Subgroups — #28374459

Order 13952.

334 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295643	C₂	1	1	yes	maximal, characteristic
2	#20295636	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295332	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295328	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295320	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295309	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295284	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295265	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295253	C₂	1	1	yes	maximal, characteristic
10	#20295247	C₂	1	1	yes	maximal, characteristic, complemented
11	#20295246	C₂	1	1	yes	maximal, characteristic, complemented
12	#20295235	C₂	1	1	yes	maximal, characteristic, complemented
13	#20295229	C₂	1	1	yes	maximal, characteristic, complemented
14	#20295054	C₂	1	1	yes	maximal, characteristic, complemented
15	#468×C₁₀₉	C₂	1	1	yes	maximal, characteristic
16	C₂×Dic₄₃₆	C₂²	1	1	yes	characteristic
17	C₂×#13182019	C₂²	1	1	yes	characteristic
18	C₂×#13182017	C₂²	1	1	yes	characteristic
19	C₂²×#11724784	C₂²	1	1	yes	characteristic
20	#16812698	C₂²	1	1	yes	characteristic
21	#16812697	C₂²	1	1	yes	characteristic
22	#16812692	—	2	1	no	—
23	#16812692	—	2	1	no	—
24	#16812692	—	2	1	no	—
25	#16812690	—	2	1	no	—
26	#16812690	C₂²	1	1	yes	characteristic
27	#16812689	—	2	1	no	—
28	#16812686	—	2	1	no	—
29	#16812686	C₂²	1	1	yes	characteristic
30	#16812680	—	2	1	no	—
31	#16812678	C₂²	1	1	yes	characteristic
32	#16812678	C₂²	1	1	yes	characteristic
33	#16812676	—	2	1	no	—
34	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
35	#16812664	C₂²	1	1	yes	characteristic
36	#16812664	C₂²	1	1	yes	characteristic
37	D₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
38	#16812658	C₂²	1	1	yes	characteristic
39	#16812632	C₂²	1	1	yes	characteristic
40	#16812631	C₂²	1	1	yes	characteristic
41	#16812630	C₂²	1	1	yes	characteristic
42	#16812628	C₂²	1	1	yes	characteristic
43	#16812628	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	#16812623	C₂²	1	1	yes	characteristic
49	#16812622	C₂²	1	1	yes	characteristic
50	#16812622	C₂²	1	1	yes	characteristic
51	#16812620	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
53	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
54	C₂²⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
55	#121×C₁₀₉	C₂²	1	1	yes	characteristic
56	#107×C₁₀₉	C₂²	1	1	yes	characteristic
57	#103×C₁₀₉	C₂²	1	1	yes	characteristic
58	#102×C₁₀₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₁₈	—	2	1	no	—
60	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
61	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
62	C₂²×Dic₁₀₉	—	2	1	no	—
63	Dic₄₃₆	—	4	1	no	—
64	#13182019	—	4	1	no	—
65	#13182017	—	4	1	no	—
66	C₂×#11724784	—	4	1	no	—
67	C₂×#11724784	C₂³	1	1	yes	characteristic
68	C₂×#11724784	C₂³	1	1	yes	characteristic
69	C₄⋊Dic₁₀₉	—	2	1	no	—
70	C₄⋊Dic₁₀₉	—	2	1	no	—
71	C₄⋊Dic₁₀₉	D₄	1	1	yes	characteristic
72	C₄⋊Dic₁₀₉	D₄	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	#13182008	—	2	1	no	—
81	#13182008	—	2	1	no	—
82	#13182008	—	2	1	no	—
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	C₂²⋊Dic₁₀₉	D₄	1	1	yes	characteristic
89	C₂²⋊Dic₁₀₉	D₄	1	1	yes	characteristic
90	C₄×Dic₁₀₉	—	2	1	no	—
91	C₄×Dic₁₀₉	—	2	1	no	—
92	C₄×Dic₁₀₉	—	2	1	no	—
93	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
94	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
95	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
96	C₂×D₄×C₁₀₉	—	2	1	no	—
97	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
98	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
99	SD₁₆×C₁₀₉	—	4	1	no	—
100	M₁₆×C₁₀₉	—	4	1	no	—
101	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
102	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
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	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
107	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
108	C₂³×C₁₀₉	—	2	1	no	—
109	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
110	Q₈×C₁₀₉	—	2	1	no	—
111	Q₈×C₁₀₉	—	2	1	no	—
112	D₄×C₁₀₉	—	2	1	no	—
113	D₄×C₁₀₉	—	2	1	no	—
114	D₄×C₁₀₉	—	8	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	—	2	1	no	—
120	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
122	C₂×Dic₁₀₉	—	2	1	no	—
123	C₂×Dic₁₀₉	—	2	1	no	—
124	C₂×Dic₁₀₉	—	2	1	no	—
125	C₂×Dic₁₀₉	—	2	1	no	—
126	C₂×Dic₁₀₉	—	8	1	no	—
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	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
131	Dic₂₁₈	—	2	1	no	—
132	Dic₂₁₈	—	2	1	no	—
133	Dic₂₁₈	—	2	1	no	—
134	Dic₂₁₈	—	2	1	no	—
135	Dic₂₁₈	—	8	1	no	—
136	C₈×C₁₀₉	—	2	1	no	—
137	C₈×C₁₀₉	—	2	1	no	—
138	#11724784	—	2	1	no	—
139	#11724784	—	2	1	no	—
140	C₂²×C₁₀₉	—	2	1	no	—
141	C₂²×C₁₀₉	—	2	1	no	—
142	C₂²×C₁₀₉	—	2	1	no	—
143	C₂²×C₁₀₉	—	4	1	no	—
144	C₂²×C₁₀₉	—	4	1	no	—
145	C₂²×C₁₀₉	—	4	1	no	—
146	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
147	C₄×C₁₀₉	—	2	1	no	—
148	C₄×C₁₀₉	—	4	1	no	—
149	C₄×C₁₀₉	—	4	1	no	—
150	C₄×C₁₀₉	—	4	1	no	—
151	C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
152	C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
153	Dic₁₀₉	—	2	1	no	—
154	Dic₁₀₉	—	2	1	no	—
155	Dic₁₀₉	—	2	1	no	—
156	Dic₁₀₉	—	2	1	no	—
157	Dic₁₀₉	—	4	1	no	—
158	Dic₁₀₉	—	4	1	no	—
159	Dic₁₀₉	—	4	1	no	—
160	Dic₁₀₉	—	4	1	no	—
161	C₂×C₁₀₉	—	4	1	no	—
162	C₂×C₁₀₉	—	8	1	no	—
163	C₂×C₁₀₉	#537	1	1	yes	characteristic
164	C₂×C₁₀₉	#577	1	1	yes	characteristic
165	C₂×C₁₀₉	#578	1	1	yes	characteristic
166	#3258	—	109	1	no	maximal, Sylow
167	C₁₀₉	#3258	1	1	yes	characteristic, complemented, Sylow
168	#542	—	109	1	no	—
169	#541	—	109	1	no	—
170	#479	—	109	1	no	—
171	#478	—	109	1	no	—
172	#477	—	109	1	no	—
173	#475	—	109	1	no	—
174	#470	—	109	1	no	—
175	#468	D₁₀₉	1	1	yes	characteristic
176	#467	—	109	1	no	—
177	#465	—	109	1	no	—
178	#464	—	109	1	no	—
179	#464	—	109	1	no	—
180	#462	—	109	1	no	—
181	#461	—	109	1	no	—
182	#424	—	109	1	no	—
183	C₂×Q₁₆	—	109	1	no	—
184	C₂×Q₁₆	—	109	1	no	—
185	C₂×SD₁₆	—	109	1	no	—
186	C₂×SD₁₆	C₂×D₁₀₉	1	1	yes	characteristic
187	C₂×M₁₆	C₂×D₁₀₉	1	1	yes	characteristic
188	C₂²×C₈	—	109	1	no	—
189	C₄⋊Q₈	—	109	1	no	—
190	C₄⋊Q₈	—	109	1	no	—
191	#126	—	218	1	no	—
192	#126	—	218	1	no	—
193	#126	—	218	1	no	—
194	#125	—	109	1	no	—
195	#125	—	218	1	no	—
196	#125	—	218	1	no	—
197	#124	—	109	1	no	—
198	#124	—	218	1	no	—
199	#123	—	218	1	no	—
200	C₂²⋊Q₈	—	109	1	no	—
201	C₂²⋊Q₈	—	109	1	no	—
202	C₂²⋊Q₈	—	218	1	no	—
203	C₂²⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic
204	#121	C₂×D₁₀₉	1	1	yes	characteristic
205	C₄×Q₈	—	109	1	no	—
206	C₄×Q₈	—	109	1	no	—
207	C₄×Q₈	—	109	1	no	—
208	C₄×D₄	—	109	1	no	—
209	#117	—	109	1	no	—
210	#107	—	109	1	no	—
211	#107	C₂×D₁₀₉	1	1	yes	characteristic
212	#106	—	109	1	no	—
213	#106	—	109	1	no	—
214	C₄⋊C₈	—	109	1	no	—
215	C₄⋊C₈	—	109	1	no	—
216	C₄⋊C₈	—	109	1	no	—
217	C₄⋊C₈	—	109	1	no	—
218	#103	—	109	1	no	—
219	#103	—	109	1	no	—
220	#103	—	109	1	no	—
221	#103	—	109	1	no	—
222	#103	—	109	1	no	—
223	#103	C₂×D₁₀₉	1	1	yes	characteristic
224	#102	—	109	1	no	—
225	#102	C₂×D₁₀₉	1	1	yes	characteristic
226	C₂×Q₈	—	109	1	no	—
227	C₂×Q₈	—	109	1	no	—
228	C₂×Q₈	—	218	1	no	—
229	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
230	C₂×D₄	—	2	1	no	—
231	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
232	C₂²×C₄	—	218	1	no	—
233	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
234	Q₁₆	—	436	1	no	—
235	Q₁₆	—	436	1	no	—
236	SD₁₆	—	4	1	no	—
237	SD₁₆	—	436	1	no	—
238	M₁₆	—	4	1	no	—
239	C₂×C₈	—	109	1	no	—
240	C₂×C₈	—	109	1	no	—
241	C₂×C₈	—	436	1	no	—
242	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
243	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	109	1	no	—
246	C₄⋊C₄	—	109	1	no	—
247	C₄⋊C₄	—	109	1	no	—
248	C₄⋊C₄	—	109	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	—	218	1	no	—
255	C₄⋊C₄	—	218	1	no	—
256	C₄⋊C₄	—	218	1	no	—
257	C₄⋊C₄	—	218	1	no	—
258	C₄⋊C₄	—	218	1	no	—
259	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
260	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
261	C₂²⋊C₄	—	2	1	no	—
262	C₂²⋊C₄	—	2	1	no	—
263	C₂²⋊C₄	—	109	1	no	—
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	C₂²⋊C₄	—	218	1	no	—
270	C₄²	—	109	1	no	—
271	C₄²	—	109	1	no	—
272	C₄²	—	218	1	no	—
273	C₄²	—	218	1	no	—
274	C₄²	—	218	1	no	—
275	C₂³	—	2	1	no	—
276	C₂³	D₄×D₁₀₉	1	1	yes	characteristic
277	Q₈	—	2	1	no	—
278	Q₈	—	2	1	no	—
279	Q₈	—	872	1	no	—
280	Q₈	—	218	1	no	—
281	Q₈	—	218	1	no	—
282	Q₈	—	218	1	no	—
283	Q₈	—	218	1	no	—
284	D₄	—	2	1	no	—
285	D₄	—	2	1	no	—
286	D₄	—	8	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	872	1	no	—
293	C₂×C₄	—	109	1	no	—
294	C₂×C₄	—	109	1	no	—
295	C₂×C₄	—	109	1	no	—
296	C₂×C₄	—	109	1	no	—
297	C₂×C₄	—	218	1	no	—
298	C₂×C₄	—	218	1	no	—
299	C₂×C₄	—	218	1	no	—
300	C₂×C₄	—	218	1	no	—
301	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
302	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
303	C₈	—	2	1	no	—
304	C₈	—	2	1	no	—
305	C₈	—	218	1	no	—
306	C₈	—	218	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	4	1	no	—
311	C₂²	—	4	1	no	—
312	C₂²	—	4	1	no	—
313	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
314	C₄	—	2	1	no	—
315	C₄	—	4	1	no	—
316	C₄	—	4	1	no	—
317	C₄	—	4	1	no	—
318	C₄	—	436	1	no	—
319	C₄	—	436	1	no	—
320	C₄	—	436	1	no	—
321	C₄	—	436	1	no	—
322	C₄	—	218	1	no	—
323	C₄	—	218	1	no	—
324	C₄	—	218	1	no	—
325	C₄	—	218	1	no	—
326	C₄	#16812744	1	1	yes	characteristic
327	C₄	#16812746	1	1	yes	characteristic
328	C₂	—	4	1	no	—
329	C₂	—	8	1	no	—
330	C₂	#20295603	1	1	yes	characteristic
331	C₂	#20295836	1	1	yes	characteristic
332	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	#468×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	#121×C₁₀₉	C₂²
hypercenter	#468	D₁₀₉
hypocenter	C₁₀₉	#3258
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #468

Complements

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

N	H ≅ G/N	type
2. #20295636	C₂	semidirect
3. #20295332	C₂	semidirect
4. #20295328	C₂	semidirect
5. #20295320	C₂	semidirect
6. #20295309	C₂	semidirect
7. #20295284	C₂	semidirect
8. #20295265	C₂	semidirect
10. #20295247	C₂	semidirect
11. #20295246	C₂	semidirect
12. #20295235	C₂	semidirect
13. #20295229	C₂	semidirect
14. #20295054	C₂	semidirect
167. C₁₀₉	#3258	semidirect

Maximal subgroups

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

subgroup	index
#20295643	2
#20295636	2
#20295332	2
#20295328	2
#20295320	2
#20295309	2
#20295284	2
#20295265	2
#20295253	2
#20295247	2
#20295246	2
#20295235	2
#20295229	2
#20295054	2
#468×C₁₀₉	2
#3258	109