Subgroups — #28375612

Order 13952.

374 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	#20295744	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295677	C₂	1	1	yes	maximal, characteristic
3	#20295430	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295363	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295361	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295352	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295350	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295310	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295305	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295273	C₂	1	1	yes	maximal, characteristic, complemented
11	#20295272	C₂	1	1	yes	maximal, characteristic, complemented
12	#20295113	C₂	1	1	yes	maximal, characteristic, complemented
13	#20295102	C₂	1	1	yes	maximal, characteristic, complemented
14	#20295065	C₂	1	1	yes	maximal, characteristic, complemented
15	#428×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13182029	—	2	1	no	—
17	C₂×#13182018	C₂²	1	1	yes	characteristic
18	C₂×#13182012	C₂²	1	1	yes	characteristic
19	#16812698	C₂²	1	1	yes	characteristic
20	#16812698	C₂²	1	1	yes	characteristic
21	#16812693	—	2	1	no	—
22	#16812692	—	2	1	no	—
23	#16812685	—	2	1	no	—
24	#16812684	—	2	1	no	—
25	#16812682	—	2	1	no	—
26	#16812679	C₂²	1	1	yes	characteristic
27	#16812678	C₂²	1	1	yes	characteristic
28	#16812677	—	2	1	no	—
29	#16812674	C₂²	1	1	yes	characteristic
30	#16812673	C₂²	1	1	yes	characteristic
31	C₄×Dic₂₁₈	—	2	1	no	—
32	#16812664	C₂²	1	1	yes	characteristic
33	#16812661	C₂²	1	1	yes	characteristic
34	C₄×C₂²⋊D₁₀₉	C₂²	1	1	yes	characteristic
35	#16812657	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₁₀₉	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	#16812630	C₂²	1	1	yes	characteristic
42	#16812628	C₂²	1	1	yes	characteristic
43	#16812624	C₂²	1	1	yes	characteristic
44	#16812622	C₂²	1	1	yes	characteristic
45	#16812622	C₂²	1	1	yes	characteristic
46	#16812621	C₂²	1	1	yes	characteristic
47	#16812619	C₂²	1	1	yes	characteristic
48	#16812619	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×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
53	#117×C₁₀₉	C₂²	1	1	yes	characteristic
54	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
55	#107×C₁₀₉	C₂²	1	1	yes	characteristic
56	#107×C₁₀₉	C₂²	1	1	yes	characteristic
57	#106×C₁₀₉	C₂²	1	1	yes	characteristic
58	#106×C₁₀₉	C₂²	1	1	yes	characteristic
59	#13182029	—	4	4	no	—
60	C₂×Dic₂₁₈	—	2	1	no	—
61	C₂×Dic₂₁₈	—	2	1	no	—
62	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
63	C₂×D₄₃₆	—	2	1	no	—
64	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
65	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
66	C₂×C₂²⋊D₁₀₉	D₄	1	1	yes	characteristic
67	C₂×C₄×D₁₀₉	—	2	1	no	—
68	C₂×C₄×D₁₀₉	—	2	1	no	—
69	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
70	#13182018	—	2	2	no	—
71	#13182012	—	2	2	no	—
72	C₂×#11724784	C₂³	1	1	yes	characteristic
73	C₂×#11724784	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₁₀₉	—	2	1	no	—
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	—	2	1	no	—
81	#13182008	—	2	1	no	—
82	#13182008	D₄	1	1	yes	characteristic
83	C₂²⋊Dic₁₀₉	—	2	1	no	—
84	C₂²⋊Dic₁₀₉	D₄	1	1	yes	characteristic
85	#13182006	—	2	1	no	—
86	#13182006	—	2	1	no	—
87	#13182006	—	2	1	no	—
88	#13182006	—	2	1	no	—
89	#13182006	—	2	1	no	—
90	#13182006	D₄	1	1	yes	characteristic
91	C₄×Dic₁₀₉	—	2	1	no	—
92	C₄×Dic₁₀₉	—	2	1	no	—
93	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
94	C₂²×C₄×C₁₀₉	—	2	1	no	—
95	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
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	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
104	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
105	C₄²×C₁₀₉	—	2	1	no	—
106	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
107	C₂²×D₁₀₉	—	2	1	no	—
108	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₁₀₉	—	2	1	no	—
110	C₂×C₄×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	—	4	2	no	—
116	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
118	C₂²⋊D₁₀₉	—	2	2	no	—
119	C₂²⋊D₁₀₉	—	4	2	no	—
120	C₂×Dic₁₀₉	—	2	1	no	—
121	C₂×Dic₁₀₉	—	2	1	no	—
122	C₂×Dic₁₀₉	—	2	1	no	—
123	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
124	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
125	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
126	D₄₃₆	—	2	1	no	—
127	D₄₃₆	—	2	1	no	—
128	D₄₃₆	—	4	2	no	—
129	C₄×D₁₀₉	—	2	1	no	—
130	C₄×D₁₀₉	—	2	1	no	—
131	C₄×D₁₀₉	—	4	2	no	—
132	C₄×D₁₀₉	—	4	2	no	—
133	Dic₂₁₈	—	2	1	no	—
134	Dic₂₁₈	—	2	1	no	—
135	Dic₂₁₈	—	4	2	no	—
136	Dic₂₁₈	—	4	2	no	—
137	C₈×C₁₀₉	—	2	1	no	—
138	C₈×C₁₀₉	#41	1	2	yes	complemented
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	C₂²×D₄	1	1	yes	characteristic
145	C₂×D₁₀₉	—	2	1	no	—
146	C₂×D₁₀₉	—	2	1	no	—
147	C₂×D₁₀₉	—	2	1	no	—
148	C₂×D₁₀₉	—	4	1	no	—
149	C₂×D₁₀₉	—	4	1	no	—
150	C₂×D₁₀₉	—	4	1	no	—
151	C₄×C₁₀₉	—	2	1	no	—
152	C₄×C₁₀₉	—	4	1	no	—
153	C₄×C₁₀₉	—	4	1	no	—
154	C₄×C₁₀₉	—	4	1	no	—
155	C₄×C₁₀₉	—	4	1	no	—
156	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
157	C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
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	AGL(1,ℤ/8)	1	2	yes	—
164	C₂×C₁₀₉	—	4	1	no	—
165	C₂×C₁₀₉	#547	1	1	yes	characteristic
166	C₂×C₁₀₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
167	C₂×C₁₀₉	#578	1	1	yes	characteristic
168	D₁₀₉	—	4	1	no	—
169	D₁₀₉	—	8	1	no	—
170	#3353	—	109	1	no	maximal, Sylow
171	C₁₀₉	#3353	1	1	yes	characteristic, complemented, Sylow
172	#556	—	109	1	no	—
173	#547	—	109	1	no	—
174	#500	—	109	1	no	—
175	#484	—	109	1	no	—
176	#484	—	109	1	no	—
177	#482	—	109	1	no	—
178	#482	—	109	1	no	—
179	#475	—	109	1	no	—
180	#474	—	109	1	no	—
181	#468	—	109	1	no	—
182	#468	—	109	1	no	—
183	C₄×SD₁₆	—	109	1	no	—
184	#436	—	109	1	no	—
185	#428	—	109	1	no	—
186	#428	D₁₀₉	1	1	yes	characteristic, complemented
187	C₂×#41	—	218	1	no	—
188	C₂×SD₁₆	—	109	1	no	—
189	C₂×M₁₆	—	109	1	no	—
190	C₂×M₁₆	C₂×D₁₀₉	1	1	yes	characteristic
191	C₄⋊Q₈	—	109	1	no	—
192	C₄⋊Q₈	—	109	1	no	—
193	#126	—	218	1	no	—
194	#126	—	218	1	no	—
195	#124	—	218	1	no	—
196	#123	—	218	1	no	—
197	#123	—	218	1	no	—
198	C₂²⋊Q₈	—	109	1	no	—
199	C₂²⋊Q₈	—	109	1	no	—
200	C₂²⋊Q₈	—	218	1	no	—
201	#121	—	109	1	no	—
202	#121	—	109	1	no	—
203	C₄×Q₈	—	109	1	no	—
204	C₄×Q₈	—	218	1	no	—
205	C₄×D₄	—	109	1	no	—
206	C₄×D₄	—	109	1	no	—
207	#117	—	109	1	no	—
208	#117	C₂×D₁₀₉	1	1	yes	characteristic
209	C₂×C₄⋊C₄	—	109	1	no	—
210	C₂×C₄⋊C₄	C₂×D₁₀₉	1	1	yes	characteristic
211	#107	—	109	1	no	—
212	#107	—	109	1	no	—
213	#107	—	109	1	no	—
214	#107	—	109	1	no	—
215	#107	C₂×D₁₀₉	1	1	yes	characteristic
216	#107	C₂×D₁₀₉	1	1	yes	characteristic
217	#106	—	109	1	no	—
218	#106	C₂×D₁₀₉	1	1	yes	characteristic
219	#106	C₂×D₁₀₉	1	1	yes	characteristic
220	C₄⋊C₈	—	109	1	no	—
221	#103	—	109	1	no	—
222	#103	—	109	1	no	—
223	#103	—	109	1	no	—
224	#102	—	109	1	no	—
225	#102	—	109	1	no	—
226	#102	—	109	1	no	—
227	C₂²⋊C₈	—	109	1	no	—
228	C₂²⋊C₈	—	109	1	no	—
229	C₄×C₈	—	109	1	no	—
230	#41	—	436	4	no	—
231	C₂×Q₈	—	109	1	no	—
232	C₂×Q₈	—	218	1	no	—
233	C₂×Q₈	—	218	1	no	—
234	C₂×D₄	—	109	1	no	—
235	C₂×D₄	—	109	1	no	—
236	C₂×D₄	—	218	1	no	—
237	C₂×D₄	—	218	1	no	—
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	—	109	1	no	—
240	C₂²×C₄	—	218	1	no	—
241	C₂²×C₄	—	218	1	no	—
242	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
243	SD₁₆	—	218	2	no	—
244	M₁₆	—	2	2	no	—
245	M₁₆	—	218	2	no	—
246	C₂×C₈	—	109	1	no	—
247	C₂×C₈	—	109	1	no	—
248	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
249	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
250	C₄⋊C₄	—	2	1	no	—
251	C₄⋊C₄	—	109	1	no	—
252	C₄⋊C₄	—	109	1	no	—
253	C₄⋊C₄	—	109	1	no	—
254	C₄⋊C₄	—	218	1	no	—
255	C₄⋊C₄	—	218	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	C₂²×D₁₀₉	1	1	yes	characteristic
261	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
262	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
263	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
264	C₂²⋊C₄	—	2	1	no	—
265	C₂²⋊C₄	—	109	1	no	—
266	C₂²⋊C₄	—	109	1	no	—
267	C₂²⋊C₄	—	218	1	no	—
268	C₂²⋊C₄	—	218	1	no	—
269	C₂²⋊C₄	—	218	1	no	—
270	C₂²⋊C₄	—	218	1	no	—
271	C₂²⋊C₄	—	218	1	no	—
272	C₂²⋊C₄	—	218	1	no	—
273	C₄²	—	2	1	no	—
274	C₄²	—	109	1	no	—
275	C₄²	—	218	1	no	—
276	C₄²	—	218	1	no	—
277	C₂³	—	109	1	no	—
278	C₂³	—	218	1	no	—
279	C₂³	D₄×D₁₀₉	1	1	yes	characteristic
280	Q₈	—	436	2	no	—
281	Q₈	—	436	2	no	—
282	Q₈	—	218	1	no	—
283	Q₈	—	218	1	no	—
284	D₄	—	436	2	no	—
285	D₄	—	436	2	no	—
286	D₄	—	218	2	no	—
287	D₄	—	218	1	no	—
288	D₄	—	218	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	109	1	no	—
297	C₂×C₄	—	109	1	no	—
298	C₂×C₄	—	109	1	no	—
299	C₂×C₄	—	436	2	no	—
300	C₂×C₄	—	436	2	no	—
301	C₂×C₄	—	218	1	no	—
302	C₂×C₄	—	218	1	no	—
303	C₂×C₄	—	218	1	no	—
304	C₂×C₄	—	218	1	no	—
305	C₂×C₄	—	218	1	no	—
306	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
307	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	218	1	no	—
310	C₈	—	218	1	no	—
311	C₈	#13182029	1	2	yes	complemented
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	436	1	no	—
316	C₂²	—	436	1	no	—
317	C₂²	—	436	1	no	—
318	C₂²	—	218	1	no	—
319	C₂²	—	218	1	no	—
320	C₂²	—	218	1	no	—
321	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	4	1	no	—
324	C₄	—	4	1	no	—
325	C₄	—	4	1	no	—
326	C₄	—	4	1	no	—
327	C₄	—	109	2	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	C₂×#13182029	1	1	yes	characteristic
334	C₄	#16812747	1	1	yes	characteristic
335	C₂	—	4	1	no	—
336	C₂	—	872	1	no	—
337	C₂	—	436	1	no	—
338	C₂	#20295677	1	1	yes	characteristic
339	C₂	AGL(1,ℤ/8)×D₁₀₉	1	1	yes	characteristic
340	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	#428×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	#20295677	C₂
hypercenter	#428	D₁₀₉
hypocenter	C₁₀₉	#3353
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #428

Complements

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

N	H ≅ G/N	type
1. #20295744	C₂	semidirect
3. #20295430	C₂	semidirect
4. #20295363	C₂	semidirect
5. #20295361	C₂	semidirect
6. #20295352	C₂	semidirect
7. #20295350	C₂	semidirect
8. #20295310	C₂	semidirect
9. #20295305	C₂	semidirect
10. #20295273	C₂	semidirect
11. #20295272	C₂	semidirect
12. #20295113	C₂	semidirect
13. #20295102	C₂	semidirect
14. #20295065	C₂	semidirect
15. #428×C₁₀₉	C₂	semidirect
138. C₈×C₁₀₉	#41	semidirect
171. C₁₀₉	#3353	semidirect
186. #428	D₁₀₉	semidirect
311. C₈	#13182029	semidirect

Maximal subgroups

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

subgroup	index
#20295744	2
#20295677	2
#20295430	2
#20295363	2
#20295361	2
#20295352	2
#20295350	2
#20295310	2
#20295305	2
#20295273	2
#20295272	2
#20295113	2
#20295102	2
#20295065	2
#428×C₁₀₉	2
#3353	109