Subgroups — #28373304

Order 13952.

384 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295649	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295507	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295344	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295315	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295310	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295251	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295244	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295186	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295173	C₂	1	1	yes	maximal, characteristic
10	#20295137	C₂	1	1	yes	maximal, characteristic, complemented
11	#20295130	C₂	1	1	yes	maximal, characteristic, complemented
12	#20295113	C₂	1	1	yes	maximal, characteristic, complemented
13	#20295111	C₂	1	1	yes	maximal, characteristic, complemented
14	#20295084	C₂	1	1	yes	maximal, characteristic, complemented
15	#483×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13182029	C₂²	1	1	yes	characteristic
17	C₂×#13182019	C₂²	1	1	yes	characteristic
18	C₂×#13182018	C₂²	1	1	yes	characteristic
19	C₂×#13182016	C₂²	1	1	yes	characteristic
20	C₂×D₈₇₂	C₂²	1	1	yes	characteristic
21	#16812698	C₂²	1	1	yes	characteristic
22	#16812694	—	2	1	no	—
23	#16812688	C₂²	1	1	yes	characteristic
24	#16812685	—	2	1	no	—
25	#16812681	—	2	1	no	—
26	#16812679	—	2	1	no	—
27	#16812673	C₂²	1	1	yes	characteristic
28	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
29	#16812664	C₂²	1	1	yes	characteristic
30	#16812664	C₂²	1	1	yes	characteristic
31	#16812661	C₂²	1	1	yes	characteristic
32	#16812661	C₂²	1	1	yes	characteristic
33	#16812661	C₂²	1	1	yes	characteristic
34	C₄×C₂²⋊D₁₀₉	—	2	1	no	—
35	C₄×C₂²⋊D₁₀₉	—	2	1	no	—
36	#16812658	C₂²	1	1	yes	characteristic
37	#16812657	—	2	1	no	—
38	C₄⋊C₄×D₁₀₉	—	2	1	no	—
39	#16812630	C₂²	1	1	yes	characteristic
40	#16812630	C₂²	1	1	yes	characteristic
41	#16812628	C₂²	1	1	yes	characteristic
42	#16812628	C₂²	1	1	yes	characteristic
43	#16812624	C₂²	1	1	yes	characteristic
44	#16812622	C₂²	1	1	yes	characteristic
45	#16812621	C₂²	1	1	yes	characteristic
46	#16812619	C₂²	1	1	yes	characteristic
47	#16812619	C₂²	1	1	yes	characteristic
48	#16812619	C₂²	1	1	yes	characteristic
49	C₂²⋊#11724784	C₂²	1	1	yes	characteristic
50	#16812609	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
52	C₂²⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
54	#106×C₁₀₉	C₂²	1	1	yes	characteristic
55	#106×C₁₀₉	C₂²	1	1	yes	characteristic
56	#103×C₁₀₉	C₂²	1	1	yes	characteristic
57	#103×C₁₀₉	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
59	#13182029	—	2	2	no	—
60	#13182029	—	2	2	no	—
61	C₂×Dic₂₁₈	—	2	1	no	—
62	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
63	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
64	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
65	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
66	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
67	C₂×C₄×D₁₀₉	—	2	1	no	—
68	C₂×C₄×D₁₀₉	—	2	1	no	—
69	C₂×C₄×D₁₀₉	—	2	1	no	—
70	C₂×C₄×D₁₀₉	—	2	1	no	—
71	#13182019	—	2	2	no	—
72	#13182018	—	2	2	no	—
73	#13182016	—	2	2	no	—
74	D₈₇₂	—	2	2	no	—
75	C₂×#11724784	C₂³	1	1	yes	characteristic
76	C₂×#11724784	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₁₀₉	—	2	1	no	—
78	C₄⋊Dic₁₀₉	D₄	1	1	yes	characteristic
79	C₄⋊Dic₁₀₉	D₄	1	1	yes	characteristic
80	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
81	#13182008	—	2	1	no	—
82	#13182008	—	2	1	no	—
83	#13182008	—	2	1	no	—
84	C₂²⋊Dic₁₀₉	D₄	1	1	yes	characteristic
85	C₂²⋊Dic₁₀₉	D₄	1	1	yes	characteristic
86	#13182006	—	2	1	no	—
87	#13182006	—	2	1	no	—
88	#13182006	—	2	1	no	—
89	#13182006	—	2	1	no	—
90	#13182006	—	2	1	no	—
91	#13182006	—	2	1	no	—
92	C₄×Dic₁₀₉	—	2	1	no	—
93	C₄×Dic₁₀₉	—	2	1	no	—
94	C₄×Dic₁₀₉	—	2	1	no	—
95	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
96	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
97	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
98	C₂²×C₄×C₁₀₉	—	2	1	no	—
99	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
100	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
101	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₁₀₉	—	2	1	no	—
103	C₄⋊C₄×C₁₀₉	—	2	1	no	—
104	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
106	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
107	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
108	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
109	C₂²×D₁₀₉	—	2	1	no	—
110	C₂²×D₁₀₉	—	2	1	no	—
111	Q₈×C₁₀₉	—	2	1	no	—
112	Q₈×C₁₀₉	—	2	1	no	—
113	C₂×C₄×C₁₀₉	—	2	1	no	—
114	C₂×C₄×C₁₀₉	—	2	1	no	—
115	C₂×C₄×C₁₀₉	—	2	1	no	—
116	C₂×C₄×C₁₀₉	—	2	1	no	—
117	C₂×C₄×C₁₀₉	—	2	1	no	—
118	C₂×C₄×C₁₀₉	—	2	1	no	—
119	C₂×C₄×C₁₀₉	—	8	1	no	—
120	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
122	C₂²⋊D₁₀₉	—	2	4	no	—
123	C₂²⋊D₁₀₉	—	8	1	no	—
124	C₂×Dic₁₀₉	—	2	1	no	—
125	C₂×Dic₁₀₉	—	2	1	no	—
126	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
127	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
128	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
129	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
130	D₄₃₆	—	2	1	no	—
131	D₄₃₆	—	2	1	no	—
132	D₄₃₆	—	2	1	no	—
133	D₄₃₆	#41	1	2	yes	—
134	C₄×D₁₀₉	—	2	4	no	—
135	C₄×D₁₀₉	—	8	1	no	—
136	C₄×D₁₀₉	—	8	1	no	—
137	C₄×D₁₀₉	—	8	1	no	—
138	Dic₂₁₈	—	2	1	no	—
139	Dic₂₁₈	—	8	1	no	—
140	Dic₂₁₈	#41	1	2	yes	—
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	1	no	—
147	C₂²×C₁₀₉	—	2	1	no	—
148	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
149	C₂×D₁₀₉	—	2	2	no	—
150	C₂×D₁₀₉	—	4	1	no	—
151	C₂×D₁₀₉	—	4	1	no	—
152	C₂×D₁₀₉	—	4	1	no	—
153	C₂×D₁₀₉	—	4	1	no	—
154	C₂×D₁₀₉	—	4	1	no	—
155	C₄×C₁₀₉	—	2	1	no	—
156	C₄×C₁₀₉	—	4	1	no	—
157	C₄×C₁₀₉	—	4	1	no	—
158	C₄×C₁₀₉	—	4	1	no	—
159	C₄×C₁₀₉	—	4	1	no	—
160	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
161	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
162	Dic₁₀₉	—	2	2	no	—
163	Dic₁₀₉	—	2	1	no	—
164	Dic₁₀₉	—	2	1	no	—
165	Dic₁₀₉	—	2	1	no	—
166	Dic₁₀₉	—	2	1	no	—
167	Dic₁₀₉	—	4	1	no	—
168	C₂×C₁₀₉	—	4	1	no	—
169	C₂×C₁₀₉	#525	1	1	yes	characteristic
170	C₂×C₁₀₉	#577	1	1	yes	characteristic
171	C₂×C₁₀₉	#577	1	1	yes	characteristic
172	D₁₀₉	—	4	2	no	—
173	D₁₀₉	—	8	1	no	—
174	#3125	—	109	1	no	maximal, Sylow
175	C₁₀₉	#3125	1	1	yes	characteristic, complemented, Sylow
176	#542	—	109	1	no	—
177	#520	—	109	1	no	—
178	#483	D₁₀₉	1	1	yes	characteristic, complemented
179	#481	—	109	1	no	—
180	#476	—	109	1	no	—
181	#475	—	109	1	no	—
182	#464	—	109	1	no	—
183	#463	—	109	1	no	—
184	#452	—	109	1	no	—
185	#449	—	109	1	no	—
186	#442	—	109	1	no	—
187	#441	—	109	1	no	—
188	C₄×SD₁₆	—	109	1	no	—
189	C₄×SD₁₆	—	109	1	no	—
190	#433	—	109	1	no	—
191	C₂×#41	—	109	1	no	—
192	C₂×Q₁₆	—	109	1	no	—
193	C₂×SD₁₆	—	109	1	no	—
194	C₂×SD₁₆	—	109	1	no	—
195	C₂×D₈	—	109	1	no	—
196	C₄⋊Q₈	—	109	1	no	—
197	#126	—	218	1	no	—
198	#124	—	109	1	no	—
199	#124	—	218	1	no	—
200	#123	—	218	1	no	—
201	C₂²⋊Q₈	—	218	1	no	—
202	C₂²⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic
203	#121	—	109	1	no	—
204	C₄×Q₈	—	109	1	no	—
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	—	218	1	no	—
211	C₄×D₄	—	218	1	no	—
212	#117	—	109	1	no	—
213	#117	—	218	1	no	—
214	C₂×C₄⋊C₄	—	218	1	no	—
215	C₂×C₄⋊C₄	C₂×D₁₀₉	1	1	yes	characteristic
216	#106	—	109	1	no	—
217	#106	—	109	1	no	—
218	#106	C₂×D₁₀₉	1	1	yes	characteristic
219	#106	C₂×D₁₀₉	1	1	yes	characteristic
220	C₄⋊C₈	—	109	1	no	—
221	C₄⋊C₈	—	109	1	no	—
222	#103	—	109	1	no	—
223	#103	—	109	1	no	—
224	#103	C₂×D₁₀₉	1	1	yes	characteristic
225	#103	C₂×D₁₀₉	1	1	yes	characteristic
226	#102	—	109	1	no	—
227	#102	—	109	1	no	—
228	#102	—	109	1	no	—
229	#102	—	109	1	no	—
230	C₂²⋊C₈	—	109	1	no	—
231	C₂²⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
232	#97	—	109	1	no	—
233	C₄×C₈	—	109	1	no	—
234	#41	—	218	2	no	—
235	#41	—	218	2	no	—
236	C₂×Q₈	—	109	1	no	—
237	C₂×Q₈	—	218	1	no	—
238	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
239	C₂×D₄	—	109	1	no	—
240	C₂×D₄	—	109	1	no	—
241	C₂×D₄	—	218	1	no	—
242	C₂×D₄	—	218	1	no	—
243	C₂²×C₄	—	2	1	no	—
244	C₂²×C₄	—	218	1	no	—
245	C₂²×C₄	—	218	1	no	—
246	C₂²×C₄	—	218	1	no	—
247	C₂²×C₄	—	218	1	no	—
248	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
249	Q₁₆	—	218	2	no	—
250	SD₁₆	—	218	2	no	—
251	SD₁₆	—	218	2	no	—
252	D₈	—	218	2	no	—
253	C₂×C₈	—	109	1	no	—
254	C₂×C₈	—	109	1	no	—
255	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
256	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
257	C₄⋊C₄	—	2	1	no	—
258	C₄⋊C₄	—	2	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	C₂²×D₁₀₉	1	1	yes	characteristic
267	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
268	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
269	C₂²⋊C₄	—	2	1	no	—
270	C₂²⋊C₄	—	109	1	no	—
271	C₂²⋊C₄	—	109	1	no	—
272	C₂²⋊C₄	—	218	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	C₄²	—	109	1	no	—
279	C₄²	—	109	1	no	—
280	C₄²	—	218	1	no	—
281	C₄²	—	218	1	no	—
282	C₄²	—	218	1	no	—
283	C₂³	—	218	1	no	—
284	C₂³	—	218	1	no	—
285	C₂³	D₄×D₁₀₉	1	1	yes	characteristic
286	Q₈	—	2	1	no	—
287	Q₈	—	2	1	no	—
288	Q₈	—	872	1	no	—
289	Q₈	—	109	2	no	—
290	Q₈	—	218	1	no	—
291	D₄	—	872	1	no	—
292	D₄	—	109	2	no	—
293	D₄	—	218	4	no	—
294	D₄	—	218	1	no	—
295	D₄	—	218	1	no	—
296	D₄	—	218	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
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	—	8	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	4	no	—
312	C₂×C₄	—	218	1	no	—
313	C₂×C₄	—	218	1	no	—
314	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
315	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
316	C₈	—	2	1	no	—
317	C₈	—	2	1	no	—
318	C₈	—	218	1	no	—
319	C₈	—	218	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	436	1	no	—
324	C₂²	—	436	1	no	—
325	C₂²	—	436	1	no	—
326	C₂²	—	436	1	no	—
327	C₂²	—	436	1	no	—
328	C₂²	—	218	2	no	—
329	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
330	C₄	—	2	1	no	—
331	C₄	—	4	1	no	—
332	C₄	—	4	1	no	—
333	C₄	—	4	1	no	—
334	C₄	—	4	1	no	—
335	C₄	—	436	1	no	—
336	C₄	—	218	2	no	—
337	C₄	—	218	1	no	—
338	C₄	—	218	1	no	—
339	C₄	—	218	1	no	—
340	C₄	—	218	1	no	—
341	C₄	#41×D₁₀₉	1	1	yes	characteristic
342	C₄	#41×D₁₀₉	1	1	yes	characteristic
343	C₂	—	4	1	no	—
344	C₂	—	872	1	no	—
345	C₂	—	436	2	no	—
346	C₂	#20295534	1	1	yes	characteristic
347	C₂	#20295832	1	1	yes	characteristic
348	C₂	#20295835	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#483×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	#20295173	C₂
hypercenter	#483	D₁₀₉
hypocenter	C₁₀₉	#3125
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #483

Complements

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

N	H ≅ G/N	type
1. #20295649	C₂	semidirect
2. #20295507	C₂	semidirect
3. #20295344	C₂	semidirect
4. #20295315	C₂	semidirect
5. #20295310	C₂	semidirect
6. #20295251	C₂	semidirect
7. #20295244	C₂	semidirect
8. #20295186	C₂	semidirect
10. #20295137	C₂	semidirect
11. #20295130	C₂	semidirect
12. #20295113	C₂	semidirect
13. #20295111	C₂	semidirect
14. #20295084	C₂	semidirect
15. #483×C₁₀₉	C₂	semidirect
175. C₁₀₉	#3125	semidirect
178. #483	D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295649	2
#20295507	2
#20295344	2
#20295315	2
#20295310	2
#20295251	2
#20295244	2
#20295186	2
#20295173	2
#20295137	2
#20295130	2
#20295113	2
#20295111	2
#20295084	2
#483×C₁₀₉	2
#3125	109