋊C₄	—	218	1	no	—
283	C₂²⋊C₄	—	218	1	no	—
284	C₂²⋊C₄	—	218	1	no	—
285	C₂²⋊C₄	—	218	1	no	—
286	C₄²	—	2	1	no	—
287	C₄²	—	218	1	no	—
288	C₄²	—	218	1	no	—
289	C₄²	—	218	1	no	—
290	C₄²	—	218	1	no	—
291	C₂³	—	2	1	no	—
292	C₂³	—	109	1	no	—
293	C₂³	—	109	1	no	—
294	C₂³	—	436	1	no	—
295	C₂³	—	218	2	no	—
296	C₂³	—	218	2	no	—
297	C₂³	D₄×D₁₀₉	1	1	yes	characteristic
298	Q₈	—	218	1	no	—
299	Q₈	—	218	1	no	—
300	D₄	—	2	1	no	—
301	D₄	—	2	1	no	—
302	D₄	—	4	2	no	—
303	D₄	—	436	2	no	—
304	D₄	—	436	2	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	109	1	no	—
311	C₂×C₄	—	109	2	no	—
312	C₂×C₄	—	109	2	no	—
313	C₂×C₄	—	109	1	no	—
314	C₂×C₄	—	436	2	no	—
315	C₂×C₄	—	436	2	no	—
316	C₂×C₄	—	436	2	no	—
317	C₂×C₄	—	218	2	no	—
318	C₂×C₄	—	218	2	no	—
319	C₂×C₄	—	218	2	no	—
320	C₂×C₄	—	218	1	no	—
321	C₂×C₄	—	218	1	no	—
322	C₂×C₄	—	218	1	no	—
323	C₂×C₄	—	218	1	no	—
324	C₂×C₄	—	218	1	no	—
325	C₂×C₄	—	218	1	no	—
326	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
327	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
328	C₈	—	2	1	no	—
329	C₈	—	2	1	no	—
330	C₈	—	218	1	no	—
331	C₈	—	218	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	4	1	no	—
336	C₂²	—	4	1	no	—
337	C₂²	—	4	1	no	—
338	C₂²	—	109	2	no	—
339	C₂²	—	109	2	no	—
340	C₂²	—	436	4	no	—
341	C₂²	—	218	1	no	—
342	C₂²	—	218	1	no	—
343	C₂²	—	218	1	no	—
344	C₂²	—	218	1	no	—
345	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
346	C₄	—	2	1	no	—
347	C₄	—	4	1	no	—
348	C₄	—	4	1	no	—
349	C₄	—	4	1	no	—
350	C₄	—	109	2	no	—
351	C₄	—	436	1	no	—
352	C₄	—	436	1	no	—
353	C₄	—	436	1	no	—
354	C₄	—	436	1	no	—
355	C₄	—	218	1	no	—
356	C₄	#41×D₁₀₉	1	1	yes	characteristic
357	C₄	#41×D₁₀₉	1	1	yes	characteristic
358	C₂	—	4	1	no	—
359	C₂	—	8	1	no	—
360	C₂	—	218	2	no	—
361	C₂	—	218	2	no	—
362	C₂	#123×D₁₀₉	1	1	yes	characteristic
363	C₂	#135×D₁₀₉	1	1	yes	characteristic
364	C₂	#20295822	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#482×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	#20295536	C₂
hypercenter	#482	D₁₀₉
hypocenter	C₁₀₉	#3091
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #482

Complements

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

N	H ≅ G/N	type
2. #117×D₁₀₉	C₂	semidirect
3. #20295368	C₂	semidirect
4. #20295367	C₂	semidirect
5. #20295363	C₂	semidirect
6. #20295358	C₂	semidirect
7. #20295347	C₂	semidirect
8. #20295343	C₂	semidirect
9. #20295338	C₂	semidirect
10. #20295063	C₂	semidirect
11. #20295061	C₂	semidirect
12. #20295019	C₂	semidirect
13. #20295019	C₂	semidirect
14. C₂²⋊C₈×D₁₀₉	C₂	semidirect
15. #482×C₁₀₉	C₂	semidirect
37. #16812633	C₂²	semidirect
38. #16812632	C₂²	semidirect
39. #16812632	C₂²	semidirect
40. #16812631	C₂²	semidirect
41. #16812630	C₂²	semidirect
42. #16812630	C₂²	semidirect
43. #16812624	C₂²	semidirect
44. #16812624	C₂²	semidirect
45. #16812622	C₂²	semidirect
46. #16812622	C₂²	semidirect
47. #16812620	C₂²	semidirect
48. #16812620	C₂²	semidirect
54. #107×C₁₀₉	C₂²	semidirect
55. #106×C₁₀₉	C₂²	semidirect
56. #102×C₁₀₉	C₂²	semidirect
57. #102×C₁₀₉	C₂²	semidirect
183. C₁₀₉	#3091	semidirect
191. #482	D₁₀₉	semidirect
225. #107	C₂×D₁₀₉	semidirect
229. #106	C₂×D₁₀₉	semidirect
236. #102	C₂×D₁₀₉	semidirect
237. #102	C₂×D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295536	2
#117×D₁₀₉	2
#20295368	2
#20295367	2
#20295363	2
#20295358	2
#20295347	2
#20295343	2
#20295338	2
#20295063	2
#20295061	2
#20295019	2
C₂²⋊C₈×D₁₀₉	2
#482×C₁₀₉	2
#3091	109