Subgroups — #28374868

Order 13952.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295715	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295664	C₂	1	1	yes	maximal, characteristic
3	#20295382	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295355	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295350	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295328	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295266	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295255	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295251	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295188	C₂	1	1	yes	maximal, characteristic, complemented
11	#20295178	C₂	1	1	yes	maximal, characteristic, complemented
12	#20295117	C₂	1	1	yes	maximal, characteristic, complemented
13	C₈×C₂²⋊D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
14	#102×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
15	#416×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×D₄×D₁₀₉	C₂²	1	1	yes	characteristic
17	C₂×#13182019	C₂²	1	1	yes	characteristic, complemented
18	C₂×#13182018	C₂²	1	1	yes	characteristic
19	C₂×#13182017	C₂²	1	1	yes	characteristic
20	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
21	#16812692	—	2	1	no	—
22	#16812691	—	2	1	no	—
23	#16812690	C₂²	1	1	yes	characteristic, complemented
24	#16812688	C₂²	1	1	yes	characteristic
25	#16812682	—	2	1	no	—
26	#16812681	—	2	1	no	—
27	#16812679	C₂²	1	1	yes	characteristic
28	#16812678	C₂²	1	1	yes	characteristic, complemented
29	#16812675	C₂²	1	1	yes	characteristic
30	#16812673	C₂²	1	1	yes	characteristic
31	#16812671	—	2	1	no	—
32	#16812667	—	2	1	no	—
33	#16812664	C₂²	1	1	yes	characteristic, complemented
34	D₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
35	C₄×C₂²⋊D₁₀₉	C₂²	1	1	yes	characteristic
36	#16812656	—	2	1	no	—
37	C₄⋊C₄×D₁₀₉	C₂²	1	1	yes	characteristic
38	C₂×#13182006	—	2	1	no	—
39	#16812632	C₂²	1	1	yes	characteristic, complemented
40	#16812631	C₂²	1	1	yes	characteristic, complemented
41	#16812630	C₂²	1	1	yes	characteristic, complemented
42	#16812628	C₂²	1	1	yes	characteristic, complemented
43	#16812624	C₂²	1	1	yes	characteristic, complemented
44	#16812623	C₂²	1	1	yes	characteristic, complemented
45	#16812621	C₂²	1	1	yes	characteristic
46	#16812620	C₂²	1	1	yes	characteristic
47	#16812619	C₂²	1	1	yes	characteristic
48	#16812619	C₂²	1	1	yes	characteristic
49	C₂²⋊#11724784	C₂²	1	1	yes	characteristic, complemented
50	#16812611	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic, complemented
52	C₂×#41×C₁₀₉	C₂²	1	1	yes	characteristic
53	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
54	#117×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
55	#103×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
56	#103×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×D₄₃₆	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
63	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
64	C₂×C₂²⋊D₁₀₉	D₄	1	1	yes	characteristic
65	D₄×D₁₀₉	—	2	2	no	—
66	D₄×D₁₀₉	—	4	1	no	—
67	C₂²×Dic₁₀₉	—	2	1	no	—
68	C₂×C₄×D₁₀₉	—	2	1	no	—
69	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
70	#13182019	—	4	1	no	—
71	#13182018	—	4	1	no	—
72	#13182017	—	2	2	no	—
73	C₂×#11724784	C₂³	1	1	yes	characteristic
74	C₂×#11724784	C₂³	1	1	yes	characteristic
75	C₈×D₁₀₉	—	4	1	no	—
76	C₄⋊Dic₁₀₉	—	2	1	no	—
77	C₄⋊Dic₁₀₉	—	2	1	no	—
78	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
79	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
80	#13182008	—	2	1	no	—
81	#13182008	—	2	1	no	—
82	#13182008	—	2	1	no	—
83	#13182008	—	2	1	no	—
84	#13182008	—	2	1	no	—
85	#13182008	D₄	1	1	yes	characteristic
86	C₂²⋊Dic₁₀₉	—	2	1	no	—
87	C₂²⋊Dic₁₀₉	—	2	1	no	—
88	C₂²⋊Dic₁₀₉	D₄	1	1	yes	characteristic
89	#13182006	—	2	1	no	—
90	#13182006	—	2	1	no	—
91	#13182006	—	2	1	no	—
92	#13182006	—	2	1	no	—
93	#13182006	—	2	1	no	—
94	#13182006	D₄	1	1	yes	characteristic
95	C₄×Dic₁₀₉	—	2	1	no	—
96	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
97	#41×C₁₀₉	—	2	2	no	—
98	#41×C₁₀₉	—	4	1	no	—
99	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
100	C₂×D₄×C₁₀₉	—	2	1	no	—
101	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
102	C₂²×C₄×C₁₀₉	—	2	1	no	—
103	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
104	C₂×C₈×C₁₀₉	—	4	1	no	—
105	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
106	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
107	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
108	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
109	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
110	C₄²×C₁₀₉	—	2	1	no	—
111	C₂³×C₁₀₉	—	2	1	no	—
112	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
113	C₂²×D₁₀₉	—	2	1	no	—
114	C₂²×D₁₀₉	—	4	2	no	—
115	C₂²×D₁₀₉	—	4	2	no	—
116	C₂²×D₁₀₉	—	8	1	no	—
117	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
118	Q₈×C₁₀₉	—	2	1	no	—
119	Q₈×C₁₀₉	—	2	1	no	—
120	D₄×C₁₀₉	—	2	1	no	—
121	D₄×C₁₀₉	—	4	2	no	—
122	D₄×C₁₀₉	#41	1	2	yes	—
123	C₂×C₄×C₁₀₉	—	2	1	no	—
124	C₂×C₄×C₁₀₉	—	2	1	no	—
125	C₂×C₄×C₁₀₉	—	2	1	no	—
126	C₂×C₄×C₁₀₉	—	2	1	no	—
127	C₂×C₄×C₁₀₉	—	2	1	no	—
128	C₂×C₄×C₁₀₉	—	4	2	no	—
129	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
130	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
131	C₂²⋊D₁₀₉	—	2	2	no	—
132	C₂²⋊D₁₀₉	—	4	2	no	—
133	C₂²⋊D₁₀₉	—	8	1	no	—
134	C₂×Dic₁₀₉	—	2	1	no	—
135	C₂×Dic₁₀₉	—	2	1	no	—
136	C₂×Dic₁₀₉	—	2	1	no	—
137	C₂×Dic₁₀₉	—	8	1	no	—
138	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
139	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
140	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
141	D₄₃₆	—	2	1	no	—
142	D₄₃₆	—	2	1	no	—
143	C₄×D₁₀₉	—	2	1	no	—
144	C₄×D₁₀₉	—	2	1	no	—
145	C₄×D₁₀₉	—	4	2	no	—
146	Dic₂₁₈	—	2	1	no	—
147	Dic₂₁₈	—	2	1	no	—
148	C₈×C₁₀₉	—	2	1	no	—
149	C₈×C₁₀₉	—	2	1	no	—
150	#11724784	—	2	1	no	—
151	#11724784	—	2	1	no	—
152	C₂²×C₁₀₉	—	2	2	no	—
153	C₂²×C₁₀₉	—	2	1	no	—
154	C₂²×C₁₀₉	—	2	1	no	—
155	C₂²×C₁₀₉	—	2	1	no	—
156	C₂²×C₁₀₉	—	4	1	no	—
157	C₂²×C₁₀₉	—	4	1	no	—
158	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
159	C₂×D₁₀₉	—	2	1	no	—
160	C₂×D₁₀₉	—	2	1	no	—
161	C₂×D₁₀₉	—	4	1	no	—
162	C₂×D₁₀₉	—	4	1	no	—
163	C₂×D₁₀₉	—	4	1	no	—
164	C₂×D₁₀₉	—	8	4	no	—
165	C₂×D₁₀₉	#135	1	2	yes	—
166	C₄×C₁₀₉	—	2	1	no	—
167	C₄×C₁₀₉	—	4	1	no	—
168	C₄×C₁₀₉	—	4	1	no	—
169	C₄×C₁₀₉	—	4	1	no	—
170	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
171	C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
172	Dic₁₀₉	—	2	1	no	—
173	Dic₁₀₉	—	2	1	no	—
174	Dic₁₀₉	—	2	1	no	—
175	Dic₁₀₉	—	4	1	no	—
176	Dic₁₀₉	—	4	1	no	—
177	Dic₁₀₉	—	4	1	no	—
178	C₂×C₁₀₉	—	4	2	no	—
179	C₂×C₁₀₉	—	4	1	no	—
180	C₂×C₁₀₉	#546	1	1	yes	characteristic
181	C₂×C₁₀₉	C₂×#135	1	1	yes	characteristic
182	C₂×C₁₀₉	#577	1	1	yes	characteristic
183	D₁₀₉	—	2	2	no	—
184	D₁₀₉	—	8	1	no	—
185	#3296	—	109	1	no	maximal, Sylow
186	C₁₀₉	#3296	1	1	yes	characteristic, complemented, Sylow
187	#552	—	109	1	no	—
188	#546	—	109	1	no	—
189	#488	—	109	1	no	—
190	#483	—	109	1	no	—
191	#482	—	109	1	no	—
192	#478	—	109	1	no	—
193	#467	—	109	1	no	—
194	#465	—	109	1	no	—
195	#464	—	109	1	no	—
196	#452	—	109	1	no	—
197	#450	—	109	1	no	—
198	C₄×SD₁₆	—	109	1	no	—
199	C₈×D₄	—	109	1	no	—
200	#416	D₁₀₉	1	1	yes	characteristic, complemented
201	C₂×#102	—	109	1	no	—
202	C₂×#41	C₂×D₁₀₉	1	1	yes	characteristic
203	C₂²×D₄	—	109	1	no	—
204	C₂×Q₁₆	—	109	1	no	—
205	C₂×SD₁₆	—	109	1	no	—
206	C₂×SD₁₆	—	109	1	no	—
207	C₂²×C₈	—	109	1	no	—
208	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic, complemented
209	#126	—	218	1	no	—
210	#125	—	109	1	no	—
211	#125	—	218	1	no	—
212	#124	—	109	1	no	—
213	#123	—	218	1	no	—
214	#123	—	218	1	no	—
215	C₂²⋊Q₈	—	109	1	no	—
216	C₂²⋊Q₈	—	109	1	no	—
217	C₂²⋊Q₈	—	109	1	no	—
218	#121	—	109	1	no	—
219	#121	—	218	1	no	—
220	#120	—	218	1	no	—
221	C₄×Q₈	—	109	1	no	—
222	C₄×D₄	—	109	1	no	—
223	C₄×D₄	—	109	1	no	—
224	#117	—	218	1	no	—
225	#117	C₂×D₁₀₉	1	1	yes	characteristic, complemented
226	C₂×C₄⋊C₄	—	109	1	no	—
227	C₂×C₂²⋊C₄	—	218	1	no	—
228	#107	—	109	1	no	—
229	#106	—	109	1	no	—
230	#106	—	109	1	no	—
231	C₄⋊C₈	—	109	1	no	—
232	#103	—	109	1	no	—
233	#103	—	109	1	no	—
234	#103	C₂×D₁₀₉	1	1	yes	characteristic, complemented
235	#103	C₂×D₁₀₉	1	1	yes	characteristic, complemented
236	#102	—	109	1	no	—
237	#102	—	109	1	no	—
238	#102	—	109	1	no	—
239	#102	—	109	1	no	—
240	#102	C₂×D₁₀₉	1	1	yes	characteristic
241	#102	C₂×D₁₀₉	1	1	yes	characteristic
242	C₂²⋊C₈	—	109	1	no	—
243	C₂²⋊C₈	—	109	1	no	—
244	C₄×C₈	—	109	1	no	—
245	C₂⁴	—	218	1	no	—
246	#41	—	2	2	no	—
247	#41	—	4	1	no	—
248	C₂×Q₈	—	109	1	no	—
249	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
250	C₂×D₄	—	2	1	no	—
251	C₂×D₄	—	109	1	no	—
252	C₂×D₄	—	109	1	no	—
253	C₂×D₄	—	436	1	no	—
254	C₂×D₄	—	218	1	no	—
255	C₂×D₄	—	218	1	no	—
256	C₂×D₄	—	218	2	no	—
257	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
258	C₂²×C₄	—	2	1	no	—
259	C₂²×C₄	—	109	1	no	—
260	C₂²×C₄	—	218	1	no	—
261	C₂²×C₄	—	218	1	no	—
262	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
263	Q₁₆	—	436	1	no	—
264	SD₁₆	—	436	1	no	—
265	SD₁₆	—	218	2	no	—
266	C₂×C₈	—	4	1	no	—
267	C₂×C₈	—	109	1	no	—
268	C₂×C₈	—	109	1	no	—
269	C₂×C₈	—	436	1	no	—
270	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
271	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
272	C₄⋊C₄	—	109	1	no	—
273	C₄⋊C₄	—	109	1	no	—
274	C₄⋊C₄	—	109	1	no	—
275	C₄⋊C₄	—	218	1	no	—
276	C₄⋊C₄	—	218	1	no	—
277	C₄⋊C₄	—	218	1	no	—
278	C₄⋊C₄	—	218	1	no	—
279	C₄⋊C₄	—	218	1	no	—
280	C₄⋊C₄	—	218	1	no	—
281	C₄⋊C₄	—	218	1	no	—
282	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
283	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
284	C₂²⋊C₄	—	2	1	no	—
285	C₂²⋊C₄	—	109	1	no	—
286	C₂²⋊C₄	—	109	1	no	—
287	C₂²⋊C₄	—	218	1	no	—
288	C₂²⋊C₄	—	218	1	no	—
289	C₂²⋊C₄	—	218	1	no	—
290	C₂²⋊C₄	—	218	1	no	—
291	C₂²⋊C₄	—	218	1	no	—
292	C₂²⋊C₄	—	218	1	no	—
293	C₂²⋊C₄	—	218	1	no	—
294	C₄²	—	2	1	no	—
295	C₄²	—	109	1	no	—
296	C₄²	—	218	1	no	—
297	C₂³	—	2	1	no	—
298	C₂³	—	872	1	no	—
299	C₂³	—	109	1	no	—
300	C₂³	—	436	2	no	—
301	C₂³	—	436	2	no	—
302	C₂³	—	218	1	no	—
303	C₂³	D₄×D₁₀₉	1	1	yes	characteristic
304	Q₈	—	2	1	no	—
305	Q₈	—	2	1	no	—
306	Q₈	—	218	1	no	—
307	Q₈	—	218	1	no	—
308	D₄	—	2	1	no	—
309	D₄	—	4	2	no	—
310	D₄	—	872	1	no	—
311	D₄	—	436	2	no	—
312	D₄	—	218	2	no	—
313	D₄	—	218	1	no	—
314	D₄	—	218	1	no	—
315	D₄	#13182029	1	2	yes	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	872	1	no	—
323	C₂×C₄	—	109	1	no	—
324	C₂×C₄	—	109	1	no	—
325	C₂×C₄	—	109	1	no	—
326	C₂×C₄	—	436	2	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	D₄×D₁₀₉	1	1	yes	characteristic
333	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
334	C₈	—	2	1	no	—
335	C₈	—	2	1	no	—
336	C₈	—	218	1	no	—
337	C₈	—	218	1	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	2	no	—
342	C₂²	—	4	1	no	—
343	C₂²	—	4	1	no	—
344	C₂²	—	872	4	no	—
345	C₂²	—	109	2	no	—
346	C₂²	—	436	1	no	—
347	C₂²	—	436	1	no	—
348	C₂²	—	436	1	no	—
349	C₂²	—	218	1	no	—
350	C₂²	—	218	1	no	—
351	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
352	C₄	—	2	1	no	—
353	C₄	—	4	1	no	—
354	C₄	—	4	1	no	—
355	C₄	—	4	1	no	—
356	C₄	—	436	1	no	—
357	C₄	—	436	1	no	—
358	C₄	—	436	1	no	—
359	C₄	—	218	1	no	—
360	C₄	—	218	1	no	—
361	C₄	—	218	1	no	—
362	C₄	C₂×#13182029	1	1	yes	characteristic
363	C₄	#16812744	1	1	yes	characteristic
364	C₂	—	4	1	no	—
365	C₂	—	4	2	no	—
366	C₂	—	872	1	no	—
367	C₂	—	218	2	no	—
368	C₂	#20295666	1	1	yes	characteristic
369	C₂	#135×D₁₀₉	1	1	yes	characteristic
370	C₂	#20295833	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#416×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	#20295664	C₂
hypercenter	#416	D₁₀₉
hypocenter	C₁₀₉	#3296
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #416

Complements

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

N	H ≅ G/N	type
1. #20295715	C₂	semidirect
3. #20295382	C₂	semidirect
4. #20295355	C₂	semidirect
5. #20295350	C₂	semidirect
6. #20295328	C₂	semidirect
7. #20295266	C₂	semidirect
8. #20295255	C₂	semidirect
9. #20295251	C₂	semidirect
10. #20295188	C₂	semidirect
11. #20295178	C₂	semidirect
12. #20295117	C₂	semidirect
13. C₈×C₂²⋊D₁₀₉	C₂	semidirect
14. #102×D₁₀₉	C₂	semidirect
15. #416×C₁₀₉	C₂	semidirect
17. C₂×#13182019	C₂²	semidirect
23. #16812690	C₂²	semidirect
28. #16812678	C₂²	semidirect
33. #16812664	C₂²	semidirect
39. #16812632	C₂²	semidirect
40. #16812631	C₂²	semidirect
41. #16812630	C₂²	semidirect
42. #16812628	C₂²	semidirect
43. #16812624	C₂²	semidirect
44. #16812623	C₂²	semidirect
49. C₂²⋊#11724784	C₂²	semidirect
51. C₈×Dic₁₀₉	C₂²	semidirect
53. C₂²×C₈×C₁₀₉	C₂²	semidirect
54. #117×C₁₀₉	C₂²	semidirect
55. #103×C₁₀₉	C₂²	semidirect
56. #103×C₁₀₉	C₂²	semidirect
186. C₁₀₉	#3296	semidirect
200. #416	D₁₀₉	semidirect
208. C₂²×C₈	C₂×D₁₀₉	semidirect
225. #117	C₂×D₁₀₉	semidirect
234. #103	C₂×D₁₀₉	semidirect
235. #103	C₂×D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295715	2
#20295664	2
#20295382	2
#20295355	2
#20295350	2
#20295328	2
#20295266	2
#20295255	2
#20295251	2
#20295188	2
#20295178	2
#20295117	2
C₈×C₂²⋊D₁₀₉	2
#102×D₁₀₉	2
#416×C₁₀₉	2
#3296	109