Subgroups — #28373273

Order 13952.

418 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295630	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295498	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295349	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295348	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295327	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295326	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295218	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295158	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295158	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295138	C₂	1	1	yes	maximal, characteristic, complemented
11	#20295135	C₂	1	1	yes	maximal, characteristic, complemented
12	D₈×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
13	#20295106	C₂	1	1	yes	maximal, characteristic, complemented
14	#20295086	C₂	1	1	yes	maximal, characteristic, complemented
15	#459×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×D₄×D₁₀₉	C₂²	1	1	yes	characteristic
17	C₂×#13182014	C₂²	1	1	yes	characteristic
18	C₂×#13182014	C₂²	1	1	yes	characteristic
19	#16812691	C₂²	1	1	yes	characteristic, complemented
20	#16812680	—	2	1	no	—
21	#16812680	—	2	1	no	—
22	#16812671	—	2	1	no	—
23	#16812671	—	2	1	no	—
24	#16812670	C₂²	1	1	yes	characteristic
25	#16812670	C₂²	1	1	yes	characteristic
26	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
27	D₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
28	D₄×Dic₁₀₉	C₂²	1	1	yes	characteristic, complemented
29	D₄×Dic₁₀₉	C₂²	1	1	yes	characteristic, complemented
30	#16812661	C₂²	1	1	yes	characteristic
31	C₄×C₂²⋊D₁₀₉	—	2	1	no	—
32	#16812659	—	2	1	no	—
33	#16812657	C₂²	1	1	yes	characteristic
34	#16812656	C₂²	1	1	yes	characteristic
35	C₂×#13182006	—	2	1	no	—
36	C₂²⋊C₄×D₁₀₉	—	2	1	no	—
37	#16812633	C₂²	1	1	yes	characteristic, complemented
38	#16812632	C₂²	1	1	yes	characteristic, complemented
39	#16812631	C₂²	1	1	yes	characteristic, complemented
40	#16812630	C₂²	1	1	yes	characteristic, complemented
41	C₄⋊#11724784	C₂²	1	1	yes	characteristic, complemented
42	#16812621	C₂²	1	1	yes	characteristic
43	#16812621	C₂²	1	1	yes	characteristic
44	#16812620	C₂²	1	1	yes	characteristic
45	#16812620	C₂²	1	1	yes	characteristic
46	#16812620	C₂²	1	1	yes	characteristic, complemented
47	#16812620	C₂²	1	1	yes	characteristic, complemented
48	#16812611	C₂²	1	1	yes	characteristic
49	#16812611	C₂²	1	1	yes	characteristic
50	#16812609	C₂²	1	1	yes	characteristic, complemented
51	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic, complemented
52	C₂×D₈×C₁₀₉	C₂²	1	1	yes	characteristic
53	C₂×D₈×C₁₀₉	C₂²	1	1	yes	characteristic
54	#127×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
55	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
57	#102×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
58	#102×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
59	C₂³×D₁₀₉	—	2	1	no	—
60	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
61	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
62	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
63	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
64	D₄×D₁₀₉	—	2	2	no	—
65	D₄×D₁₀₉	—	2	2	no	—
66	C₂²×Dic₁₀₉	—	2	1	no	—
67	C₂²×Dic₁₀₉	—	2	1	no	—
68	C₂²×Dic₁₀₉	—	2	1	no	—
69	C₂×C₄×D₁₀₉	—	2	1	no	—
70	C₂×C₄×D₁₀₉	—	2	1	no	—
71	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
72	#13182014	—	2	2	no	—
73	#13182014	—	2	2	no	—
74	C₂×#11724784	C₂³	1	1	yes	characteristic
75	C₂×#11724784	C₂³	1	1	yes	characteristic
76	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
79	#13182008	—	2	1	no	—
80	#13182008	—	2	1	no	—
81	#13182008	D₄	1	1	yes	characteristic
82	#13182008	D₄	1	1	yes	characteristic
83	C₂²⋊Dic₁₀₉	—	2	1	no	—
84	C₂²⋊Dic₁₀₉	—	2	1	no	—
85	C₂²⋊Dic₁₀₉	—	2	1	no	—
86	C₂²⋊Dic₁₀₉	—	2	1	no	—
87	C₂²⋊Dic₁₀₉	—	2	1	no	—
88	#13182006	—	2	1	no	—
89	#13182006	—	2	1	no	—
90	#13182006	—	2	1	no	—
91	#13182006	D₄	1	1	yes	characteristic
92	#13182006	D₄	1	1	yes	characteristic
93	C₄×Dic₁₀₉	—	2	1	no	—
94	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
95	C₂×D₄×C₁₀₉	—	2	1	no	—
96	C₂×D₄×C₁₀₉	—	2	1	no	—
97	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
98	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
99	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
100	C₂²×C₄×C₁₀₉	—	2	1	no	—
101	D₈×C₁₀₉	—	2	2	no	—
102	D₈×C₁₀₉	—	2	2	no	—
103	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
104	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
106	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
107	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
108	C₂³×C₁₀₉	—	2	1	no	—
109	C₂³×C₁₀₉	—	2	1	no	—
110	C₂³×C₁₀₉	—	2	1	no	—
111	C₂²×D₁₀₉	—	2	1	no	—
112	C₂²×D₁₀₉	—	2	4	no	—
113	C₂²×D₁₀₉	—	8	1	no	—
114	C₂²×D₁₀₉	—	8	1	no	—
115	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
116	D₄×C₁₀₉	—	2	1	no	—
117	D₄×C₁₀₉	—	2	1	no	—
118	D₄×C₁₀₉	—	2	1	no	—
119	D₄×C₁₀₉	—	2	1	no	—
120	D₄×C₁₀₉	—	2	1	no	—
121	D₄×C₁₀₉	—	8	1	no	—
122	D₄×C₁₀₉	—	8	1	no	—
123	D₄×C₁₀₉	#41	1	2	yes	—
124	C₂×C₄×C₁₀₉	—	2	1	no	—
125	C₂×C₄×C₁₀₉	—	8	1	no	—
126	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
129	C₂²⋊D₁₀₉	—	2	4	no	—
130	C₂²⋊D₁₀₉	—	8	1	no	—
131	C₂²⋊D₁₀₉	—	8	1	no	—
132	C₂×Dic₁₀₉	—	2	1	no	—
133	C₂×Dic₁₀₉	—	2	1	no	—
134	C₂×Dic₁₀₉	—	2	1	no	—
135	C₂×Dic₁₀₉	—	8	1	no	—
136	C₂×Dic₁₀₉	—	8	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	#41	1	2	yes	—
143	C₄×D₁₀₉	—	2	1	no	—
144	C₄×D₁₀₉	—	2	1	no	—
145	C₄×D₁₀₉	—	8	1	no	—
146	C₄×D₁₀₉	—	8	1	no	—
147	C₈×C₁₀₉	—	2	1	no	—
148	C₈×C₁₀₉	—	2	1	no	—
149	#11724784	—	2	1	no	—
150	#11724784	—	2	1	no	—
151	C₂²×C₁₀₉	—	2	2	no	—
152	C₂²×C₁₀₉	—	4	1	no	—
153	C₂²×C₁₀₉	—	4	1	no	—
154	C₂²×C₁₀₉	—	4	1	no	—
155	C₂²×C₁₀₉	—	4	1	no	—
156	C₂²×C₁₀₉	—	4	1	no	—
157	C₂²×C₁₀₉	—	4	1	no	—
158	C₂²×C₁₀₉	—	4	1	no	—
159	C₂²×C₁₀₉	—	4	1	no	—
160	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
161	C₂×D₁₀₉	—	2	1	no	—
162	C₂×D₁₀₉	—	2	2	no	—
163	C₂×D₁₀₉	—	2	1	no	—
164	C₂×D₁₀₉	—	2	1	no	—
165	C₂×D₁₀₉	—	4	1	no	—
166	C₂×D₁₀₉	—	4	1	no	—
167	C₂×D₁₀₉	—	8	4	no	—
168	C₄×C₁₀₉	—	2	1	no	—
169	C₄×C₁₀₉	—	2	1	no	—
170	C₄×C₁₀₉	—	4	1	no	—
171	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
172	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
173	Dic₁₀₉	—	2	1	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	C₂×C₁₀₉	—	4	2	no	—
180	C₂×C₁₀₉	—	8	1	no	—
181	C₂×C₁₀₉	—	8	1	no	—
182	C₂×C₁₀₉	#525	1	1	yes	characteristic
183	C₂×C₁₀₉	#576	1	1	yes	characteristic
184	C₂×C₁₀₉	#576	1	1	yes	characteristic
185	D₁₀₉	—	4	1	no	—
186	D₁₀₉	—	4	2	no	—
187	#3122	—	109	1	no	maximal, Sylow
188	C₁₀₉	#3122	1	1	yes	characteristic, complemented, Sylow
189	#540	—	109	1	no	—
190	#518	—	109	1	no	—
191	#482	—	109	1	no	—
192	#482	—	109	1	no	—
193	#478	—	109	1	no	—
194	#478	—	109	1	no	—
195	#459	—	109	1	no	—
196	#459	D₁₀₉	1	1	yes	characteristic, complemented
197	#447	—	109	1	no	—
198	#447	—	109	1	no	—
199	#442	—	109	1	no	—
200	#442	—	109	1	no	—
201	C₄×D₈	—	109	1	no	—
202	C₄×D₈	—	109	1	no	—
203	#433	—	109	1	no	—
204	C₂²×D₄	—	109	1	no	—
205	C₂×D₈	—	109	1	no	—
206	C₂×D₈	—	109	1	no	—
207	C₂×D₈	C₂×D₁₀₉	1	1	yes	characteristic
208	C₂×D₈	C₂×D₁₀₉	1	1	yes	characteristic
209	#127	C₂×D₁₀₉	1	1	yes	characteristic, complemented
210	#125	—	109	1	no	—
211	#123	—	218	1	no	—
212	#123	—	218	1	no	—
213	#121	—	109	1	no	—
214	#121	—	109	1	no	—
215	#121	—	218	1	no	—
216	#121	—	218	1	no	—
217	C₄×D₄	—	109	1	no	—
218	C₄×D₄	—	109	1	no	—
219	C₄×D₄	—	109	1	no	—
220	C₄×D₄	—	109	1	no	—
221	C₄×D₄	—	109	1	no	—
222	C₄×D₄	—	218	1	no	—
223	C₄×D₄	—	218	1	no	—
224	C₄×D₄	C₂×D₁₀₉	1	1	yes	characteristic
225	#117	—	109	1	no	—
226	#117	—	109	1	no	—
227	C₂×C₂²⋊C₄	—	218	1	no	—
228	C₂×C₂²⋊C₄	—	218	1	no	—
229	#107	—	109	1	no	—
230	#107	—	109	1	no	—
231	#106	—	109	1	no	—
232	#106	—	109	1	no	—
233	C₄⋊C₈	—	109	1	no	—
234	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic, complemented
235	#102	—	109	1	no	—
236	#102	—	109	1	no	—
237	#102	—	109	1	no	—
238	#102	—	109	1	no	—
239	#102	—	109	1	no	—
240	#102	—	109	1	no	—
241	#102	C₂×D₁₀₉	1	1	yes	characteristic, complemented
242	#102	C₂×D₁₀₉	1	1	yes	characteristic, complemented
243	C₂²⋊C₈	—	109	1	no	—
244	C₂²⋊C₈	—	109	1	no	—
245	#97	—	109	1	no	—
246	C₄×C₈	—	109	1	no	—
247	C₂⁴	—	218	1	no	—
248	C₂×D₄	—	2	1	no	—
249	C₂×D₄	—	2	1	no	—
250	C₂×D₄	—	109	1	no	—
251	C₂×D₄	—	218	1	no	—
252	C₂×D₄	—	218	2	no	—
253	C₂×D₄	—	218	1	no	—
254	C₂×D₄	—	218	1	no	—
255	C₂×D₄	—	218	2	no	—
256	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
257	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
258	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
259	C₂²×C₄	—	2	1	no	—
260	C₂²×C₄	—	109	1	no	—
261	C₂²×C₄	—	218	1	no	—
262	C₂²×C₄	—	218	1	no	—
263	C₂²×C₄	—	218	1	no	—
264	C₂²×C₄	—	218	1	no	—
265	C₂²×C₄	—	218	1	no	—
266	D₈	—	2	2	no	—
267	D₈	—	2	2	no	—
268	D₈	—	218	2	no	—
269	D₈	—	218	2	no	—
270	C₂×C₈	—	109	1	no	—
271	C₂×C₈	—	109	1	no	—
272	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
273	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
274	C₄⋊C₄	—	109	1	no	—
275	C₄⋊C₄	—	109	1	no	—
276	C₄⋊C₄	—	109	1	no	—
277	C₄⋊C₄	—	109	1	no	—
278	C₄⋊C₄	—	109	1	no	—
279	C₄⋊C₄	—	218	1	no	—
280	C₄⋊C₄	—	218	1	no	—
281	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
282	C₂²⋊C₄	—	2	1	no	—
283	C₂²⋊C₄	—	109	1	no	—
284	C₂²⋊C₄	—	109	1	no	—
285	C₂²⋊C₄	—	218	1	no	—
286	C₂²⋊C₄	—	218	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	C₄²	—	109	1	no	—
294	C₄²	—	218	1	no	—
295	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
296	C₂³	—	2	1	no	—
297	C₂³	—	2	1	no	—
298	C₂³	—	2	1	no	—
299	C₂³	—	872	1	no	—
300	C₂³	—	872	1	no	—
301	C₂³	—	109	1	no	—
302	C₂³	—	218	4	no	—
303	C₂³	—	218	1	no	—
304	D₄	—	2	1	no	—
305	D₄	—	2	1	no	—
306	D₄	—	2	1	no	—
307	D₄	—	2	1	no	—
308	D₄	—	2	1	no	—
309	D₄	—	8	1	no	—
310	D₄	—	8	1	no	—
311	D₄	—	872	1	no	—
312	D₄	—	872	1	no	—
313	D₄	—	109	2	no	—
314	D₄	—	218	4	no	—
315	D₄	—	218	1	no	—
316	D₄	#13182028	1	2	yes	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	8	1	no	—
319	C₂×C₄	—	872	1	no	—
320	C₂×C₄	—	872	1	no	—
321	C₂×C₄	—	872	1	no	—
322	C₂×C₄	—	872	1	no	—
323	C₂×C₄	—	872	1	no	—
324	C₂×C₄	—	109	1	no	—
325	C₂×C₄	—	109	1	no	—
326	C₂×C₄	—	109	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	D₄×D₁₀₉	1	1	yes	characteristic
333	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
334	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
335	C₈	—	2	1	no	—
336	C₈	—	2	1	no	—
337	C₈	—	218	1	no	—
338	C₈	—	218	1	no	—
339	C₂²	—	2	2	no	—
340	C₂²	—	4	1	no	—
341	C₂²	—	4	1	no	—
342	C₂²	—	4	1	no	—
343	C₂²	—	4	1	no	—
344	C₂²	—	4	1	no	—
345	C₂²	—	4	1	no	—
346	C₂²	—	4	1	no	—
347	C₂²	—	4	1	no	—
348	C₂²	—	872	4	no	—
349	C₂²	—	436	1	no	—
350	C₂²	—	436	1	no	—
351	C₂²	—	218	1	no	—
352	C₂²	—	218	2	no	—
353	C₂²	—	218	1	no	—
354	C₂²	—	218	1	no	—
355	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
356	C₄	—	2	1	no	—
357	C₄	—	2	1	no	—
358	C₄	—	4	1	no	—
359	C₄	—	436	1	no	—
360	C₄	—	436	1	no	—
361	C₄	—	436	1	no	—
362	C₄	—	218	1	no	—
363	C₄	—	218	1	no	—
364	C₄	—	218	1	no	—
365	C₄	#41×D₁₀₉	1	1	yes	characteristic
366	C₄	C₂×#13182028	1	1	yes	characteristic
367	C₂	—	4	2	no	—
368	C₂	—	8	1	no	—
369	C₂	—	8	1	no	—
370	C₂	—	436	1	no	—
371	C₂	—	436	2	no	—
372	C₂	#20295536	1	1	yes	characteristic
373	C₂	#20295827	1	1	yes	characteristic
374	C₂	#20295830	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#459×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
hypercenter	#459	D₁₀₉
hypocenter	C₁₀₉	#3122
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #459

Complements

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

N	H ≅ G/N	type
1. #20295630	C₂	semidirect
2. #20295498	C₂	semidirect
3. #20295349	C₂	semidirect
4. #20295348	C₂	semidirect
5. #20295327	C₂	semidirect
6. #20295326	C₂	semidirect
7. #20295218	C₂	semidirect
8. #20295158	C₂	semidirect
9. #20295158	C₂	semidirect
10. #20295138	C₂	semidirect
11. #20295135	C₂	semidirect
12. D₈×Dic₁₀₉	C₂	semidirect
13. #20295106	C₂	semidirect
14. #20295086	C₂	semidirect
15. #459×C₁₀₉	C₂	semidirect
19. #16812691	C₂²	semidirect
28. D₄×Dic₁₀₉	C₂²	semidirect
29. D₄×Dic₁₀₉	C₂²	semidirect
37. #16812633	C₂²	semidirect
38. #16812632	C₂²	semidirect
39. #16812631	C₂²	semidirect
40. #16812630	C₂²	semidirect
41. C₄⋊#11724784	C₂²	semidirect
46. #16812620	C₂²	semidirect
47. #16812620	C₂²	semidirect
50. #16812609	C₂²	semidirect
51. C₈×Dic₁₀₉	C₂²	semidirect
54. #127×C₁₀₉	C₂²	semidirect
56. C₄⋊C₈×C₁₀₉	C₂²	semidirect
57. #102×C₁₀₉	C₂²	semidirect
58. #102×C₁₀₉	C₂²	semidirect
188. C₁₀₉	#3122	semidirect
196. #459	D₁₀₉	semidirect
209. #127	C₂×D₁₀₉	semidirect
234. C₄⋊C₈	C₂×D₁₀₉	semidirect
241. #102	C₂×D₁₀₉	semidirect
242. #102	C₂×D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295630	2
#20295498	2
#20295349	2
#20295348	2
#20295327	2
#20295326	2
#20295218	2
#20295158	2
#20295138	2
#20295135	2
D₈×Dic₁₀₉	2
#20295106	2
#20295086	2
#459×C₁₀₉	2
#3122	109