Subgroups — #28375633

Order 13952.

360 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295728	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295723	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295426	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295349	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295347	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295338	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295337	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295328	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295324	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295276	C₂	1	1	yes	maximal, characteristic, complemented
11	D₈×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#20295102	C₂	1	1	yes	maximal, characteristic, complemented
13	#20295068	C₂	1	1	yes	maximal, characteristic, complemented
14	#20295065	C₂	1	1	yes	maximal, characteristic, complemented
15	#469×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13182012	C₂²	1	1	yes	characteristic
17	#16812692	—	2	1	no	—
18	#16812692	—	2	1	no	—
19	#16812690	C₂²	1	1	yes	characteristic
20	#16812690	C₂²	1	1	yes	characteristic
21	#16812683	—	2	1	no	—
22	#16812680	—	2	1	no	—
23	#16812671	—	2	1	no	—
24	#16812670	—	2	1	no	—
25	#16812670	C₂²	1	1	yes	characteristic
26	#16812670	C₂²	1	1	yes	characteristic
27	D₄×Dic₁₀₉	—	2	1	no	—
28	D₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
29	D₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
30	C₄×C₂²⋊D₁₀₉	C₂²	1	1	yes	characteristic
31	#16812659	—	2	1	no	—
32	#16812658	C₂²	1	1	yes	characteristic
33	#16812657	C₂²	1	1	yes	characteristic
34	C₂×C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
35	C₄⋊C₄×D₁₀₉	C₂²	1	1	yes	characteristic
36	#16812633	C₂²	1	1	yes	characteristic
37	#16812633	C₂²	1	1	yes	characteristic
38	#16812632	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	#16812628	C₂²	1	1	yes	characteristic
45	#16812620	C₂²	1	1	yes	characteristic
46	#16812620	C₂²	1	1	yes	characteristic
47	#16812620	C₂²	1	1	yes	characteristic
48	#16812620	C₂²	1	1	yes	characteristic
49	C₂²⋊#11724784	C₂²	1	1	yes	characteristic
50	#16812611	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
52	C₂×D₈×C₁₀₉	C₂²	1	1	yes	characteristic
53	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
54	#121×C₁₀₉	C₂²	1	1	yes	characteristic
55	#121×C₁₀₉	C₂²	1	1	yes	characteristic
56	#106×C₁₀₉	C₂²	1	1	yes	characteristic
57	#102×C₁₀₉	C₂²	1	1	yes	characteristic
58	#102×C₁₀₉	C₂²	1	1	yes	characteristic
59	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
60	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
61	C₂×C₂²⋊D₁₀₉	D₄	1	1	yes	characteristic
62	C₂²×Dic₁₀₉	—	2	1	no	—
63	C₂²×Dic₁₀₉	—	2	1	no	—
64	C₂²×Dic₁₀₉	—	2	1	no	—
65	C₂×C₄×D₁₀₉	—	2	1	no	—
66	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
67	#13182012	—	2	2	no	—
68	C₂×#11724784	C₂³	1	1	yes	characteristic
69	C₂×#11724784	C₂³	1	1	yes	characteristic
70	C₄⋊Dic₁₀₉	—	2	1	no	—
71	C₄⋊Dic₁₀₉	—	2	1	no	—
72	C₄⋊Dic₁₀₉	—	2	1	no	—
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	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
77	#13182008	—	2	1	no	—
78	#13182008	—	2	1	no	—
79	#13182008	—	2	1	no	—
80	#13182008	D₄	1	1	yes	characteristic
81	C₂²⋊Dic₁₀₉	—	2	1	no	—
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	D₄	1	1	yes	characteristic
87	#13182006	—	2	1	no	—
88	#13182006	D₄	1	1	yes	characteristic
89	C₄×Dic₁₀₉	—	2	1	no	—
90	C₄×Dic₁₀₉	—	2	1	no	—
91	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
92	C₂×D₄×C₁₀₉	—	2	1	no	—
93	C₂×D₄×C₁₀₉	—	2	1	no	—
94	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
95	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
96	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
97	D₈×C₁₀₉	—	2	2	no	—
98	M₁₆×C₁₀₉	—	2	2	no	—
99	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
100	C₂×C₈×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	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
104	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
105	C₂³×C₁₀₉	—	2	1	no	—
106	C₂³×C₁₀₉	—	2	1	no	—
107	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
108	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
109	D₄×C₁₀₉	—	2	1	no	—
110	D₄×C₁₀₉	—	2	1	no	—
111	D₄×C₁₀₉	—	2	1	no	—
112	D₄×C₁₀₉	—	2	1	no	—
113	D₄×C₁₀₉	—	4	2	no	—
114	D₄×C₁₀₉	—	8	1	no	—
115	C₂×C₄×C₁₀₉	—	2	1	no	—
116	C₂×C₄×C₁₀₉	—	2	1	no	—
117	C₂×C₄×C₁₀₉	—	2	1	no	—
118	C₂×C₄×C₁₀₉	—	2	1	no	—
119	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
121	C₂²⋊D₁₀₉	—	4	1	no	—
122	C₂²⋊D₁₀₉	—	4	2	no	—
123	C₂²⋊D₁₀₉	—	8	1	no	—
124	C₂×Dic₁₀₉	—	2	1	no	—
125	C₂×Dic₁₀₉	—	2	1	no	—
126	C₂×Dic₁₀₉	—	2	1	no	—
127	C₂×Dic₁₀₉	—	2	1	no	—
128	C₂×Dic₁₀₉	—	8	1	no	—
129	C₂×Dic₁₀₉	—	8	1	no	—
130	C₂×Dic₁₀₉	—	8	1	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	C₄×D₁₀₉	—	2	1	no	—
135	C₄×D₁₀₉	—	2	1	no	—
136	C₄×D₁₀₉	—	8	1	no	—
137	C₈×C₁₀₉	—	2	1	no	—
138	C₈×C₁₀₉	#41	1	2	yes	—
139	#11724784	—	2	1	no	—
140	#11724784	—	2	1	no	—
141	C₂²×C₁₀₉	—	2	1	no	—
142	C₂²×C₁₀₉	—	2	1	no	—
143	C₂²×C₁₀₉	—	2	1	no	—
144	C₂²×C₁₀₉	—	4	1	no	—
145	C₂²×C₁₀₉	—	4	1	no	—
146	C₂²×C₁₀₉	—	4	1	no	—
147	C₂²×C₁₀₉	—	4	1	no	—
148	C₂²×C₁₀₉	—	4	1	no	—
149	C₂²×C₁₀₉	—	4	1	no	—
150	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
151	C₂×D₁₀₉	—	2	1	no	—
152	C₂×D₁₀₉	—	2	1	no	—
153	C₂×D₁₀₉	—	2	1	no	—
154	C₄×C₁₀₉	—	2	1	no	—
155	C₄×C₁₀₉	—	4	1	no	—
156	C₄×C₁₀₉	—	4	1	no	—
157	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
158	C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
159	Dic₁₀₉	—	2	1	no	—
160	Dic₁₀₉	—	2	1	no	—
161	Dic₁₀₉	—	2	1	no	—
162	Dic₁₀₉	—	4	1	no	—
163	Dic₁₀₉	—	4	1	no	—
164	Dic₁₀₉	—	4	1	no	—
165	Dic₁₀₉	—	4	1	no	—
166	C₂×C₁₀₉	—	4	1	no	—
167	C₂×C₁₀₉	—	8	1	no	—
168	C₂×C₁₀₉	—	8	1	no	—
169	C₂×C₁₀₉	#547	1	1	yes	characteristic
170	C₂×C₁₀₉	#575	1	1	yes	characteristic
171	C₂×C₁₀₉	#576	1	1	yes	characteristic
172	D₁₀₉	—	4	1	no	—
173	#3355	—	109	1	no	maximal, Sylow
174	C₁₀₉	#3355	1	1	yes	characteristic, complemented, Sylow
175	#553	—	109	1	no	—
176	#553	—	109	1	no	—
177	#499	—	109	1	no	—
178	#482	—	109	1	no	—
179	#482	—	109	1	no	—
180	#480	—	109	1	no	—
181	#480	—	109	1	no	—
182	#478	—	109	1	no	—
183	#478	—	109	1	no	—
184	#469	—	109	1	no	—
185	#469	D₁₀₉	1	1	yes	characteristic, complemented
186	C₄×D₈	—	109	1	no	—
187	#436	—	109	1	no	—
188	#428	—	109	1	no	—
189	#428	—	109	1	no	—
190	C₂×D₈	C₂×D₁₀₉	1	1	yes	characteristic
191	C₂×M₁₆	—	109	1	no	—
192	C₂×M₁₆	C₂×D₁₀₉	1	1	yes	characteristic
193	#126	—	218	1	no	—
194	#126	—	218	1	no	—
195	#125	—	109	1	no	—
196	#125	—	109	1	no	—
197	#123	—	218	1	no	—
198	#123	—	218	1	no	—
199	#121	—	109	1	no	—
200	#121	—	109	1	no	—
201	#121	—	218	1	no	—
202	#121	—	218	1	no	—
203	#121	C₂×D₁₀₉	1	1	yes	characteristic
204	#121	C₂×D₁₀₉	1	1	yes	characteristic
205	C₄×D₄	—	109	1	no	—
206	C₄×D₄	—	109	1	no	—
207	C₄×D₄	—	109	1	no	—
208	C₄×D₄	—	218	1	no	—
209	C₄×D₄	—	218	1	no	—
210	#117	—	109	1	no	—
211	#117	—	109	1	no	—
212	C₂×C₄⋊C₄	—	109	1	no	—
213	C₂×C₄⋊C₄	—	109	1	no	—
214	#107	—	109	1	no	—
215	#107	—	109	1	no	—
216	#107	—	109	1	no	—
217	#107	—	109	1	no	—
218	#107	—	109	1	no	—
219	#106	—	109	1	no	—
220	#106	—	109	1	no	—
221	#106	—	109	1	no	—
222	#106	C₂×D₁₀₉	1	1	yes	characteristic
223	C₄⋊C₈	—	109	1	no	—
224	#102	—	109	1	no	—
225	#102	—	109	1	no	—
226	#102	—	109	1	no	—
227	#102	—	109	1	no	—
228	#102	C₂×D₁₀₉	1	1	yes	characteristic
229	#102	C₂×D₁₀₉	1	1	yes	characteristic
230	C₂²⋊C₈	—	109	1	no	—
231	C₂²⋊C₈	—	109	1	no	—
232	C₄×C₈	—	109	1	no	—
233	C₂×D₄	—	2	1	no	—
234	C₂×D₄	—	2	1	no	—
235	C₂×D₄	—	109	1	no	—
236	C₂×D₄	—	218	1	no	—
237	C₂×D₄	—	218	1	no	—
238	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
239	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
240	C₂²×C₄	—	109	1	no	—
241	C₂²×C₄	—	218	1	no	—
242	C₂²×C₄	—	218	1	no	—
243	C₂²×C₄	—	218	1	no	—
244	C₂²×C₄	—	218	1	no	—
245	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
246	D₈	—	2	2	no	—
247	M₁₆	—	2	2	no	—
248	M₁₆	—	218	2	no	—
249	C₂×C₈	—	109	1	no	—
250	C₂×C₈	—	109	1	no	—
251	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
252	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
253	C₄⋊C₄	—	109	1	no	—
254	C₄⋊C₄	—	109	1	no	—
255	C₄⋊C₄	—	109	1	no	—
256	C₄⋊C₄	—	109	1	no	—
257	C₄⋊C₄	—	109	1	no	—
258	C₄⋊C₄	—	218	1	no	—
259	C₄⋊C₄	—	218	1	no	—
260	C₄⋊C₄	—	218	1	no	—
261	C₄⋊C₄	—	218	1	no	—
262	C₄⋊C₄	—	218	1	no	—
263	C₄⋊C₄	—	218	1	no	—
264	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
265	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
266	C₂²⋊C₄	—	2	1	no	—
267	C₂²⋊C₄	—	2	1	no	—
268	C₂²⋊C₄	—	109	1	no	—
269	C₂²⋊C₄	—	109	1	no	—
270	C₂²⋊C₄	—	218	1	no	—
271	C₂²⋊C₄	—	218	1	no	—
272	C₂²⋊C₄	—	218	1	no	—
273	C₂²⋊C₄	—	218	1	no	—
274	C₂²⋊C₄	—	218	1	no	—
275	C₂²⋊C₄	—	218	1	no	—
276	C₄²	—	109	1	no	—
277	C₄²	—	218	1	no	—
278	C₄²	—	218	1	no	—
279	C₂³	—	2	1	no	—
280	C₂³	—	2	1	no	—
281	C₂³	—	109	1	no	—
282	C₂³	D₄×D₁₀₉	1	1	yes	characteristic
283	D₄	—	2	1	no	—
284	D₄	—	2	1	no	—
285	D₄	—	2	1	no	—
286	D₄	—	2	1	no	—
287	D₄	—	4	2	no	—
288	D₄	—	8	1	no	—
289	D₄	—	872	1	no	—
290	D₄	—	436	2	no	—
291	D₄	—	436	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	872	1	no	—
297	C₂×C₄	—	872	1	no	—
298	C₂×C₄	—	872	1	no	—
299	C₂×C₄	—	872	1	no	—
300	C₂×C₄	—	109	1	no	—
301	C₂×C₄	—	109	1	no	—
302	C₂×C₄	—	109	1	no	—
303	C₂×C₄	—	218	1	no	—
304	C₂×C₄	—	218	1	no	—
305	C₂×C₄	—	218	1	no	—
306	C₂×C₄	—	218	1	no	—
307	C₂×C₄	—	218	1	no	—
308	C₂×C₄	—	218	1	no	—
309	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
310	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
311	C₈	—	2	1	no	—
312	C₈	—	218	1	no	—
313	C₈	—	218	1	no	—
314	C₈	#13182028	1	2	yes	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	4	1	no	—
320	C₂²	—	4	1	no	—
321	C₂²	—	4	1	no	—
322	C₂²	—	4	1	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	218	1	no	—
325	C₂²	—	218	1	no	—
326	C₂²	—	218	1	no	—
327	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	4	1	no	—
330	C₄	—	4	1	no	—
331	C₄	—	436	1	no	—
332	C₄	—	436	1	no	—
333	C₄	—	436	1	no	—
334	C₄	—	436	1	no	—
335	C₄	—	218	1	no	—
336	C₄	—	218	1	no	—
337	C₄	—	218	1	no	—
338	C₄	C₂×#13182028	1	1	yes	characteristic
339	C₄	#16812746	1	1	yes	characteristic
340	C₂	—	4	1	no	—
341	C₂	—	8	1	no	—
342	C₂	—	8	1	no	—
343	C₂	—	436	1	no	—
344	C₂	#20295676	1	1	yes	characteristic
345	C₂	#20295822	1	1	yes	characteristic
346	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	#469×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
hypercenter	#469	D₁₀₉
hypocenter	C₁₀₉	#3355
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #469

Complements

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

N	H ≅ G/N	type
1. #20295728	C₂	semidirect
2. #20295723	C₂	semidirect
3. #20295426	C₂	semidirect
4. #20295349	C₂	semidirect
5. #20295347	C₂	semidirect
6. #20295338	C₂	semidirect
7. #20295337	C₂	semidirect
8. #20295328	C₂	semidirect
9. #20295324	C₂	semidirect
10. #20295276	C₂	semidirect
11. D₈×Dic₁₀₉	C₂	semidirect
12. #20295102	C₂	semidirect
13. #20295068	C₂	semidirect
14. #20295065	C₂	semidirect
15. #469×C₁₀₉	C₂	semidirect
174. C₁₀₉	#3355	semidirect
185. #469	D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295728	2
#20295723	2
#20295426	2
#20295349	2
#20295347	2
#20295338	2
#20295337	2
#20295328	2
#20295324	2
#20295276	2
D₈×Dic₁₀₉	2
#20295102	2
#20295068	2
#20295065	2
#469×C₁₀₉	2
#3355	109