Subgroups — #28375602

Order 13952.

336 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295718	C₂	1	1	yes	maximal, characteristic
2	#20295710	C₂	1	1	yes	maximal, characteristic
3	#20295426	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295365	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295364	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295360	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295359	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295335	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295331	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295281	C₂	1	1	yes	maximal, characteristic, complemented
11	Q₁₆×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#20295102	C₂	1	1	yes	maximal, characteristic
13	#20295068	C₂	1	1	yes	maximal, characteristic, complemented
14	#20295065	C₂	1	1	yes	maximal, characteristic, complemented
15	#470×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13182012	C₂²	1	1	yes	characteristic
17	#16812694	—	2	1	no	—
18	#16812692	—	2	1	no	—
19	#16812690	—	2	1	no	—
20	#16812690	C₂²	1	1	yes	characteristic
21	#16812690	C₂²	1	1	yes	characteristic
22	#16812689	—	2	1	no	—
23	#16812683	—	2	1	no	—
24	#16812681	—	2	1	no	—
25	#16812675	C₂²	1	1	yes	characteristic
26	#16812675	C₂²	1	1	yes	characteristic
27	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
28	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
29	C₄×C₂²⋊D₁₀₉	C₂²	1	1	yes	characteristic
30	#16812658	C₂²	1	1	yes	characteristic
31	#16812657	—	2	1	no	—
32	#16812657	C₂²	1	1	yes	characteristic
33	#16812655	—	2	1	no	—
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	#16812623	C₂²	1	1	yes	characteristic
46	#16812623	C₂²	1	1	yes	characteristic
47	#16812623	C₂²	1	1	yes	characteristic
48	#16812623	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×Q₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
53	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
54	C₂²⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
55	C₂²⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
56	#106×C₁₀₉	C₂²	1	1	yes	characteristic
57	#103×C₁₀₉	C₂²	1	1	yes	characteristic
58	#103×C₁₀₉	C₂²	1	1	yes	characteristic
59	C₂×C₂²⋊D₁₀₉	D₄	1	1	yes	characteristic
60	C₂²×Dic₁₀₉	—	2	1	no	—
61	C₂×C₄×D₁₀₉	—	2	1	no	—
62	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
63	#13182012	—	2	2	no	—
64	C₂×#11724784	C₂³	1	1	yes	characteristic
65	C₂×#11724784	C₂³	1	1	yes	characteristic
66	C₄⋊Dic₁₀₉	—	2	1	no	—
67	C₄⋊Dic₁₀₉	—	2	1	no	—
68	C₄⋊Dic₁₀₉	—	2	1	no	—
69	C₄⋊Dic₁₀₉	—	2	1	no	—
70	C₄⋊Dic₁₀₉	—	2	1	no	—
71	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
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	#13182008	D₄	1	1	yes	characteristic
81	C₂²⋊Dic₁₀₉	—	2	1	no	—
82	C₂²⋊Dic₁₀₉	D₄	1	1	yes	characteristic
83	#13182006	—	2	1	no	—
84	#13182006	—	2	1	no	—
85	#13182006	—	2	1	no	—
86	#13182006	D₄	1	1	yes	characteristic
87	C₄×Dic₁₀₉	—	2	1	no	—
88	C₄×Dic₁₀₉	—	2	1	no	—
89	C₄×Dic₁₀₉	—	2	1	no	—
90	C₄×Dic₁₀₉	—	2	1	no	—
91	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
92	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
93	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
94	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
95	Q₁₆×C₁₀₉	—	2	2	no	—
96	M₁₆×C₁₀₉	—	2	2	no	—
97	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
98	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
99	C₄⋊C₄×C₁₀₉	—	2	1	no	—
100	C₄⋊C₄×C₁₀₉	—	2	1	no	—
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	C₂×D₄	1	1	yes	characteristic
106	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
107	Q₈×C₁₀₉	—	2	1	no	—
108	Q₈×C₁₀₉	—	2	1	no	—
109	Q₈×C₁₀₉	—	2	1	no	—
110	Q₈×C₁₀₉	—	2	1	no	—
111	C₂×C₄×C₁₀₉	—	2	1	no	—
112	C₂×C₄×C₁₀₉	—	2	1	no	—
113	C₂×C₄×C₁₀₉	—	2	1	no	—
114	C₂×C₄×C₁₀₉	—	2	1	no	—
115	C₂×C₄×C₁₀₉	—	2	1	no	—
116	C₂×C₄×C₁₀₉	—	2	1	no	—
117	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
119	C₂²⋊D₁₀₉	—	4	1	no	—
120	C₂×Dic₁₀₉	—	2	1	no	—
121	C₂×Dic₁₀₉	—	2	1	no	—
122	C₂×Dic₁₀₉	—	2	1	no	—
123	C₂×Dic₁₀₉	—	2	1	no	—
124	C₂×Dic₁₀₉	—	4	2	no	—
125	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
126	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
127	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
128	C₄×D₁₀₉	—	2	1	no	—
129	C₄×D₁₀₉	—	2	1	no	—
130	C₄×D₁₀₉	—	4	2	no	—
131	C₈×C₁₀₉	—	2	1	no	—
132	C₈×C₁₀₉	#41	1	2	yes	—
133	#11724784	—	2	1	no	—
134	#11724784	—	2	1	no	—
135	C₂²×C₁₀₉	—	2	1	no	—
136	C₂²×C₁₀₉	—	2	1	no	—
137	C₂²×C₁₀₉	—	2	1	no	—
138	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
139	C₂×D₁₀₉	—	2	1	no	—
140	C₂×D₁₀₉	—	2	1	no	—
141	C₂×D₁₀₉	—	2	1	no	—
142	C₄×C₁₀₉	—	2	1	no	—
143	C₄×C₁₀₉	—	4	1	no	—
144	C₄×C₁₀₉	—	4	1	no	—
145	C₄×C₁₀₉	—	4	1	no	—
146	C₄×C₁₀₉	—	4	1	no	—
147	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
148	C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
149	Dic₁₀₉	—	2	1	no	—
150	Dic₁₀₉	—	2	1	no	—
151	Dic₁₀₉	—	2	1	no	—
152	Dic₁₀₉	—	4	1	no	—
153	Dic₁₀₉	—	4	1	no	—
154	Dic₁₀₉	—	4	1	no	—
155	Dic₁₀₉	—	4	1	no	—
156	C₂×C₁₀₉	—	4	1	no	—
157	C₂×C₁₀₉	#547	1	1	yes	characteristic
158	C₂×C₁₀₉	#575	1	1	yes	characteristic
159	C₂×C₁₀₉	#578	1	1	yes	characteristic
160	D₁₀₉	—	4	1	no	—
161	#3352	—	109	1	no	maximal, Sylow
162	C₁₀₉	#3352	1	1	yes	characteristic, complemented, Sylow
163	#552	—	109	1	no	—
164	#552	—	109	1	no	—
165	#499	—	109	1	no	—
166	#485	—	109	1	no	—
167	#485	—	109	1	no	—
168	#484	—	109	1	no	—
169	#484	—	109	1	no	—
170	#479	—	109	1	no	—
171	#479	—	109	1	no	—
172	#470	—	109	1	no	—
173	#470	D₁₀₉	1	1	yes	characteristic, complemented
174	C₄×Q₁₆	—	109	1	no	—
175	#436	—	109	1	no	—
176	#428	—	109	1	no	—
177	#428	—	109	1	no	—
178	C₂×Q₁₆	C₂×D₁₀₉	1	1	yes	characteristic
179	C₂×M₁₆	—	109	1	no	—
180	C₂×M₁₆	C₂×D₁₀₉	1	1	yes	characteristic
181	#126	—	218	1	no	—
182	#126	—	218	1	no	—
183	#125	—	109	1	no	—
184	#125	—	109	1	no	—
185	#125	—	218	1	no	—
186	#125	—	218	1	no	—
187	#123	—	218	1	no	—
188	#123	—	218	1	no	—
189	C₂²⋊Q₈	—	109	1	no	—
190	C₂²⋊Q₈	—	109	1	no	—
191	C₂²⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic
192	C₂²⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic
193	C₄×Q₈	—	109	1	no	—
194	C₄×Q₈	—	109	1	no	—
195	C₄×D₄	—	109	1	no	—
196	#117	—	109	1	no	—
197	#117	—	109	1	no	—
198	#117	—	218	1	no	—
199	#117	—	218	1	no	—
200	C₂×C₄⋊C₄	—	109	1	no	—
201	C₂×C₄⋊C₄	—	109	1	no	—
202	#107	—	109	1	no	—
203	#107	—	109	1	no	—
204	#107	—	109	1	no	—
205	#107	—	109	1	no	—
206	#107	—	109	1	no	—
207	#106	—	109	1	no	—
208	#106	—	109	1	no	—
209	#106	—	109	1	no	—
210	#106	C₂×D₁₀₉	1	1	yes	characteristic
211	C₄⋊C₈	—	109	1	no	—
212	#103	—	109	1	no	—
213	#103	—	109	1	no	—
214	#103	—	109	1	no	—
215	#103	—	109	1	no	—
216	#103	C₂×D₁₀₉	1	1	yes	characteristic
217	#103	C₂×D₁₀₉	1	1	yes	characteristic
218	C₂²⋊C₈	—	109	1	no	—
219	C₂²⋊C₈	—	109	1	no	—
220	C₄×C₈	—	109	1	no	—
221	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
222	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
223	C₂×D₄	—	109	1	no	—
224	C₂²×C₄	—	109	1	no	—
225	C₂²×C₄	—	218	1	no	—
226	C₂²×C₄	—	218	1	no	—
227	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
228	Q₁₆	—	2	2	no	—
229	M₁₆	—	2	2	no	—
230	M₁₆	—	218	2	no	—
231	C₂×C₈	—	109	1	no	—
232	C₂×C₈	—	109	1	no	—
233	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
234	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
235	C₄⋊C₄	—	2	1	no	—
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	—	109	1	no	—
238	C₄⋊C₄	—	109	1	no	—
239	C₄⋊C₄	—	109	1	no	—
240	C₄⋊C₄	—	109	1	no	—
241	C₄⋊C₄	—	109	1	no	—
242	C₄⋊C₄	—	218	1	no	—
243	C₄⋊C₄	—	218	1	no	—
244	C₄⋊C₄	—	218	1	no	—
245	C₄⋊C₄	—	218	1	no	—
246	C₄⋊C₄	—	218	1	no	—
247	C₄⋊C₄	—	218	1	no	—
248	C₄⋊C₄	—	218	1	no	—
249	C₄⋊C₄	—	218	1	no	—
250	C₄⋊C₄	—	218	1	no	—
251	C₄⋊C₄	—	218	1	no	—
252	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
253	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
254	C₂²⋊C₄	—	2	1	no	—
255	C₂²⋊C₄	—	2	1	no	—
256	C₂²⋊C₄	—	109	1	no	—
257	C₂²⋊C₄	—	109	1	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	C₄²	—	109	1	no	—
263	C₄²	—	218	1	no	—
264	C₄²	—	218	1	no	—
265	C₄²	—	218	1	no	—
266	C₄²	—	218	1	no	—
267	C₂³	—	109	1	no	—
268	C₂³	D₄×D₁₀₉	1	1	yes	characteristic
269	Q₈	—	2	1	no	—
270	Q₈	—	2	1	no	—
271	Q₈	—	2	1	no	—
272	Q₈	—	2	1	no	—
273	D₄	—	436	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	109	1	no	—
281	C₂×C₄	—	109	1	no	—
282	C₂×C₄	—	109	1	no	—
283	C₂×C₄	—	436	2	no	—
284	C₂×C₄	—	436	2	no	—
285	C₂×C₄	—	218	1	no	—
286	C₂×C₄	—	218	1	no	—
287	C₂×C₄	—	218	1	no	—
288	C₂×C₄	—	218	1	no	—
289	C₂×C₄	—	218	1	no	—
290	C₂×C₄	—	218	1	no	—
291	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
292	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
293	C₈	—	2	1	no	—
294	C₈	—	218	1	no	—
295	C₈	—	218	1	no	—
296	C₈	#13182028	1	2	yes	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	218	1	no	—
301	C₂²	—	218	1	no	—
302	C₂²	—	218	1	no	—
303	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
304	C₄	—	2	1	no	—
305	C₄	—	4	1	no	—
306	C₄	—	4	1	no	—
307	C₄	—	4	1	no	—
308	C₄	—	4	1	no	—
309	C₄	—	436	1	no	—
310	C₄	—	436	1	no	—
311	C₄	—	436	1	no	—
312	C₄	—	436	1	no	—
313	C₄	—	218	1	no	—
314	C₄	—	218	1	no	—
315	C₄	—	218	1	no	—
316	C₄	C₂×#13182028	1	1	yes	characteristic
317	C₄	#16812746	1	1	yes	characteristic
318	C₂	—	4	1	no	—
319	C₂	—	436	1	no	—
320	C₂	#20295676	1	1	yes	characteristic
321	C₂	#20295825	1	1	yes	characteristic
322	C₂	#20295842	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#470×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂×C₂²⋊D₁₀₉	D₄
hypercenter	#470	D₁₀₉
hypocenter	C₁₀₉	#3352
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #470

Complements

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

N	H ≅ G/N	type
3. #20295426	C₂	semidirect
4. #20295365	C₂	semidirect
5. #20295364	C₂	semidirect
6. #20295360	C₂	semidirect
7. #20295359	C₂	semidirect
8. #20295335	C₂	semidirect
9. #20295331	C₂	semidirect
10. #20295281	C₂	semidirect
11. Q₁₆×Dic₁₀₉	C₂	semidirect
13. #20295068	C₂	semidirect
14. #20295065	C₂	semidirect
15. #470×C₁₀₉	C₂	semidirect
162. C₁₀₉	#3352	semidirect
173. #470	D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295718	2
#20295710	2
#20295426	2
#20295365	2
#20295364	2
#20295360	2
#20295359	2
#20295335	2
#20295331	2
#20295281	2
Q₁₆×Dic₁₀₉	2
#20295102	2
#20295068	2
#20295065	2
#470×C₁₀₉	2
#3352	109