Subgroups — #28372560

Order 13952.

458 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	C₂×#16812728	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₁₆×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295596	C₂	1	1	yes	maximal, characteristic
4	#20295502	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295247	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295246	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295245	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295237	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295236	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295235	C₂	1	1	yes	maximal, characteristic, complemented
11	#20295234	C₂	1	1	yes	maximal, characteristic, complemented
12	#20295051	C₂	1	1	yes	maximal, characteristic, complemented
13	#20295033	C₂	1	1	yes	maximal, characteristic, complemented
14	#20295019	C₂	1	1	yes	maximal, characteristic
15	#464×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13182028	C₂²	1	1	yes	characteristic
17	C₂×Q₈×D₁₀₉	C₂²	1	1	yes	characteristic
18	C₂×Q₈×D₁₀₉	C₂²	1	1	yes	characteristic
19	#16812728	—	2	4	no	—
20	C₂×Dic₄₃₆	C₂²	1	1	yes	characteristic
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×#13182019	C₂²	1	1	yes	characteristic
25	Q₁₆×D₁₀₉	—	2	4	no	—
26	C₂×#13182017	C₂²	1	1	yes	characteristic
27	C₂×#13182012	C₂²	1	1	yes	characteristic
28	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
29	#16812699	C₂²	1	1	yes	characteristic
30	#16812697	C₂²	1	1	yes	characteristic
31	#16812686	C₂²	1	1	yes	characteristic
32	#16812681	—	2	1	no	—
33	#16812681	—	2	1	no	—
34	#16812676	—	2	1	no	—
35	#16812676	—	2	1	no	—
36	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
37	C₄×Dic₂₁₈	—	2	1	no	—
38	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
39	#16812664	—	2	1	no	—
40	#16812664	C₂²	1	1	yes	characteristic
41	#16812657	—	2	1	no	—
42	#16812657	C₂²	1	1	yes	characteristic
43	#16812656	—	2	1	no	—
44	#16812656	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	#16812624	C₂²	1	1	yes	characteristic
50	#16812623	C₂²	1	1	yes	characteristic
51	#16812622	C₂²	1	1	yes	characteristic
52	#16812622	C₂²	1	1	yes	characteristic
53	#16812620	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	#13182028	—	2	2	no	—
62	#13182028	—	2	2	no	—
63	C₂×Dic₂₁₈	—	2	1	no	—
64	C₂×Dic₂₁₈	—	2	1	no	—
65	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
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	—	2	2	no	—
70	Q₈×D₁₀₉	—	2	2	no	—
71	Q₈×D₁₀₉	D₄	1	4	yes	—
72	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
73	C₂²×Dic₁₀₉	—	2	1	no	—
74	C₂×C₄×D₁₀₉	—	2	1	no	—
75	C₂×C₄×D₁₀₉	—	2	1	no	—
76	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
77	Dic₄₃₆	—	2	2	no	—
78	Dic₄₃₆	—	2	2	no	—
79	#13182019	—	2	2	no	—
80	#13182019	—	2	2	no	—
81	#13182019	—	2	2	no	—
82	#13182017	—	2	2	no	—
83	#13182012	—	2	2	no	—
84	C₂×#11724784	C₂³	1	1	yes	characteristic
85	C₂×#11724784	C₂³	1	1	yes	characteristic
86	C₈×D₁₀₉	—	2	2	no	—
87	C₄⋊Dic₁₀₉	—	2	1	no	—
88	C₄⋊Dic₁₀₉	—	2	1	no	—
89	C₄⋊Dic₁₀₉	—	2	1	no	—
90	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
91	#13182008	—	2	1	no	—
92	#13182008	—	2	1	no	—
93	#13182008	—	2	1	no	—
94	#13182008	—	2	1	no	—
95	#13182008	D₄	1	1	yes	characteristic
96	#13182008	D₄	1	1	yes	characteristic
97	C₂²⋊Dic₁₀₉	—	2	1	no	—
98	C₂²⋊Dic₁₀₉	—	2	1	no	—
99	#13182006	—	2	1	no	—
100	#13182006	D₄	1	1	yes	characteristic
101	#13182006	D₄	1	1	yes	characteristic
102	C₄×Dic₁₀₉	—	2	1	no	—
103	C₄×Dic₁₀₉	—	2	1	no	—
104	C₄×Dic₁₀₉	—	2	1	no	—
105	C₄×Dic₁₀₉	—	2	1	no	—
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	C₂×D₄	1	1	yes	characteristic
122	Q₈×C₁₀₉	—	2	1	no	—
123	Q₈×C₁₀₉	—	2	1	no	—
124	Q₈×C₁₀₉	—	2	1	no	—
125	Q₈×C₁₀₉	C₂×D₄	1	2	yes	—
126	D₄×C₁₀₉	—	2	1	no	—
127	D₄×C₁₀₉	—	2	1	no	—
128	C₂×C₄×C₁₀₉	—	2	1	no	—
129	C₂×C₄×C₁₀₉	—	2	1	no	—
130	C₂×C₄×C₁₀₉	—	2	1	no	—
131	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
132	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
134	C₂²⋊D₁₀₉	—	4	2	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	—	4	2	no	—
140	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
141	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
142	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
143	C₄×D₁₀₉	—	2	4	no	—
144	C₄×D₁₀₉	—	4	2	no	—
145	C₄×D₁₀₉	C₂×D₄	1	2	yes	—
146	C₄×D₁₀₉	#41	1	2	yes	—
147	Dic₂₁₈	—	2	1	no	—
148	Dic₂₁₈	—	2	4	no	—
149	Dic₂₁₈	—	2	1	no	—
150	Dic₂₁₈	—	2	1	no	—
151	Dic₂₁₈	—	2	1	no	—
152	Dic₂₁₈	—	2	1	no	—
153	Dic₂₁₈	—	4	2	no	—
154	Dic₂₁₈	C₂×D₄	1	2	yes	—
155	C₈×C₁₀₉	—	2	1	no	—
156	C₈×C₁₀₉	—	2	1	no	—
157	#11724784	—	2	1	no	—
158	#11724784	—	2	1	no	—
159	C₂²×C₁₀₉	—	4	1	no	—
160	C₂²×C₁₀₉	—	4	1	no	—
161	C₂²×C₁₀₉	—	4	1	no	—
162	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
163	C₂×D₁₀₉	—	2	1	no	—
164	C₂×D₁₀₉	—	2	1	no	—
165	C₂×D₁₀₉	#121	1	2	yes	—
166	C₄×C₁₀₉	—	2	2	no	—
167	C₄×C₁₀₉	—	2	1	no	—
168	C₄×C₁₀₉	—	2	1	no	—
169	C₄×C₁₀₉	—	4	1	no	—
170	C₄×C₁₀₉	—	4	1	no	—
171	C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
172	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
173	Dic₁₀₉	—	2	2	no	—
174	Dic₁₀₉	—	2	1	no	—
175	Dic₁₀₉	—	2	1	no	—
176	Dic₁₀₉	—	4	1	no	—
177	Dic₁₀₉	—	4	1	no	—
178	Dic₁₀₉	—	4	1	no	—
179	Dic₁₀₉	#121	1	2	yes	—
180	C₂×C₁₀₉	—	8	1	no	—
181	C₂×C₁₀₉	C₂×#121	1	1	yes	characteristic
182	C₂×C₁₀₉	#578	1	1	yes	characteristic
183	C₂×C₁₀₉	#578	1	1	yes	characteristic
184	D₁₀₉	—	2	2	no	—
185	#3044	—	109	1	no	maximal, Sylow
186	C₁₀₉	#3044	1	1	yes	characteristic, complemented, Sylow
187	C₂×#137	—	109	1	no	—
188	C₂²×Q₁₆	—	109	1	no	—
189	#536	—	109	1	no	—
190	#519	—	109	1	no	—
191	#464	—	109	1	no	—
192	#464	—	109	1	no	—
193	#464	—	109	1	no	—
194	#464	D₁₀₉	1	1	yes	characteristic, complemented
195	#462	—	109	1	no	—
196	#462	—	109	1	no	—
197	#462	—	109	1	no	—
198	#462	—	109	1	no	—
199	#424	—	109	1	no	—
200	#419	—	109	1	no	—
201	#416	—	109	1	no	—
202	C₂×#41	—	109	1	no	—
203	C₂²×Q₈	—	109	1	no	—
204	C₂²×Q₈	—	109	1	no	—
205	#137	—	218	4	no	—
206	C₂×Q₁₆	—	109	1	no	—
207	C₂×Q₁₆	—	109	1	no	—
208	C₂×Q₁₆	—	109	1	no	—
209	C₂×Q₁₆	—	109	1	no	—
210	C₂×Q₁₆	—	109	1	no	—
211	C₂×Q₁₆	—	218	4	no	—
212	C₂×Q₁₆	C₂×D₁₀₉	1	1	yes	characteristic
213	C₂×SD₁₆	—	109	1	no	—
214	C₂×SD₁₆	C₂×D₁₀₉	1	1	yes	characteristic
215	C₂×M₁₆	—	109	1	no	—
216	C₂²×C₈	—	109	1	no	—
217	C₄⋊Q₈	—	109	1	no	—
218	C₄⋊Q₈	—	109	1	no	—
219	#124	—	109	1	no	—
220	#124	C₂×D₁₀₉	1	1	yes	characteristic
221	#123	—	218	1	no	—
222	#123	—	218	1	no	—
223	C₂²⋊Q₈	—	218	1	no	—
224	C₂²⋊Q₈	—	218	1	no	—
225	C₄×Q₈	—	109	1	no	—
226	C₄×Q₈	—	109	1	no	—
227	C₄×Q₈	—	109	1	no	—
228	C₄×Q₈	—	218	1	no	—
229	C₄×Q₈	—	218	1	no	—
230	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
231	#117	—	109	1	no	—
232	#117	—	109	1	no	—
233	#117	—	218	1	no	—
234	#117	—	218	1	no	—
235	C₄⋊C₈	—	109	1	no	—
236	C₄⋊C₈	—	109	1	no	—
237	C₄⋊C₈	—	109	1	no	—
238	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
239	#103	—	109	1	no	—
240	#103	—	109	1	no	—
241	#103	—	109	1	no	—
242	#103	—	109	1	no	—
243	#103	—	109	1	no	—
244	#103	C₂×D₁₀₉	1	1	yes	characteristic
245	#102	—	109	1	no	—
246	#102	C₂×D₁₀₉	1	1	yes	characteristic
247	#41	—	218	2	no	—
248	#41	—	218	2	no	—
249	C₂×Q₈	—	109	4	no	—
250	C₂×Q₈	—	109	1	no	—
251	C₂×Q₈	—	109	1	no	—
252	C₂×Q₈	—	109	1	no	—
253	C₂×Q₈	—	218	1	no	—
254	C₂×Q₈	—	218	1	no	—
255	C₂×Q₈	—	218	2	no	—
256	C₂×Q₈	—	218	2	no	—
257	C₂×Q₈	—	218	2	no	—
258	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
259	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
260	C₂×D₄	—	218	1	no	—
261	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
262	C₂²×C₄	—	109	1	no	—
263	C₂²×C₄	—	218	1	no	—
264	C₂²×C₄	—	218	1	no	—
265	C₂²×C₄	—	218	1	no	—
266	Q₁₆	—	2	2	no	—
267	Q₁₆	—	218	2	no	—
268	Q₁₆	—	218	2	no	—
269	Q₁₆	—	218	2	no	—
270	Q₁₆	—	218	2	no	—
271	Q₁₆	—	218	2	no	—
272	SD₁₆	—	2	2	no	—
273	SD₁₆	—	218	2	no	—
274	M₁₆	—	218	2	no	—
275	C₂×C₈	—	109	1	no	—
276	C₂×C₈	—	109	1	no	—
277	C₂×C₈	—	218	2	no	—
278	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
279	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
280	C₄⋊C₄	—	2	1	no	—
281	C₄⋊C₄	—	109	1	no	—
282	C₄⋊C₄	—	109	1	no	—
283	C₄⋊C₄	—	109	1	no	—
284	C₄⋊C₄	—	218	1	no	—
285	C₄⋊C₄	—	218	1	no	—
286	C₄⋊C₄	—	218	1	no	—
287	C₄⋊C₄	—	218	1	no	—
288	C₄⋊C₄	—	218	1	no	—
289	C₄⋊C₄	—	218	1	no	—
290	C₄⋊C₄	—	218	1	no	—
291	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
292	C₂²⋊C₄	—	2	1	no	—
293	C₂²⋊C₄	—	2	1	no	—
294	C₂²⋊C₄	—	109	1	no	—
295	C₂²⋊C₄	—	109	1	no	—
296	C₂²⋊C₄	—	218	1	no	—
297	C₂²⋊C₄	—	218	1	no	—
298	C₂²⋊C₄	—	218	1	no	—
299	C₄²	—	2	1	no	—
300	C₄²	—	109	1	no	—
301	C₄²	—	218	1	no	—
302	C₄²	—	218	1	no	—
303	C₄²	—	218	1	no	—
304	C₄²	—	218	1	no	—
305	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
306	C₂³	—	2	1	no	—
307	C₂³	—	109	1	no	—
308	Q₈	—	2	1	no	—
309	Q₈	—	2	1	no	—
310	Q₈	—	2	1	no	—
311	Q₈	—	109	2	no	—
312	Q₈	—	436	2	no	—
313	Q₈	—	218	4	no	—
314	Q₈	—	218	1	no	—
315	Q₈	—	218	1	no	—
316	Q₈	—	218	1	no	—
317	Q₈	—	218	1	no	—
318	Q₈	—	218	1	no	—
319	Q₈	D₄×D₁₀₉	1	2	yes	—
320	D₄	—	2	1	no	—
321	D₄	—	2	1	no	—
322	D₄	—	436	2	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	109	1	no	—
327	C₂×C₄	—	109	2	no	—
328	C₂×C₄	—	109	2	no	—
329	C₂×C₄	—	109	1	no	—
330	C₂×C₄	—	109	1	no	—
331	C₂×C₄	—	436	2	no	—
332	C₂×C₄	—	436	2	no	—
333	C₂×C₄	—	218	4	no	—
334	C₂×C₄	—	218	1	no	—
335	C₂×C₄	—	218	1	no	—
336	C₂×C₄	—	218	1	no	—
337	C₂×C₄	—	218	1	no	—
338	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
339	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
340	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
341	C₈	—	2	1	no	—
342	C₈	—	2	1	no	—
343	C₈	—	218	1	no	—
344	C₈	—	218	1	no	—
345	C₂²	—	4	1	no	—
346	C₂²	—	4	1	no	—
347	C₂²	—	4	1	no	—
348	C₂²	—	109	2	no	—
349	C₂²	—	218	1	no	—
350	C₂²	—	218	1	no	—
351	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
352	C₄	—	2	2	no	—
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	4	1	no	—
356	C₄	—	4	1	no	—
357	C₄	—	109	2	no	—
358	C₄	—	436	1	no	—
359	C₄	—	436	1	no	—
360	C₄	—	436	1	no	—
361	C₄	—	218	2	no	—
362	C₄	—	218	1	no	—
363	C₄	—	218	1	no	—
364	C₄	C₂×D₄×D₁₀₉	1	1	yes	characteristic
365	C₄	#41×D₁₀₉	1	1	yes	characteristic
366	C₂	—	8	1	no	—
367	C₂	—	218	2	no	—
368	C₂	#121×D₁₀₉	1	1	yes	characteristic
369	C₂	#20295840	1	1	yes	characteristic
370	C₂	#20295842	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	C₂×#13182028	C₂²
hypercenter	#464	D₁₀₉
hypocenter	C₁₀₉	#3044
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
2. C₂×Q₁₆×D₁₀₉	C₂	semidirect
4. #20295502	C₂	semidirect
5. #20295247	C₂	semidirect
6. #20295246	C₂	semidirect
7. #20295245	C₂	semidirect
8. #20295237	C₂	semidirect
9. #20295236	C₂	semidirect
10. #20295235	C₂	semidirect
11. #20295234	C₂	semidirect
12. #20295051	C₂	semidirect
13. #20295033	C₂	semidirect
15. #464×C₁₀₉	C₂	semidirect
186. C₁₀₉	#3044	semidirect
194. #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×#16812728	2
C₂×Q₁₆×D₁₀₉	2
#20295596	2
#20295502	2
#20295247	2
#20295246	2
#20295245	2
#20295237	2
#20295236	2
#20295235	2
#20295234	2
#20295051	2
#20295033	2
#20295019	2
#464×C₁₀₉	2
#3044	109