Subgroups — #28373598

Order 13952.

380 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295563	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295558	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295426	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295407	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295382	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295379	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295266	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295261	C₂	1	1	yes	maximal, characteristic
9	#20295256	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295252	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#13182019	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#13182018	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×#13182014	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×C₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×SD₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13182019	C₂²	1	1	yes	characteristic
17	C₂×#13182018	C₂²	1	1	yes	characteristic
18	C₂×#13182014	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
20	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
21	#16812694	—	2	1	no	—
22	#16812692	—	2	1	no	—
23	#16812690	C₂²	1	1	yes	characteristic
24	#16812690	C₂²	1	1	yes	characteristic
25	#16812688	C₂²	1	1	yes	characteristic
26	#16812686	C₂²	1	1	yes	characteristic
27	#16812684	—	2	1	no	—
28	#16812681	—	2	1	no	—
29	#16812679	C₂²	1	1	yes	characteristic
30	#16812675	C₂²	1	1	yes	characteristic
31	#16812674	C₂²	1	1	yes	characteristic
32	#16812670	C₂²	1	1	yes	characteristic
33	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
34	C₄×D₄₃₆	—	2	1	no	—
35	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
36	C₄×C₂²⋊D₁₀₉	—	2	1	no	—
37	#16812658	—	2	1	no	—
38	#16812656	—	2	1	no	—
39	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
40	#16812632	C₂²	1	1	yes	characteristic
41	#16812632	C₂²	1	1	yes	characteristic
42	#16812631	C₂²	1	1	yes	characteristic
43	#16812624	C₂²	1	1	yes	characteristic
44	#16812623	C₂²	1	1	yes	characteristic
45	#16812622	C₂²	1	1	yes	characteristic
46	#16812621	C₂²	1	1	yes	characteristic
47	#16812620	C₂²	1	1	yes	characteristic
48	#16812619	C₂²	1	1	yes	characteristic
49	C₄×#11724784	C₂²	1	1	yes	characteristic
50	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
55	#106×C₁₀₉	C₂²	1	1	yes	characteristic
56	#103×C₁₀₉	C₂²	1	1	yes	characteristic
57	#102×C₁₀₉	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
60	C₂×D₄₃₆	—	2	1	no	—
61	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
63	C₂×C₄×D₁₀₉	—	2	1	no	—
64	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
65	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
66	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
67	#13182019	—	2	2	no	—
68	#13182018	—	2	2	no	—
69	#13182014	—	2	2	no	—
70	C₂×#11724784	C₂³	1	1	yes	characteristic
71	C₂×#11724784	C₂³	1	1	yes	characteristic
72	C₈×D₁₀₉	—	2	2	no	—
73	C₈×D₁₀₉	—	2	2	no	—
74	C₄⋊Dic₁₀₉	—	2	1	no	—
75	C₄⋊Dic₁₀₉	—	2	1	no	—
76	C₄⋊Dic₁₀₉	—	2	1	no	—
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	C₂²⋊Dic₁₀₉	—	2	1	no	—
84	C₂²⋊Dic₁₀₉	—	2	1	no	—
85	C₂²⋊Dic₁₀₉	—	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	—	2	1	no	—
91	#13182006	—	2	1	no	—
92	C₄×Dic₁₀₉	—	2	1	no	—
93	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
94	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
95	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
96	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
97	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
98	C₂²×C₄×C₁₀₉	—	2	1	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	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
110	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
111	Q₈×C₁₀₉	—	2	1	no	—
112	Q₈×C₁₀₉	—	2	1	no	—
113	D₄×C₁₀₉	—	2	1	no	—
114	D₄×C₁₀₉	—	2	1	no	—
115	C₂×C₄×C₁₀₉	—	2	1	no	—
116	C₂×C₄×C₁₀₉	—	2	1	no	—
117	C₂×C₄×C₁₀₉	—	2	1	no	—
118	C₂×C₄×C₁₀₉	—	4	2	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	C₂×D₄	1	1	yes	characteristic
127	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
128	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
129	D₄₃₆	—	2	1	no	—
130	D₄₃₆	—	2	1	no	—
131	D₄₃₆	—	4	2	no	—
132	C₄×D₁₀₉	—	2	2	no	—
133	C₄×D₁₀₉	—	2	2	no	—
134	C₄×D₁₀₉	—	2	1	no	—
135	C₄×D₁₀₉	—	2	1	no	—
136	C₄×D₁₀₉	—	4	2	no	—
137	Dic₂₁₈	—	2	1	no	—
138	Dic₂₁₈	—	2	1	no	—
139	C₈×C₁₀₉	—	2	1	no	—
140	C₈×C₁₀₉	#41	1	2	yes	—
141	#11724784	—	2	1	no	—
142	#11724784	#41	1	2	yes	—
143	C₂²×C₁₀₉	—	4	1	no	—
144	C₂²×C₁₀₉	—	4	1	no	—
145	C₂²×C₁₀₉	—	4	1	no	—
146	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
147	C₂×D₁₀₉	—	2	1	no	—
148	C₂×D₁₀₉	—	2	1	no	—
149	C₂×D₁₀₉	—	4	1	no	—
150	C₂×D₁₀₉	—	4	1	no	—
151	C₂×D₁₀₉	—	4	1	no	—
152	C₂×D₁₀₉	#135	1	2	yes	—
153	C₄×C₁₀₉	—	2	1	no	—
154	C₄×C₁₀₉	—	4	1	no	—
155	C₄×C₁₀₉	—	4	1	no	—
156	C₄×C₁₀₉	—	4	1	no	—
157	C₄×C₁₀₉	#135	1	2	yes	—
158	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
159	C₄×C₁₀₉	C₂×#41	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	C₂×C₁₀₉	—	8	1	no	—
167	C₂×C₁₀₉	#532	1	1	yes	characteristic
168	C₂×C₁₀₉	C₂×#135	1	1	yes	characteristic
169	C₂×C₁₀₉	C₂×#135	1	1	yes	characteristic
170	D₁₀₉	—	2	2	no	—
171	D₁₀₉	—	8	1	no	—
172	#3168	—	109	1	no	maximal, Sylow
173	C₁₀₉	#3168	1	1	yes	characteristic, complemented, Sylow
174	#529	—	109	1	no	—
175	#529	—	109	1	no	—
176	#499	—	109	1	no	—
177	#495	—	109	1	no	—
178	#488	—	109	1	no	—
179	#488	—	109	1	no	—
180	#467	—	109	1	no	—
181	#466	—	109	1	no	—
182	#465	—	109	1	no	—
183	#465	—	109	1	no	—
184	C₄×Q₁₆	—	109	1	no	—
185	C₄×SD₁₆	—	109	1	no	—
186	C₄×SD₁₆	D₁₀₉	1	1	yes	characteristic, complemented
187	C₄×D₈	—	109	1	no	—
188	C₂×C₄×C₈	—	109	1	no	—
189	C₂×Q₁₆	—	109	1	no	—
190	C₂×SD₁₆	—	109	1	no	—
191	C₂×SD₁₆	C₂×D₁₀₉	1	1	yes	characteristic
192	C₂×D₈	—	109	1	no	—
193	C₂²×C₈	—	109	1	no	—
194	C₂²×C₈	—	109	1	no	—
195	#126	—	218	1	no	—
196	#126	—	218	1	no	—
197	#125	—	109	1	no	—
198	#125	—	109	1	no	—
199	#124	—	109	1	no	—
200	#124	—	109	1	no	—
201	#123	—	218	1	no	—
202	#123	—	218	1	no	—
203	C₂²⋊Q₈	—	109	1	no	—
204	C₂²⋊Q₈	—	109	1	no	—
205	#121	—	109	1	no	—
206	#121	—	109	1	no	—
207	C₄×Q₈	—	109	1	no	—
208	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
209	C₄×D₄	—	109	1	no	—
210	C₄×D₄	—	218	1	no	—
211	C₄×D₄	—	218	1	no	—
212	C₄×D₄	C₂×D₁₀₉	1	1	yes	characteristic
213	#117	—	218	1	no	—
214	#117	—	218	1	no	—
215	C₂×C₄²	—	109	1	no	—
216	#107	—	109	1	no	—
217	#107	—	109	1	no	—
218	#106	—	109	1	no	—
219	#106	C₂×D₁₀₉	1	1	yes	characteristic
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	—	109	1	no	—
226	#102	—	109	1	no	—
227	#102	C₂×D₁₀₉	1	1	yes	characteristic
228	C₄×C₈	—	109	1	no	—
229	C₄×C₈	—	109	1	no	—
230	C₄×C₈	—	109	1	no	—
231	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
232	C₂×Q₈	—	109	1	no	—
233	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
234	C₂×D₄	—	109	1	no	—
235	C₂×D₄	—	218	1	no	—
236	C₂×D₄	—	218	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	—	109	1	no	—
242	C₂²×C₄	—	218	1	no	—
243	Q₁₆	—	218	2	no	—
244	SD₁₆	—	2	2	no	—
245	SD₁₆	—	218	2	no	—
246	D₈	—	218	2	no	—
247	C₂×C₈	—	109	1	no	—
248	C₂×C₈	—	109	1	no	—
249	C₂×C₈	—	218	2	no	—
250	C₂×C₈	—	218	2	no	—
251	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
252	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
253	C₄⋊C₄	—	2	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	C₂²×D₁₀₉	1	1	yes	characteristic
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	C₂²⋊C₄	—	218	1	no	—
271	C₂²⋊C₄	—	218	1	no	—
272	C₂²⋊C₄	—	218	1	no	—
273	C₂²⋊C₄	—	218	1	no	—
274	C₂²⋊C₄	—	218	1	no	—
275	C₄²	—	2	1	no	—
276	C₄²	—	109	1	no	—
277	C₄²	—	109	1	no	—
278	C₄²	—	109	1	no	—
279	C₄²	—	218	1	no	—
280	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
281	C₂³	—	2	1	no	—
282	C₂³	—	109	1	no	—
283	C₂³	—	218	1	no	—
284	Q₈	—	2	1	no	—
285	Q₈	—	2	1	no	—
286	Q₈	—	218	1	no	—
287	Q₈	—	218	1	no	—
288	D₄	—	2	1	no	—
289	D₄	—	2	1	no	—
290	D₄	—	436	2	no	—
291	D₄	—	436	2	no	—
292	D₄	—	218	1	no	—
293	D₄	—	218	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	4	2	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	—	218	2	no	—
303	C₂×C₄	—	218	1	no	—
304	C₂×C₄	—	218	1	no	—
305	C₂×C₄	—	218	1	no	—
306	C₂×C₄	—	218	2	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	D₄×D₁₀₉	1	1	yes	characteristic
311	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
312	C₈	—	2	1	no	—
313	C₈	—	109	2	no	—
314	C₈	—	218	1	no	—
315	C₈	#13182029	1	2	yes	—
316	C₂²	—	4	1	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	109	2	no	—
320	C₂²	—	436	1	no	—
321	C₂²	—	436	1	no	—
322	C₂²	—	436	1	no	—
323	C₂²	—	218	1	no	—
324	C₂²	—	218	1	no	—
325	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	4	1	no	—
328	C₄	—	4	1	no	—
329	C₄	—	4	1	no	—
330	C₄	—	436	1	no	—
331	C₄	—	436	1	no	—
332	C₄	—	436	1	no	—
333	C₄	—	218	1	no	—
334	C₄	—	218	1	no	—
335	C₄	—	218	1	no	—
336	C₄	#16812720	1	2	yes	—
337	C₄	#41×D₁₀₉	1	1	yes	characteristic
338	C₄	C₂×#13182029	1	1	yes	characteristic
339	C₂	—	8	1	no	—
340	C₂	—	872	1	no	—
341	C₂	—	218	2	no	—
342	C₂	#20295573	1	1	yes	characteristic
343	C₂	#135×D₁₀₉	1	1	yes	characteristic
344	C₂	C₂×#16812720	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	C₄×SD₁₆×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	#20295261	C₂
hypercenter	C₄×SD₁₆	D₁₀₉
hypocenter	C₁₀₉	#3168
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×SD₁₆

Complements

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

N	H ≅ G/N	type
1. #20295563	C₂	semidirect
2. #20295558	C₂	semidirect
3. #20295426	C₂	semidirect
4. #20295407	C₂	semidirect
5. #20295382	C₂	semidirect
6. #20295379	C₂	semidirect
7. #20295266	C₂	semidirect
9. #20295256	C₂	semidirect
10. #20295252	C₂	semidirect
11. C₄×#13182019	C₂	semidirect
12. C₄×#13182018	C₂	semidirect
13. C₄×#13182014	C₂	semidirect
14. C₄×C₈×D₁₀₉	C₂	semidirect
15. C₄×SD₁₆×C₁₀₉	C₂	semidirect
173. C₁₀₉	#3168	semidirect
186. C₄×SD₁₆	D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295563	2
#20295558	2
#20295426	2
#20295407	2
#20295382	2
#20295379	2
#20295266	2
#20295261	2
#20295256	2
#20295252	2
C₄×#13182019	2
C₄×#13182018	2
C₄×#13182014	2
C₄×C₈×D₁₀₉	2
C₄×SD₁₆×C₁₀₉	2
#3168	109