Subgroups — #28373278

Order 13952.

394 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295636	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295501	C₂	1	1	yes	maximal, characteristic
3	#20295368	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295364	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295328	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295324	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295229	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295229	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295177	C₂	1	1	yes	maximal, characteristic
10	#20295127	C₂	1	1	yes	maximal, characteristic, complemented
11	#20295123	C₂	1	1	yes	maximal, characteristic, complemented
12	#20295117	C₂	1	1	yes	maximal, characteristic, complemented
13	SD₁₆×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
14	#20295084	C₂	1	1	yes	maximal, characteristic, complemented
15	#450×C₁₀₉	C₂	1	1	yes	maximal, characteristic
16	C₂×#13182017	C₂²	1	1	yes	characteristic
17	C₂×#13182017	C₂²	1	1	yes	characteristic
18	#16812697	C₂²	1	1	yes	characteristic, complemented
19	#16812692	—	2	1	no	—
20	#16812692	—	2	1	no	—
21	#16812690	—	2	1	no	—
22	#16812690	C₂²	1	1	yes	characteristic, complemented
23	#16812689	—	2	1	no	—
24	#16812678	C₂²	1	1	yes	characteristic
25	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic, complemented
26	#16812664	C₂²	1	1	yes	characteristic, complemented
27	D₄×Dic₁₀₉	—	2	1	no	—
28	D₄×Dic₁₀₉	—	2	1	no	—
29	D₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
30	D₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
31	D₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
32	D₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
33	#16812658	C₂²	1	1	yes	characteristic
34	#16812658	C₂²	1	1	yes	characteristic
35	C₂×C₂²⋊Dic₁₀₉	—	2	1	no	—
36	C₂×C₂²⋊Dic₁₀₉	—	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	#16812628	C₂²	1	1	yes	characteristic, complemented
42	#16812628	C₂²	1	1	yes	characteristic, complemented
43	#16812624	C₂²	1	1	yes	characteristic, complemented
44	#16812624	C₂²	1	1	yes	characteristic, complemented
45	#16812623	C₂²	1	1	yes	characteristic, complemented
46	#16812623	C₂²	1	1	yes	characteristic, complemented
47	#16812620	C₂²	1	1	yes	characteristic
48	#16812620	C₂²	1	1	yes	characteristic
49	C₂²⋊#11724784	C₂²	1	1	yes	characteristic
50	#16812609	C₂²	1	1	yes	characteristic, complemented
51	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic, complemented
52	C₂²×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
53	C₂×SD₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
54	C₂×SD₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
55	C₂²⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
56	#102×C₁₀₉	C₂²	1	1	yes	characteristic
57	#102×C₁₀₉	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
60	C₂²×Dic₁₀₉	—	2	1	no	—
61	C₂²×Dic₁₀₉	—	2	1	no	—
62	C₂²×Dic₁₀₉	—	2	1	no	—
63	C₂²×Dic₁₀₉	—	2	1	no	—
64	#13182017	—	2	2	no	—
65	#13182017	—	2	2	no	—
66	C₂×#11724784	C₂³	1	1	yes	characteristic
67	C₂×#11724784	C₂³	1	1	yes	characteristic
68	C₄⋊Dic₁₀₉	—	2	1	no	—
69	C₄⋊Dic₁₀₉	—	2	1	no	—
70	C₄⋊Dic₁₀₉	D₄	1	1	yes	characteristic
71	C₄⋊Dic₁₀₉	D₄	1	1	yes	characteristic
72	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
75	#13182008	—	2	1	no	—
76	#13182008	—	2	1	no	—
77	#13182008	—	2	1	no	—
78	#13182008	—	2	1	no	—
79	#13182008	—	2	1	no	—
80	C₂²⋊Dic₁₀₉	—	2	1	no	—
81	C₂²⋊Dic₁₀₉	—	2	1	no	—
82	C₂²⋊Dic₁₀₉	—	2	1	no	—
83	C₂²⋊Dic₁₀₉	—	2	1	no	—
84	C₂²⋊Dic₁₀₉	—	2	1	no	—
85	C₂²⋊Dic₁₀₉	—	2	1	no	—
86	C₂²⋊Dic₁₀₉	D₄	1	1	yes	characteristic
87	C₂²⋊Dic₁₀₉	D₄	1	1	yes	characteristic
88	C₄×Dic₁₀₉	—	2	1	no	—
89	C₄×Dic₁₀₉	—	2	1	no	—
90	C₄×Dic₁₀₉	—	2	1	no	—
91	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
92	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
93	C₂⁴×C₁₀₉	—	2	1	no	—
94	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
95	C₂×D₄×C₁₀₉	—	2	2	no	—
96	C₂×D₄×C₁₀₉	—	2	2	no	—
97	C₂×D₄×C₁₀₉	—	2	1	no	—
98	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
99	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
100	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
101	SD₁₆×C₁₀₉	—	2	2	no	—
102	SD₁₆×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	—	2	1	no	—
106	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
107	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
108	C₂³×C₁₀₉	—	2	4	no	—
109	C₂³×C₁₀₉	—	2	1	no	—
110	C₂³×C₁₀₉	—	2	1	no	—
111	C₂³×C₁₀₉	—	8	1	no	—
112	C₂³×C₁₀₉	—	8	1	no	—
113	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
114	Q₈×C₁₀₉	—	2	1	no	—
115	Q₈×C₁₀₉	—	2	1	no	—
116	D₄×C₁₀₉	—	2	4	no	—
117	D₄×C₁₀₉	—	2	1	no	—
118	D₄×C₁₀₉	—	2	1	no	—
119	D₄×C₁₀₉	#41	1	2	yes	—
120	D₄×C₁₀₉	#41	1	2	yes	—
121	C₂×C₄×C₁₀₉	—	2	1	no	—
122	C₂×C₄×C₁₀₉	—	2	1	no	—
123	C₂×C₄×C₁₀₉	—	2	1	no	—
124	C₂×C₄×C₁₀₉	—	2	1	no	—
125	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
127	C₂×Dic₁₀₉	—	2	1	no	—
128	C₂×Dic₁₀₉	—	2	1	no	—
129	C₂×Dic₁₀₉	—	2	1	no	—
130	C₂×Dic₁₀₉	—	2	1	no	—
131	C₂×Dic₁₀₉	—	8	1	no	—
132	C₂×Dic₁₀₉	—	8	1	no	—
133	C₂×Dic₁₀₉	—	8	1	no	—
134	C₂×Dic₁₀₉	—	8	1	no	—
135	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
136	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
137	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
138	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
139	Dic₂₁₈	—	2	1	no	—
140	Dic₂₁₈	—	2	1	no	—
141	C₈×C₁₀₉	—	2	1	no	—
142	C₈×C₁₀₉	—	2	1	no	—
143	#11724784	—	2	1	no	—
144	#11724784	—	2	1	no	—
145	C₂²×C₁₀₉	—	2	1	no	—
146	C₂²×C₁₀₉	—	2	2	no	—
147	C₂²×C₁₀₉	—	2	2	no	—
148	C₂²×C₁₀₉	—	2	1	no	—
149	C₂²×C₁₀₉	—	2	1	no	—
150	C₂²×C₁₀₉	—	4	1	no	—
151	C₂²×C₁₀₉	—	4	1	no	—
152	C₂²×C₁₀₉	—	4	1	no	—
153	C₂²×C₁₀₉	—	4	1	no	—
154	C₂²×C₁₀₉	—	8	4	no	—
155	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
156	C₄×C₁₀₉	—	2	1	no	—
157	C₄×C₁₀₉	—	4	1	no	—
158	C₄×C₁₀₉	—	4	1	no	—
159	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
160	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
161	Dic₁₀₉	—	2	1	no	—
162	Dic₁₀₉	—	2	1	no	—
163	Dic₁₀₉	—	2	1	no	—
164	Dic₁₀₉	—	2	1	no	—
165	Dic₁₀₉	—	4	1	no	—
166	Dic₁₀₉	—	4	1	no	—
167	Dic₁₀₉	—	4	1	no	—
168	Dic₁₀₉	—	4	1	no	—
169	C₂×C₁₀₉	—	4	1	no	—
170	C₂×C₁₀₉	—	4	2	no	—
171	C₂×C₁₀₉	—	4	2	no	—
172	C₂×C₁₀₉	#525	1	1	yes	characteristic
173	C₂×C₁₀₉	#577	1	1	yes	characteristic
174	C₂×C₁₀₉	#577	1	1	yes	characteristic
175	#3123	—	109	1	no	maximal, Sylow
176	C₁₀₉	#3123	1	1	yes	characteristic, complemented, Sylow
177	#541	—	109	1	no	—
178	#518	—	109	1	no	—
179	#485	—	109	1	no	—
180	#485	—	109	1	no	—
181	#478	—	109	1	no	—
182	#478	—	109	1	no	—
183	#461	—	109	1	no	—
184	#461	—	109	1	no	—
185	#450	—	109	1	no	—
186	#450	D₁₀₉	1	1	yes	characteristic
187	#440	—	109	1	no	—
188	#440	—	109	1	no	—
189	C₄×SD₁₆	—	109	1	no	—
190	C₄×SD₁₆	—	109	1	no	—
191	#433	—	109	1	no	—
192	C₂²×D₄	C₂×D₁₀₉	1	1	yes	characteristic
193	C₂×SD₁₆	—	109	1	no	—
194	C₂×SD₁₆	—	109	1	no	—
195	C₂×SD₁₆	C₂×D₁₀₉	1	1	yes	characteristic
196	C₂×SD₁₆	C₂×D₁₀₉	1	1	yes	characteristic
197	C₄⋊Q₈	—	109	1	no	—
198	#126	—	218	1	no	—
199	#126	—	218	1	no	—
200	#125	—	109	1	no	—
201	#125	—	218	1	no	—
202	#125	—	218	1	no	—
203	C₂²⋊Q₈	—	109	1	no	—
204	C₂²⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic
205	C₄×Q₈	—	109	1	no	—
206	C₄×Q₈	—	109	1	no	—
207	C₄×D₄	—	109	1	no	—
208	C₄×D₄	—	109	1	no	—
209	C₄×D₄	—	109	1	no	—
210	C₄×D₄	—	109	1	no	—
211	C₄×D₄	—	218	1	no	—
212	C₄×D₄	—	218	1	no	—
213	#117	—	109	1	no	—
214	#117	—	109	1	no	—
215	C₂×C₂²⋊C₄	—	218	1	no	—
216	C₂×C₂²⋊C₄	—	218	1	no	—
217	#107	—	109	1	no	—
218	#107	—	109	1	no	—
219	#106	—	109	1	no	—
220	#106	—	109	1	no	—
221	C₄⋊C₈	—	109	1	no	—
222	C₄⋊C₈	—	109	1	no	—
223	#103	—	109	1	no	—
224	#103	—	109	1	no	—
225	#103	—	109	1	no	—
226	#103	—	109	1	no	—
227	#102	—	109	1	no	—
228	#102	—	109	1	no	—
229	#102	C₂×D₁₀₉	1	1	yes	characteristic
230	#102	C₂×D₁₀₉	1	1	yes	characteristic
231	C₂²⋊C₈	—	109	1	no	—
232	C₂²⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
233	#97	—	109	1	no	—
234	C₄×C₈	—	109	1	no	—
235	C₂⁴	—	2	1	no	—
236	C₂×Q₈	—	109	1	no	—
237	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
238	C₂×D₄	—	2	2	no	—
239	C₂×D₄	—	2	2	no	—
240	C₂×D₄	—	2	1	no	—
241	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
242	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
243	C₂²×C₄	—	218	1	no	—
244	C₂²×C₄	—	218	1	no	—
245	C₂²×C₄	—	218	1	no	—
246	C₂²×C₄	—	218	1	no	—
247	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
248	SD₁₆	—	2	2	no	—
249	SD₁₆	—	2	2	no	—
250	SD₁₆	—	218	2	no	—
251	SD₁₆	—	218	2	no	—
252	C₂×C₈	—	109	1	no	—
253	C₂×C₈	—	109	1	no	—
254	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
255	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
256	C₄⋊C₄	—	2	1	no	—
257	C₄⋊C₄	—	109	1	no	—
258	C₄⋊C₄	—	109	1	no	—
259	C₄⋊C₄	—	109	1	no	—
260	C₄⋊C₄	—	109	1	no	—
261	C₄⋊C₄	—	109	1	no	—
262	C₄⋊C₄	—	218	1	no	—
263	C₄⋊C₄	—	218	1	no	—
264	C₄⋊C₄	—	218	1	no	—
265	C₄⋊C₄	—	218	1	no	—
266	C₄⋊C₄	—	218	1	no	—
267	C₄⋊C₄	—	218	1	no	—
268	C₄⋊C₄	—	218	1	no	—
269	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
270	C₂²⋊C₄	—	2	1	no	—
271	C₂²⋊C₄	—	109	1	no	—
272	C₂²⋊C₄	—	109	1	no	—
273	C₂²⋊C₄	—	218	1	no	—
274	C₂²⋊C₄	—	218	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	C₄²	—	109	1	no	—
280	C₄²	—	109	1	no	—
281	C₄²	—	218	1	no	—
282	C₄²	—	218	1	no	—
283	C₄²	—	218	1	no	—
284	C₂³	—	2	4	no	—
285	C₂³	—	2	1	no	—
286	C₂³	—	2	1	no	—
287	C₂³	—	8	1	no	—
288	C₂³	—	8	1	no	—
289	C₂³	D₄×D₁₀₉	1	1	yes	characteristic
290	Q₈	—	2	1	no	—
291	Q₈	—	2	1	no	—
292	Q₈	—	218	1	no	—
293	Q₈	—	218	1	no	—
294	D₄	—	2	4	no	—
295	D₄	—	2	1	no	—
296	D₄	—	2	1	no	—
297	D₄	#13182028	1	2	yes	—
298	D₄	#13182028	1	2	yes	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	872	1	no	—
304	C₂×C₄	—	872	1	no	—
305	C₂×C₄	—	872	1	no	—
306	C₂×C₄	—	872	1	no	—
307	C₂×C₄	—	109	1	no	—
308	C₂×C₄	—	109	1	no	—
309	C₂×C₄	—	109	1	no	—
310	C₂×C₄	—	109	1	no	—
311	C₂×C₄	—	218	1	no	—
312	C₂×C₄	—	218	1	no	—
313	C₂×C₄	—	218	1	no	—
314	C₂×C₄	—	218	1	no	—
315	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
316	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
317	C₈	—	2	1	no	—
318	C₈	—	2	1	no	—
319	C₈	—	218	1	no	—
320	C₈	—	218	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	2	2	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	4	1	no	—
327	C₂²	—	4	1	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	4	1	no	—
330	C₂²	—	8	4	no	—
331	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	4	1	no	—
334	C₄	—	4	1	no	—
335	C₄	—	436	1	no	—
336	C₄	—	436	1	no	—
337	C₄	—	436	1	no	—
338	C₄	—	436	1	no	—
339	C₄	—	218	1	no	—
340	C₄	—	218	1	no	—
341	C₄	—	218	1	no	—
342	C₄	—	218	1	no	—
343	C₄	C₂×#13182028	1	1	yes	characteristic
344	C₄	C₂×#13182028	1	1	yes	characteristic
345	C₂	—	4	1	no	—
346	C₂	—	4	2	no	—
347	C₂	—	4	2	no	—
348	C₂	#20295532	1	1	yes	characteristic
349	C₂	#20295832	1	1	yes	characteristic
350	C₂	#20295836	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#450×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂²×D₄×C₁₀₉	C₂²
hypercenter	#450	D₁₀₉
hypocenter	C₁₀₉	#3123
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #450

Complements

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

N	H ≅ G/N	type
1. #20295636	C₂	semidirect
3. #20295368	C₂	semidirect
4. #20295364	C₂	semidirect
5. #20295328	C₂	semidirect
6. #20295324	C₂	semidirect
7. #20295229	C₂	semidirect
8. #20295229	C₂	semidirect
10. #20295127	C₂	semidirect
11. #20295123	C₂	semidirect
12. #20295117	C₂	semidirect
13. SD₁₆×Dic₁₀₉	C₂	semidirect
14. #20295084	C₂	semidirect
18. #16812697	C₂²	semidirect
22. #16812690	C₂²	semidirect
25. Q₈×Dic₁₀₉	C₂²	semidirect
26. #16812664	C₂²	semidirect
37. #16812633	C₂²	semidirect
38. #16812632	C₂²	semidirect
39. #16812631	C₂²	semidirect
40. #16812630	C₂²	semidirect
41. #16812628	C₂²	semidirect
42. #16812628	C₂²	semidirect
43. #16812624	C₂²	semidirect
44. #16812624	C₂²	semidirect
45. #16812623	C₂²	semidirect
46. #16812623	C₂²	semidirect
50. #16812609	C₂²	semidirect
51. C₈×Dic₁₀₉	C₂²	semidirect
176. C₁₀₉	#3123	semidirect

Maximal subgroups

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

subgroup	index
#20295636	2
#20295501	2
#20295368	2
#20295364	2
#20295328	2
#20295324	2
#20295229	2
#20295177	2
#20295127	2
#20295123	2
#20295117	2
SD₁₆×Dic₁₀₉	2
#20295084	2
#450×C₁₀₉	2
#3123	109