Subgroups — #28374485

Order 13952.

322 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295639	C₂	1	1	yes	maximal, characteristic
2	#20295634	C₂	1	1	yes	maximal, characteristic
3	#20295336	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295332	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295330	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295285	C₂	1	1	yes	maximal, characteristic
7	#20295283	C₂	1	1	yes	maximal, characteristic
8	#20295268	C₂	1	1	yes	maximal, characteristic
9	#20295266	C₂	1	1	yes	maximal, characteristic
10	#20295237	C₂	1	1	yes	maximal, characteristic, complemented
11	#20295236	C₂	1	1	yes	maximal, characteristic, complemented
12	#20295235	C₂	1	1	yes	maximal, characteristic, complemented
13	#20295235	C₂	1	1	yes	maximal, characteristic, complemented
14	#20295051	C₂	1	1	yes	maximal, characteristic
15	#479×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Dic₄₃₆	C₂²	1	1	yes	characteristic
17	C₂×Dic₄₃₆	C₂²	1	1	yes	characteristic
18	C₂×#13182019	C₂²	1	1	yes	characteristic
19	C₂×#13182019	C₂²	1	1	yes	characteristic
20	C₂×#13182012	C₂²	1	1	yes	characteristic
21	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
22	#16812699	C₂²	1	1	yes	characteristic
23	#16812697	C₂²	1	1	yes	characteristic
24	#16812694	—	2	1	no	—
25	#16812694	—	2	1	no	—
26	#16812694	—	2	1	no	—
27	#16812693	—	2	1	no	—
28	#16812691	—	2	1	no	—
29	#16812690	—	2	1	no	—
30	#16812690	—	2	1	no	—
31	#16812690	C₂²	1	1	yes	characteristic
32	#16812689	—	2	1	no	—
33	#16812679	C₂²	1	1	yes	characteristic
34	#16812679	C₂²	1	1	yes	characteristic
35	#16812679	C₂²	1	1	yes	characteristic
36	#16812675	C₂²	1	1	yes	characteristic
37	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
38	#16812664	C₂²	1	1	yes	characteristic
39	#16812664	C₂²	1	1	yes	characteristic
40	#16812656	C₂²	1	1	yes	characteristic
41	#16812632	C₂²	1	1	yes	characteristic
42	#16812630	C₂²	1	1	yes	characteristic
43	C₄⋊#11724784	C₂²	1	1	yes	characteristic
44	#16812628	C₂²	1	1	yes	characteristic
45	#16812628	C₂²	1	1	yes	characteristic
46	#16812624	C₂²	1	1	yes	characteristic
47	#16812624	C₂²	1	1	yes	characteristic
48	#16812622	C₂²	1	1	yes	characteristic
49	#16812622	C₂²	1	1	yes	characteristic
50	#16812622	C₂²	1	1	yes	characteristic
51	#16812622	C₂²	1	1	yes	characteristic
52	#125×C₁₀₉	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
54	#107×C₁₀₉	C₂²	1	1	yes	characteristic
55	#106×C₁₀₉	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
57	#103×C₁₀₉	C₂²	1	1	yes	characteristic
58	#103×C₁₀₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
60	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
61	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
62	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
63	Dic₄₃₆	—	4	1	no	—
64	Dic₄₃₆	—	4	1	no	—
65	#13182019	—	4	1	no	—
66	#13182019	—	4	1	no	—
67	#13182012	—	4	1	no	—
68	C₂×#11724784	C₂³	1	1	yes	characteristic
69	C₂×#11724784	C₂³	1	1	yes	characteristic
70	C₈×D₁₀₉	—	4	1	no	—
71	C₄⋊Dic₁₀₉	—	2	1	no	—
72	C₄⋊Dic₁₀₉	—	2	1	no	—
73	C₄⋊Dic₁₀₉	—	2	1	no	—
74	C₄⋊Dic₁₀₉	—	2	1	no	—
75	C₄⋊Dic₁₀₉	—	2	1	no	—
76	C₄⋊Dic₁₀₉	—	2	1	no	—
77	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
78	#13182008	—	2	1	no	—
79	#13182008	—	2	1	no	—
80	#13182008	—	2	1	no	—
81	#13182008	—	2	1	no	—
82	#13182008	—	2	1	no	—
83	#13182008	—	2	1	no	—
84	#13182008	—	2	1	no	—
85	#13182008	D₄	1	1	yes	characteristic
86	#13182008	D₄	1	1	yes	characteristic
87	#13182006	—	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	—	2	1	no	—
95	C₄×Dic₁₀₉	—	2	1	no	—
96	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
97	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
98	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
99	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
100	C₄⋊C₄×C₁₀₉	—	2	1	no	—
101	C₄⋊C₄×C₁₀₉	—	2	1	no	—
102	C₄⋊C₄×C₁₀₉	—	2	1	no	—
103	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
106	C₄²×C₁₀₉	—	2	1	no	—
107	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
108	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
109	Q₈×C₁₀₉	—	2	1	no	—
110	Q₈×C₁₀₉	—	2	1	no	—
111	C₂×C₄×C₁₀₉	—	2	1	no	—
112	C₂×C₄×C₁₀₉	—	2	1	no	—
113	C₂×C₄×C₁₀₉	—	2	1	no	—
114	C₂×C₄×C₁₀₉	—	2	1	no	—
115	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
118	C₂×Dic₁₀₉	—	2	1	no	—
119	C₂×Dic₁₀₉	—	2	1	no	—
120	C₂×Dic₁₀₉	—	2	1	no	—
121	C₂×Dic₁₀₉	—	2	1	no	—
122	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
123	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
124	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
125	C₄×D₁₀₉	—	2	1	no	—
126	C₄×D₁₀₉	—	2	1	no	—
127	Dic₂₁₈	—	2	1	no	—
128	Dic₂₁₈	—	2	1	no	—
129	Dic₂₁₈	—	2	1	no	—
130	Dic₂₁₈	—	2	1	no	—
131	Dic₂₁₈	—	2	1	no	—
132	Dic₂₁₈	—	2	1	no	—
133	C₈×C₁₀₉	—	2	1	no	—
134	C₈×C₁₀₉	—	2	1	no	—
135	#11724784	—	2	1	no	—
136	#11724784	—	2	1	no	—
137	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
138	C₂×D₁₀₉	—	2	1	no	—
139	C₂×D₁₀₉	—	2	1	no	—
140	C₂×D₁₀₉	—	2	1	no	—
141	C₄×C₁₀₉	—	2	1	no	—
142	C₄×C₁₀₉	—	2	1	no	—
143	C₄×C₁₀₉	—	4	1	no	—
144	C₄×C₁₀₉	—	4	1	no	—
145	C₄×C₁₀₉	—	4	1	no	—
146	C₄×C₁₀₉	—	4	1	no	—
147	C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
148	C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
149	Dic₁₀₉	—	2	1	no	—
150	Dic₁₀₉	—	2	1	no	—
151	Dic₁₀₉	—	2	1	no	—
152	Dic₁₀₉	—	4	1	no	—
153	Dic₁₀₉	—	4	1	no	—
154	Dic₁₀₉	—	4	1	no	—
155	Dic₁₀₉	—	4	1	no	—
156	C₂×C₁₀₉	#537	1	1	yes	characteristic
157	C₂×C₁₀₉	#578	1	1	yes	characteristic
158	C₂×C₁₀₉	#578	1	1	yes	characteristic
159	D₁₀₉	—	4	1	no	—
160	#3261	—	109	1	no	maximal, Sylow
161	C₁₀₉	#3261	1	1	yes	characteristic, complemented, Sylow
162	#541	—	109	1	no	—
163	#541	—	109	1	no	—
164	#479	—	109	1	no	—
165	#479	—	109	1	no	—
166	#479	—	109	1	no	—
167	#479	D₁₀₉	1	1	yes	characteristic, complemented
168	#470	—	109	1	no	—
169	#470	—	109	1	no	—
170	#467	—	109	1	no	—
171	#467	—	109	1	no	—
172	#462	—	109	1	no	—
173	#462	—	109	1	no	—
174	#462	—	109	1	no	—
175	#462	—	109	1	no	—
176	#424	—	109	1	no	—
177	C₂×Q₁₆	—	109	1	no	—
178	C₂×Q₁₆	—	109	1	no	—
179	C₂×Q₁₆	—	109	1	no	—
180	C₂×Q₁₆	—	109	1	no	—
181	C₂×M₁₆	—	109	1	no	—
182	C₂²×C₈	—	109	1	no	—
183	C₄⋊Q₈	—	109	1	no	—
184	C₄⋊Q₈	—	109	1	no	—
185	#126	—	218	1	no	—
186	#126	—	218	1	no	—
187	#126	—	218	1	no	—
188	#126	—	218	1	no	—
189	#125	—	109	1	no	—
190	#125	—	218	1	no	—
191	#125	—	218	1	no	—
192	#125	—	218	1	no	—
193	#125	—	218	1	no	—
194	#125	C₂×D₁₀₉	1	1	yes	characteristic
195	C₂²⋊Q₈	—	109	1	no	—
196	C₂²⋊Q₈	—	109	1	no	—
197	C₂²⋊Q₈	—	109	1	no	—
198	C₂²⋊Q₈	—	109	1	no	—
199	C₄×Q₈	—	109	1	no	—
200	C₄×Q₈	—	109	1	no	—
201	C₄×Q₈	—	109	1	no	—
202	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
203	#117	—	109	1	no	—
204	#107	—	109	1	no	—
205	#107	C₂×D₁₀₉	1	1	yes	characteristic
206	#106	—	109	1	no	—
207	#106	C₂×D₁₀₉	1	1	yes	characteristic
208	C₄⋊C₈	—	109	1	no	—
209	C₄⋊C₈	—	109	1	no	—
210	C₄⋊C₈	—	109	1	no	—
211	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
212	#103	—	109	1	no	—
213	#103	—	109	1	no	—
214	#103	—	109	1	no	—
215	#103	—	109	1	no	—
216	#103	—	109	1	no	—
217	#103	—	109	1	no	—
218	#103	C₂×D₁₀₉	1	1	yes	characteristic
219	#103	C₂×D₁₀₉	1	1	yes	characteristic
220	C₂×Q₈	—	109	1	no	—
221	C₂×Q₈	—	109	1	no	—
222	C₂×Q₈	—	109	1	no	—
223	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
224	C₂²×C₄	—	109	1	no	—
225	Q₁₆	—	436	1	no	—
226	Q₁₆	—	436	1	no	—
227	Q₁₆	—	436	1	no	—
228	Q₁₆	—	436	1	no	—
229	M₁₆	—	436	1	no	—
230	C₂×C₈	—	109	1	no	—
231	C₂×C₈	—	109	1	no	—
232	C₂×C₈	—	436	1	no	—
233	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
234	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
235	C₄⋊C₄	—	2	1	no	—
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	—	2	1	no	—
238	C₄⋊C₄	—	109	1	no	—
239	C₄⋊C₄	—	109	1	no	—
240	C₄⋊C₄	—	109	1	no	—
241	C₄⋊C₄	—	218	1	no	—
242	C₄⋊C₄	—	218	1	no	—
243	C₄⋊C₄	—	218	1	no	—
244	C₄⋊C₄	—	218	1	no	—
245	C₄⋊C₄	—	218	1	no	—
246	C₄⋊C₄	—	218	1	no	—
247	C₄⋊C₄	—	218	1	no	—
248	C₄⋊C₄	—	218	1	no	—
249	C₄⋊C₄	—	218	1	no	—
250	C₄⋊C₄	—	218	1	no	—
251	C₄⋊C₄	—	218	1	no	—
252	C₄⋊C₄	—	218	1	no	—
253	C₄⋊C₄	—	218	1	no	—
254	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
255	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
256	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
257	C₂²⋊C₄	—	109	1	no	—
258	C₂²⋊C₄	—	109	1	no	—
259	C₂²⋊C₄	—	218	1	no	—
260	C₂²⋊C₄	—	218	1	no	—
261	C₂²⋊C₄	—	218	1	no	—
262	C₂²⋊C₄	—	218	1	no	—
263	C₄²	—	2	1	no	—
264	C₄²	—	109	1	no	—
265	C₄²	—	218	1	no	—
266	C₄²	—	218	1	no	—
267	C₄²	—	218	1	no	—
268	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
269	C₂³	—	109	1	no	—
270	Q₈	—	2	1	no	—
271	Q₈	—	2	1	no	—
272	Q₈	—	218	1	no	—
273	Q₈	—	218	1	no	—
274	Q₈	—	218	1	no	—
275	Q₈	—	218	1	no	—
276	Q₈	—	218	1	no	—
277	Q₈	—	218	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	109	1	no	—
283	C₂×C₄	—	109	1	no	—
284	C₂×C₄	—	109	1	no	—
285	C₂×C₄	—	218	1	no	—
286	C₂×C₄	—	218	1	no	—
287	C₂×C₄	—	218	1	no	—
288	C₂×C₄	—	218	1	no	—
289	C₂×C₄	—	218	1	no	—
290	C₂×C₄	—	218	1	no	—
291	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
292	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
293	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
294	C₈	—	2	1	no	—
295	C₈	—	2	1	no	—
296	C₈	—	218	1	no	—
297	C₈	—	218	1	no	—
298	C₂²	—	218	1	no	—
299	C₂²	—	218	1	no	—
300	C₂²	—	218	1	no	—
301	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	4	1	no	—
305	C₄	—	4	1	no	—
306	C₄	—	4	1	no	—
307	C₄	—	4	1	no	—
308	C₄	—	436	1	no	—
309	C₄	—	436	1	no	—
310	C₄	—	436	1	no	—
311	C₄	—	436	1	no	—
312	C₄	—	218	1	no	—
313	C₄	—	218	1	no	—
314	C₄	—	218	1	no	—
315	C₄	#16812745	1	1	yes	characteristic
316	C₄	#16812747	1	1	yes	characteristic
317	C₂	—	436	1	no	—
318	C₂	#20295606	1	1	yes	characteristic
319	C₂	#20295840	1	1	yes	characteristic
320	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	#479×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂²×D₁₀₉	C₂×D₄
hypercenter	#479	D₁₀₉
hypocenter	C₁₀₉	#3261
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #479

Complements

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

N	H ≅ G/N	type
3. #20295336	C₂	semidirect
4. #20295332	C₂	semidirect
5. #20295330	C₂	semidirect
10. #20295237	C₂	semidirect
11. #20295236	C₂	semidirect
12. #20295235	C₂	semidirect
13. #20295235	C₂	semidirect
15. #479×C₁₀₉	C₂	semidirect
161. C₁₀₉	#3261	semidirect
167. #479	D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295639	2
#20295634	2
#20295336	2
#20295332	2
#20295330	2
#20295285	2
#20295283	2
#20295268	2
#20295266	2
#20295237	2
#20295236	2
#20295235	2
#20295051	2
#479×C₁₀₉	2
#3261	109