Subgroups — #28376117

Order 13952.

380 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295762	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295758	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295414	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295407	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295246	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295245	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295243	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295239	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295219	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295218	C₂	1	1	yes	maximal, characteristic, complemented
11	#20295154	C₂	1	1	yes	maximal, characteristic, complemented
12	#20295128	C₂	1	1	yes	maximal, characteristic, complemented
13	#20295127	C₂	1	1	yes	maximal, characteristic, complemented
14	D₈×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
15	#496×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13182019	C₂²	1	1	yes	characteristic
17	C₂×#13182018	C₂²	1	1	yes	characteristic
18	C₂×#13182017	C₂²	1	1	yes	characteristic
19	C₂×#13182014	C₂²	1	1	yes	characteristic
20	C₂×#13182014	C₂²	1	1	yes	characteristic
21	#16812695	C₂²	1	1	yes	characteristic
22	#16812688	C₂²	1	1	yes	characteristic
23	#16812686	—	2	1	no	—
24	#16812686	C₂²	1	1	yes	characteristic
25	#16812686	C₂²	1	1	yes	characteristic
26	#16812685	—	2	1	no	—
27	#16812683	—	2	1	no	—
28	#16812681	—	2	1	no	—
29	#16812680	—	2	1	no	—
30	#16812680	—	2	1	no	—
31	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
32	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
33	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
34	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
35	D₄×Dic₁₀₉	—	2	1	no	—
36	D₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
37	D₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
38	D₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
39	#16812659	—	2	1	no	—
40	#16812633	C₂²	1	1	yes	characteristic
41	C₄⋊#11724784	C₂²	1	1	yes	characteristic
42	#16812628	C₂²	1	1	yes	characteristic
43	#16812628	C₂²	1	1	yes	characteristic
44	#16812624	C₂²	1	1	yes	characteristic
45	#16812623	C₂²	1	1	yes	characteristic
46	#16812621	C₂²	1	1	yes	characteristic
47	#16812620	C₂²	1	1	yes	characteristic
48	#16812620	C₂²	1	1	yes	characteristic
49	#16812620	C₂²	1	1	yes	characteristic
50	#16812609	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
53	C₂×SD₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
54	C₂×D₈×C₁₀₉	C₂²	1	1	yes	characteristic
55	C₂×D₈×C₁₀₉	C₂²	1	1	yes	characteristic
56	#127×C₁₀₉	C₂²	1	1	yes	characteristic
57	#124×C₁₀₉	C₂²	1	1	yes	characteristic
58	#97×C₁₀₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₁₈	D₄	1	1	yes	characteristic
60	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
61	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
63	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
64	C₂²×Dic₁₀₉	—	2	1	no	—
65	C₂²×Dic₁₀₉	—	2	1	no	—
66	C₂²×Dic₁₀₉	—	2	1	no	—
67	C₂×C₄×D₁₀₉	—	2	1	no	—
68	#13182019	—	4	1	no	—
69	#13182018	—	2	2	no	—
70	#13182017	—	4	1	no	—
71	#13182014	—	4	1	no	—
72	#13182014	—	4	1	no	—
73	C₂×#11724784	C₂³	1	1	yes	characteristic
74	C₂×#11724784	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₁₀₉	—	2	1	no	—
76	C₄⋊Dic₁₀₉	D₄	1	1	yes	characteristic
77	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
79	#13182008	—	2	1	no	—
80	#13182008	D₄	1	1	yes	characteristic
81	#13182008	D₄	1	1	yes	characteristic
82	C₂²⋊Dic₁₀₉	—	2	1	no	—
83	C₂²⋊Dic₁₀₉	—	2	1	no	—
84	C₂²⋊Dic₁₀₉	—	2	1	no	—
85	C₂²⋊Dic₁₀₉	—	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	C₄×Dic₁₀₉	—	2	1	no	—
93	C₄×Dic₁₀₉	—	2	1	no	—
94	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
95	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
96	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
97	C₂×D₄×C₁₀₉	—	2	1	no	—
98	C₂×D₄×C₁₀₉	—	2	1	no	—
99	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
100	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
101	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
102	SD₁₆×C₁₀₉	—	2	2	no	—
103	SD₁₆×C₁₀₉	—	4	1	no	—
104	D₈×C₁₀₉	—	2	2	no	—
105	D₈×C₁₀₉	—	4	1	no	—
106	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
107	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
108	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
109	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
110	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
111	C₂³×C₁₀₉	—	2	1	no	—
112	C₂³×C₁₀₉	—	2	1	no	—
113	C₂³×C₁₀₉	—	2	1	no	—
114	C₂²×D₁₀₉	—	2	1	no	—
115	Q₈×C₁₀₉	—	2	1	no	—
116	Q₈×C₁₀₉	—	2	1	no	—
117	D₄×C₁₀₉	—	2	1	no	—
118	D₄×C₁₀₉	—	2	1	no	—
119	D₄×C₁₀₉	—	2	1	no	—
120	D₄×C₁₀₉	—	2	1	no	—
121	D₄×C₁₀₉	—	2	1	no	—
122	D₄×C₁₀₉	—	2	1	no	—
123	D₄×C₁₀₉	—	4	2	no	—
124	D₄×C₁₀₉	—	8	1	no	—
125	C₂×C₄×C₁₀₉	—	2	1	no	—
126	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
129	C₂²⋊D₁₀₉	—	4	2	no	—
130	C₂²⋊D₁₀₉	—	8	1	no	—
131	C₂×Dic₁₀₉	—	2	1	no	—
132	C₂×Dic₁₀₉	—	2	1	no	—
133	C₂×Dic₁₀₉	—	2	1	no	—
134	C₂×Dic₁₀₉	—	8	1	no	—
135	C₂×Dic₁₀₉	—	8	1	no	—
136	C₂×Dic₁₀₉	—	8	1	no	—
137	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
138	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
139	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
140	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
141	D₄₃₆	—	2	1	no	—
142	D₄₃₆	—	2	1	no	—
143	C₄×D₁₀₉	—	8	1	no	—
144	Dic₂₁₈	—	2	1	no	—
145	Dic₂₁₈	—	2	1	no	—
146	Dic₂₁₈	—	4	1	no	—
147	C₈×C₁₀₉	—	2	1	no	—
148	C₈×C₁₀₉	#41	1	2	yes	—
149	#11724784	—	2	1	no	—
150	#11724784	—	2	1	no	—
151	C₂²×C₁₀₉	—	4	1	no	—
152	C₂²×C₁₀₉	—	4	1	no	—
153	C₂²×C₁₀₉	—	4	1	no	—
154	C₂²×C₁₀₉	—	4	1	no	—
155	C₂²×C₁₀₉	—	4	1	no	—
156	C₂²×C₁₀₉	—	4	1	no	—
157	C₂²×C₁₀₉	—	4	1	no	—
158	C₂²×C₁₀₉	—	4	1	no	—
159	C₂²×C₁₀₉	—	4	1	no	—
160	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
161	C₂×D₁₀₉	—	4	1	no	—
162	C₂×D₁₀₉	—	4	1	no	—
163	C₂×D₁₀₉	—	4	1	no	—
164	C₄×C₁₀₉	—	2	1	no	—
165	C₄×C₁₀₉	—	2	1	no	—
166	C₄×C₁₀₉	—	4	1	no	—
167	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
168	C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
169	Dic₁₀₉	—	2	1	no	—
170	Dic₁₀₉	—	2	1	no	—
171	Dic₁₀₉	—	2	1	no	—
172	Dic₁₀₉	—	2	1	no	—
173	Dic₁₀₉	—	4	1	no	—
174	Dic₁₀₉	—	4	1	no	—
175	Dic₁₀₉	—	4	1	no	—
176	C₂×C₁₀₉	—	8	1	no	—
177	C₂×C₁₀₉	—	8	1	no	—
178	C₂×C₁₀₉	—	8	1	no	—
179	C₂×C₁₀₉	#550	1	1	yes	characteristic
180	C₂×C₁₀₉	#575	1	1	yes	characteristic
181	C₂×C₁₀₉	#576	1	1	yes	characteristic
182	D₁₀₉	—	8	1	no	—
183	#3403	—	109	1	no	maximal, Sylow
184	C₁₀₉	#3403	1	1	yes	characteristic, complemented, Sylow
185	#559	—	109	1	no	—
186	#559	—	109	1	no	—
187	#496	—	109	1	no	—
188	#496	D₁₀₉	1	1	yes	characteristic, complemented
189	#495	—	109	1	no	—
190	#464	—	109	1	no	—
191	#464	—	109	1	no	—
192	#463	—	109	1	no	—
193	#463	—	109	1	no	—
194	#459	—	109	1	no	—
195	#459	—	109	1	no	—
196	#446	—	109	1	no	—
197	#440	—	109	1	no	—
198	#440	—	109	1	no	—
199	C₄×D₈	—	109	1	no	—
200	C₂×Q₁₆	—	109	1	no	—
201	C₂×SD₁₆	—	109	1	no	—
202	C₂×SD₁₆	—	109	1	no	—
203	C₂×SD₁₆	C₂×D₁₀₉	1	1	yes	characteristic
204	C₂×SD₁₆	C₂×D₁₀₉	1	1	yes	characteristic
205	C₂×D₈	—	109	1	no	—
206	C₂×D₈	—	109	1	no	—
207	C₂×D₈	C₂×D₁₀₉	1	1	yes	characteristic
208	C₂×D₈	C₂×D₁₀₉	1	1	yes	characteristic
209	#127	—	109	1	no	—
210	#127	C₂×D₁₀₉	1	1	yes	characteristic
211	#124	—	109	1	no	—
212	#124	—	109	1	no	—
213	#124	—	109	1	no	—
214	#124	—	218	1	no	—
215	#124	—	218	1	no	—
216	#124	C₂×D₁₀₉	1	1	yes	characteristic
217	#123	—	218	1	no	—
218	#123	—	218	1	no	—
219	#123	—	218	1	no	—
220	#123	—	218	1	no	—
221	C₄×Q₈	—	109	1	no	—
222	C₄×Q₈	—	109	1	no	—
223	C₄×Q₈	—	109	1	no	—
224	C₄×D₄	—	109	1	no	—
225	C₄×D₄	—	109	1	no	—
226	C₄×D₄	—	109	1	no	—
227	C₄×D₄	—	109	1	no	—
228	C₄×D₄	—	218	1	no	—
229	C₄×D₄	—	218	1	no	—
230	#107	—	109	1	no	—
231	C₄⋊C₈	—	109	1	no	—
232	C₄⋊C₈	—	109	1	no	—
233	C₄⋊C₈	—	109	1	no	—
234	#103	—	109	1	no	—
235	#103	—	109	1	no	—
236	#102	—	109	1	no	—
237	#102	—	109	1	no	—
238	#102	—	109	1	no	—
239	#102	—	109	1	no	—
240	#97	—	109	1	no	—
241	#97	C₂×D₁₀₉	1	1	yes	characteristic
242	C₄×C₈	—	109	1	no	—
243	C₂×Q₈	—	109	1	no	—
244	C₂×Q₈	—	109	1	no	—
245	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
246	C₂×D₄	—	2	1	no	—
247	C₂×D₄	—	2	1	no	—
248	C₂×D₄	—	109	1	no	—
249	C₂×D₄	—	218	1	no	—
250	C₂×D₄	—	218	1	no	—
251	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
252	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
253	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
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	Q₁₆	—	436	1	no	—
259	SD₁₆	—	2	2	no	—
260	SD₁₆	—	4	1	no	—
261	SD₁₆	—	436	1	no	—
262	SD₁₆	—	218	2	no	—
263	D₈	—	2	2	no	—
264	D₈	—	4	1	no	—
265	D₈	—	436	1	no	—
266	D₈	—	436	1	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	—	109	1	no	—
272	C₄⋊C₄	—	109	1	no	—
273	C₄⋊C₄	—	109	1	no	—
274	C₄⋊C₄	—	109	1	no	—
275	C₄⋊C₄	—	109	1	no	—
276	C₄⋊C₄	—	218	1	no	—
277	C₄⋊C₄	—	218	1	no	—
278	C₂²⋊C₄	—	2	1	no	—
279	C₂²⋊C₄	—	2	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	C₂²⋊C₄	—	218	1	no	—
287	C₂²⋊C₄	—	218	1	no	—
288	C₂²⋊C₄	—	218	1	no	—
289	C₂²⋊C₄	—	218	1	no	—
290	C₄²	—	109	1	no	—
291	C₄²	—	109	1	no	—
292	C₄²	—	218	1	no	—
293	C₄²	—	218	1	no	—
294	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
295	C₂³	—	2	1	no	—
296	C₂³	—	2	1	no	—
297	C₂³	—	2	1	no	—
298	C₂³	—	218	1	no	—
299	Q₈	—	2	1	no	—
300	Q₈	—	2	1	no	—
301	Q₈	—	436	1	no	—
302	Q₈	—	218	1	no	—
303	Q₈	—	218	1	no	—
304	D₄	—	2	1	no	—
305	D₄	—	2	1	no	—
306	D₄	—	2	1	no	—
307	D₄	—	2	1	no	—
308	D₄	—	2	1	no	—
309	D₄	—	2	1	no	—
310	D₄	—	4	2	no	—
311	D₄	—	8	1	no	—
312	D₄	—	872	1	no	—
313	D₄	—	436	2	no	—
314	D₄	—	218	1	no	—
315	D₄	—	218	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	872	1	no	—
318	C₂×C₄	—	872	1	no	—
319	C₂×C₄	—	872	1	no	—
320	C₂×C₄	—	872	1	no	—
321	C₂×C₄	—	109	1	no	—
322	C₂×C₄	—	109	1	no	—
323	C₂×C₄	—	109	1	no	—
324	C₂×C₄	—	109	1	no	—
325	C₂×C₄	—	218	1	no	—
326	C₂×C₄	—	218	1	no	—
327	C₂×C₄	—	218	1	no	—
328	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
329	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
330	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
331	C₈	—	2	1	no	—
332	C₈	—	218	1	no	—
333	C₈	—	218	1	no	—
334	C₈	#13182028	1	2	yes	—
335	C₂²	—	4	1	no	—
336	C₂²	—	4	1	no	—
337	C₂²	—	4	1	no	—
338	C₂²	—	4	1	no	—
339	C₂²	—	4	1	no	—
340	C₂²	—	4	1	no	—
341	C₂²	—	4	1	no	—
342	C₂²	—	4	1	no	—
343	C₂²	—	4	1	no	—
344	C₂²	—	436	1	no	—
345	C₂²	—	436	1	no	—
346	C₂²	—	436	1	no	—
347	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	4	1	no	—
351	C₄	—	436	1	no	—
352	C₄	—	436	1	no	—
353	C₄	—	436	1	no	—
354	C₄	—	218	1	no	—
355	C₄	—	218	1	no	—
356	C₄	—	218	1	no	—
357	C₄	—	218	1	no	—
358	C₄	C₂×#13182028	1	1	yes	characteristic
359	C₄	#16812744	1	1	yes	characteristic
360	C₂	—	8	1	no	—
361	C₂	—	8	1	no	—
362	C₂	—	8	1	no	—
363	C₂	—	872	1	no	—
364	C₂	#20295696	1	1	yes	characteristic
365	C₂	#20295822	1	1	yes	characteristic
366	C₂	#20295830	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#496×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
hypercenter	#496	D₁₀₉
hypocenter	C₁₀₉	#3403
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #496

Complements

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

N	H ≅ G/N	type
1. #20295762	C₂	semidirect
2. #20295758	C₂	semidirect
3. #20295414	C₂	semidirect
4. #20295407	C₂	semidirect
5. #20295246	C₂	semidirect
6. #20295245	C₂	semidirect
7. #20295243	C₂	semidirect
8. #20295239	C₂	semidirect
9. #20295219	C₂	semidirect
10. #20295218	C₂	semidirect
11. #20295154	C₂	semidirect
12. #20295128	C₂	semidirect
13. #20295127	C₂	semidirect
14. D₈×Dic₁₀₉	C₂	semidirect
15. #496×C₁₀₉	C₂	semidirect
184. C₁₀₉	#3403	semidirect
188. #496	D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295762	2
#20295758	2
#20295414	2
#20295407	2
#20295246	2
#20295245	2
#20295243	2
#20295239	2
#20295219	2
#20295218	2
#20295154	2
#20295128	2
#20295127	2
D₈×Dic₁₀₉	2
#496×C₁₀₉	2
#3403	109