Subgroups — #28373524

Order 13952.

532 conjugacy classes of subgroups.

The subgroup lattice has 370 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	C₂×#16812720	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₁₆×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295601	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295596	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295420	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295419	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295417	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295408	C₂	1	1	yes	maximal, characteristic, complemented
11	#20295407	C₂	1	1	yes	maximal, characteristic, complemented
12	#20295405	C₂	1	1	yes	maximal, characteristic, complemented
13	#20295404	C₂	1	1	yes	maximal, characteristic, complemented
14	#20294983	C₂	1	1	yes	maximal, characteristic, complemented
15	#497×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	C₂×Q₈×D₁₀₉	C₂²	1	1	yes	characteristic
20	#16812731	—	2	4	no	—
21	#16812728	—	2	4	no	—
22	#16812720	—	2	4	no	—
23	C₂×Dic₄₃₆	C₂²	1	1	yes	characteristic
24	C₂×Dic₄₃₆	C₂²	1	1	yes	characteristic
25	C₂×#13182019	C₂²	1	1	yes	characteristic
26	C₂×#13182019	C₂²	1	1	yes	characteristic
27	C₂×#13182019	C₂²	1	1	yes	characteristic
28	C₂×#13182019	C₂²	1	1	yes	characteristic
29	C₂×#13182019	C₂²	1	1	yes	characteristic
30	Q₁₆×D₁₀₉	—	2	4	no	—
31	C₂×#13182018	C₂²	1	1	yes	characteristic
32	C₂×#13182018	C₂²	1	1	yes	characteristic
33	C₂×#13182017	C₂²	1	1	yes	characteristic
34	C₂×#13182016	C₂²	1	1	yes	characteristic
35	C₂×#13182014	C₂²	1	1	yes	characteristic
36	C₂×#13182012	C₂²	1	1	yes	characteristic
37	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
38	#16812699	C₂²	1	1	yes	characteristic
39	#16812697	C₂²	1	1	yes	characteristic
40	#16812697	C₂²	1	1	yes	characteristic
41	#16812688	C₂²	1	1	yes	characteristic
42	#16812687	C₂²	1	1	yes	characteristic
43	#16812686	C₂²	1	1	yes	characteristic
44	#16812684	—	2	2	no	—
45	#16812681	—	2	2	no	—
46	#16812677	—	2	2	no	—
47	#16812676	—	2	2	no	—
48	#16812656	C₂²	1	1	yes	characteristic
49	#16812609	C₂²	1	1	yes	characteristic
50	C₄×#11724784	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
52	C₂×Q₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
53	C₂×Q₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
54	C₂×SD₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
55	C₂×SD₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
56	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
57	#124×C₁₀₉	C₂²	1	1	yes	characteristic
58	#97×C₁₀₉	C₂²	1	1	yes	characteristic
59	#13182030	—	2	2	no	—
60	#13182030	—	2	2	no	—
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	—	2	2	no	—
72	C₂×D₄₃₆	—	2	1	no	—
73	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
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	C₂³	1	1	yes	characteristic
80	Dic₄₃₆	—	2	2	no	—
81	Dic₄₃₆	—	2	2	no	—
82	#13182019	—	2	2	no	—
83	#13182019	—	2	2	no	—
84	#13182019	—	2	2	no	—
85	#13182019	—	2	2	no	—
86	#13182019	—	2	2	no	—
87	#13182018	—	2	2	no	—
88	#13182018	—	2	2	no	—
89	#13182017	—	2	2	no	—
90	#13182016	—	2	2	no	—
91	#13182014	—	2	2	no	—
92	#13182012	D₄	1	4	yes	—
93	C₂×#11724784	C₂³	1	1	yes	characteristic
94	C₂×#11724784	C₂³	1	1	yes	characteristic
95	C₈×D₁₀₉	D₄	1	4	yes	—
96	C₄⋊Dic₁₀₉	—	2	2	no	—
97	#13182008	—	2	2	no	—
98	#13182008	—	2	2	no	—
99	#13182008	D₄	1	2	yes	—
100	C₂²⋊Dic₁₀₉	—	2	2	no	—
101	#13182006	—	2	2	no	—
102	#13182006	—	2	2	no	—
103	#13182006	D₄	1	2	yes	—
104	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
105	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
106	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
107	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
108	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
109	Q₁₆×C₁₀₉	—	2	2	no	—
110	Q₁₆×C₁₀₉	—	2	2	no	—
111	SD₁₆×C₁₀₉	—	2	2	no	—
112	SD₁₆×C₁₀₉	—	2	2	no	—
113	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
114	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
115	C₄⋊C₄×C₁₀₉	—	2	2	no	—
116	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
117	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
118	C₂³×C₁₀₉	—	2	1	no	—
119	C₂²×D₁₀₉	—	2	1	no	—
120	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
121	Q₈×C₁₀₉	—	2	1	no	—
122	Q₈×C₁₀₉	—	2	1	no	—
123	Q₈×C₁₀₉	—	2	1	no	—
124	Q₈×C₁₀₉	—	2	1	no	—
125	Q₈×C₁₀₉	—	2	1	no	—
126	Q₈×C₁₀₉	—	2	1	no	—
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	2	yes	—
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	—	4	2	no	—
139	C₂×Dic₁₀₉	C₂×D₄	1	2	yes	—
140	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
141	D₄₃₆	—	2	1	no	—
142	D₄₃₆	—	2	1	no	—
143	D₄₃₆	—	4	2	no	—
144	C₄×D₁₀₉	—	4	2	no	—
145	C₄×D₁₀₉	—	4	2	no	—
146	C₄×D₁₀₉	—	4	2	no	—
147	C₄×D₁₀₉	C₂×D₄	1	2	yes	—
148	C₄×D₁₀₉	C₂×D₄	1	2	yes	—
149	Dic₂₁₈	—	2	1	no	—
150	Dic₂₁₈	—	2	1	no	—
151	Dic₂₁₈	—	2	1	no	—
152	Dic₂₁₈	—	2	1	no	—
153	Dic₂₁₈	—	2	1	no	—
154	Dic₂₁₈	—	2	1	no	—
155	Dic₂₁₈	—	4	2	no	—
156	Dic₂₁₈	—	4	2	no	—
157	C₈×C₁₀₉	C₂×D₄	1	2	yes	—
158	C₈×C₁₀₉	C₂×D₄	1	2	yes	—
159	#11724784	C₂×D₄	1	2	yes	—
160	#11724784	C₂×D₄	1	2	yes	—
161	C₂²×C₁₀₉	—	4	1	no	—
162	C₂²×C₁₀₉	—	4	2	no	—
163	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
164	C₂×D₁₀₉	—	2	2	no	—
165	C₂×D₁₀₉	—	4	1	no	—
166	C₂×D₁₀₉	—	4	2	no	—
167	C₂×D₁₀₉	#127	1	2	yes	—
168	C₄×C₁₀₉	—	2	2	no	—
169	C₄×C₁₀₉	—	4	1	no	—
170	C₄×C₁₀₉	—	4	1	no	—
171	C₄×C₁₀₉	—	4	1	no	—
172	C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
173	C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
174	Dic₁₀₉	—	2	2	no	—
175	Dic₁₀₉	—	4	1	no	—
176	Dic₁₀₉	—	4	1	no	—
177	Dic₁₀₉	—	4	1	no	—
178	Dic₁₀₉	#127	1	2	yes	—
179	C₂×C₁₀₉	—	8	1	no	—
180	C₂×C₁₀₉	C₂×#127	1	1	yes	characteristic
181	C₂×C₁₀₉	#578	1	2	yes	—
182	D₁₀₉	—	2	2	no	—
183	D₁₀₉	—	8	1	no	—
184	#3156	—	109	1	no	maximal, Sylow
185	C₁₀₉	#3156	1	1	yes	characteristic, complemented, Sylow
186	C₂×#137	—	109	1	no	—
187	C₂×#137	—	109	1	no	—
188	C₂×#135	—	109	1	no	—
189	C₂²×Q₁₆	—	109	1	no	—
190	#536	—	109	1	no	—
191	#536	—	109	1	no	—
192	#497	—	109	1	no	—
193	#497	—	109	1	no	—
194	#497	—	109	1	no	—
195	#497	D₁₀₉	1	1	yes	characteristic, complemented
196	#495	—	109	1	no	—
197	#495	—	109	1	no	—
198	#494	—	109	1	no	—
199	#494	—	109	1	no	—
200	#405	—	109	1	no	—
201	C₂×#41	—	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	#137	—	218	4	no	—
207	#135	—	218	4	no	—
208	C₂×Q₁₆	—	109	1	no	—
209	C₂×Q₁₆	—	109	1	no	—
210	C₂×Q₁₆	—	109	1	no	—
211	C₂×Q₁₆	—	109	1	no	—
212	C₂×Q₁₆	—	109	1	no	—
213	C₂×Q₁₆	—	109	1	no	—
214	C₂×Q₁₆	—	109	1	no	—
215	C₂×Q₁₆	—	218	4	no	—
216	C₂×Q₁₆	C₂×D₁₀₉	1	1	yes	characteristic
217	C₂×Q₁₆	C₂×D₁₀₉	1	1	yes	characteristic
218	C₂×SD₁₆	—	109	1	no	—
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×SD₁₆	C₂×D₁₀₉	1	1	yes	characteristic
224	C₂×D₈	—	109	1	no	—
225	C₂×M₁₆	—	109	1	no	—
226	C₂²×C₈	—	109	1	no	—
227	C₄⋊Q₈	—	109	1	no	—
228	C₄⋊Q₈	—	109	1	no	—
229	C₄⋊Q₈	—	109	1	no	—
230	C₄⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic
231	#124	—	109	1	no	—
232	#124	—	109	1	no	—
233	#124	—	109	1	no	—
234	#124	C₂×D₁₀₉	1	1	yes	characteristic
235	#123	—	218	2	no	—
236	#123	—	218	2	no	—
237	C₂²⋊Q₈	—	218	2	no	—
238	C₂²⋊Q₈	—	218	2	no	—
239	#117	—	109	1	no	—
240	#97	—	109	1	no	—
241	#97	C₂×D₁₀₉	1	1	yes	characteristic
242	C₄×C₈	—	109	1	no	—
243	C₄×C₈	—	109	1	no	—
244	#41	—	218	2	no	—
245	#41	—	218	2	no	—
246	#41	—	218	2	no	—
247	#41	—	218	2	no	—
248	C₂×Q₈	—	109	1	no	—
249	C₂×Q₈	—	109	1	no	—
250	C₂×Q₈	—	109	1	no	—
251	C₂×Q₈	—	218	1	no	—
252	C₂×Q₈	—	218	1	no	—
253	C₂×Q₈	—	218	2	no	—
254	C₂×Q₈	—	218	2	no	—
255	C₂×Q₈	—	218	2	no	—
256	C₂×Q₈	—	218	2	no	—
257	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
258	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
259	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
260	C₂×D₄	—	109	1	no	—
261	C₂×D₄	—	218	1	no	—
262	C₂×D₄	—	218	1	no	—
263	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
264	C₂²×C₄	—	109	1	no	—
265	C₂²×C₄	—	218	1	no	—
266	C₂²×C₄	—	218	1	no	—
267	C₂²×C₄	—	218	1	no	—
268	C₂²×C₄	—	218	1	no	—
269	Q₁₆	—	2	2	no	—
270	Q₁₆	—	2	2	no	—
271	Q₁₆	—	218	2	no	—
272	Q₁₆	—	218	2	no	—
273	Q₁₆	—	218	2	no	—
274	Q₁₆	—	218	2	no	—
275	Q₁₆	—	218	2	no	—
276	Q₁₆	—	218	2	no	—
277	Q₁₆	—	218	2	no	—
278	SD₁₆	—	2	2	no	—
279	SD₁₆	—	2	2	no	—
280	SD₁₆	—	218	2	no	—
281	SD₁₆	—	218	2	no	—
282	SD₁₆	—	218	2	no	—
283	SD₁₆	—	218	2	no	—
284	D₈	—	218	2	no	—
285	M₁₆	—	109	4	no	—
286	C₂×C₈	—	109	1	no	—
287	C₂×C₈	—	109	1	no	—
288	C₂×C₈	—	109	4	no	—
289	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
290	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	109	2	no	—
293	C₄⋊C₄	—	218	2	no	—
294	C₄⋊C₄	—	218	2	no	—
295	C₄⋊C₄	—	218	2	no	—
296	C₂²⋊C₄	—	2	2	no	—
297	C₂²⋊C₄	—	109	2	no	—
298	C₂²⋊C₄	—	218	2	no	—
299	C₂²⋊C₄	—	218	2	no	—
300	C₂²⋊C₄	—	218	2	no	—
301	C₄²	—	109	1	no	—
302	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
303	C₂³	—	2	1	no	—
304	C₂³	—	109	1	no	—
305	C₂³	—	218	1	no	—
306	Q₈	—	2	1	no	—
307	Q₈	—	2	1	no	—
308	Q₈	—	2	1	no	—
309	Q₈	—	2	1	no	—
310	Q₈	—	2	1	no	—
311	Q₈	—	2	1	no	—
312	Q₈	—	436	2	no	—
313	Q₈	—	436	2	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	—	218	1	no	—
320	D₄	—	2	1	no	—
321	D₄	—	2	1	no	—
322	D₄	—	436	2	no	—
323	D₄	—	436	2	no	—
324	D₄	—	218	1	no	—
325	D₄	—	218	1	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	109	1	no	—
330	C₂×C₄	—	109	2	no	—
331	C₂×C₄	—	109	2	no	—
332	C₂×C₄	—	109	2	no	—
333	C₂×C₄	—	436	2	no	—
334	C₂×C₄	—	436	2	no	—
335	C₂×C₄	—	436	2	no	—
336	C₂×C₄	—	436	2	no	—
337	C₂×C₄	—	218	1	no	—
338	C₂×C₄	—	218	1	no	—
339	C₂×C₄	—	218	1	no	—
340	C₂×C₄	D₄×D₁₀₉	1	2	yes	—
341	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
342	C₈	—	109	2	no	—
343	C₈	—	109	2	no	—
344	C₈	D₄×D₁₀₉	1	2	yes	—
345	C₈	D₄×D₁₀₉	1	2	yes	—
346	C₂²	—	4	1	no	—
347	C₂²	—	4	2	no	—
348	C₂²	—	109	2	no	—
349	C₂²	—	436	1	no	—
350	C₂²	—	436	2	no	—
351	C₂²	—	218	2	no	—
352	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
353	C₄	—	2	2	no	—
354	C₄	—	4	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	C₂×D₄×D₁₀₉	1	1	yes	characteristic
363	C₄	C₂×D₄×D₁₀₉	1	1	yes	characteristic
364	C₂	—	8	1	no	—
365	C₂	—	872	1	no	—
366	C₂	—	218	2	no	—
367	C₂	#127×D₁₀₉	1	1	yes	characteristic
368	C₂	#20295842	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#497×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂×#16812720	C₂
hypercenter	#497	D₁₀₉
hypocenter	C₁₀₉	#3156
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #497

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. C₂×Q₁₆×D₁₀₉	C₂	semidirect
5. #20295601	C₂	semidirect
6. #20295596	C₂	semidirect
7. #20295420	C₂	semidirect
8. #20295419	C₂	semidirect
9. #20295417	C₂	semidirect
10. #20295408	C₂	semidirect
11. #20295407	C₂	semidirect
12. #20295405	C₂	semidirect
13. #20295404	C₂	semidirect
14. #20294983	C₂	semidirect
15. #497×C₁₀₉	C₂	semidirect
185. C₁₀₉	#3156	semidirect
195. #497	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
C₂×#16812720	2
C₂×Q₁₆×D₁₀₉	2
#20295601	2
#20295596	2
#20295420	2
#20295419	2
#20295417	2
#20295408	2
#20295407	2
#20295405	2
#20295404	2
#20294983	2
#497×C₁₀₉	2
#3156	109