Subgroups — #28373286

Order 13952.

394 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295639	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295498	C₂	1	1	yes	maximal, characteristic
3	#20295370	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295369	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295323	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295232	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295231	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295178	C₂	1	1	yes	maximal, characteristic
9	#20295178	C₂	1	1	yes	maximal, characteristic
10	#20295125	C₂	1	1	yes	maximal, characteristic, complemented
11	#20295123	C₂	1	1	yes	maximal, characteristic, complemented
12	#20295117	C₂	1	1	yes	maximal, characteristic, complemented
13	#20295113	C₂	1	1	yes	maximal, characteristic, complemented
14	#20295086	C₂	1	1	yes	maximal, characteristic, complemented
15	#478×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×D₄×D₁₀₉	C₂²	1	1	yes	characteristic
17	C₂×#13182018	C₂²	1	1	yes	characteristic
18	C₂×#13182018	C₂²	1	1	yes	characteristic
19	C₂×#13182017	C₂²	1	1	yes	characteristic
20	C₂×#13182017	C₂²	1	1	yes	characteristic
21	#16812699	C₂²	1	1	yes	characteristic, complemented
22	#16812694	—	2	1	no	—
23	#16812694	—	2	1	no	—
24	#16812690	—	2	1	no	—
25	#16812690	—	2	1	no	—
26	#16812679	C₂²	1	1	yes	characteristic
27	#16812679	C₂²	1	1	yes	characteristic
28	#16812664	C₂²	1	1	yes	characteristic, complemented
29	#16812664	C₂²	1	1	yes	characteristic, complemented
30	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
31	D₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
32	#16812661	C₂²	1	1	yes	characteristic
33	C₄×C₂²⋊D₁₀₉	—	2	1	no	—
34	#16812659	—	2	1	no	—
35	#16812657	C₂²	1	1	yes	characteristic
36	#16812656	C₂²	1	1	yes	characteristic
37	C₂×#13182006	—	2	1	no	—
38	C₂²⋊C₄×D₁₀₉	—	2	1	no	—
39	#16812632	C₂²	1	1	yes	characteristic, complemented
40	#16812630	C₂²	1	1	yes	characteristic, complemented
41	C₄⋊#11724784	C₂²	1	1	yes	characteristic, complemented
42	#16812624	C₂²	1	1	yes	characteristic, complemented
43	#16812624	C₂²	1	1	yes	characteristic, complemented
44	#16812622	C₂²	1	1	yes	characteristic, complemented
45	#16812622	C₂²	1	1	yes	characteristic, complemented
46	#16812619	C₂²	1	1	yes	characteristic
47	#16812619	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, complemented
51	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic, complemented
52	#125×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
53	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
54	#107×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
55	#106×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
56	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
57	#102×C₁₀₉	C₂²	1	1	yes	characteristic
58	#102×C₁₀₉	C₂²	1	1	yes	characteristic
59	C₂³×D₁₀₉	—	2	1	no	—
60	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
61	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
62	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
64	D₄×D₁₀₉	—	2	2	no	—
65	D₄×D₁₀₉	—	2	2	no	—
66	C₂²×Dic₁₀₉	—	2	1	no	—
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	#13182018	—	2	2	no	—
72	#13182017	—	2	2	no	—
73	#13182017	—	2	2	no	—
74	C₂×#11724784	C₂³	1	1	yes	characteristic
75	C₂×#11724784	C₂³	1	1	yes	characteristic
76	C₄⋊Dic₁₀₉	—	2	1	no	—
77	C₄⋊Dic₁₀₉	—	2	1	no	—
78	C₄⋊Dic₁₀₉	—	2	1	no	—
79	C₄⋊Dic₁₀₉	—	2	1	no	—
80	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
81	#13182008	—	2	1	no	—
82	#13182008	—	2	1	no	—
83	#13182008	D₄	1	1	yes	characteristic
84	#13182008	D₄	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
92	#13182006	D₄	1	1	yes	characteristic
93	C₄×Dic₁₀₉	—	2	1	no	—
94	C₄×Dic₁₀₉	—	2	1	no	—
95	C₄×Dic₁₀₉	—	2	1	no	—
96	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
97	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
98	C₂²×C₄×C₁₀₉	—	2	1	no	—
99	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
100	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₁₀₉	—	2	1	no	—
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	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
106	C₂²⋊C₄×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	—	2	4	no	—
111	C₂²×D₁₀₉	—	8	1	no	—
112	C₂²×D₁₀₉	—	8	1	no	—
113	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
114	D₄×C₁₀₉	—	2	1	no	—
115	D₄×C₁₀₉	#41	1	2	yes	—
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	—	8	1	no	—
120	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
123	C₂²⋊D₁₀₉	—	2	4	no	—
124	C₂×Dic₁₀₉	—	2	1	no	—
125	C₂×Dic₁₀₉	—	2	1	no	—
126	C₂×Dic₁₀₉	—	2	1	no	—
127	C₂×Dic₁₀₉	—	8	1	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	D₄₃₆	—	2	1	no	—
132	D₄₃₆	#41	1	2	yes	—
133	C₄×D₁₀₉	—	2	1	no	—
134	C₄×D₁₀₉	—	2	1	no	—
135	C₄×D₁₀₉	—	8	1	no	—
136	C₄×D₁₀₉	—	8	1	no	—
137	Dic₂₁₈	—	2	1	no	—
138	Dic₂₁₈	—	2	1	no	—
139	Dic₂₁₈	—	2	1	no	—
140	Dic₂₁₈	—	2	1	no	—
141	C₈×C₁₀₉	—	2	1	no	—
142	C₈×C₁₀₉	—	2	1	no	—
143	#11724784	—	2	1	no	—
144	#11724784	—	2	1	no	—
145	C₂²×C₁₀₉	—	2	2	no	—
146	C₂²×C₁₀₉	—	4	1	no	—
147	C₂²×C₁₀₉	—	4	1	no	—
148	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
149	C₂×D₁₀₉	—	2	1	no	—
150	C₂×D₁₀₉	—	2	2	no	—
151	C₂×D₁₀₉	—	2	1	no	—
152	C₂×D₁₀₉	—	2	1	no	—
153	C₂×D₁₀₉	—	4	1	no	—
154	C₂×D₁₀₉	—	4	1	no	—
155	C₂×D₁₀₉	—	8	4	no	—
156	C₄×C₁₀₉	—	2	1	no	—
157	C₄×C₁₀₉	—	2	1	no	—
158	C₄×C₁₀₉	—	4	1	no	—
159	C₄×C₁₀₉	—	4	1	no	—
160	C₄×C₁₀₉	—	4	1	no	—
161	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
162	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
163	Dic₁₀₉	—	2	1	no	—
164	Dic₁₀₉	—	2	1	no	—
165	Dic₁₀₉	—	2	1	no	—
166	Dic₁₀₉	—	4	1	no	—
167	Dic₁₀₉	—	4	1	no	—
168	Dic₁₀₉	—	4	1	no	—
169	C₂×C₁₀₉	—	4	2	no	—
170	C₂×C₁₀₉	#525	1	1	yes	characteristic
171	C₂×C₁₀₉	#577	1	1	yes	characteristic
172	C₂×C₁₀₉	#577	1	1	yes	characteristic
173	D₁₀₉	—	4	1	no	—
174	D₁₀₉	—	4	2	no	—
175	#3123	—	109	1	no	maximal, Sylow
176	C₁₀₉	#3123	1	1	yes	characteristic, complemented, Sylow
177	#541	—	109	1	no	—
178	#518	—	109	1	no	—
179	#485	—	109	1	no	—
180	#485	—	109	1	no	—
181	#478	—	109	1	no	—
182	#478	D₁₀₉	1	1	yes	characteristic, complemented
183	#461	—	109	1	no	—
184	#461	—	109	1	no	—
185	#450	—	109	1	no	—
186	#450	—	109	1	no	—
187	#440	—	109	1	no	—
188	#440	—	109	1	no	—
189	C₄×SD₁₆	—	109	1	no	—
190	C₄×SD₁₆	—	109	1	no	—
191	#433	—	109	1	no	—
192	C₂²×D₄	—	109	1	no	—
193	C₂×SD₁₆	—	109	1	no	—
194	C₂×SD₁₆	—	109	1	no	—
195	C₂×SD₁₆	—	109	1	no	—
196	C₂×SD₁₆	—	109	1	no	—
197	C₄⋊Q₈	—	109	1	no	—
198	#126	—	218	1	no	—
199	#126	—	218	1	no	—
200	#125	—	218	1	no	—
201	#125	—	218	1	no	—
202	#125	C₂×D₁₀₉	1	1	yes	characteristic, complemented
203	C₂²⋊Q₈	—	109	1	no	—
204	C₂²⋊Q₈	—	109	1	no	—
205	C₄×Q₈	—	109	1	no	—
206	C₄×Q₈	—	109	1	no	—
207	C₄×D₄	—	109	1	no	—
208	C₄×D₄	—	109	1	no	—
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	—	109	1	no	—
214	#117	—	109	1	no	—
215	C₂×C₂²⋊C₄	—	218	1	no	—
216	C₂×C₂²⋊C₄	—	218	1	no	—
217	#107	—	109	1	no	—
218	#107	C₂×D₁₀₉	1	1	yes	characteristic, complemented
219	#106	—	109	1	no	—
220	#106	C₂×D₁₀₉	1	1	yes	characteristic, complemented
221	C₄⋊C₈	—	109	1	no	—
222	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic, complemented
223	#103	—	109	1	no	—
224	#103	—	109	1	no	—
225	#103	—	109	1	no	—
226	#103	—	109	1	no	—
227	#102	—	109	1	no	—
228	#102	—	109	1	no	—
229	#102	C₂×D₁₀₉	1	1	yes	characteristic
230	#102	C₂×D₁₀₉	1	1	yes	characteristic
231	C₂²⋊C₈	—	109	1	no	—
232	C₂²⋊C₈	—	109	1	no	—
233	#97	—	109	1	no	—
234	C₄×C₈	—	109	1	no	—
235	C₂⁴	—	218	1	no	—
236	C₂×Q₈	—	109	1	no	—
237	C₂×Q₈	—	109	1	no	—
238	C₂×D₄	—	109	1	no	—
239	C₂×D₄	—	218	1	no	—
240	C₂×D₄	—	218	2	no	—
241	C₂×D₄	—	218	2	no	—
242	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
243	C₂²×C₄	—	2	1	no	—
244	C₂²×C₄	—	109	1	no	—
245	C₂²×C₄	—	218	1	no	—
246	C₂²×C₄	—	218	1	no	—
247	C₂²×C₄	—	218	1	no	—
248	SD₁₆	—	218	2	no	—
249	SD₁₆	—	218	2	no	—
250	SD₁₆	—	218	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	—	2	1	no	—
258	C₄⋊C₄	—	109	1	no	—
259	C₄⋊C₄	—	109	1	no	—
260	C₄⋊C₄	—	109	1	no	—
261	C₄⋊C₄	—	218	1	no	—
262	C₄⋊C₄	—	218	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	C₂²×D₁₀₉	1	1	yes	characteristic
268	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
269	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
270	C₂²⋊C₄	—	2	1	no	—
271	C₂²⋊C₄	—	109	1	no	—
272	C₂²⋊C₄	—	109	1	no	—
273	C₂²⋊C₄	—	218	1	no	—
274	C₂²⋊C₄	—	218	1	no	—
275	C₂²⋊C₄	—	218	1	no	—
276	C₂²⋊C₄	—	218	1	no	—
277	C₂²⋊C₄	—	218	1	no	—
278	C₂²⋊C₄	—	218	1	no	—
279	C₄²	—	109	1	no	—
280	C₄²	—	218	1	no	—
281	C₄²	—	218	1	no	—
282	C₄²	—	218	1	no	—
283	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
284	C₂³	—	2	1	no	—
285	C₂³	—	872	1	no	—
286	C₂³	—	872	1	no	—
287	C₂³	—	109	1	no	—
288	C₂³	—	218	4	no	—
289	C₂³	—	218	1	no	—
290	Q₈	—	218	1	no	—
291	Q₈	—	218	1	no	—
292	Q₈	—	218	1	no	—
293	Q₈	—	218	1	no	—
294	D₄	—	2	1	no	—
295	D₄	—	109	2	no	—
296	D₄	—	218	4	no	—
297	D₄	—	218	1	no	—
298	D₄	#13182030	1	2	yes	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	8	1	no	—
303	C₂×C₄	—	872	1	no	—
304	C₂×C₄	—	872	1	no	—
305	C₂×C₄	—	872	1	no	—
306	C₂×C₄	—	109	1	no	—
307	C₂×C₄	—	109	1	no	—
308	C₂×C₄	—	109	1	no	—
309	C₂×C₄	—	218	1	no	—
310	C₂×C₄	—	218	1	no	—
311	C₂×C₄	—	218	1	no	—
312	C₂×C₄	—	218	1	no	—
313	C₂×C₄	—	218	1	no	—
314	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
315	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
316	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
317	C₈	—	2	1	no	—
318	C₈	—	2	1	no	—
319	C₈	—	218	1	no	—
320	C₈	—	218	1	no	—
321	C₂²	—	2	2	no	—
322	C₂²	—	4	1	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	872	4	no	—
325	C₂²	—	436	1	no	—
326	C₂²	—	436	1	no	—
327	C₂²	—	218	1	no	—
328	C₂²	—	218	2	no	—
329	C₂²	—	218	1	no	—
330	C₂²	—	218	1	no	—
331	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	4	1	no	—
335	C₄	—	4	1	no	—
336	C₄	—	4	1	no	—
337	C₄	—	436	1	no	—
338	C₄	—	436	1	no	—
339	C₄	—	436	1	no	—
340	C₄	—	218	1	no	—
341	C₄	—	218	1	no	—
342	C₄	—	218	1	no	—
343	C₄	#41×D₁₀₉	1	1	yes	characteristic
344	C₄	C₂×#13182030	1	1	yes	characteristic
345	C₂	—	4	2	no	—
346	C₂	—	436	1	no	—
347	C₂	—	436	2	no	—
348	C₂	#20295539	1	1	yes	characteristic
349	C₂	#20295833	1	1	yes	characteristic
350	C₂	#20295836	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#478×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂×D₄×D₁₀₉	C₂²
hypercenter	#478	D₁₀₉
hypocenter	C₁₀₉	#3123
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #478

Complements

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

N	H ≅ G/N	type
1. #20295639	C₂	semidirect
3. #20295370	C₂	semidirect
4. #20295369	C₂	semidirect
5. #20295323	C₂	semidirect
6. #20295232	C₂	semidirect
7. #20295231	C₂	semidirect
10. #20295125	C₂	semidirect
11. #20295123	C₂	semidirect
12. #20295117	C₂	semidirect
13. #20295113	C₂	semidirect
14. #20295086	C₂	semidirect
15. #478×C₁₀₉	C₂	semidirect
21. #16812699	C₂²	semidirect
28. #16812664	C₂²	semidirect
29. #16812664	C₂²	semidirect
39. #16812632	C₂²	semidirect
40. #16812630	C₂²	semidirect
41. C₄⋊#11724784	C₂²	semidirect
42. #16812624	C₂²	semidirect
43. #16812624	C₂²	semidirect
44. #16812622	C₂²	semidirect
45. #16812622	C₂²	semidirect
50. #16812609	C₂²	semidirect
51. C₈×Dic₁₀₉	C₂²	semidirect
52. #125×C₁₀₉	C₂²	semidirect
54. #107×C₁₀₉	C₂²	semidirect
55. #106×C₁₀₉	C₂²	semidirect
56. C₄⋊C₈×C₁₀₉	C₂²	semidirect
176. C₁₀₉	#3123	semidirect
182. #478	D₁₀₉	semidirect
202. #125	C₂×D₁₀₉	semidirect
218. #107	C₂×D₁₀₉	semidirect
220. #106	C₂×D₁₀₉	semidirect
222. C₄⋊C₈	C₂×D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295639	2
#20295498	2
#20295370	2
#20295369	2
#20295323	2
#20295232	2
#20295231	2
#20295178	2
#20295125	2
#20295123	2
#20295117	2
#20295113	2
#20295086	2
#478×C₁₀₉	2
#3123	109