Subgroups — #28373187

Order 13952.

386 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295578	C₂	1	1	yes	maximal, characteristic, complemented
2	#41×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
3	#20295368	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295364	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295319	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295314	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295234	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295229	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295188	C₂	1	1	yes	maximal, characteristic
10	#20295133	C₂	1	1	yes	maximal, characteristic, complemented
11	#20295130	C₂	1	1	yes	maximal, characteristic, complemented
12	#20295127	C₂	1	1	yes	maximal, characteristic, complemented
13	#20295123	C₂	1	1	yes	maximal, characteristic, complemented
14	#20295080	C₂	1	1	yes	maximal, characteristic, complemented
15	#452×C₁₀₉	C₂	1	1	yes	maximal, characteristic
16	C₂×#13182019	C₂²	1	1	yes	characteristic
17	C₂×#13182017	C₂²	1	1	yes	characteristic
18	#16812698	C₂²	1	1	yes	characteristic
19	#16812697	C₂²	1	1	yes	characteristic
20	#16812689	—	2	1	no	—
21	#16812678	C₂²	1	1	yes	characteristic
22	#16812676	—	2	1	no	—
23	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
24	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
25	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
26	#16812664	—	2	1	no	—
27	#16812664	C₂²	1	1	yes	characteristic
28	D₄×Dic₁₀₉	—	2	1	no	—
29	D₄×Dic₁₀₉	—	2	1	no	—
30	D₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
31	D₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
32	#16812658	—	2	1	no	—
33	#16812658	C₂²	1	1	yes	characteristic
34	#16812655	—	2	1	no	—
35	C₂×C₄×Dic₁₀₉	—	2	1	no	—
36	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
37	#16812633	C₂²	1	1	yes	characteristic
38	#16812632	C₂²	1	1	yes	characteristic
39	#16812631	C₂²	1	1	yes	characteristic
40	#16812630	C₂²	1	1	yes	characteristic
41	#16812628	C₂²	1	1	yes	characteristic
42	#16812628	C₂²	1	1	yes	characteristic
43	#16812624	C₂²	1	1	yes	characteristic
44	#16812624	C₂²	1	1	yes	characteristic
45	#16812623	C₂²	1	1	yes	characteristic
46	#16812623	C₂²	1	1	yes	characteristic
47	#16812623	C₂²	1	1	yes	characteristic
48	#16812620	C₂²	1	1	yes	characteristic
49	C₂²⋊#11724784	C₂²	1	1	yes	characteristic
50	#16812609	C₂²	1	1	yes	characteristic
51	#16812609	C₂²	1	1	yes	characteristic
52	C₂×#41×C₁₀₉	C₂²	1	1	yes	characteristic
53	C₂×Q₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
54	C₂×SD₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
55	C₂²⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
56	#103×C₁₀₉	C₂²	1	1	yes	characteristic
57	#102×C₁₀₉	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₁₈	—	2	1	no	—
60	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
61	C₂²×Dic₁₀₉	—	2	1	no	—
62	C₂²×Dic₁₀₉	—	2	1	no	—
63	C₂²×Dic₁₀₉	D₄	1	1	yes	characteristic
64	C₂²×Dic₁₀₉	D₄	1	1	yes	characteristic
65	#13182019	—	2	2	no	—
66	#13182017	—	2	2	no	—
67	C₂×#11724784	C₂³	1	1	yes	characteristic
68	C₂×#11724784	C₂³	1	1	yes	characteristic
69	C₄⋊Dic₁₀₉	—	2	1	no	—
70	C₄⋊Dic₁₀₉	—	2	1	no	—
71	C₄⋊Dic₁₀₉	—	2	1	no	—
72	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
75	#13182008	—	2	1	no	—
76	#13182008	—	2	1	no	—
77	#13182008	—	2	1	no	—
78	#13182008	—	2	1	no	—
79	#13182008	—	2	1	no	—
80	C₂²⋊Dic₁₀₉	—	2	1	no	—
81	C₂²⋊Dic₁₀₉	—	2	1	no	—
82	C₂²⋊Dic₁₀₉	—	2	1	no	—
83	C₂²⋊Dic₁₀₉	—	2	1	no	—
84	C₄×Dic₁₀₉	—	2	1	no	—
85	C₄×Dic₁₀₉	—	2	1	no	—
86	C₄×Dic₁₀₉	—	2	1	no	—
87	C₄×Dic₁₀₉	—	2	1	no	—
88	C₄×Dic₁₀₉	—	2	1	no	—
89	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
90	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
91	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
92	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
93	#41×C₁₀₉	—	2	2	no	—
94	#41×C₁₀₉	—	2	2	no	—
95	C₂×Q₈×C₁₀₉	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	C₂³	1	1	yes	characteristic
99	C₂²×C₄×C₁₀₉	—	2	1	no	—
100	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
101	Q₁₆×C₁₀₉	—	2	2	no	—
102	SD₁₆×C₁₀₉	—	2	2	no	—
103	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
104	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₁₀₉	—	2	1	no	—
106	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
107	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
108	C₂³×C₁₀₉	—	2	1	no	—
109	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
110	Q₈×C₁₀₉	—	2	1	no	—
111	Q₈×C₁₀₉	—	2	1	no	—
112	Q₈×C₁₀₉	—	2	1	no	—
113	Q₈×C₁₀₉	#41	1	2	yes	—
114	D₄×C₁₀₉	—	2	4	no	—
115	D₄×C₁₀₉	—	2	1	no	—
116	D₄×C₁₀₉	#41	1	2	yes	—
117	C₂×C₄×C₁₀₉	—	2	4	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	C₂×Dic₁₀₉	—	2	2	no	—
126	C₂×Dic₁₀₉	—	2	1	no	—
127	C₂×Dic₁₀₉	—	2	1	no	—
128	C₂×Dic₁₀₉	—	2	1	no	—
129	C₂×Dic₁₀₉	—	2	1	no	—
130	C₂×Dic₁₀₉	—	4	1	no	—
131	C₂×Dic₁₀₉	—	4	2	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	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
136	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
137	Dic₂₁₈	—	2	1	no	—
138	Dic₂₁₈	—	2	1	no	—
139	Dic₂₁₈	—	4	2	no	—
140	C₈×C₁₀₉	—	2	1	no	—
141	C₈×C₁₀₉	—	2	1	no	—
142	#11724784	—	2	1	no	—
143	#11724784	—	2	1	no	—
144	C₂²×C₁₀₉	—	2	1	no	—
145	C₂²×C₁₀₉	—	2	2	no	—
146	C₂²×C₁₀₉	—	2	1	no	—
147	C₂²×C₁₀₉	—	2	1	no	—
148	C₂²×C₁₀₉	—	4	1	no	—
149	C₂²×C₁₀₉	—	4	1	no	—
150	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
151	C₄×C₁₀₉	—	2	1	no	—
152	C₄×C₁₀₉	—	2	2	no	—
153	C₄×C₁₀₉	—	4	1	no	—
154	C₄×C₁₀₉	—	4	1	no	—
155	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
156	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
157	Dic₁₀₉	—	2	1	no	—
158	Dic₁₀₉	—	2	1	no	—
159	Dic₁₀₉	—	2	1	no	—
160	Dic₁₀₉	—	4	1	no	—
161	Dic₁₀₉	—	4	1	no	—
162	Dic₁₀₉	—	4	1	no	—
163	Dic₁₀₉	—	4	1	no	—
164	Dic₁₀₉	#137	1	2	yes	—
165	C₂×C₁₀₉	—	4	1	no	—
166	C₂×C₁₀₉	—	4	2	no	—
167	C₂×C₁₀₉	#525	1	1	yes	characteristic
168	C₂×C₁₀₉	C₂×#137	1	1	yes	characteristic
169	C₂×C₁₀₉	#575	1	1	yes	characteristic
170	#3116	—	109	1	no	maximal, Sylow
171	C₁₀₉	#3116	1	1	yes	characteristic, complemented, Sylow
172	#533	—	109	1	no	—
173	C₄×#41	—	109	1	no	—
174	#485	—	109	1	no	—
175	#485	—	109	1	no	—
176	#477	—	109	1	no	—
177	#476	—	109	1	no	—
178	#462	—	109	1	no	—
179	#461	—	109	1	no	—
180	#452	—	109	1	no	—
181	#452	D₁₀₉	1	1	yes	characteristic
182	#441	—	109	1	no	—
183	#441	—	109	1	no	—
184	#440	—	109	1	no	—
185	#440	—	109	1	no	—
186	#432	—	109	1	no	—
187	C₂×#41	C₂×D₁₀₉	1	1	yes	characteristic
188	C₂×Q₁₆	—	109	1	no	—
189	C₂×Q₁₆	C₂×D₁₀₉	1	1	yes	characteristic
190	C₂×SD₁₆	—	109	1	no	—
191	C₂×SD₁₆	C₂×D₁₀₉	1	1	yes	characteristic
192	C₄⋊Q₈	—	109	1	no	—
193	C₄⋊Q₈	—	109	1	no	—
194	#125	—	218	1	no	—
195	C₂²⋊Q₈	—	109	1	no	—
196	C₂²⋊Q₈	—	218	1	no	—
197	C₂²⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic
198	C₄×Q₈	—	109	1	no	—
199	C₄×Q₈	—	109	1	no	—
200	C₄×Q₈	—	109	1	no	—
201	C₄×Q₈	—	109	1	no	—
202	C₄×Q₈	—	218	1	no	—
203	C₄×D₄	—	109	1	no	—
204	C₄×D₄	—	109	1	no	—
205	C₄×D₄	—	218	1	no	—
206	C₄×D₄	—	218	1	no	—
207	#117	—	109	1	no	—
208	#117	—	218	1	no	—
209	#117	—	218	1	no	—
210	C₂×C₄²	—	109	1	no	—
211	C₂×C₄²	—	218	1	no	—
212	#107	—	109	1	no	—
213	#107	—	109	1	no	—
214	#106	—	109	1	no	—
215	#106	—	109	1	no	—
216	C₄⋊C₈	—	109	1	no	—
217	C₄⋊C₈	—	109	1	no	—
218	#103	—	109	1	no	—
219	#103	—	109	1	no	—
220	#103	—	109	1	no	—
221	#103	—	109	1	no	—
222	#103	—	109	1	no	—
223	#103	C₂×D₁₀₉	1	1	yes	characteristic
224	#102	—	109	1	no	—
225	#102	C₂×D₁₀₉	1	1	yes	characteristic
226	C₂²⋊C₈	—	109	1	no	—
227	C₂²⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
228	#97	—	109	1	no	—
229	#97	—	109	1	no	—
230	#41	—	2	2	no	—
231	#41	—	2	2	no	—
232	C₂×Q₈	—	109	1	no	—
233	C₂×Q₈	—	218	1	no	—
234	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
235	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
236	C₂×D₄	—	2	1	no	—
237	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	—	109	1	no	—
240	C₂²×C₄	—	109	1	no	—
241	C₂²×C₄	—	218	1	no	—
242	C₂²×C₄	—	218	1	no	—
243	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
244	Q₁₆	—	2	2	no	—
245	Q₁₆	—	218	2	no	—
246	SD₁₆	—	2	2	no	—
247	SD₁₆	—	218	2	no	—
248	C₂×C₈	—	109	1	no	—
249	C₂×C₈	—	109	1	no	—
250	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
251	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	—	109	1	no	—
254	C₄⋊C₄	—	109	1	no	—
255	C₄⋊C₄	—	109	1	no	—
256	C₄⋊C₄	—	218	1	no	—
257	C₄⋊C₄	—	218	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	C₂²⋊C₄	—	2	1	no	—
266	C₂²⋊C₄	—	218	1	no	—
267	C₂²⋊C₄	—	218	1	no	—
268	C₂²⋊C₄	—	218	1	no	—
269	C₂²⋊C₄	—	218	1	no	—
270	C₄²	—	109	1	no	—
271	C₄²	—	109	1	no	—
272	C₄²	—	109	1	no	—
273	C₄²	—	109	1	no	—
274	C₄²	—	218	1	no	—
275	C₄²	—	218	1	no	—
276	C₄²	—	218	1	no	—
277	C₄²	—	218	1	no	—
278	C₄²	—	218	1	no	—
279	C₂³	—	2	1	no	—
280	C₂³	D₄×D₁₀₉	1	1	yes	characteristic
281	Q₈	—	2	1	no	—
282	Q₈	—	2	1	no	—
283	Q₈	—	2	1	no	—
284	Q₈	—	436	2	no	—
285	Q₈	—	218	1	no	—
286	Q₈	—	218	1	no	—
287	Q₈	#13182028	1	2	yes	—
288	D₄	—	2	4	no	—
289	D₄	—	2	1	no	—
290	D₄	#13182028	1	2	yes	—
291	C₂×C₄	—	2	4	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	—	2	1	no	—
297	C₂×C₄	—	109	1	no	—
298	C₂×C₄	—	109	1	no	—
299	C₂×C₄	—	109	1	no	—
300	C₂×C₄	—	109	1	no	—
301	C₂×C₄	—	436	2	no	—
302	C₂×C₄	—	436	2	no	—
303	C₂×C₄	—	436	1	no	—
304	C₂×C₄	—	218	2	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	—	2	1	no	—
313	C₈	—	218	1	no	—
314	C₈	—	218	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	2	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	4	1	no	—
320	C₂²	—	4	1	no	—
321	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	2	no	—
324	C₄	—	4	1	no	—
325	C₄	—	4	1	no	—
326	C₄	—	109	2	no	—
327	C₄	—	436	1	no	—
328	C₄	—	436	1	no	—
329	C₄	—	436	1	no	—
330	C₄	—	436	1	no	—
331	C₄	—	218	1	no	—
332	C₄	—	218	1	no	—
333	C₄	—	218	1	no	—
334	C₄	C₂×#13182028	1	1	yes	characteristic
335	C₄	C₂×#13182028	1	1	yes	characteristic
336	C₂	—	4	1	no	—
337	C₂	—	4	2	no	—
338	C₂	#20295532	1	1	yes	characteristic
339	C₂	#137×D₁₀₉	1	1	yes	characteristic
340	C₂	#20295823	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#452×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂×#41×C₁₀₉	C₂²
hypercenter	#452	D₁₀₉
hypocenter	C₁₀₉	#3116
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #452

Complements

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

N	H ≅ G/N	type
1. #20295578	C₂	semidirect
3. #20295368	C₂	semidirect
4. #20295364	C₂	semidirect
5. #20295319	C₂	semidirect
6. #20295314	C₂	semidirect
7. #20295234	C₂	semidirect
8. #20295229	C₂	semidirect
10. #20295133	C₂	semidirect
11. #20295130	C₂	semidirect
12. #20295127	C₂	semidirect
13. #20295123	C₂	semidirect
14. #20295080	C₂	semidirect
171. C₁₀₉	#3116	semidirect

Maximal subgroups

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

subgroup	index
#20295578	2
#41×Dic₁₀₉	2
#20295368	2
#20295364	2
#20295319	2
#20295314	2
#20295234	2
#20295229	2
#20295188	2
#20295133	2
#20295130	2
#20295127	2
#20295123	2
#20295080	2
#452×C₁₀₉	2
#3116	109