Subgroups — #28375756

Order 13952.

382 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	#20295752	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295696	C₂	1	1	yes	maximal, characteristic
3	#20295390	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295389	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295371	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295328	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295327	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295304	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295224	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295224	C₂	1	1	yes	maximal, characteristic, complemented
11	#20295155	C₂	1	1	yes	maximal, characteristic, complemented
12	#20295127	C₂	1	1	yes	maximal, characteristic, complemented
13	#20295124	C₂	1	1	yes	maximal, characteristic, complemented
14	SD₁₆×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
15	#460×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13182028	—	2	1	no	—
17	C₂×#13182016	C₂²	1	1	yes	characteristic
18	#16812698	—	2	1	no	—
19	#16812698	C₂²	1	1	yes	characteristic
20	#16812695	C₂²	1	1	yes	characteristic
21	#16812695	C₂²	1	1	yes	characteristic
22	#16812691	—	2	1	no	—
23	#16812691	C₂²	1	1	yes	characteristic
24	#16812690	—	2	1	no	—
25	#16812690	C₂²	1	1	yes	characteristic
26	#16812671	—	2	1	no	—
27	#16812671	—	2	1	no	—
28	#16812670	—	2	1	no	—
29	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
30	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
31	C₄×Dic₂₁₈	—	2	1	no	—
32	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
33	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
34	D₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
35	D₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
36	#16812633	C₂²	1	1	yes	characteristic
37	#16812632	C₂²	1	1	yes	characteristic
38	#16812631	C₂²	1	1	yes	characteristic
39	#16812628	C₂²	1	1	yes	characteristic
40	#16812628	C₂²	1	1	yes	characteristic
41	#16812623	C₂²	1	1	yes	characteristic
42	#16812621	C₂²	1	1	yes	characteristic
43	#16812621	C₂²	1	1	yes	characteristic
44	#16812620	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	#16812619	C₂²	1	1	yes	characteristic
49	#16812610	C₂²	1	1	yes	characteristic
50	#16812609	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
53	C₂×SD₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
54	#127×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	#102×C₁₀₉	C₂²	1	1	yes	characteristic
58	#102×C₁₀₉	C₂²	1	1	yes	characteristic
59	#13182028	—	4	4	no	—
60	C₂×Dic₂₁₈	—	2	1	no	—
61	C₂×Dic₂₁₈	D₄	1	1	yes	characteristic
62	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
64	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
65	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
66	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
67	C₂²×Dic₁₀₉	—	2	1	no	—
68	C₂²×Dic₁₀₉	—	2	1	no	—
69	C₂×C₄×D₁₀₉	—	2	1	no	—
70	#13182016	—	2	2	no	—
71	C₂×#11724784	C₂³	1	1	yes	characteristic
72	C₂×#11724784	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₁₀₉	—	2	1	no	—
74	C₄⋊Dic₁₀₉	—	2	1	no	—
75	C₄⋊Dic₁₀₉	—	2	1	no	—
76	C₄⋊Dic₁₀₉	D₄	1	1	yes	characteristic
77	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
79	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
80	#13182008	—	2	1	no	—
81	#13182008	—	2	1	no	—
82	#13182008	—	2	1	no	—
83	#13182008	—	2	1	no	—
84	#13182008	D₄	1	1	yes	characteristic
85	#13182008	D₄	1	1	yes	characteristic
86	C₂²⋊Dic₁₀₉	—	2	1	no	—
87	C₂²⋊Dic₁₀₉	—	2	1	no	—
88	#13182006	—	2	1	no	—
89	C₄×Dic₁₀₉	—	2	1	no	—
90	C₄×Dic₁₀₉	—	2	1	no	—
91	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
92	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
93	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
94	C₂×D₄×C₁₀₉	—	2	1	no	—
95	C₂×D₄×C₁₀₉	—	2	1	no	—
96	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
97	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
98	SD₁₆×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	C₄²×C₁₀₉	—	2	1	no	—
105	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
106	C₂³×C₁₀₉	—	2	1	no	—
107	C₂³×C₁₀₉	—	2	1	no	—
108	C₂²×D₁₀₉	—	2	1	no	—
109	Q₈×C₁₀₉	—	2	1	no	—
110	Q₈×C₁₀₉	#41	1	2	yes	complemented
111	D₄×C₁₀₉	—	2	1	no	—
112	D₄×C₁₀₉	—	2	1	no	—
113	D₄×C₁₀₉	—	2	1	no	—
114	D₄×C₁₀₉	—	2	1	no	—
115	D₄×C₁₀₉	—	4	2	no	—
116	D₄×C₁₀₉	—	8	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	C₂²⋊D₁₀₉	—	4	2	no	—
124	C₂²⋊D₁₀₉	—	8	1	no	—
125	C₂²⋊D₁₀₉	—	8	1	no	—
126	C₂×Dic₁₀₉	—	2	1	no	—
127	C₂×Dic₁₀₉	—	2	1	no	—
128	C₂×Dic₁₀₉	—	2	1	no	—
129	C₂×Dic₁₀₉	—	4	2	no	—
130	C₂×Dic₁₀₉	—	4	2	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	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
135	D₄₃₆	—	2	1	no	—
136	D₄₃₆	—	2	1	no	—
137	C₄×D₁₀₉	—	4	2	no	—
138	Dic₂₁₈	—	2	2	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	—	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	—	4	1	no	—
152	C₂×D₁₀₉	—	4	1	no	—
153	C₂×D₁₀₉	—	4	1	no	—
154	C₄×C₁₀₉	—	2	2	no	—
155	C₄×C₁₀₉	—	2	1	no	—
156	C₄×C₁₀₉	—	2	1	no	—
157	C₄×C₁₀₉	—	4	1	no	—
158	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
159	C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
160	Dic₁₀₉	—	2	1	no	—
161	Dic₁₀₉	—	2	1	no	—
162	Dic₁₀₉	—	2	1	no	—
163	Dic₁₀₉	—	4	1	no	—
164	Dic₁₀₉	—	4	1	no	—
165	Dic₁₀₉	—	4	1	no	—
166	Dic₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
167	C₂×C₁₀₉	—	8	1	no	—
168	C₂×C₁₀₉	—	8	1	no	—
169	C₂×C₁₀₉	#548	1	1	yes	characteristic
170	C₂×C₁₀₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
171	C₂×C₁₀₉	#577	1	1	yes	characteristic
172	D₁₀₉	—	8	1	no	—
173	#3368	—	109	1	no	maximal, Sylow
174	C₁₀₉	#3368	1	1	yes	characteristic, complemented, Sylow
175	#557	—	109	1	no	—
176	#550	—	109	1	no	—
177	#490	—	109	1	no	—
178	#490	—	109	1	no	—
179	#486	—	109	1	no	—
180	#478	—	109	1	no	—
181	#478	—	109	1	no	—
182	#474	—	109	1	no	—
183	#460	—	109	1	no	—
184	#460	—	109	1	no	—
185	#460	D₁₀₉	1	1	yes	characteristic, complemented
186	#446	—	109	1	no	—
187	#440	—	109	1	no	—
188	#440	—	109	1	no	—
189	C₄×SD₁₆	—	109	1	no	—
190	C₂×#41	—	218	1	no	—
191	C₂×SD₁₆	—	109	1	no	—
192	C₂×SD₁₆	C₂×D₁₀₉	1	1	yes	characteristic
193	C₂×SD₁₆	C₂×D₁₀₉	1	1	yes	characteristic
194	C₄⋊Q₈	—	109	1	no	—
195	C₄⋊Q₈	—	218	1	no	—
196	#127	—	109	1	no	—
197	#127	—	109	1	no	—
198	#127	C₂×D₁₀₉	1	1	yes	characteristic
199	#125	—	109	1	no	—
200	#125	—	109	1	no	—
201	#125	—	218	1	no	—
202	#125	—	218	1	no	—
203	#121	—	218	1	no	—
204	#121	—	218	1	no	—
205	#121	—	218	1	no	—
206	C₄×Q₈	—	109	1	no	—
207	C₄×Q₈	—	109	1	no	—
208	C₄×Q₈	—	109	1	no	—
209	C₄×Q₈	—	218	1	no	—
210	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
211	C₄×D₄	—	109	1	no	—
212	C₄×D₄	—	109	1	no	—
213	C₄×D₄	—	109	1	no	—
214	#107	—	109	1	no	—
215	#107	—	109	1	no	—
216	#106	—	109	1	no	—
217	C₄⋊C₈	—	109	1	no	—
218	C₄⋊C₈	—	109	1	no	—
219	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
220	#103	—	109	1	no	—
221	#102	—	109	1	no	—
222	#102	—	109	1	no	—
223	#102	—	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	#97	—	109	1	no	—
231	#97	—	109	1	no	—
232	C₄×C₈	—	109	1	no	—
233	#41	—	436	4	no	—
234	C₂×Q₈	—	109	1	no	—
235	C₂×Q₈	—	218	1	no	—
236	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
237	C₂×D₄	—	2	1	no	—
238	C₂×D₄	—	2	1	no	—
239	C₂×D₄	—	109	1	no	—
240	C₂×D₄	—	218	1	no	—
241	C₂×D₄	—	218	1	no	—
242	C₂×D₄	—	218	1	no	—
243	C₂×D₄	—	218	1	no	—
244	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
245	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
246	C₂²×C₄	—	218	1	no	—
247	C₂²×C₄	—	218	1	no	—
248	C₂²×C₄	—	218	1	no	—
249	SD₁₆	—	2	2	no	—
250	SD₁₆	—	2	2	no	—
251	SD₁₆	—	218	2	no	—
252	C₂×C₈	—	109	1	no	—
253	C₂×C₈	—	109	1	no	—
254	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
255	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
256	C₄⋊C₄	—	2	1	no	—
257	C₄⋊C₄	—	109	1	no	—
258	C₄⋊C₄	—	109	1	no	—
259	C₄⋊C₄	—	109	1	no	—
260	C₄⋊C₄	—	109	1	no	—
261	C₄⋊C₄	—	109	1	no	—
262	C₄⋊C₄	—	109	1	no	—
263	C₄⋊C₄	—	218	1	no	—
264	C₄⋊C₄	—	218	1	no	—
265	C₄⋊C₄	—	218	1	no	—
266	C₄⋊C₄	—	218	1	no	—
267	C₄⋊C₄	—	218	1	no	—
268	C₄⋊C₄	—	218	1	no	—
269	C₄⋊C₄	—	218	1	no	—
270	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
271	C₂²⋊C₄	—	218	1	no	—
272	C₂²⋊C₄	—	218	1	no	—
273	C₂²⋊C₄	—	218	1	no	—
274	C₄²	—	2	1	no	—
275	C₄²	—	109	1	no	—
276	C₄²	—	109	1	no	—
277	C₄²	—	218	1	no	—
278	C₄²	—	218	1	no	—
279	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
280	C₂³	—	2	1	no	—
281	C₂³	—	2	1	no	—
282	C₂³	—	218	1	no	—
283	Q₈	—	2	1	no	—
284	Q₈	—	436	2	no	—
285	Q₈	—	218	2	no	—
286	Q₈	#13182028	1	2	yes	complemented
287	D₄	—	2	1	no	—
288	D₄	—	2	1	no	—
289	D₄	—	2	1	no	—
290	D₄	—	2	1	no	—
291	D₄	—	4	2	no	—
292	D₄	—	8	1	no	—
293	D₄	—	872	1	no	—
294	D₄	—	872	1	no	—
295	D₄	—	436	2	no	—
296	D₄	—	436	2	no	—
297	D₄	—	218	1	no	—
298	D₄	—	218	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	109	1	no	—
302	C₂×C₄	—	109	1	no	—
303	C₂×C₄	—	109	1	no	—
304	C₂×C₄	—	109	1	no	—
305	C₂×C₄	—	436	2	no	—
306	C₂×C₄	—	436	2	no	—
307	C₂×C₄	—	436	2	no	—
308	C₂×C₄	—	218	1	no	—
309	C₂×C₄	—	218	1	no	—
310	C₂×C₄	—	218	1	no	—
311	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
312	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
313	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
314	C₈	—	2	1	no	—
315	C₈	—	2	1	no	—
316	C₈	—	218	1	no	—
317	C₈	—	218	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	—	436	1	no	—
325	C₂²	—	436	1	no	—
326	C₂²	—	436	1	no	—
327	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
328	C₄	—	2	2	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	4	1	no	—
332	C₄	—	109	2	no	—
333	C₄	—	436	1	no	—
334	C₄	—	436	1	no	—
335	C₄	—	436	1	no	—
336	C₄	—	218	1	no	—
337	C₄	—	218	1	no	—
338	C₄	—	218	1	no	—
339	C₄	C₂×#13182028	1	1	yes	characteristic
340	C₄	#16812746	1	1	yes	characteristic
341	C₂	—	8	1	no	—
342	C₂	—	8	1	no	—
343	C₂	—	872	1	no	—
344	C₂	#20295682	1	1	yes	characteristic
345	C₂	AGL(1,ℤ/8)×D₁₀₉	1	1	yes	characteristic
346	C₂	#20295832	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#460×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	#20295696	C₂
hypercenter	#460	D₁₀₉
hypocenter	C₁₀₉	#3368
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #460

Complements

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

N	H ≅ G/N	type
1. #20295752	C₂	semidirect
3. #20295390	C₂	semidirect
4. #20295389	C₂	semidirect
5. #20295371	C₂	semidirect
6. #20295328	C₂	semidirect
7. #20295327	C₂	semidirect
8. #20295304	C₂	semidirect
9. #20295224	C₂	semidirect
10. #20295224	C₂	semidirect
11. #20295155	C₂	semidirect
12. #20295127	C₂	semidirect
13. #20295124	C₂	semidirect
14. SD₁₆×Dic₁₀₉	C₂	semidirect
15. #460×C₁₀₉	C₂	semidirect
110. Q₈×C₁₀₉	#41	semidirect
174. C₁₀₉	#3368	semidirect
185. #460	D₁₀₉	semidirect
286. Q₈	#13182028	semidirect

Maximal subgroups

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

subgroup	index
#20295752	2
#20295696	2
#20295390	2
#20295389	2
#20295371	2
#20295328	2
#20295327	2
#20295304	2
#20295224	2
#20295155	2
#20295127	2
#20295124	2
SD₁₆×Dic₁₀₉	2
#460×C₁₀₉	2
#3368	109