Subgroups — #28373588

Order 13952.

412 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295580	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295573	C₂	1	1	yes	maximal, characteristic
3	#20295423	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295397	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295375	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295371	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295258	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295256	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295254	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295252	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#13182018	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#13182016	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×#13182017	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×#13182029	—	2	1	no	—
17	C₂×#13182018	C₂²	1	1	yes	characteristic
18	C₂×#13182017	C₂²	1	1	yes	characteristic
19	C₂×#13182016	C₂²	1	1	yes	characteristic
20	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
21	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
22	#16812698	C₂²	1	1	yes	characteristic
23	#16812698	C₂²	1	1	yes	characteristic
24	#16812697	C₂²	1	1	yes	characteristic
25	#16812695	C₂²	1	1	yes	characteristic
26	#16812689	—	2	1	no	—
27	#16812685	—	2	1	no	—
28	#16812679	C₂²	1	1	yes	characteristic
29	#16812677	—	2	1	no	—
30	#16812675	C₂²	1	1	yes	characteristic
31	#16812674	C₂²	1	1	yes	characteristic
32	#16812672	—	2	1	no	—
33	#16812670	C₂²	1	1	yes	characteristic
34	C₄×Dic₂₁₈	—	2	1	no	—
35	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
36	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
37	C₄×C₂²⋊D₁₀₉	—	2	1	no	—
38	#16812656	—	2	1	no	—
39	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
40	#16812631	C₂²	1	1	yes	characteristic
41	#16812630	C₂²	1	1	yes	characteristic
42	#16812630	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	#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	—	2	1	no	—
67	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
68	C₂×C₄×D₁₀₉	—	2	1	no	—
69	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
70	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
71	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
72	#13182018	—	2	2	no	—
73	#13182017	—	2	2	no	—
74	#13182016	—	2	2	no	—
75	C₂×#11724784	C₂³	1	1	yes	characteristic
76	C₂×#11724784	C₂³	1	1	yes	characteristic
77	C₈×D₁₀₉	—	2	2	no	—
78	C₈×D₁₀₉	—	2	2	no	—
79	C₄⋊Dic₁₀₉	—	2	1	no	—
80	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
81	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
82	#13182008	—	2	1	no	—
83	#13182008	—	2	1	no	—
84	#13182008	—	2	1	no	—
85	#13182008	—	2	1	no	—
86	#13182008	—	2	1	no	—
87	#13182008	—	2	1	no	—
88	C₂²⋊Dic₁₀₉	—	2	1	no	—
89	#13182006	—	2	1	no	—
90	#13182006	—	2	1	no	—
91	#13182006	—	2	1	no	—
92	#13182006	—	2	1	no	—
93	C₄×Dic₁₀₉	—	2	1	no	—
94	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
95	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
96	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
97	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
98	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
99	C₂²×C₄×C₁₀₉	—	2	1	no	—
100	SD₁₆×C₁₀₉	—	2	2	no	—
101	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
102	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₁₀₉	—	2	1	no	—
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	—	2	1	no	—
108	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
109	C₂³×C₁₀₉	—	2	1	no	—
110	C₂²×D₁₀₉	—	2	1	no	—
111	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
112	Q₈×C₁₀₉	—	2	1	no	—
113	Q₈×C₁₀₉	—	2	1	no	—
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	—	4	2	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	—	4	2	no	—
124	C₂²⋊D₁₀₉	—	4	2	no	—
125	C₂²⋊D₁₀₉	—	4	2	no	—
126	C₂×Dic₁₀₉	—	2	1	no	—
127	C₂×Dic₁₀₉	—	2	1	no	—
128	C₂×Dic₁₀₉	—	2	1	no	—
129	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
131	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
132	D₄₃₆	—	2	1	no	—
133	D₄₃₆	—	2	1	no	—
134	D₄₃₆	—	4	2	no	—
135	C₄×D₁₀₉	—	2	1	no	—
136	C₄×D₁₀₉	—	2	1	no	—
137	C₄×D₁₀₉	—	2	2	no	—
138	C₄×D₁₀₉	—	4	2	no	—
139	C₄×D₁₀₉	SD₁₆	1	4	yes	complemented
140	Dic₂₁₈	—	2	1	no	—
141	Dic₂₁₈	—	2	1	no	—
142	Dic₂₁₈	—	4	2	no	—
143	Dic₂₁₈	—	4	2	no	—
144	C₈×C₁₀₉	—	2	1	no	—
145	C₈×C₁₀₉	#41	1	2	yes	complemented
146	#11724784	—	2	1	no	—
147	#11724784	#41	1	2	yes	complemented
148	C₂²×C₁₀₉	—	4	1	no	—
149	C₂²×C₁₀₉	—	4	1	no	—
150	C₂²×C₁₀₉	—	4	1	no	—
151	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
152	C₂×D₁₀₉	—	2	1	no	—
153	C₂×D₁₀₉	—	2	1	no	—
154	C₂×D₁₀₉	—	4	1	no	—
155	C₂×D₁₀₉	—	4	1	no	—
156	C₂×D₁₀₉	—	4	1	no	—
157	C₂×D₁₀₉	C₂×SD₁₆	1	2	yes	—
158	C₄×C₁₀₉	—	2	1	no	—
159	C₄×C₁₀₉	—	4	1	no	—
160	C₄×C₁₀₉	—	4	1	no	—
161	C₄×C₁₀₉	—	4	1	no	—
162	C₄×C₁₀₉	C₂×SD₁₆	1	2	yes	—
163	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
164	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
165	Dic₁₀₉	—	2	1	no	—
166	Dic₁₀₉	—	4	1	no	—
167	Dic₁₀₉	—	4	1	no	—
168	Dic₁₀₉	—	4	1	no	—
169	Dic₁₀₉	C₂×SD₁₆	1	2	yes	—
170	Dic₁₀₉	#135	1	2	yes	—
171	C₂×C₁₀₉	—	8	1	no	—
172	C₂×C₁₀₉	#532	1	1	yes	characteristic
173	C₂×C₁₀₉	C₂²×SD₁₆	1	1	yes	characteristic
174	C₂×C₁₀₉	C₂×#135	1	1	yes	characteristic
175	D₁₀₉	—	2	2	no	—
176	D₁₀₉	—	8	1	no	—
177	#3167	—	109	1	no	maximal, Sylow
178	C₁₀₉	#3167	1	1	yes	characteristic, complemented, Sylow
179	#533	—	109	1	no	—
180	#532	—	109	1	no	—
181	#498	—	109	1	no	—
182	#492	—	109	1	no	—
183	#487	—	109	1	no	—
184	#486	—	109	1	no	—
185	#465	—	109	1	no	—
186	#465	—	109	1	no	—
187	#465	—	109	1	no	—
188	#465	—	109	1	no	—
189	C₄×SD₁₆	—	109	1	no	—
190	C₄×SD₁₆	—	109	1	no	—
191	C₄×SD₁₆	—	109	1	no	—
192	C₄×SD₁₆	D₁₀₉	1	1	yes	characteristic, complemented
193	C₂×C₄×C₈	—	109	1	no	—
194	C₂×#41	—	218	1	no	—
195	C₂×SD₁₆	—	109	1	no	—
196	C₂×SD₁₆	—	109	1	no	—
197	C₂×SD₁₆	—	109	1	no	—
198	C₂×SD₁₆	C₂×D₁₀₉	1	1	yes	characteristic
199	C₂²×C₈	—	109	1	no	—
200	C₂²×C₈	—	109	1	no	—
201	C₄⋊Q₈	—	109	1	no	—
202	C₄⋊Q₈	—	109	1	no	—
203	C₄⋊Q₈	—	109	1	no	—
204	#127	—	109	1	no	—
205	#125	—	218	1	no	—
206	#124	—	218	1	no	—
207	C₂²⋊Q₈	—	109	1	no	—
208	C₂²⋊Q₈	—	109	1	no	—
209	C₂²⋊Q₈	—	218	1	no	—
210	#121	—	109	1	no	—
211	#121	—	109	1	no	—
212	#121	—	218	1	no	—
213	C₄×Q₈	—	109	1	no	—
214	C₄×Q₈	—	218	1	no	—
215	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
216	C₄×D₄	—	109	1	no	—
217	C₄×D₄	—	218	1	no	—
218	C₄×D₄	C₂×D₁₀₉	1	1	yes	characteristic
219	#117	—	218	1	no	—
220	C₂×C₄²	—	109	1	no	—
221	#106	—	109	1	no	—
222	#106	—	109	1	no	—
223	#106	—	109	1	no	—
224	#106	C₂×D₁₀₉	1	1	yes	characteristic
225	#103	—	109	1	no	—
226	#103	—	109	1	no	—
227	#103	—	109	1	no	—
228	#103	C₂×D₁₀₉	1	1	yes	characteristic
229	#102	—	109	1	no	—
230	#102	—	109	1	no	—
231	#102	—	109	1	no	—
232	#102	C₂×D₁₀₉	1	1	yes	characteristic
233	C₄×C₈	—	109	1	no	—
234	C₄×C₈	—	109	1	no	—
235	C₄×C₈	—	109	1	no	—
236	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
237	#41	—	436	4	no	—
238	C₂×Q₈	—	109	1	no	—
239	C₂×Q₈	—	218	1	no	—
240	C₂×Q₈	—	218	1	no	—
241	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
242	C₂×D₄	—	109	1	no	—
243	C₂×D₄	—	218	1	no	—
244	C₂×D₄	—	218	1	no	—
245	C₂×D₄	—	218	1	no	—
246	C₂×D₄	—	218	1	no	—
247	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
248	C₂²×C₄	—	2	1	no	—
249	C₂²×C₄	—	109	1	no	—
250	C₂²×C₄	—	109	1	no	—
251	C₂²×C₄	—	109	1	no	—
252	C₂²×C₄	—	218	1	no	—
253	SD₁₆	—	2	2	no	—
254	SD₁₆	—	218	2	no	—
255	SD₁₆	—	218	2	no	—
256	SD₁₆	—	218	2	no	—
257	C₂×C₈	—	109	1	no	—
258	C₂×C₈	—	109	1	no	—
259	C₂×C₈	—	218	2	no	—
260	C₂×C₈	—	218	2	no	—
261	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
262	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
263	C₄⋊C₄	—	2	1	no	—
264	C₄⋊C₄	—	109	1	no	—
265	C₄⋊C₄	—	109	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	—	218	1	no	—
271	C₄⋊C₄	—	218	1	no	—
272	C₄⋊C₄	—	218	1	no	—
273	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
274	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
275	C₂²⋊C₄	—	2	1	no	—
276	C₂²⋊C₄	—	218	1	no	—
277	C₂²⋊C₄	—	218	1	no	—
278	C₂²⋊C₄	—	218	1	no	—
279	C₂²⋊C₄	—	218	1	no	—
280	C₂²⋊C₄	—	218	1	no	—
281	C₄²	—	2	1	no	—
282	C₄²	—	109	1	no	—
283	C₄²	—	109	1	no	—
284	C₄²	—	109	1	no	—
285	C₄²	—	218	1	no	—
286	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
287	C₂³	—	2	1	no	—
288	C₂³	—	109	1	no	—
289	C₂³	—	218	1	no	—
290	Q₈	—	2	1	no	—
291	Q₈	—	2	1	no	—
292	Q₈	—	436	2	no	—
293	Q₈	—	436	2	no	—
294	Q₈	—	218	1	no	—
295	Q₈	—	218	1	no	—
296	D₄	—	2	1	no	—
297	D₄	—	2	1	no	—
298	D₄	—	436	2	no	—
299	D₄	—	436	2	no	—
300	D₄	—	436	2	no	—
301	D₄	—	436	2	no	—
302	D₄	—	218	1	no	—
303	D₄	—	218	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	109	1	no	—
309	C₂×C₄	—	109	1	no	—
310	C₂×C₄	—	109	4	no	—
311	C₂×C₄	—	109	1	no	—
312	C₂×C₄	—	436	2	no	—
313	C₂×C₄	—	218	2	no	—
314	C₂×C₄	—	218	1	no	—
315	C₂×C₄	—	218	1	no	—
316	C₂×C₄	—	218	1	no	—
317	C₂×C₄	—	218	1	no	—
318	C₂×C₄	—	218	1	no	—
319	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
320	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
321	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
322	C₈	—	2	1	no	—
323	C₈	—	109	2	no	—
324	C₈	—	218	1	no	—
325	C₈	#13182029	1	2	yes	complemented
326	C₂²	—	4	1	no	—
327	C₂²	—	4	1	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	109	2	no	—
330	C₂²	—	436	1	no	—
331	C₂²	—	436	1	no	—
332	C₂²	—	436	1	no	—
333	C₂²	—	218	1	no	—
334	C₂²	—	218	1	no	—
335	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	4	1	no	—
338	C₄	—	4	1	no	—
339	C₄	—	4	1	no	—
340	C₄	—	109	2	no	—
341	C₄	—	109	2	no	—
342	C₄	—	436	1	no	—
343	C₄	—	436	1	no	—
344	C₄	—	436	1	no	—
345	C₄	—	218	1	no	—
346	C₄	SD₁₆×D₁₀₉	1	2	yes	—
347	C₄	#41×D₁₀₉	1	1	yes	characteristic
348	C₄	C₂×#13182029	1	1	yes	characteristic
349	C₂	—	8	1	no	—
350	C₂	—	872	1	no	—
351	C₂	—	218	2	no	—
352	C₂	#20295573	1	1	yes	characteristic
353	C₂	C₂×SD₁₆×D₁₀₉	1	1	yes	characteristic
354	C₂	#135×D₁₀₉	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	#20295573	C₂
hypercenter	C₄×SD₁₆	D₁₀₉
hypocenter	C₁₀₉	#3167
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. #20295580	C₂	semidirect
3. #20295423	C₂	semidirect
4. #20295397	C₂	semidirect
5. #20295375	C₂	semidirect
6. #20295371	C₂	semidirect
7. #20295258	C₂	semidirect
8. #20295256	C₂	semidirect
9. #20295254	C₂	semidirect
10. #20295252	C₂	semidirect
11. C₄×#13182018	C₂	semidirect
12. C₄×#13182016	C₂	semidirect
13. C₄×#13182017	C₂	semidirect
14. C₄×C₈×D₁₀₉	C₂	semidirect
15. C₄×SD₁₆×C₁₀₉	C₂	semidirect
139. C₄×D₁₀₉	SD₁₆	semidirect
145. C₈×C₁₀₉	#41	semidirect
147. #11724784	#41	semidirect
178. C₁₀₉	#3167	semidirect
192. C₄×SD₁₆	D₁₀₉	semidirect
325. C₈	#13182029	semidirect

Maximal subgroups

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

subgroup	index
#20295580	2
#20295573	2
#20295423	2
#20295397	2
#20295375	2
#20295371	2
#20295258	2
#20295256	2
#20295254	2
#20295252	2
C₄×#13182018	2
C₄×#13182016	2
C₄×#13182017	2
C₄×C₈×D₁₀₉	2
C₄×SD₁₆×C₁₀₉	2
#3167	109