Subgroups — #28366333

Order 13952.

432 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₉⋊#401	C₂	1	1	yes	maximal, characteristic, complemented
2	#20294968	C₂	1	1	yes	maximal, characteristic, complemented
3	#20294965	C₂	1	1	yes	maximal, characteristic, complemented
4	#20294964	C₂	1	1	yes	maximal, characteristic, complemented
5	#20294955	C₂	1	1	yes	maximal, characteristic
6	#20294953	C₂	1	1	yes	maximal, characteristic
7	#20294952	C₂	1	1	yes	maximal, characteristic
8	#20294948	C₂	1	1	yes	maximal, characteristic
9	C₄×#13182008	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₄⋊Dic₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#13182006	C₂	1	1	yes	maximal, characteristic
12	C₄×#13182006	C₂	1	1	yes	maximal, characteristic
13	C₄×#13182006	C₂	1	1	yes	maximal, characteristic
14	#20294872	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×C₄⋊C₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
19	C₂×#13182008	C₂²	1	1	yes	characteristic
20	C₂×#13182008	C₂²	1	1	yes	characteristic
21	C₂×#13182006	C₂²	1	1	yes	characteristic
22	C₂×#13182006	C₂²	1	1	yes	characteristic
23	C₂×#13182006	C₂²	1	1	yes	characteristic
24	C₂×#13182006	C₂²	1	1	yes	characteristic
25	C₂×#13182006	C₂²	1	1	yes	characteristic
26	C₂×#13182006	C₂²	1	1	yes	characteristic
27	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
28	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
29	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
30	#16812606	C₂²	1	1	yes	characteristic
31	#16812606	C₂²	1	1	yes	characteristic
32	#16812606	C₂²	1	1	yes	characteristic
33	#16812606	C₂²	1	1	yes	characteristic
34	#16812606	C₂²	1	1	yes	characteristic
35	#16812606	C₂²	1	1	yes	characteristic
36	#16812606	C₂²	1	1	yes	characteristic
37	#16812606	C₂²	1	1	yes	characteristic
38	#16812606	C₂²	1	1	yes	characteristic
39	#16812606	C₂²	1	1	yes	characteristic
40	#16812606	C₂²	1	1	yes	characteristic
41	#16812606	C₂²	1	1	yes	characteristic
42	#16812606	C₂²	1	1	yes	characteristic
43	#16812606	C₂²	1	1	yes	characteristic
44	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
45	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
46	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
47	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
48	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
49	#95×C₁₀₉	C₂²	1	1	yes	characteristic
50	#95×C₁₀₉	C₂²	1	1	yes	characteristic
51	C₂³×D₁₀₉	C₂³	1	1	yes	characteristic
52	C₂²×Dic₁₀₉	C₂³	1	1	yes	characteristic
53	C₂²×Dic₁₀₉	C₂³	1	1	yes	characteristic
54	C₂²×Dic₁₀₉	C₂³	1	1	yes	characteristic
55	C₂²×Dic₁₀₉	C₂³	1	1	yes	characteristic
56	C₂²×Dic₁₀₉	C₂³	1	1	yes	characteristic
57	C₂²×Dic₁₀₉	C₂³	1	1	yes	characteristic
58	C₂²×Dic₁₀₉	C₂³	1	1	yes	characteristic
59	C₂×C₄×D₁₀₉	—	2	2	no	—
60	C₂×C₄×D₁₀₉	—	2	2	no	—
61	C₂×C₄×D₁₀₉	—	2	2	no	—
62	C₄⋊Dic₁₀₉	—	2	2	no	—
63	C₄⋊Dic₁₀₉	—	2	2	no	—
64	#13182008	—	2	2	no	—
65	#13182008	—	2	2	no	—
66	#13182006	—	2	2	no	—
67	#13182006	—	2	2	no	—
68	#13182006	—	2	2	no	—
69	#13182006	—	2	2	no	—
70	#13182006	—	2	2	no	—
71	#13182006	—	2	2	no	—
72	C₄×Dic₁₀₉	—	2	2	no	—
73	C₄×Dic₁₀₉	—	2	2	no	—
74	C₄×Dic₁₀₉	—	2	2	no	—
75	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
76	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
77	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
78	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
79	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
80	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
81	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
82	C₄⋊C₄×C₁₀₉	—	2	2	no	—
83	C₄⋊C₄×C₁₀₉	—	2	2	no	—
84	C₄²×C₁₀₉	—	2	2	no	—
85	C₄²×C₁₀₉	—	2	2	no	—
86	C₄²×C₁₀₉	—	2	2	no	—
87	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
88	C₂²×D₁₀₉	—	2	1	no	—
89	C₂²×D₁₀₉	—	2	1	no	—
90	C₂²×D₁₀₉	—	2	1	no	—
91	C₂²×D₁₀₉	—	2	1	no	—
92	C₂²×D₁₀₉	—	2	1	no	—
93	C₂²×D₁₀₉	—	2	1	no	—
94	C₂²×D₁₀₉	#41	1	2	yes	—
95	C₂×C₄×C₁₀₉	—	2	1	no	—
96	C₂×C₄×C₁₀₉	—	2	1	no	—
97	C₂×C₄×C₁₀₉	—	2	1	no	—
98	C₂×C₄×C₁₀₉	—	2	1	no	—
99	C₂×C₄×C₁₀₉	—	2	1	no	—
100	C₂×C₄×C₁₀₉	—	2	1	no	—
101	C₂×C₄×C₁₀₉	—	2	1	no	—
102	C₂×C₄×C₁₀₉	—	2	1	no	—
103	C₂×C₄×C₁₀₉	—	2	1	no	—
104	C₂×C₄×C₁₀₉	—	2	1	no	—
105	C₂×C₄×C₁₀₉	—	2	1	no	—
106	C₂×C₄×C₁₀₉	—	2	1	no	—
107	C₂×C₄×C₁₀₉	—	2	1	no	—
108	C₂×C₄×C₁₀₉	—	2	1	no	—
109	C₂×C₄×C₁₀₉	—	2	1	no	—
110	C₂×C₄×C₁₀₉	—	2	1	no	—
111	C₂×C₄×C₁₀₉	#41	1	2	yes	—
112	C₂×C₄×C₁₀₉	#41	1	2	yes	—
113	C₂×C₄×C₁₀₉	#41	1	2	yes	—
114	C₂×C₄×C₁₀₉	#41	1	2	yes	—
115	C₂×C₄×C₁₀₉	#41	1	2	yes	—
116	C₂×Dic₁₀₉	—	2	1	no	—
117	C₂×Dic₁₀₉	—	2	1	no	—
118	C₂×Dic₁₀₉	—	2	1	no	—
119	C₂×Dic₁₀₉	—	2	1	no	—
120	C₂×Dic₁₀₉	—	2	1	no	—
121	C₂×Dic₁₀₉	—	2	1	no	—
122	C₂×Dic₁₀₉	—	2	1	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	—	2	1	no	—
128	C₂×Dic₁₀₉	—	2	1	no	—
129	C₂×Dic₁₀₉	—	2	1	no	—
130	C₂×Dic₁₀₉	—	2	1	no	—
131	C₂×Dic₁₀₉	—	2	1	no	—
132	C₂×Dic₁₀₉	—	2	1	no	—
133	C₂×Dic₁₀₉	—	2	1	no	—
134	C₂×Dic₁₀₉	—	2	1	no	—
135	C₂×Dic₁₀₉	—	2	1	no	—
136	C₂×Dic₁₀₉	#41	1	2	yes	—
137	C₄×D₁₀₉	—	4	4	no	—
138	C₂²×C₁₀₉	C₂×#41	1	1	yes	characteristic
139	C₂²×C₁₀₉	C₂×#41	1	1	yes	characteristic
140	C₂²×C₁₀₉	C₂×#41	1	1	yes	characteristic
141	C₂²×C₁₀₉	C₂×#41	1	1	yes	characteristic
142	C₂²×C₁₀₉	C₂×#41	1	1	yes	characteristic
143	C₂²×C₁₀₉	C₂×#41	1	1	yes	characteristic
144	C₂²×C₁₀₉	C₂×#41	1	1	yes	characteristic
145	C₂×D₁₀₉	—	2	2	no	—
146	C₂×D₁₀₉	—	2	2	no	—
147	C₂×D₁₀₉	—	2	2	no	—
148	C₂×D₁₀₉	—	4	1	no	—
149	C₂×D₁₀₉	—	4	1	no	—
150	C₂×D₁₀₉	—	4	1	no	—
151	C₂×D₁₀₉	—	4	1	no	—
152	C₄×C₁₀₉	—	2	2	no	—
153	C₄×C₁₀₉	—	2	2	no	—
154	C₄×C₁₀₉	—	4	1	no	—
155	C₄×C₁₀₉	—	4	1	no	—
156	C₄×C₁₀₉	—	4	1	no	—
157	C₄×C₁₀₉	—	4	1	no	—
158	C₄×C₁₀₉	#126	1	4	yes	—
159	Dic₁₀₉	—	2	2	no	—
160	Dic₁₀₉	—	4	1	no	—
161	Dic₁₀₉	—	4	1	no	—
162	Dic₁₀₉	—	4	1	no	—
163	Dic₁₀₉	—	4	1	no	—
164	Dic₁₀₉	—	4	1	no	—
165	Dic₁₀₉	—	4	1	no	—
166	C₂×C₁₀₉	C₂×#126	1	1	yes	characteristic
167	C₂×C₁₀₉	#529	1	1	yes	characteristic
168	C₂×C₁₀₉	#529	1	1	yes	characteristic
169	C₂×C₁₀₉	#529	1	1	yes	characteristic
170	C₂×C₁₀₉	#529	1	1	yes	characteristic
171	C₂×C₁₀₉	#529	1	1	yes	characteristic
172	C₂×C₁₀₉	#529	1	1	yes	characteristic
173	D₁₀₉	—	4	2	no	—
174	#2401	—	109	1	no	maximal, Sylow
175	C₁₀₉	#2401	1	1	yes	characteristic, complemented, Sylow
176	#401	—	109	1	no	—
177	#400	—	109	1	no	—
178	#400	—	109	1	no	—
179	#400	—	109	1	no	—
180	#397	—	109	1	no	—
181	#397	—	109	1	no	—
182	#397	—	109	1	no	—
183	#396	—	109	1	no	—
184	C₄×C₄⋊C₄	—	109	1	no	—
185	C₄×C₄⋊C₄	—	109	1	no	—
186	C₄×C₄⋊C₄	D₁₀₉	1	1	yes	characteristic, complemented
187	C₄×C₂²⋊C₄	—	109	1	no	—
188	C₄×C₂²⋊C₄	—	109	1	no	—
189	C₄×C₂²⋊C₄	—	109	1	no	—
190	#376	—	109	1	no	—
191	C₂³×C₄	—	109	1	no	—
192	C₂×C₄⋊C₄	—	109	1	no	—
193	C₂×C₄⋊C₄	—	109	1	no	—
194	C₂×C₄⋊C₄	—	109	1	no	—
195	C₂×C₄⋊C₄	—	109	1	no	—
196	C₂×C₄⋊C₄	C₂×D₁₀₉	1	1	yes	characteristic
197	C₂×C₄⋊C₄	C₂×D₁₀₉	1	1	yes	characteristic
198	C₂×C₂²⋊C₄	—	109	1	no	—
199	C₂×C₂²⋊C₄	—	109	1	no	—
200	C₂×C₂²⋊C₄	—	109	1	no	—
201	C₂×C₂²⋊C₄	—	109	1	no	—
202	C₂×C₂²⋊C₄	—	109	1	no	—
203	C₂×C₂²⋊C₄	—	109	1	no	—
204	C₂×C₄²	—	109	1	no	—
205	C₂×C₄²	—	109	1	no	—
206	C₂×C₄²	—	109	1	no	—
207	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
208	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
209	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
210	#95	—	109	1	no	—
211	#95	—	109	1	no	—
212	#95	—	109	1	no	—
213	#95	—	109	1	no	—
214	#95	—	109	1	no	—
215	#95	—	109	1	no	—
216	#95	—	109	1	no	—
217	#95	—	109	1	no	—
218	#95	—	109	1	no	—
219	#95	—	109	1	no	—
220	#95	—	109	1	no	—
221	#95	—	109	1	no	—
222	#95	—	109	1	no	—
223	#95	—	109	1	no	—
224	#95	C₂×D₁₀₉	1	1	yes	characteristic
225	#95	C₂×D₁₀₉	1	1	yes	characteristic
226	C₂⁴	—	109	1	no	—
227	C₂²×C₄	—	109	1	no	—
228	C₂²×C₄	—	109	1	no	—
229	C₂²×C₄	—	109	1	no	—
230	C₂²×C₄	—	109	1	no	—
231	C₂²×C₄	—	109	1	no	—
232	C₂²×C₄	—	109	1	no	—
233	C₂²×C₄	—	109	1	no	—
234	C₂²×C₄	—	218	2	no	—
235	C₂²×C₄	—	218	2	no	—
236	C₂²×C₄	—	218	2	no	—
237	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
238	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
239	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
240	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
241	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
242	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
243	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	218	2	no	—
247	C₄⋊C₄	—	218	2	no	—
248	C₄⋊C₄	—	218	2	no	—
249	C₄⋊C₄	—	218	2	no	—
250	C₂²⋊C₄	—	218	2	no	—
251	C₂²⋊C₄	—	218	2	no	—
252	C₂²⋊C₄	—	218	2	no	—
253	C₂²⋊C₄	—	218	2	no	—
254	C₂²⋊C₄	—	218	2	no	—
255	C₂²⋊C₄	—	218	2	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	218	2	no	—
260	C₄²	—	218	2	no	—
261	C₄²	—	218	2	no	—
262	C₂³	—	109	2	no	—
263	C₂³	—	218	1	no	—
264	C₂³	—	218	1	no	—
265	C₂³	—	218	1	no	—
266	C₂³	—	218	1	no	—
267	C₂³	—	218	1	no	—
268	C₂³	—	218	1	no	—
269	C₂³	C₂³×D₁₀₉	1	1	yes	characteristic
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	109	2	no	—
287	C₂×C₄	—	436	4	no	—
288	C₂×C₄	—	218	1	no	—
289	C₂×C₄	—	218	1	no	—
290	C₂×C₄	—	218	1	no	—
291	C₂×C₄	—	218	1	no	—
292	C₂×C₄	—	218	1	no	—
293	C₂×C₄	—	218	1	no	—
294	C₂×C₄	—	218	1	no	—
295	C₂×C₄	—	218	1	no	—
296	C₂×C₄	—	218	1	no	—
297	C₂×C₄	—	218	1	no	—
298	C₂×C₄	—	218	1	no	—
299	C₂×C₄	—	218	1	no	—
300	C₂×C₄	—	218	1	no	—
301	C₂×C₄	—	218	1	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	#13182028	1	2	yes	—
309	C₂×C₄	#13182029	1	2	yes	—
310	C₂×C₄	#13182029	1	2	yes	—
311	C₂×C₄	#13182029	1	2	yes	—
312	C₂×C₄	#13182030	1	2	yes	—
313	C₂²	—	436	1	no	—
314	C₂²	—	436	1	no	—
315	C₂²	—	436	1	no	—
316	C₂²	—	436	1	no	—
317	C₂²	—	218	2	no	—
318	C₂²	—	218	2	no	—
319	C₂²	—	218	2	no	—
320	C₂²	#41×D₁₀₉	1	1	yes	characteristic
321	C₂²	#41×D₁₀₉	1	1	yes	characteristic
322	C₂²	C₂×#13182028	1	1	yes	characteristic
323	C₂²	C₂×#13182029	1	1	yes	characteristic
324	C₂²	C₂×#13182029	1	1	yes	characteristic
325	C₂²	C₂×#13182029	1	1	yes	characteristic
326	C₂²	C₂×#13182030	1	1	yes	characteristic
327	C₄	—	2	2	no	—
328	C₄	—	2	2	no	—
329	C₄	—	4	1	no	—
330	C₄	—	4	1	no	—
331	C₄	—	4	1	no	—
332	C₄	—	4	1	no	—
333	C₄	—	436	1	no	—
334	C₄	—	436	1	no	—
335	C₄	—	436	1	no	—
336	C₄	—	436	1	no	—
337	C₄	—	436	1	no	—
338	C₄	—	436	1	no	—
339	C₄	—	218	2	no	—
340	C₄	#16812694	1	4	yes	—
341	C₂	—	436	2	no	—
342	C₂	C₂×#16812694	1	1	yes	characteristic
343	C₂	#20295554	1	1	yes	characteristic
344	C₂	#20295556	1	1	yes	characteristic
345	C₂	#20295558	1	1	yes	characteristic
346	C₂	#20295560	1	1	yes	characteristic
347	C₂	#20295563	1	1	yes	characteristic
348	C₂	#20295563	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₀₉
commutator, socle, cosocle	C₂³×C₁₀₉	C₂⁴
Frattini	C₂³	C₂³×D₁₀₉
Fitting	C₄×C₄⋊C₄×C₁₀₉	C₂
prime radical	C₂³×D₁₀₉	C₂³
hypercenter	C₄×C₄⋊C₄	D₁₀₉
hypocenter	C₁₀₉	#2401
LCS	G ▷ C₂³×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₁₀₉⋊#401	C₂	semidirect
2. #20294968	C₂	semidirect
3. #20294965	C₂	semidirect
4. #20294964	C₂	semidirect
9. C₄×#13182008	C₂	semidirect
10. C₄×C₄⋊Dic₁₀₉	C₂	semidirect
14. #20294872	C₂	semidirect
15. C₄×C₄⋊C₄×C₁₀₉	C₂	semidirect
175. C₁₀₉	#2401	semidirect
186. C₄×C₄⋊C₄	D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₁₀₉⋊#401	2
#20294968	2
#20294965	2
#20294964	2
#20294955	2
#20294953	2
#20294952	2
#20294948	2
C₄×#13182008	2
C₄×C₄⋊Dic₁₀₉	2
C₄×#13182006	2
#20294872	2
C₄×C₄⋊C₄×C₁₀₉	2
#2401	109