Subgroups — #28372502

Order 13952.

484 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812731	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16812728	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295575	C₂	1	1	yes	maximal, characteristic
4	C₄×Q₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295249	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295248	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295245	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295235	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295234	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295225	C₂	1	1	yes	maximal, characteristic, complemented
11	#20295224	C₂	1	1	yes	maximal, characteristic, complemented
12	#20295048	C₂	1	1	yes	maximal, characteristic, complemented
13	#20295026	C₂	1	1	yes	maximal, characteristic, complemented
14	#20295024	C₂	1	1	yes	maximal, characteristic, complemented
15	#464×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13182030	C₂²	1	1	yes	characteristic
17	C₂×#13182028	C₂²	1	1	yes	characteristic
18	C₂×Q₈×D₁₀₉	C₂²	1	1	yes	characteristic
19	#16812731	—	2	4	no	—
20	#16812728	—	2	4	no	—
21	C₂×Dic₄₃₆	C₂²	1	1	yes	characteristic
22	C₂×#13182019	C₂²	1	1	yes	characteristic
23	C₂×#13182019	C₂²	1	1	yes	characteristic
24	C₂×#13182018	C₂²	1	1	yes	characteristic
25	C₂×#13182017	C₂²	1	1	yes	characteristic
26	C₂×#13182016	C₂²	1	1	yes	characteristic
27	C₂×#13182012	C₂²	1	1	yes	characteristic
28	C₂×#13182012	C₂²	1	1	yes	characteristic
29	#16812697	C₂²	1	1	yes	characteristic
30	#16812695	C₂²	1	1	yes	characteristic
31	#16812687	C₂²	1	1	yes	characteristic
32	#16812672	—	2	1	no	—
33	#16812672	—	2	1	no	—
34	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
35	C₄×Dic₂₁₈	—	2	1	no	—
36	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
37	#16812664	—	2	1	no	—
38	#16812664	C₂²	1	1	yes	characteristic
39	#16812659	—	2	1	no	—
40	#16812659	—	2	1	no	—
41	C₄⋊C₄×D₁₀₉	—	2	1	no	—
42	C₄⋊C₄×D₁₀₉	C₂²	1	1	yes	characteristic
43	C₄²×D₁₀₉	—	2	1	no	—
44	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
45	C₄⋊#11724784	C₂²	1	1	yes	characteristic
46	#16812628	C₂²	1	1	yes	characteristic
47	#16812628	C₂²	1	1	yes	characteristic
48	#16812624	C₂²	1	1	yes	characteristic
49	#16812623	C₂²	1	1	yes	characteristic
50	#16812622	C₂²	1	1	yes	characteristic
51	#16812621	C₂²	1	1	yes	characteristic
52	#16812620	C₂²	1	1	yes	characteristic
53	#16812619	C₂²	1	1	yes	characteristic
54	C₂×Q₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
55	C₂×SD₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
56	#124×C₁₀₉	C₂²	1	1	yes	characteristic
57	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
58	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
59	#103×C₁₀₉	C₂²	1	1	yes	characteristic
60	#102×C₁₀₉	C₂²	1	1	yes	characteristic
61	#13182030	—	2	2	no	—
62	#13182030	—	2	2	no	—
63	#13182028	—	2	2	no	—
64	#13182028	—	2	2	no	—
65	C₂×Dic₂₁₈	—	2	1	no	—
66	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
67	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
68	Q₈×D₁₀₉	—	2	2	no	—
69	Q₈×D₁₀₉	D₄	1	4	yes	complemented
70	C₂×D₄₃₆	—	2	1	no	—
71	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
72	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
73	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
74	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
75	C₂²×Dic₁₀₉	—	2	1	no	—
76	C₂×C₄×D₁₀₉	—	2	1	no	—
77	C₂×C₄×D₁₀₉	—	2	1	no	—
78	C₂×C₄×D₁₀₉	—	2	1	no	—
79	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
80	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
81	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
82	Dic₄₃₆	—	2	2	no	—
83	#13182019	—	2	2	no	—
84	#13182019	—	2	2	no	—
85	#13182018	—	2	2	no	—
86	#13182017	—	2	2	no	—
87	#13182016	—	2	2	no	—
88	#13182012	—	2	2	no	—
89	#13182012	—	2	2	no	—
90	C₂×#11724784	C₂³	1	1	yes	characteristic
91	C₂×#11724784	C₂³	1	1	yes	characteristic
92	C₄⋊Dic₁₀₉	—	2	1	no	—
93	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
94	#13182008	—	2	1	no	—
95	#13182008	—	2	1	no	—
96	#13182008	—	2	1	no	—
97	#13182008	—	2	1	no	—
98	#13182008	—	2	1	no	—
99	#13182006	—	2	1	no	—
100	#13182006	—	2	1	no	—
101	C₄×Dic₁₀₉	—	2	1	no	—
102	C₄×Dic₁₀₉	—	2	1	no	—
103	C₄×Dic₁₀₉	—	2	1	no	—
104	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
105	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
106	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
107	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
108	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
109	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
110	Q₁₆×C₁₀₉	—	2	2	no	—
111	SD₁₆×C₁₀₉	—	2	2	no	—
112	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
113	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
114	C₄⋊C₄×C₁₀₉	—	2	1	no	—
115	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
116	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
117	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
118	C₄²×C₁₀₉	—	2	1	no	—
119	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
120	C₂³×C₁₀₉	—	2	1	no	—
121	C₂²×D₁₀₉	—	2	1	no	—
122	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
123	Q₈×C₁₀₉	—	2	1	no	—
124	Q₈×C₁₀₉	—	2	1	no	—
125	Q₈×C₁₀₉	—	2	1	no	—
126	Q₈×C₁₀₉	C₂×D₄	1	2	yes	—
127	D₄×C₁₀₉	—	2	1	no	—
128	D₄×C₁₀₉	—	2	1	no	—
129	C₂×C₄×C₁₀₉	—	2	1	no	—
130	C₂×C₄×C₁₀₉	—	2	1	no	—
131	C₂×C₄×C₁₀₉	—	2	1	no	—
132	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
134	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
135	C₂²⋊D₁₀₉	—	4	2	no	—
136	C₂²⋊D₁₀₉	—	4	2	no	—
137	C₂²⋊D₁₀₉	—	4	2	no	—
138	C₂×Dic₁₀₉	—	2	1	no	—
139	C₂×Dic₁₀₉	—	2	1	no	—
140	C₂×Dic₁₀₉	—	2	1	no	—
141	C₂×Dic₁₀₉	—	4	2	no	—
142	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
143	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
144	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
145	D₄₃₆	—	2	1	no	—
146	D₄₃₆	—	2	1	no	—
147	D₄₃₆	—	4	2	no	—
148	C₄×D₁₀₉	—	2	4	no	—
149	C₄×D₁₀₉	—	2	2	no	—
150	C₄×D₁₀₉	—	4	1	no	—
151	C₄×D₁₀₉	—	4	2	no	—
152	C₄×D₁₀₉	—	4	2	no	—
153	C₄×D₁₀₉	C₂×D₄	1	2	yes	—
154	C₄×D₁₀₉	#41	1	2	yes	—
155	Dic₂₁₈	—	2	1	no	—
156	Dic₂₁₈	—	2	4	no	—
157	Dic₂₁₈	—	2	1	no	—
158	Dic₂₁₈	—	2	1	no	—
159	Dic₂₁₈	C₂×D₄	1	2	yes	—
160	C₈×C₁₀₉	—	2	1	no	—
161	C₈×C₁₀₉	—	2	1	no	—
162	#11724784	—	2	1	no	—
163	#11724784	—	2	1	no	—
164	C₂²×C₁₀₉	—	4	1	no	—
165	C₂²×C₁₀₉	—	4	1	no	—
166	C₂²×C₁₀₉	—	4	1	no	—
167	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
168	C₂×D₁₀₉	—	2	1	no	—
169	C₂×D₁₀₉	—	2	1	no	—
170	C₂×D₁₀₉	—	4	1	no	—
171	C₂×D₁₀₉	—	4	1	no	—
172	C₂×D₁₀₉	—	4	1	no	—
173	C₂×D₁₀₉	#121	1	2	yes	—
174	C₄×C₁₀₉	—	2	2	no	—
175	C₄×C₁₀₉	—	2	1	no	—
176	C₄×C₁₀₉	—	2	1	no	—
177	C₄×C₁₀₉	—	4	1	no	—
178	C₄×C₁₀₉	—	4	1	no	—
179	C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
180	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
181	Dic₁₀₉	—	2	2	no	—
182	Dic₁₀₉	—	2	1	no	—
183	Dic₁₀₉	—	4	1	no	—
184	Dic₁₀₉	—	4	1	no	—
185	Dic₁₀₉	#121	1	2	yes	—
186	Dic₁₀₉	#137	1	2	yes	complemented
187	C₂×C₁₀₉	—	8	1	no	—
188	C₂×C₁₀₉	C₂×#121	1	1	yes	characteristic
189	C₂×C₁₀₉	C₂×#137	1	1	yes	characteristic
190	C₂×C₁₀₉	#575	1	1	yes	characteristic
191	D₁₀₉	—	2	2	no	—
192	D₁₀₉	—	8	1	no	—
193	#3037	—	109	1	no	maximal, Sylow
194	C₁₀₉	#3037	1	1	yes	characteristic, complemented, Sylow
195	C₂×#137	—	109	1	no	—
196	C₂×#137	—	109	1	no	—
197	#532	—	109	1	no	—
198	C₂×C₄×Q₈	—	109	1	no	—
199	#464	—	109	1	no	—
200	#464	—	109	1	no	—
201	#464	—	109	1	no	—
202	#464	D₁₀₉	1	1	yes	characteristic, complemented
203	#462	—	109	1	no	—
204	#462	—	109	1	no	—
205	#460	—	109	1	no	—
206	#460	—	109	1	no	—
207	#423	—	109	1	no	—
208	#417	—	109	1	no	—
209	#417	—	109	1	no	—
210	C₂×#41	—	109	1	no	—
211	C₂×#41	—	109	1	no	—
212	C₂²×Q₈	—	109	1	no	—
213	#137	—	218	4	no	—
214	#137	—	218	4	no	—
215	C₂×Q₁₆	—	109	1	no	—
216	C₂×Q₁₆	—	109	1	no	—
217	C₂×Q₁₆	—	109	1	no	—
218	C₂×Q₁₆	C₂×D₁₀₉	1	1	yes	characteristic
219	C₂×SD₁₆	—	109	1	no	—
220	C₂×SD₁₆	—	109	1	no	—
221	C₂×SD₁₆	—	109	1	no	—
222	C₂×SD₁₆	C₂×D₁₀₉	1	1	yes	characteristic
223	C₂×M₁₆	—	109	1	no	—
224	C₂×M₁₆	—	109	1	no	—
225	C₄⋊Q₈	—	109	1	no	—
226	#127	—	109	1	no	—
227	#124	—	109	1	no	—
228	#124	C₂×D₁₀₉	1	1	yes	characteristic
229	#121	—	218	1	no	—
230	#121	—	218	1	no	—
231	C₄×Q₈	—	109	1	no	—
232	C₄×Q₈	—	109	1	no	—
233	C₄×Q₈	—	109	1	no	—
234	C₄×Q₈	—	218	1	no	—
235	C₄×Q₈	—	218	1	no	—
236	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
237	C₄×D₄	—	218	1	no	—
238	C₄×D₄	—	218	1	no	—
239	C₂×C₄⋊C₄	—	109	1	no	—
240	C₂×C₄⋊C₄	—	218	1	no	—
241	C₂×C₄²	—	109	1	no	—
242	C₂×C₄²	—	218	1	no	—
243	C₄⋊C₈	—	109	1	no	—
244	C₄⋊C₈	—	109	1	no	—
245	C₄⋊C₈	—	109	1	no	—
246	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
247	#103	—	109	1	no	—
248	#103	—	109	1	no	—
249	#103	—	109	1	no	—
250	#103	C₂×D₁₀₉	1	1	yes	characteristic
251	#102	—	109	1	no	—
252	#102	—	109	1	no	—
253	#102	—	109	1	no	—
254	#102	C₂×D₁₀₉	1	1	yes	characteristic
255	#41	—	218	2	no	—
256	#41	—	218	2	no	—
257	#41	—	218	2	no	—
258	#41	—	218	2	no	—
259	C₂×Q₈	—	109	4	no	—
260	C₂×Q₈	—	109	1	no	—
261	C₂×Q₈	—	109	1	no	—
262	C₂×Q₈	—	218	1	no	—
263	C₂×Q₈	—	218	2	no	—
264	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
265	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
266	C₂×D₄	—	109	1	no	—
267	C₂×D₄	—	218	1	no	—
268	C₂×D₄	—	218	1	no	—
269	C₂×D₄	—	218	1	no	—
270	C₂×D₄	—	218	1	no	—
271	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
272	C₂²×C₄	—	109	1	no	—
273	C₂²×C₄	—	109	1	no	—
274	C₂²×C₄	—	109	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	Q₁₆	—	2	2	no	—
280	Q₁₆	—	218	2	no	—
281	Q₁₆	—	218	2	no	—
282	Q₁₆	—	218	2	no	—
283	SD₁₆	—	2	2	no	—
284	SD₁₆	—	218	2	no	—
285	SD₁₆	—	218	2	no	—
286	SD₁₆	—	218	2	no	—
287	M₁₆	—	218	2	no	—
288	M₁₆	—	218	2	no	—
289	C₂×C₈	—	109	1	no	—
290	C₂×C₈	—	109	1	no	—
291	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
292	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
293	C₄⋊C₄	—	2	1	no	—
294	C₄⋊C₄	—	109	1	no	—
295	C₄⋊C₄	—	218	1	no	—
296	C₄⋊C₄	—	218	1	no	—
297	C₄⋊C₄	—	218	1	no	—
298	C₄⋊C₄	—	218	1	no	—
299	C₄⋊C₄	—	218	1	no	—
300	C₄⋊C₄	—	218	1	no	—
301	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
302	C₂²⋊C₄	—	2	1	no	—
303	C₂²⋊C₄	—	2	1	no	—
304	C₂²⋊C₄	—	218	1	no	—
305	C₂²⋊C₄	—	218	1	no	—
306	C₄²	—	2	1	no	—
307	C₄²	—	109	1	no	—
308	C₄²	—	109	1	no	—
309	C₄²	—	109	1	no	—
310	C₄²	—	218	1	no	—
311	C₄²	—	218	1	no	—
312	C₄²	—	218	1	no	—
313	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
314	C₂³	—	2	1	no	—
315	C₂³	—	109	1	no	—
316	C₂³	—	218	1	no	—
317	Q₈	—	2	1	no	—
318	Q₈	—	2	1	no	—
319	Q₈	—	2	1	no	—
320	Q₈	—	109	2	no	—
321	Q₈	—	218	4	no	—
322	Q₈	—	218	1	no	—
323	Q₈	—	218	1	no	—
324	Q₈	—	218	1	no	—
325	Q₈	D₄×D₁₀₉	1	2	yes	—
326	D₄	—	2	1	no	—
327	D₄	—	2	1	no	—
328	D₄	—	436	2	no	—
329	D₄	—	436	2	no	—
330	D₄	—	436	2	no	—
331	D₄	—	436	2	no	—
332	D₄	—	218	1	no	—
333	D₄	—	218	1	no	—
334	C₂×C₄	—	2	1	no	—
335	C₂×C₄	—	2	1	no	—
336	C₂×C₄	—	2	1	no	—
337	C₂×C₄	—	109	1	no	—
338	C₂×C₄	—	109	1	no	—
339	C₂×C₄	—	109	2	no	—
340	C₂×C₄	—	109	2	no	—
341	C₂×C₄	—	109	1	no	—
342	C₂×C₄	—	436	1	no	—
343	C₂×C₄	—	436	2	no	—
344	C₂×C₄	—	436	2	no	—
345	C₂×C₄	—	436	2	no	—
346	C₂×C₄	—	218	2	no	—
347	C₂×C₄	—	218	4	no	—
348	C₂×C₄	—	218	1	no	—
349	C₂×C₄	—	218	1	no	—
350	C₂×C₄	—	218	1	no	—
351	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
352	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
353	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
354	C₈	—	2	1	no	—
355	C₈	—	2	1	no	—
356	C₈	—	218	1	no	—
357	C₈	—	218	1	no	—
358	C₂²	—	4	1	no	—
359	C₂²	—	4	1	no	—
360	C₂²	—	4	1	no	—
361	C₂²	—	109	2	no	—
362	C₂²	—	436	1	no	—
363	C₂²	—	436	1	no	—
364	C₂²	—	436	1	no	—
365	C₂²	—	218	1	no	—
366	C₂²	—	218	1	no	—
367	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
368	C₄	—	2	2	no	—
369	C₄	—	2	1	no	—
370	C₄	—	2	1	no	—
371	C₄	—	4	1	no	—
372	C₄	—	4	1	no	—
373	C₄	—	109	2	no	—
374	C₄	—	109	2	no	—
375	C₄	—	436	1	no	—
376	C₄	—	436	1	no	—
377	C₄	—	218	2	no	—
378	C₄	—	218	1	no	—
379	C₄	C₂×D₄×D₁₀₉	1	1	yes	characteristic
380	C₄	#41×D₁₀₉	1	1	yes	characteristic
381	C₂	—	8	1	no	—
382	C₂	—	872	1	no	—
383	C₂	—	218	2	no	—
384	C₂	#121×D₁₀₉	1	1	yes	characteristic
385	C₂	#137×D₁₀₉	1	1	yes	characteristic
386	C₂	#20295823	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#464×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	#20295575	C₂
hypercenter	#464	D₁₀₉
hypocenter	C₁₀₉	#3037
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #464

Complements

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

N	H ≅ G/N	type
1. C₂×#16812731	C₂	semidirect
2. C₂×#16812728	C₂	semidirect
4. C₄×Q₈×D₁₀₉	C₂	semidirect
5. #20295249	C₂	semidirect
6. #20295248	C₂	semidirect
7. #20295245	C₂	semidirect
8. #20295235	C₂	semidirect
9. #20295234	C₂	semidirect
10. #20295225	C₂	semidirect
11. #20295224	C₂	semidirect
12. #20295048	C₂	semidirect
13. #20295026	C₂	semidirect
14. #20295024	C₂	semidirect
15. #464×C₁₀₉	C₂	semidirect
69. Q₈×D₁₀₉	D₄	semidirect
186. Dic₁₀₉	#137	semidirect
194. C₁₀₉	#3037	semidirect
202. #464	D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16812731	2
C₂×#16812728	2
#20295575	2
C₄×Q₈×D₁₀₉	2
#20295249	2
#20295248	2
#20295245	2
#20295235	2
#20295234	2
#20295225	2
#20295224	2
#20295048	2
#20295026	2
#20295024	2
#464×C₁₀₉	2
#3037	109