Subgroups — #28373202

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	#20295555	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295360	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295343	C₂	1	1	yes	maximal, characteristic
5	#20295334	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295333	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295245	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295180	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295163	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	#20295082	C₂	1	1	yes	maximal, characteristic, complemented
15	#464×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₁₀₉	C₂²	1	1	yes	characteristic
17	C₂×#13182019	C₂²	1	1	yes	characteristic
18	C₂×#13182017	C₂²	1	1	yes	characteristic
19	#16812692	—	2	1	no	—
20	#16812691	C₂²	1	1	yes	characteristic
21	#16812681	—	2	1	no	—
22	#16812675	C₂²	1	1	yes	characteristic
23	#16812670	C₂²	1	1	yes	characteristic
24	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
25	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
26	C₄×Dic₂₁₈	—	2	1	no	—
27	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
28	#16812664	—	2	1	no	—
29	#16812664	C₂²	1	1	yes	characteristic
30	D₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
31	#16812659	—	2	1	no	—
32	#16812655	—	2	1	no	—
33	C₄⋊C₄×D₁₀₉	—	2	1	no	—
34	C₄⋊C₄×D₁₀₉	C₂²	1	1	yes	characteristic
35	C₄²×D₁₀₉	—	2	1	no	—
36	C₄²×D₁₀₉	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	C₄⋊#11724784	C₂²	1	1	yes	characteristic
42	#16812624	C₂²	1	1	yes	characteristic
43	#16812624	C₂²	1	1	yes	characteristic
44	#16812623	C₂²	1	1	yes	characteristic
45	#16812623	C₂²	1	1	yes	characteristic
46	#16812623	C₂²	1	1	yes	characteristic
47	#16812620	C₂²	1	1	yes	characteristic
48	#16812611	C₂²	1	1	yes	characteristic
49	#16812611	C₂²	1	1	yes	characteristic
50	#16812609	C₂²	1	1	yes	characteristic
51	#16812609	C₂²	1	1	yes	characteristic
52	C₂×Q₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
53	C₂×SD₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
54	#124×C₁₀₉	C₂²	1	1	yes	characteristic
55	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
57	#103×C₁₀₉	C₂²	1	1	yes	characteristic
58	#102×C₁₀₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₁₈	—	2	1	no	—
60	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
61	Q₈×D₁₀₉	—	2	2	no	—
62	Q₈×D₁₀₉	—	2	2	no	—
63	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
64	C₂²×Dic₁₀₉	—	2	1	no	—
65	C₂×C₄×D₁₀₉	—	2	1	no	—
66	C₂×C₄×D₁₀₉	—	2	1	no	—
67	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
68	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
69	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
70	#13182019	—	2	2	no	—
71	#13182017	—	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	—	2	1	no	—
76	C₄⋊Dic₁₀₉	C₂³	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	—	2	1	no	—
81	#13182008	—	2	1	no	—
82	#13182008	—	2	1	no	—
83	#13182008	—	2	1	no	—
84	#13182008	—	2	1	no	—
85	C₂²⋊Dic₁₀₉	—	2	1	no	—
86	C₂²⋊Dic₁₀₉	—	2	1	no	—
87	#13182006	—	2	1	no	—
88	C₄×Dic₁₀₉	—	2	1	no	—
89	C₄×Dic₁₀₉	—	2	1	no	—
90	C₄×Dic₁₀₉	—	2	1	no	—
91	C₄×Dic₁₀₉	—	2	1	no	—
92	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
93	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
94	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
95	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
96	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
97	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
98	Q₁₆×C₁₀₉	—	2	2	no	—
99	SD₁₆×C₁₀₉	—	2	2	no	—
100	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
101	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₁₀₉	—	2	1	no	—
103	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
104	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
105	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
106	C₄²×C₁₀₉	—	2	1	no	—
107	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
108	C₂³×C₁₀₉	—	2	1	no	—
109	C₂²×D₁₀₉	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	1	no	—
115	D₄×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	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
122	C₂²⋊D₁₀₉	—	4	2	no	—
123	C₂×Dic₁₀₉	—	2	1	no	—
124	C₂×Dic₁₀₉	—	2	1	no	—
125	C₂×Dic₁₀₉	—	2	1	no	—
126	C₂×Dic₁₀₉	—	2	1	no	—
127	C₂×Dic₁₀₉	—	4	2	no	—
128	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
129	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
131	C₄×D₁₀₉	—	2	4	no	—
132	C₄×D₁₀₉	—	2	2	no	—
133	C₄×D₁₀₉	—	2	1	no	—
134	C₄×D₁₀₉	—	2	1	no	—
135	C₄×D₁₀₉	—	4	1	no	—
136	C₄×D₁₀₉	—	4	2	no	—
137	Dic₂₁₈	—	2	4	no	—
138	Dic₂₁₈	—	2	1	no	—
139	Dic₂₁₈	#41	1	2	yes	—
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	—	4	1	no	—
145	C₂²×C₁₀₉	—	4	1	no	—
146	C₂²×C₁₀₉	—	4	1	no	—
147	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
148	C₂×D₁₀₉	—	2	1	no	—
149	C₂×D₁₀₉	—	2	1	no	—
150	C₂×D₁₀₉	#137	1	2	yes	—
151	C₄×C₁₀₉	—	2	2	no	—
152	C₄×C₁₀₉	—	2	1	no	—
153	C₄×C₁₀₉	—	2	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	C₂×#41	1	1	yes	characteristic
158	Dic₁₀₉	—	2	1	no	—
159	Dic₁₀₉	—	2	2	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	C₂×C₁₀₉	—	8	1	no	—
166	C₂×C₁₀₉	#525	1	1	yes	characteristic
167	C₂×C₁₀₉	C₂×#137	1	1	yes	characteristic
168	C₂×C₁₀₉	#575	1	1	yes	characteristic
169	D₁₀₉	—	2	2	no	—
170	#3117	—	109	1	no	maximal, Sylow
171	C₁₀₉	#3117	1	1	yes	characteristic, complemented, Sylow
172	#529	—	109	1	no	—
173	C₂×C₄×Q₈	—	109	1	no	—
174	#484	—	109	1	no	—
175	#481	—	109	1	no	—
176	#479	—	109	1	no	—
177	#479	—	109	1	no	—
178	#464	—	109	1	no	—
179	#464	D₁₀₉	1	1	yes	characteristic, complemented
180	#451	—	109	1	no	—
181	#448	—	109	1	no	—
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	C₂²×Q₈	—	109	1	no	—
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	#126	—	218	1	no	—
193	#125	—	109	1	no	—
194	#124	C₂×D₁₀₉	1	1	yes	characteristic
195	#123	—	218	1	no	—
196	C₂²⋊Q₈	—	109	1	no	—
197	#121	—	109	1	no	—
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×Q₈	—	218	1	no	—
204	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
205	C₄×D₄	—	109	1	no	—
206	C₄×D₄	—	218	1	no	—
207	#117	—	218	1	no	—
208	C₂×C₄⋊C₄	—	109	1	no	—
209	C₂×C₄⋊C₄	—	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	C₂×D₁₀₉	1	1	yes	characteristic
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	—	109	1	no	—
228	#97	—	109	1	no	—
229	#97	—	109	1	no	—
230	C₂×Q₈	—	109	1	no	—
231	C₂×Q₈	—	218	1	no	—
232	C₂×Q₈	—	218	2	no	—
233	C₂×Q₈	—	218	2	no	—
234	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
235	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
236	C₂×D₄	—	218	1	no	—
237	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
238	C₂²×C₄	—	109	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	—	218	1	no	—
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	—	2	1	no	—
267	C₂²⋊C₄	—	218	1	no	—
268	C₂²⋊C₄	—	218	1	no	—
269	C₂²⋊C₄	—	218	1	no	—
270	C₄²	—	2	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	C₂²×D₁₀₉	1	1	yes	characteristic
279	C₂³	—	2	1	no	—
280	C₂³	—	109	1	no	—
281	Q₈	—	2	1	no	—
282	Q₈	—	2	1	no	—
283	Q₈	—	2	1	no	—
284	Q₈	—	109	2	no	—
285	Q₈	—	218	4	no	—
286	Q₈	—	218	1	no	—
287	Q₈	#13182028	1	2	yes	—
288	D₄	—	2	1	no	—
289	D₄	—	2	1	no	—
290	D₄	—	436	2	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	109	1	no	—
295	C₂×C₄	—	109	1	no	—
296	C₂×C₄	—	109	1	no	—
297	C₂×C₄	—	436	1	no	—
298	C₂×C₄	—	436	2	no	—
299	C₂×C₄	—	436	2	no	—
300	C₂×C₄	—	218	2	no	—
301	C₂×C₄	—	218	4	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	—	218	1	no	—
307	C₂×C₄	—	218	1	no	—
308	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
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	—	4	1	no	—
316	C₂²	—	4	1	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	109	2	no	—
319	C₂²	—	218	1	no	—
320	C₂²	—	218	1	no	—
321	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
322	C₄	—	2	2	no	—
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	4	1	no	—
326	C₄	—	4	1	no	—
327	C₄	—	436	1	no	—
328	C₄	—	436	1	no	—
329	C₄	—	436	1	no	—
330	C₄	—	218	1	no	—
331	C₄	—	218	2	no	—
332	C₄	—	218	1	no	—
333	C₄	—	218	1	no	—
334	C₄	#41×D₁₀₉	1	1	yes	characteristic
335	C₄	C₂×#13182028	1	1	yes	characteristic
336	C₂	—	8	1	no	—
337	C₂	—	218	2	no	—
338	C₂	#20295536	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	#464×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	#16812670	C₂²
hypercenter	#464	D₁₀₉
hypocenter	C₁₀₉	#3117
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #464

Complements

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

N	H ≅ G/N	type
2. C₄×Q₈×D₁₀₉	C₂	semidirect
3. #20295360	C₂	semidirect
5. #20295334	C₂	semidirect
6. #20295333	C₂	semidirect
7. #20295245	C₂	semidirect
8. #20295180	C₂	semidirect
10. #20295133	C₂	semidirect
11. #20295130	C₂	semidirect
12. #20295127	C₂	semidirect
13. #20295123	C₂	semidirect
14. #20295082	C₂	semidirect
15. #464×C₁₀₉	C₂	semidirect
171. C₁₀₉	#3117	semidirect
179. #464	D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295555	2
C₄×Q₈×D₁₀₉	2
#20295360	2
#20295343	2
#20295334	2
#20295333	2
#20295245	2
#20295180	2
#20295163	2
#20295133	2
#20295130	2
#20295127	2
#20295123	2
#20295082	2
#464×C₁₀₉	2
#3117	109