Subgroups — #28373174

Order 13952.

450 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295563	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×D₄×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295355	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295341	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295329	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295328	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295244	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295243	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295175	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295161	C₂	1	1	yes	maximal, characteristic
11	#20295139	C₂	1	1	yes	maximal, characteristic, complemented
12	#20295136	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×#13182016	C₂²	1	1	yes	characteristic
19	C₂×D₈₇₂	C₂²	1	1	yes	characteristic
20	C₂×#13182014	C₂²	1	1	yes	characteristic
21	#16812694	—	2	1	no	—
22	#16812690	C₂²	1	1	yes	characteristic, complemented
23	#16812688	C₂²	1	1	yes	characteristic, complemented
24	#16812684	—	2	1	no	—
25	#16812675	C₂²	1	1	yes	characteristic
26	#16812674	C₂²	1	1	yes	characteristic
27	C₄×D₄₃₆	—	2	1	no	—
28	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
29	C₄×D₄₃₆	C₂²	1	1	yes	characteristic, complemented
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	#16812632	C₂²	1	1	yes	characteristic, complemented
39	#16812631	C₂²	1	1	yes	characteristic, complemented
40	#16812630	C₂²	1	1	yes	characteristic, complemented
41	#16812628	C₂²	1	1	yes	characteristic, complemented
42	#16812628	C₂²	1	1	yes	characteristic, complemented
43	#16812623	C₂²	1	1	yes	characteristic, complemented
44	#16812621	C₂²	1	1	yes	characteristic
45	#16812621	C₂²	1	1	yes	characteristic, complemented
46	#16812620	C₂²	1	1	yes	characteristic
47	#16812619	C₂²	1	1	yes	characteristic
48	#16812619	C₂²	1	1	yes	characteristic, complemented
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×D₄₃₆	—	2	1	no	—
61	C₂×D₄₃₆	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	—	2	1	no	—
71	C₂×C₄×D₁₀₉	—	4	4	no	—
72	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
73	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
74	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
75	#13182016	—	2	2	no	—
76	D₈₇₂	—	2	2	no	—
77	#13182014	—	2	2	no	—
78	C₂×#11724784	C₂³	1	1	yes	characteristic
79	C₂×#11724784	C₂³	1	1	yes	characteristic
80	C₄⋊Dic₁₀₉	—	2	1	no	—
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	#13182006	—	2	1	no	—
88	#13182006	—	2	1	no	—
89	#13182006	—	2	1	no	—
90	#13182006	—	2	1	no	—
91	#13182006	—	2	1	no	—
92	#13182006	—	2	1	no	—
93	#13182006	—	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	—	2	1	no	—
113	C₂²×D₁₀₉	—	4	2	no	—
114	C₂²×D₁₀₉	—	4	2	no	—
115	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
116	Q₈×C₁₀₉	—	2	1	no	—
117	Q₈×C₁₀₉	—	2	1	no	—
118	D₄×C₁₀₉	—	2	1	no	—
119	D₄×C₁₀₉	#41	1	2	yes	—
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	—	4	2	no	—
124	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
127	C₂²⋊D₁₀₉	—	2	4	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	—	2	1	no	—
136	D₄₃₆	—	2	1	no	—
137	D₄₃₆	—	4	2	no	—
138	D₄₃₆	#41	1	2	yes	—
139	C₄×D₁₀₉	—	2	2	no	—
140	C₄×D₁₀₉	—	2	1	no	—
141	C₄×D₁₀₉	—	2	1	no	—
142	C₄×D₁₀₉	—	4	1	no	—
143	C₄×D₁₀₉	—	4	2	no	—
144	C₄×D₁₀₉	—	4	2	no	—
145	C₄×D₁₀₉	—	4	2	no	—
146	C₄×D₁₀₉	—	4	2	no	—
147	C₈×C₁₀₉	—	2	1	no	—
148	C₈×C₁₀₉	—	2	1	no	—
149	#11724784	—	2	1	no	—
150	#11724784	—	2	1	no	—
151	C₂²×C₁₀₉	—	2	2	no	—
152	C₂²×C₁₀₉	—	4	1	no	—
153	C₂²×C₁₀₉	—	4	1	no	—
154	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
155	C₂×D₁₀₉	—	2	1	no	—
156	C₂×D₁₀₉	—	2	1	no	—
157	C₂×D₁₀₉	—	2	2	no	—
158	C₂×D₁₀₉	—	4	8	no	—
159	C₂×D₁₀₉	—	4	1	no	—
160	C₂×D₁₀₉	—	4	1	no	—
161	C₂×D₁₀₉	—	4	1	no	—
162	C₂×D₁₀₉	—	4	1	no	—
163	C₂×D₁₀₉	—	4	1	no	—
164	C₂×D₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
165	C₄×C₁₀₉	—	2	1	no	—
166	C₄×C₁₀₉	—	2	1	no	—
167	C₄×C₁₀₉	—	4	1	no	—
168	C₄×C₁₀₉	—	4	1	no	—
169	C₄×C₁₀₉	—	4	1	no	—
170	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
171	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
172	Dic₁₀₉	—	2	1	no	—
173	Dic₁₀₉	—	2	1	no	—
174	Dic₁₀₉	—	2	1	no	—
175	Dic₁₀₉	—	4	1	no	—
176	Dic₁₀₉	—	4	1	no	—
177	C₂×C₁₀₉	—	4	2	no	—
178	C₂×C₁₀₉	#525	1	1	yes	characteristic
179	C₂×C₁₀₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
180	C₂×C₁₀₉	#575	1	1	yes	characteristic
181	D₁₀₉	—	2	2	no	—
182	D₁₀₉	—	4	2	no	—
183	D₁₀₉	—	8	1	no	—
184	#3114	—	109	1	no	maximal, Sylow
185	C₁₀₉	#3114	1	1	yes	characteristic, complemented, Sylow
186	#529	—	109	1	no	—
187	C₂×C₄×D₄	—	109	1	no	—
188	#483	—	109	1	no	—
189	#480	—	109	1	no	—
190	#478	—	109	1	no	—
191	#478	—	109	1	no	—
192	#463	—	109	1	no	—
193	#463	—	109	1	no	—
194	#450	—	109	1	no	—
195	#447	—	109	1	no	—
196	#442	—	109	1	no	—
197	#442	—	109	1	no	—
198	#440	—	109	1	no	—
199	#440	D₁₀₉	1	1	yes	characteristic, complemented
200	#432	—	109	1	no	—
201	C₂²×D₄	—	109	1	no	—
202	C₂³×C₄	—	218	1	no	—
203	C₂×SD₁₆	—	109	1	no	—
204	C₂×SD₁₆	C₂×D₁₀₉	1	1	yes	characteristic
205	C₂×D₈	—	109	1	no	—
206	C₂×D₈	—	109	1	no	—
207	#126	—	218	1	no	—
208	#125	—	109	1	no	—
209	#124	—	109	1	no	—
210	#123	—	218	1	no	—
211	C₂²⋊Q₈	—	109	1	no	—
212	#121	—	109	1	no	—
213	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic, complemented
214	C₄×D₄	—	109	1	no	—
215	C₄×D₄	—	109	1	no	—
216	C₄×D₄	—	109	1	no	—
217	C₄×D₄	—	109	1	no	—
218	C₄×D₄	—	218	1	no	—
219	C₄×D₄	—	218	1	no	—
220	C₄×D₄	—	218	1	no	—
221	C₄×D₄	C₂×D₁₀₉	1	1	yes	characteristic
222	#117	—	218	1	no	—
223	C₂×C₄⋊C₄	—	109	1	no	—
224	C₂×C₂²⋊C₄	—	218	1	no	—
225	C₂×C₄²	—	109	1	no	—
226	#107	—	109	1	no	—
227	#107	C₂×D₁₀₉	1	1	yes	characteristic, complemented
228	#106	—	109	1	no	—
229	#106	—	109	1	no	—
230	C₄⋊C₈	—	109	1	no	—
231	C₄⋊C₈	—	109	1	no	—
232	#103	—	109	1	no	—
233	#103	C₂×D₁₀₉	1	1	yes	characteristic, complemented
234	#102	—	109	1	no	—
235	#102	—	109	1	no	—
236	#102	—	109	1	no	—
237	#102	—	109	1	no	—
238	#102	—	109	1	no	—
239	#102	C₂×D₁₀₉	1	1	yes	characteristic
240	C₂²⋊C₈	—	109	1	no	—
241	C₂²⋊C₈	—	109	1	no	—
242	#97	—	109	1	no	—
243	#97	C₂×D₁₀₉	1	1	yes	characteristic, complemented
244	C₂⁴	—	218	1	no	—
245	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
246	C₂×D₄	—	109	1	no	—
247	C₂×D₄	—	109	1	no	—
248	C₂×D₄	—	218	1	no	—
249	C₂×D₄	—	218	1	no	—
250	C₂×D₄	—	218	2	no	—
251	C₂×D₄	—	218	2	no	—
252	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
253	C₂²×C₄	—	2	1	no	—
254	C₂²×C₄	—	109	1	no	—
255	C₂²×C₄	—	109	1	no	—
256	C₂²×C₄	—	109	1	no	—
257	C₂²×C₄	—	436	4	no	—
258	C₂²×C₄	—	218	1	no	—
259	C₂²×C₄	—	218	1	no	—
260	C₂²×C₄	—	218	1	no	—
261	C₂²×C₄	—	218	1	no	—
262	C₂²×C₄	—	218	1	no	—
263	SD₁₆	—	2	2	no	—
264	SD₁₆	—	218	2	no	—
265	D₈	—	218	2	no	—
266	D₈	—	218	2	no	—
267	C₂×C₈	—	109	1	no	—
268	C₂×C₈	—	109	1	no	—
269	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
270	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	109	1	no	—
273	C₄⋊C₄	—	109	1	no	—
274	C₄⋊C₄	—	218	1	no	—
275	C₄⋊C₄	—	218	1	no	—
276	C₄⋊C₄	—	218	1	no	—
277	C₄⋊C₄	—	218	1	no	—
278	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
279	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
280	C₂²⋊C₄	—	2	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	C₂²⋊C₄	—	218	1	no	—
287	C₂²⋊C₄	—	218	1	no	—
288	C₂²⋊C₄	—	218	1	no	—
289	C₄²	—	2	1	no	—
290	C₄²	—	109	1	no	—
291	C₄²	—	109	1	no	—
292	C₄²	—	109	1	no	—
293	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
294	C₂³	—	2	1	no	—
295	C₂³	—	109	1	no	—
296	C₂³	—	436	2	no	—
297	C₂³	—	436	2	no	—
298	C₂³	—	218	4	no	—
299	C₂³	—	218	1	no	—
300	C₂³	—	218	1	no	—
301	Q₈	—	2	1	no	—
302	Q₈	—	2	1	no	—
303	D₄	—	2	1	no	—
304	D₄	—	109	2	no	—
305	D₄	—	436	2	no	—
306	D₄	—	218	4	no	—
307	D₄	—	218	1	no	—
308	D₄	—	218	1	no	—
309	D₄	—	218	1	no	—
310	D₄	#13182029	1	2	yes	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	109	1	no	—
316	C₂×C₄	—	109	1	no	—
317	C₂×C₄	—	109	1	no	—
318	C₂×C₄	—	436	2	no	—
319	C₂×C₄	—	436	2	no	—
320	C₂×C₄	—	436	1	no	—
321	C₂×C₄	—	436	2	no	—
322	C₂×C₄	—	436	2	no	—
323	C₂×C₄	—	436	2	no	—
324	C₂×C₄	—	218	2	no	—
325	C₂×C₄	—	218	1	no	—
326	C₂×C₄	—	218	1	no	—
327	C₂×C₄	—	218	1	no	—
328	C₂×C₄	—	218	1	no	—
329	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
330	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
331	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
332	C₈	—	2	1	no	—
333	C₈	—	2	1	no	—
334	C₈	—	218	1	no	—
335	C₈	—	218	1	no	—
336	C₂²	—	2	2	no	—
337	C₂²	—	4	1	no	—
338	C₂²	—	4	1	no	—
339	C₂²	—	109	2	no	—
340	C₂²	—	436	8	no	—
341	C₂²	—	436	1	no	—
342	C₂²	—	436	1	no	—
343	C₂²	—	436	1	no	—
344	C₂²	—	436	1	no	—
345	C₂²	—	436	1	no	—
346	C₂²	—	218	2	no	—
347	C₂²	—	218	1	no	—
348	C₂²	—	218	1	no	—
349	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	4	1	no	—
353	C₄	—	4	1	no	—
354	C₄	—	4	1	no	—
355	C₄	—	436	1	no	—
356	C₄	—	436	1	no	—
357	C₄	—	218	1	no	—
358	C₄	—	218	1	no	—
359	C₄	—	218	1	no	—
360	C₄	#41×D₁₀₉	1	1	yes	characteristic
361	C₄	C₂×#13182029	1	1	yes	characteristic
362	C₂	—	4	2	no	—
363	C₂	—	872	1	no	—
364	C₂	—	436	2	no	—
365	C₂	—	218	2	no	—
366	C₂	#20295535	1	1	yes	characteristic
367	C₂	AGL(1,ℤ/8)×D₁₀₉	1	1	yes	characteristic
368	C₂	#20295825	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	#20295161	C₂
hypercenter	#440	D₁₀₉
hypocenter	C₁₀₉	#3114
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. #20295563	C₂	semidirect
2. C₄×D₄×D₁₀₉	C₂	semidirect
3. #20295355	C₂	semidirect
4. #20295341	C₂	semidirect
5. #20295329	C₂	semidirect
6. #20295328	C₂	semidirect
7. #20295244	C₂	semidirect
8. #20295243	C₂	semidirect
9. #20295175	C₂	semidirect
11. #20295139	C₂	semidirect
12. #20295136	C₂	semidirect
13. #20295124	C₂	semidirect
14. #20295081	C₂	semidirect
15. #440×C₁₀₉	C₂	semidirect
22. #16812690	C₂²	semidirect
23. #16812688	C₂²	semidirect
29. C₄×D₄₃₆	C₂²	semidirect
38. #16812632	C₂²	semidirect
39. #16812631	C₂²	semidirect
40. #16812630	C₂²	semidirect
41. #16812628	C₂²	semidirect
42. #16812628	C₂²	semidirect
43. #16812623	C₂²	semidirect
45. #16812621	C₂²	semidirect
48. #16812619	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
185. C₁₀₉	#3114	semidirect
199. #440	D₁₀₉	semidirect
213. C₄×Q₈	C₂×D₁₀₉	semidirect
227. #107	C₂×D₁₀₉	semidirect
233. #103	C₂×D₁₀₉	semidirect
243. #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
#20295563	2
C₄×D₄×D₁₀₉	2
#20295355	2
#20295341	2
#20295329	2
#20295328	2
#20295244	2
#20295243	2
#20295175	2
#20295161	2
#20295139	2
#20295136	2
#20295124	2
#20295081	2
#440×C₁₀₉	2
#3114	109