Subgroups — #28371339

Order 13952.

556 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812689	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄²×D₁₀₉	C₂	1	1	yes	maximal, characteristic
3	#20294923	C₂	1	1	yes	maximal, characteristic, complemented
4	#20294921	C₂	1	1	yes	maximal, characteristic, complemented
5	#20294913	C₂	1	1	yes	maximal, characteristic, complemented
6	#20294910	C₂	1	1	yes	maximal, characteristic, complemented
7	#20294909	C₂	1	1	yes	maximal, characteristic, complemented
8	#20294900	C₂	1	1	yes	maximal, characteristic
9	#20294900	C₂	1	1	yes	maximal, characteristic
10	#20294897	C₂	1	1	yes	maximal, characteristic
11	#20294897	C₂	1	1	yes	maximal, characteristic
12	#20294890	C₂	1	1	yes	maximal, characteristic, complemented
13	#20294885	C₂	1	1	yes	maximal, characteristic
14	#20294885	C₂	1	1	yes	maximal, characteristic
15	#387×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
18	C₂²×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
19	#16812689	—	2	4	no	—
20	C₂×C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
22	C₂×#13182008	C₂²	1	1	yes	characteristic
23	C₂×#13182008	C₂²	1	1	yes	characteristic
24	C₂×#13182008	C₂²	1	1	yes	characteristic
25	C₂×#13182008	C₂²	1	1	yes	characteristic
26	C₂×#13182008	C₂²	1	1	yes	characteristic
27	C₂×#13182008	C₂²	1	1	yes	characteristic
28	C₂×#13182008	C₂²	1	1	yes	characteristic
29	C₂×#13182008	C₂²	1	1	yes	characteristic
30	C₂×#13182006	C₂²	1	1	yes	characteristic
31	C₂×#13182006	C₂²	1	1	yes	characteristic
32	C₂×#13182006	C₂²	1	1	yes	characteristic
33	C₂×#13182006	C₂²	1	1	yes	characteristic
34	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
35	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
36	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
37	C₄²×D₁₀₉	—	2	4	no	—
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	#16812606	C₂²	1	1	yes	characteristic
45	#16812606	C₂²	1	1	yes	characteristic
46	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
47	C₂×C₄⋊C₄×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	#95×C₁₀₉	C₂²	1	1	yes	characteristic
52	#95×C₁₀₉	C₂²	1	1	yes	characteristic
53	C₂³×D₁₀₉	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	C₂²×Dic₁₀₉	C₂³	1	1	yes	characteristic
60	C₂²×Dic₁₀₉	C₂³	1	1	yes	characteristic
61	C₂×C₄×D₁₀₉	—	2	2	no	—
62	C₂×C₄×D₁₀₉	—	2	2	no	—
63	C₂×C₄×D₁₀₉	—	2	2	no	—
64	C₂×C₄×D₁₀₉	—	2	2	no	—
65	C₂×C₄×D₁₀₉	—	2	2	no	—
66	C₂×C₄×D₁₀₉	—	2	2	no	—
67	C₂×C₄×D₁₀₉	—	2	2	no	—
68	C₂×C₄×D₁₀₉	—	2	2	no	—
69	C₂×C₄×D₁₀₉	Q₈	1	4	yes	—
70	C₄⋊Dic₁₀₉	—	2	2	no	—
71	C₄⋊Dic₁₀₉	—	2	2	no	—
72	#13182008	—	2	2	no	—
73	#13182008	—	2	2	no	—
74	#13182008	—	2	2	no	—
75	#13182008	—	2	2	no	—
76	#13182008	—	2	2	no	—
77	#13182008	—	2	2	no	—
78	#13182008	—	2	2	no	—
79	#13182008	—	2	2	no	—
80	#13182006	—	2	2	no	—
81	#13182006	—	2	2	no	—
82	#13182006	—	2	2	no	—
83	#13182006	—	2	2	no	—
84	C₄×Dic₁₀₉	—	2	2	no	—
85	C₄×Dic₁₀₉	—	2	2	no	—
86	C₄×Dic₁₀₉	D₄	1	4	yes	—
87	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
88	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
89	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
90	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
91	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
92	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
93	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
94	C₄⋊C₄×C₁₀₉	—	2	2	no	—
95	C₄⋊C₄×C₁₀₉	—	2	2	no	—
96	C₄²×C₁₀₉	—	2	2	no	—
97	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
98	C₂²×D₁₀₉	—	2	1	no	—
99	C₂²×D₁₀₉	—	2	1	no	—
100	C₂²×D₁₀₉	—	2	1	no	—
101	C₂²×D₁₀₉	—	2	1	no	—
102	C₂²×D₁₀₉	C₂×Q₈	1	2	yes	—
103	C₂²×D₁₀₉	#41	1	2	yes	—
104	C₂²×D₁₀₉	#41	1	2	yes	—
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	—	2	1	no	—
112	C₂×C₄×C₁₀₉	—	2	1	no	—
113	C₂×C₄×C₁₀₉	—	2	1	no	—
114	C₂×C₄×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	—	2	1	no	—
119	C₂×C₄×C₁₀₉	—	2	1	no	—
120	C₂×C₄×C₁₀₉	—	2	1	no	—
121	C₂×C₄×C₁₀₉	—	2	1	no	—
122	C₂×C₄×C₁₀₉	—	2	1	no	—
123	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
124	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
125	C₂×C₄×C₁₀₉	#41	1	2	yes	—
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	—	2	1	no	—
137	C₂×Dic₁₀₉	—	2	1	no	—
138	C₂×Dic₁₀₉	—	2	1	no	—
139	C₂×Dic₁₀₉	—	2	1	no	—
140	C₂×Dic₁₀₉	—	2	1	no	—
141	C₂×Dic₁₀₉	—	2	1	no	—
142	C₂×Dic₁₀₉	C₂×D₄	1	2	yes	—
143	C₂×Dic₁₀₉	C₂×D₄	1	2	yes	—
144	C₂×Dic₁₀₉	C₂×Q₈	1	2	yes	—
145	C₂×Dic₁₀₉	#41	1	2	yes	—
146	C₂×Dic₁₀₉	#41	1	2	yes	—
147	C₄×D₁₀₉	—	2	8	no	—
148	C₄×D₁₀₉	—	2	8	no	—
149	C₄×D₁₀₉	—	2	8	no	—
150	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
151	C₂²×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
152	C₂²×C₁₀₉	C₂×#41	1	1	yes	characteristic
153	C₂²×C₁₀₉	C₂×#41	1	1	yes	characteristic
154	C₂²×C₁₀₉	C₂×#41	1	1	yes	characteristic
155	C₂²×C₁₀₉	C₂×#41	1	1	yes	characteristic
156	C₂²×C₁₀₉	C₂×#41	1	1	yes	characteristic
157	C₂×D₁₀₉	—	2	2	no	—
158	C₂×D₁₀₉	—	2	2	no	—
159	C₂×D₁₀₉	—	2	2	no	—
160	C₂×D₁₀₉	—	2	2	no	—
161	C₂×D₁₀₉	—	2	2	no	—
162	C₂×D₁₀₉	—	2	2	no	—
163	C₂×D₁₀₉	#125	1	4	yes	complemented
164	C₄×C₁₀₉	—	2	2	no	—
165	C₄×C₁₀₉	—	2	2	no	—
166	C₄×C₁₀₉	—	2	2	no	—
167	C₄×C₁₀₉	—	4	1	no	—
168	C₄×C₁₀₉	—	4	1	no	—
169	C₄×C₁₀₉	—	4	1	no	—
170	C₄×C₁₀₉	—	4	1	no	—
171	Dic₁₀₉	—	2	2	no	—
172	Dic₁₀₉	—	2	2	no	—
173	Dic₁₀₉	—	4	1	no	—
174	Dic₁₀₉	—	4	1	no	—
175	Dic₁₀₉	—	4	1	no	—
176	Dic₁₀₉	—	4	1	no	—
177	Dic₁₀₉	C₂²⋊Q₈	1	4	yes	—
178	C₂×C₁₀₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
179	C₂×C₁₀₉	#525	1	1	yes	characteristic
180	C₂×C₁₀₉	#525	1	1	yes	characteristic
181	C₂×C₁₀₉	C₂×#125	1	1	yes	characteristic
182	C₂×C₁₀₉	#529	1	1	yes	characteristic
183	C₂×C₁₀₉	#529	1	1	yes	characteristic
184	C₂×C₁₀₉	#533	1	1	yes	characteristic
185	D₁₀₉	—	2	4	no	—
186	#2851	—	109	1	no	maximal, Sylow
187	C₁₀₉	#2851	1	1	yes	characteristic, complemented, Sylow
188	C₂×#125	—	109	1	no	—
189	C₂²×C₄²	—	109	1	no	—
190	#389	—	109	1	no	—
191	#389	—	109	1	no	—
192	#387	—	109	1	no	—
193	#387	—	109	1	no	—
194	#387	—	109	1	no	—
195	#387	D₁₀₉	1	1	yes	characteristic, complemented
196	#385	—	109	1	no	—
197	#385	—	109	1	no	—
198	#385	—	109	1	no	—
199	#385	—	109	1	no	—
200	#382	—	109	1	no	—
201	#380	—	109	1	no	—
202	#380	—	109	1	no	—
203	C₂³×C₄	—	109	1	no	—
204	C₂³×C₄	—	109	1	no	—
205	C₂³×C₄	—	109	1	no	—
206	#125	—	218	4	no	—
207	C₂×C₄⋊C₄	—	109	1	no	—
208	C₂×C₄⋊C₄	—	109	1	no	—
209	C₂×C₄⋊C₄	—	109	1	no	—
210	C₂×C₄⋊C₄	—	109	1	no	—
211	C₂×C₄⋊C₄	—	109	1	no	—
212	C₂×C₄⋊C₄	—	109	1	no	—
213	C₂×C₄⋊C₄	—	109	1	no	—
214	C₂×C₄⋊C₄	—	109	1	no	—
215	C₂×C₄⋊C₄	—	109	1	no	—
216	C₂×C₄⋊C₄	—	109	1	no	—
217	C₂×C₄⋊C₄	C₂×D₁₀₉	1	1	yes	characteristic
218	C₂×C₄⋊C₄	C₂×D₁₀₉	1	1	yes	characteristic
219	C₂×C₂²⋊C₄	—	109	1	no	—
220	C₂×C₂²⋊C₄	—	109	1	no	—
221	C₂×C₂²⋊C₄	—	109	1	no	—
222	C₂×C₂²⋊C₄	—	109	1	no	—
223	C₂×C₄²	—	109	1	no	—
224	C₂×C₄²	—	109	1	no	—
225	C₂×C₄²	—	109	1	no	—
226	C₂×C₄²	—	218	4	no	—
227	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
228	#95	—	109	1	no	—
229	#95	—	109	1	no	—
230	#95	—	109	1	no	—
231	#95	—	109	1	no	—
232	#95	—	109	1	no	—
233	#95	—	109	1	no	—
234	#95	—	109	1	no	—
235	#95	—	109	1	no	—
236	#95	C₂×D₁₀₉	1	1	yes	characteristic
237	#95	C₂×D₁₀₉	1	1	yes	characteristic
238	#95	C₂×D₁₀₉	1	1	yes	characteristic
239	#95	C₂×D₁₀₉	1	1	yes	characteristic
240	C₂⁴	—	109	1	no	—
241	C₂²×C₄	—	109	1	no	—
242	C₂²×C₄	—	109	1	no	—
243	C₂²×C₄	—	109	1	no	—
244	C₂²×C₄	—	109	4	no	—
245	C₂²×C₄	—	109	1	no	—
246	C₂²×C₄	—	109	1	no	—
247	C₂²×C₄	—	109	1	no	—
248	C₂²×C₄	—	109	1	no	—
249	C₂²×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	C₂²×C₄	—	218	2	no	—
257	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
258	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
259	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
260	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
261	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
262	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
263	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	218	2	no	—
267	C₄⋊C₄	—	218	2	no	—
268	C₄⋊C₄	—	218	2	no	—
269	C₄⋊C₄	—	218	2	no	—
270	C₄⋊C₄	—	218	2	no	—
271	C₄⋊C₄	—	218	2	no	—
272	C₄⋊C₄	—	218	2	no	—
273	C₄⋊C₄	—	218	2	no	—
274	C₄⋊C₄	—	218	2	no	—
275	C₄⋊C₄	—	218	2	no	—
276	C₂²⋊C₄	—	218	2	no	—
277	C₂²⋊C₄	—	218	2	no	—
278	C₂²⋊C₄	—	218	2	no	—
279	C₂²⋊C₄	—	218	2	no	—
280	C₄²	—	2	2	no	—
281	C₄²	—	109	4	no	—
282	C₄²	—	218	2	no	—
283	C₄²	—	218	2	no	—
284	C₂³	—	109	2	no	—
285	C₂³	—	109	2	no	—
286	C₂³	—	109	2	no	—
287	C₂³	—	218	1	no	—
288	C₂³	—	218	1	no	—
289	C₂³	—	218	1	no	—
290	C₂³	—	218	1	no	—
291	C₂³	C₂³×D₁₀₉	1	1	yes	characteristic
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	109	2	no	—
311	C₂×C₄	—	109	2	no	—
312	C₂×C₄	—	109	2	no	—
313	C₂×C₄	—	109	2	no	—
314	C₂×C₄	—	109	2	no	—
315	C₂×C₄	—	218	8	no	—
316	C₂×C₄	—	218	8	no	—
317	C₂×C₄	—	218	8	no	—
318	C₂×C₄	—	218	1	no	—
319	C₂×C₄	—	218	1	no	—
320	C₂×C₄	—	218	1	no	—
321	C₂×C₄	—	218	1	no	—
322	C₂×C₄	—	218	1	no	—
323	C₂×C₄	—	218	1	no	—
324	C₂×C₄	—	218	1	no	—
325	C₂×C₄	—	218	1	no	—
326	C₂×C₄	—	218	1	no	—
327	C₂×C₄	—	218	1	no	—
328	C₂×C₄	—	218	1	no	—
329	C₂×C₄	—	218	1	no	—
330	C₂×C₄	—	218	1	no	—
331	C₂×C₄	—	218	1	no	—
332	C₂×C₄	—	218	1	no	—
333	C₂×C₄	—	218	1	no	—
334	C₂×C₄	D₄×D₁₀₉	1	2	yes	—
335	C₂×C₄	Q₈×D₁₀₉	1	2	yes	—
336	C₂×C₄	#13182029	1	2	yes	—
337	C₂²	—	109	4	no	—
338	C₂²	—	218	2	no	—
339	C₂²	—	218	2	no	—
340	C₂²	—	218	2	no	—
341	C₂²	—	218	2	no	—
342	C₂²	—	218	2	no	—
343	C₂²	—	218	2	no	—
344	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
345	C₂²	C₂×Q₈×D₁₀₉	1	1	yes	characteristic
346	C₂²	#41×D₁₀₉	1	1	yes	characteristic
347	C₂²	#41×D₁₀₉	1	1	yes	characteristic
348	C₂²	#41×D₁₀₉	1	1	yes	characteristic
349	C₂²	#41×D₁₀₉	1	1	yes	characteristic
350	C₂²	C₂×#13182029	1	1	yes	characteristic
351	C₄	—	2	2	no	—
352	C₄	—	2	2	no	—
353	C₄	—	2	2	no	—
354	C₄	—	4	1	no	—
355	C₄	—	4	1	no	—
356	C₄	—	4	1	no	—
357	C₄	—	4	1	no	—
358	C₄	—	109	4	no	—
359	C₄	—	436	1	no	—
360	C₄	—	436	1	no	—
361	C₄	—	436	1	no	—
362	C₄	—	436	1	no	—
363	C₄	—	218	2	no	—
364	C₄	—	218	2	no	—
365	C₂	—	218	4	no	—
366	C₂	C₂²⋊Q₈×D₁₀₉	1	1	yes	characteristic
367	C₂	#20295534	1	1	yes	characteristic
368	C₂	#20295535	1	1	yes	characteristic
369	C₂	#125×D₁₀₉	1	1	yes	characteristic
370	C₂	#20295554	1	1	yes	characteristic
371	C₂	#20295559	1	1	yes	characteristic
372	C₂	#20295580	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₉
commutator, socle, cosocle	C₂³×C₁₀₉	C₂⁴
Fitting	#387×C₁₀₉	C₂
prime radical	C₂³×D₁₀₉	C₂³
hypercenter	#387	D₁₀₉
hypocenter	C₁₀₉	#2851
LCS	G ▷ C₂³×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ #387

Complements

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

N	H ≅ G/N	type
1. C₂×#16812689	C₂	semidirect
3. #20294923	C₂	semidirect
4. #20294921	C₂	semidirect
5. #20294913	C₂	semidirect
6. #20294910	C₂	semidirect
7. #20294909	C₂	semidirect
12. #20294890	C₂	semidirect
15. #387×C₁₀₉	C₂	semidirect
163. C₂×D₁₀₉	#125	semidirect
187. C₁₀₉	#2851	semidirect
195. #387	D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16812689	2
C₂×C₄²×D₁₀₉	2
#20294923	2
#20294921	2
#20294913	2
#20294910	2
#20294909	2
#20294900	2
#20294897	2
#20294890	2
#20294885	2
#387×C₁₀₉	2
#2851	109