Subgroups — #28373129

Order 13952.

444 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295580	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×D₄×D₁₀₉	C₂	1	1	yes	maximal, characteristic
3	#20295363	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295360	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295305	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295304	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295230	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295229	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295178	C₂	1	1	yes	maximal, characteristic
10	#20295175	C₂	1	1	yes	maximal, characteristic
11	#20295128	C₂	1	1	yes	maximal, characteristic, complemented
12	#20295126	C₂	1	1	yes	maximal, characteristic, complemented
13	#20295124	C₂	1	1	yes	maximal, characteristic, complemented
14	#20295081	C₂	1	1	yes	maximal, characteristic, complemented
15	#440×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×D₄×D₁₀₉	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₁₀₉	—	2	1	no	—
18	C₂×#13182018	C₂²	1	1	yes	characteristic
19	C₂×#13182017	C₂²	1	1	yes	characteristic
20	C₂×#13182016	C₂²	1	1	yes	characteristic
21	#16812698	C₂²	1	1	yes	characteristic, complemented
22	#16812697	C₂²	1	1	yes	characteristic, complemented
23	#16812689	—	2	1	no	—
24	#16812679	C₂²	1	1	yes	characteristic
25	#16812677	—	2	1	no	—
26	#16812675	C₂²	1	1	yes	characteristic
27	C₄×Dic₂₁₈	—	2	1	no	—
28	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic, complemented
29	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
30	D₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
31	#16812661	C₂²	1	1	yes	characteristic
32	C₄×C₂²⋊D₁₀₉	—	2	1	no	—
33	#16812659	—	2	1	no	—
34	#16812656	—	2	1	no	—
35	C₄⋊C₄×D₁₀₉	C₂²	1	1	yes	characteristic
36	C₂²⋊C₄×D₁₀₉	—	2	1	no	—
37	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
38	#16812633	C₂²	1	1	yes	characteristic, complemented
39	#16812632	C₂²	1	1	yes	characteristic, complemented
40	#16812632	C₂²	1	1	yes	characteristic, complemented
41	#16812628	C₂²	1	1	yes	characteristic, complemented
42	#16812628	C₂²	1	1	yes	characteristic, complemented
43	#16812624	C₂²	1	1	yes	characteristic, complemented
44	#16812623	C₂²	1	1	yes	characteristic, complemented
45	#16812622	C₂²	1	1	yes	characteristic, complemented
46	#16812621	C₂²	1	1	yes	characteristic
47	#16812620	C₂²	1	1	yes	characteristic
48	#16812619	C₂²	1	1	yes	characteristic
49	#16812611	C₂²	1	1	yes	characteristic
50	#16812611	C₂²	1	1	yes	characteristic
51	#16812610	C₂²	1	1	yes	characteristic, complemented
52	C₂×SD₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
54	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
55	#107×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
56	#103×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
57	#102×C₁₀₉	C₂²	1	1	yes	characteristic
58	#97×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
59	C₂³×D₁₀₉	—	2	1	no	—
60	C₂×Dic₂₁₈	—	2	1	no	—
61	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
62	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
64	D₄×D₁₀₉	—	2	2	no	—
65	D₄×D₁₀₉	—	2	2	no	—
66	C₂²×Dic₁₀₉	—	2	1	no	—
67	C₂×C₄×D₁₀₉	—	2	1	no	—
68	C₂×C₄×D₁₀₉	—	2	1	no	—
69	C₂×C₄×D₁₀₉	—	2	1	no	—
70	C₂×C₄×D₁₀₉	—	4	4	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	#13182018	—	2	2	no	—
75	#13182017	—	2	2	no	—
76	#13182016	—	2	2	no	—
77	C₂×#11724784	C₂³	1	1	yes	characteristic
78	C₂×#11724784	C₂³	1	1	yes	characteristic
79	C₄⋊Dic₁₀₉	—	2	1	no	—
80	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
81	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
82	#13182008	—	2	1	no	—
83	#13182008	—	2	1	no	—
84	#13182008	—	2	1	no	—
85	#13182008	—	2	1	no	—
86	#13182008	—	2	1	no	—
87	#13182008	—	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	D₄	1	1	yes	characteristic
95	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
96	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
97	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
98	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
99	C₂²×C₄×C₁₀₉	—	2	1	no	—
100	SD₁₆×C₁₀₉	—	2	2	no	—
101	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
102	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₁₀₉	—	2	1	no	—
104	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
106	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
107	C₄²×C₁₀₉	—	2	1	no	—
108	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
109	C₂³×C₁₀₉	—	2	1	no	—
110	C₂²×D₁₀₉	—	2	4	no	—
111	C₂²×D₁₀₉	—	2	1	no	—
112	C₂²×D₁₀₉	—	4	2	no	—
113	C₂²×D₁₀₉	—	4	2	no	—
114	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
115	Q₈×C₁₀₉	—	2	1	no	—
116	Q₈×C₁₀₉	—	2	1	no	—
117	D₄×C₁₀₉	—	2	1	no	—
118	D₄×C₁₀₉	#41	1	2	yes	—
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	—	4	2	no	—
123	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
126	C₂²⋊D₁₀₉	—	2	4	no	—
127	C₂×Dic₁₀₉	—	2	1	no	—
128	C₂×Dic₁₀₉	—	2	1	no	—
129	C₂×Dic₁₀₉	—	2	1	no	—
130	C₂×Dic₁₀₉	—	4	2	no	—
131	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
132	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
133	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
134	D₄₃₆	—	2	1	no	—
135	D₄₃₆	#41	1	2	yes	—
136	C₄×D₁₀₉	—	2	2	no	—
137	C₄×D₁₀₉	—	2	1	no	—
138	C₄×D₁₀₉	—	2	1	no	—
139	C₄×D₁₀₉	—	2	2	no	—
140	C₄×D₁₀₉	—	4	2	no	—
141	C₄×D₁₀₉	—	4	2	no	—
142	C₄×D₁₀₉	—	4	2	no	—
143	Dic₂₁₈	—	2	1	no	—
144	Dic₂₁₈	—	2	1	no	—
145	Dic₂₁₈	—	4	2	no	—
146	C₈×C₁₀₉	—	2	1	no	—
147	C₈×C₁₀₉	—	2	1	no	—
148	#11724784	—	2	1	no	—
149	#11724784	—	2	1	no	—
150	C₂²×C₁₀₉	—	2	2	no	—
151	C₂²×C₁₀₉	—	4	1	no	—
152	C₂²×C₁₀₉	—	4	1	no	—
153	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
154	C₂×D₁₀₉	—	2	1	no	—
155	C₂×D₁₀₉	—	2	1	no	—
156	C₂×D₁₀₉	—	2	2	no	—
157	C₂×D₁₀₉	—	4	8	no	—
158	C₂×D₁₀₉	—	4	1	no	—
159	C₂×D₁₀₉	—	4	1	no	—
160	C₂×D₁₀₉	#137	1	2	yes	—
161	C₄×C₁₀₉	—	2	1	no	—
162	C₄×C₁₀₉	—	2	1	no	—
163	C₄×C₁₀₉	—	4	1	no	—
164	C₄×C₁₀₉	—	4	1	no	—
165	C₄×C₁₀₉	—	4	1	no	—
166	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
167	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
168	Dic₁₀₉	—	2	1	no	—
169	Dic₁₀₉	—	2	1	no	—
170	Dic₁₀₉	—	4	1	no	—
171	Dic₁₀₉	—	4	1	no	—
172	Dic₁₀₉	—	4	1	no	—
173	Dic₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
174	C₂×C₁₀₉	—	4	2	no	—
175	C₂×C₁₀₉	#525	1	1	yes	characteristic
176	C₂×C₁₀₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
177	C₂×C₁₀₉	C₂×#137	1	1	yes	characteristic
178	D₁₀₉	—	2	2	no	—
179	D₁₀₉	—	4	2	no	—
180	#3110	—	109	1	no	maximal, Sylow
181	C₁₀₉	#3110	1	1	yes	characteristic, complemented, Sylow
182	#533	—	109	1	no	—
183	C₂×C₄×D₄	—	109	1	no	—
184	#484	—	109	1	no	—
185	#484	—	109	1	no	—
186	#474	—	109	1	no	—
187	#474	—	109	1	no	—
188	#461	—	109	1	no	—
189	#461	—	109	1	no	—
190	#450	—	109	1	no	—
191	#450	—	109	1	no	—
192	#440	—	109	1	no	—
193	#440	—	109	1	no	—
194	#440	—	109	1	no	—
195	#440	D₁₀₉	1	1	yes	characteristic, complemented
196	#432	—	109	1	no	—
197	C₂²×D₄	—	109	1	no	—
198	C₂³×C₄	—	218	1	no	—
199	C₂×SD₁₆	—	109	1	no	—
200	C₂×SD₁₆	—	109	1	no	—
201	C₂×SD₁₆	—	109	1	no	—
202	C₂×SD₁₆	C₂×D₁₀₉	1	1	yes	characteristic
203	C₄⋊Q₈	—	109	1	no	—
204	C₄⋊Q₈	—	109	1	no	—
205	#125	—	218	1	no	—
206	C₂²⋊Q₈	—	109	1	no	—
207	C₂²⋊Q₈	—	109	1	no	—
208	C₂²⋊Q₈	—	218	1	no	—
209	C₄×Q₈	—	109	1	no	—
210	C₄×Q₈	—	218	1	no	—
211	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic, complemented
212	C₄×D₄	—	109	1	no	—
213	C₄×D₄	—	109	1	no	—
214	C₄×D₄	—	109	1	no	—
215	C₄×D₄	—	218	1	no	—
216	C₄×D₄	—	218	1	no	—
217	C₄×D₄	C₂×D₁₀₉	1	1	yes	characteristic
218	#117	—	218	1	no	—
219	C₂×C₄⋊C₄	—	109	1	no	—
220	C₂×C₂²⋊C₄	—	218	1	no	—
221	C₂×C₄²	—	109	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	C₄⋊C₈	—	109	1	no	—
227	C₄⋊C₈	—	109	1	no	—
228	#103	—	109	1	no	—
229	#103	—	109	1	no	—
230	#103	—	109	1	no	—
231	#103	C₂×D₁₀₉	1	1	yes	characteristic, complemented
232	#102	—	109	1	no	—
233	#102	—	109	1	no	—
234	#102	—	109	1	no	—
235	#102	C₂×D₁₀₉	1	1	yes	characteristic
236	C₂²⋊C₈	—	109	1	no	—
237	C₂²⋊C₈	—	109	1	no	—
238	#97	—	109	1	no	—
239	#97	C₂×D₁₀₉	1	1	yes	characteristic, complemented
240	C₂⁴	—	218	1	no	—
241	C₂×Q₈	—	109	1	no	—
242	C₂×Q₈	—	218	1	no	—
243	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
244	C₂×D₄	—	109	1	no	—
245	C₂×D₄	—	218	1	no	—
246	C₂×D₄	—	218	2	no	—
247	C₂×D₄	—	218	2	no	—
248	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	—	109	1	no	—
251	C₂²×C₄	—	109	1	no	—
252	C₂²×C₄	—	109	1	no	—
253	C₂²×C₄	—	436	4	no	—
254	C₂²×C₄	—	218	1	no	—
255	C₂²×C₄	—	218	1	no	—
256	C₂²×C₄	—	218	1	no	—
257	C₂²×C₄	—	218	1	no	—
258	SD₁₆	—	2	2	no	—
259	SD₁₆	—	218	2	no	—
260	SD₁₆	—	218	2	no	—
261	SD₁₆	—	218	2	no	—
262	C₂×C₈	—	109	1	no	—
263	C₂×C₈	—	109	1	no	—
264	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
265	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
266	C₄⋊C₄	—	2	1	no	—
267	C₄⋊C₄	—	109	1	no	—
268	C₄⋊C₄	—	109	1	no	—
269	C₄⋊C₄	—	218	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	—	218	1	no	—
275	C₄⋊C₄	—	218	1	no	—
276	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
277	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
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	C₄²	—	2	1	no	—
285	C₄²	—	109	1	no	—
286	C₄²	—	109	1	no	—
287	C₄²	—	109	1	no	—
288	C₄²	—	218	1	no	—
289	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
290	C₂³	—	2	1	no	—
291	C₂³	—	109	1	no	—
292	C₂³	—	436	2	no	—
293	C₂³	—	436	2	no	—
294	C₂³	—	218	4	no	—
295	C₂³	—	218	1	no	—
296	Q₈	—	2	1	no	—
297	Q₈	—	2	1	no	—
298	Q₈	—	436	2	no	—
299	Q₈	—	218	1	no	—
300	Q₈	—	218	1	no	—
301	D₄	—	2	1	no	—
302	D₄	—	109	2	no	—
303	D₄	—	218	4	no	—
304	D₄	—	218	1	no	—
305	D₄	#13182029	1	2	yes	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	109	1	no	—
311	C₂×C₄	—	109	1	no	—
312	C₂×C₄	—	109	1	no	—
313	C₂×C₄	—	436	2	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	1	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	D₄×D₁₀₉	1	1	yes	characteristic
325	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
326	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
327	C₈	—	2	1	no	—
328	C₈	—	2	1	no	—
329	C₈	—	218	1	no	—
330	C₈	—	218	1	no	—
331	C₂²	—	2	2	no	—
332	C₂²	—	4	1	no	—
333	C₂²	—	4	1	no	—
334	C₂²	—	109	2	no	—
335	C₂²	—	436	8	no	—
336	C₂²	—	436	1	no	—
337	C₂²	—	436	1	no	—
338	C₂²	—	218	2	no	—
339	C₂²	—	218	1	no	—
340	C₂²	—	218	1	no	—
341	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	4	1	no	—
345	C₄	—	4	1	no	—
346	C₄	—	4	1	no	—
347	C₄	—	109	2	no	—
348	C₄	—	436	1	no	—
349	C₄	—	436	1	no	—
350	C₄	—	436	1	no	—
351	C₄	—	218	1	no	—
352	C₄	—	218	1	no	—
353	C₄	#41×D₁₀₉	1	1	yes	characteristic
354	C₄	C₂×#13182029	1	1	yes	characteristic
355	C₂	—	4	2	no	—
356	C₂	—	436	2	no	—
357	C₂	—	218	2	no	—
358	C₂	#20295535	1	1	yes	characteristic
359	C₂	AGL(1,ℤ/8)×D₁₀₉	1	1	yes	characteristic
360	C₂	#137×D₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#440×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂×D₄×D₁₀₉	C₂²
hypercenter	#440	D₁₀₉
hypocenter	C₁₀₉	#3110
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #440

Complements

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

N	H ≅ G/N	type
1. #20295580	C₂	semidirect
3. #20295363	C₂	semidirect
4. #20295360	C₂	semidirect
5. #20295305	C₂	semidirect
6. #20295304	C₂	semidirect
7. #20295230	C₂	semidirect
8. #20295229	C₂	semidirect
11. #20295128	C₂	semidirect
12. #20295126	C₂	semidirect
13. #20295124	C₂	semidirect
14. #20295081	C₂	semidirect
15. #440×C₁₀₉	C₂	semidirect
21. #16812698	C₂²	semidirect
22. #16812697	C₂²	semidirect
28. C₄×Dic₂₁₈	C₂²	semidirect
38. #16812633	C₂²	semidirect
39. #16812632	C₂²	semidirect
40. #16812632	C₂²	semidirect
41. #16812628	C₂²	semidirect
42. #16812628	C₂²	semidirect
43. #16812624	C₂²	semidirect
44. #16812623	C₂²	semidirect
45. #16812622	C₂²	semidirect
51. #16812610	C₂²	semidirect
53. C₄×Q₈×C₁₀₉	C₂²	semidirect
55. #107×C₁₀₉	C₂²	semidirect
56. #103×C₁₀₉	C₂²	semidirect
58. #97×C₁₀₉	C₂²	semidirect
181. C₁₀₉	#3110	semidirect
195. #440	D₁₀₉	semidirect
211. C₄×Q₈	C₂×D₁₀₉	semidirect
225. #107	C₂×D₁₀₉	semidirect
231. #103	C₂×D₁₀₉	semidirect
239. #97	C₂×D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295580	2
C₄×D₄×D₁₀₉	2
#20295363	2
#20295360	2
#20295305	2
#20295304	2
#20295230	2
#20295229	2
#20295178	2
#20295175	2
#20295128	2
#20295126	2
#20295124	2
#20295081	2
#440×C₁₀₉	2
#3110	109