Subgroups — #28376640

Order 13952.

1242 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295762	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295737	C₂	1	2	yes	maximal, complemented
3	#20295723	C₂	1	4	yes	maximal, complemented
4	#20295706	C₂	1	4	yes	maximal, complemented
5	#20295673	C₂	1	4	yes	maximal, complemented
6	#20295661	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295558	C₂	1	4	yes	maximal, complemented
8	#20295533	C₂	1	4	yes	maximal, complemented
9	C₄×#13182029	C₂	1	2	yes	maximal, complemented
10	C₄×D₄×D₁₀₉	C₂	1	4	yes	maximal, complemented
11	#532×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13182029	C₂²	1	2	yes	—
13	C₂×D₄×D₁₀₉	C₂²	1	4	yes	—
14	C₂²×C₄×D₁₀₉	C₂²	1	4	yes	complemented
15	#16812692	C₂²	1	8	yes	complemented
16	#16812690	C₂²	1	4	yes	complemented
17	#16812686	C₂²	1	4	yes	complemented
18	#16812681	C₂²	1	8	yes	complemented
19	#16812680	C₂²	1	4	yes	complemented
20	#16812680	C₂²	1	4	yes	complemented
21	#16812679	C₂²	1	4	yes	complemented
22	#16812675	C₂²	1	8	yes	complemented
23	#16812670	C₂²	1	4	yes	—
24	#16812670	C₂²	1	8	yes	complemented
25	#16812667	C₂²	1	4	yes	—
26	#16812667	C₂²	1	4	yes	complemented
27	C₄×Dic₂₁₈	C₂²	1	2	yes	complemented
28	C₄×Dic₂₁₈	C₂²	1	2	yes	complemented
29	C₄×D₄₃₆	C₂²	1	2	yes	—
30	C₄×D₄₃₆	C₂²	1	2	yes	—
31	D₄×Dic₁₀₉	C₂²	1	4	yes	complemented
32	#16812661	C₂²	1	4	yes	—
33	C₄×C₂²⋊D₁₀₉	C₂²	1	4	yes	complemented
34	C₄×C₂²⋊D₁₀₉	C₂²	1	8	yes	complemented
35	C₄×C₂²⋊D₁₀₉	C₂²	1	4	yes	complemented
36	#16812659	C₂²	1	8	yes	complemented
37	#16812658	C₂²	1	2	yes	complemented
38	#16812658	C₂²	1	4	yes	complemented
39	#16812656	C₂²	1	4	yes	complemented
40	C₄⋊C₄×D₁₀₉	C₂²	1	4	yes	complemented
41	C₂²⋊C₄×D₁₀₉	C₂²	1	8	yes	complemented
42	C₄²×D₁₀₉	C₂²	1	4	yes	complemented
43	C₂×#41×C₁₀₉	C₂²	1	2	yes	complemented
44	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
45	#127×C₁₀₉	C₂²	1	1	yes	characteristic
46	#124×C₁₀₉	C₂²	1	2	yes	complemented
47	#121×C₁₀₉	C₂²	1	4	yes	complemented
48	C₄×D₄×C₁₀₉	C₂²	1	4	yes	complemented
49	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
50	C₂³×D₁₀₉	C₂³	1	4	yes	—
51	#13182029	—	2	8	no	—
52	C₂×Dic₂₁₈	C₂³	1	2	yes	—
53	C₂×D₄₃₆	C₂³	1	2	yes	—
54	C₂×C₂²⋊D₁₀₉	C₂³	1	8	yes	—
55	C₂×C₂²⋊D₁₀₉	C₂³	1	4	yes	—
56	D₄×D₁₀₉	—	2	16	no	—
57	C₂²×Dic₁₀₉	C₂³	1	4	yes	—
58	C₂×C₄×D₁₀₉	—	4	8	no	—
59	C₂×C₄×D₁₀₉	C₂³	1	4	yes	—
60	C₂×C₄×D₁₀₉	C₂³	1	4	yes	—
61	C₂×C₄×D₁₀₉	C₂³	1	4	yes	—
62	C₂×C₄×D₁₀₉	C₂³	1	8	yes	—
63	C₄⋊Dic₁₀₉	C₂³	1	2	yes	—
64	C₄⋊Dic₁₀₉	C₂³	1	2	yes	—
65	C₄⋊Dic₁₀₉	C₂³	1	8	yes	—
66	#13182008	C₂³	1	8	yes	—
67	#13182008	C₂³	1	4	yes	—
68	#13182008	C₂³	1	4	yes	—
69	C₂²⋊Dic₁₀₉	C₂³	1	8	yes	—
70	C₂²⋊Dic₁₀₉	C₂³	1	4	yes	—
71	C₂²⋊Dic₁₀₉	C₂³	1	4	yes	—
72	C₂²⋊Dic₁₀₉	C₂³	1	4	yes	—
73	#13182006	C₂³	1	4	yes	—
74	#13182006	C₂³	1	8	yes	—
75	#13182006	C₂³	1	4	yes	—
76	C₄×Dic₁₀₉	C₂³	1	4	yes	—
77	C₄×Dic₁₀₉	C₂³	1	4	yes	—
78	C₄×Dic₁₀₉	C₂³	1	4	yes	—
79	#41×C₁₀₉	—	4	4	no	—
80	C₂×Q₈×C₁₀₉	C₂³	1	2	yes	—
81	C₂×D₄×C₁₀₉	C₂³	1	4	yes	—
82	C₂×D₄×C₁₀₉	C₂³	1	4	yes	—
83	C₂×D₄×C₁₀₉	C₂³	1	2	yes	—
84	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
85	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
86	C₂²×C₄×C₁₀₉	C₂³	1	4	yes	—
87	C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
88	C₂²⋊C₄×C₁₀₉	C₂³	1	8	yes	—
89	C₄²×C₁₀₉	C₂³	1	2	yes	—
90	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
91	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
92	C₂³×C₁₀₉	C₂⁴	1	4	yes	—
93	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
94	C₂²×D₁₀₉	—	2	16	no	—
95	C₂²×D₁₀₉	—	4	4	no	—
96	C₂²×D₁₀₉	C₂⁴	1	4	yes	—
97	Q₈×C₁₀₉	—	4	2	no	—
98	D₄×C₁₀₉	—	2	8	no	—
99	D₄×C₁₀₉	—	4	4	no	—
100	D₄×C₁₀₉	—	4	2	no	—
101	C₂×C₄×C₁₀₉	—	2	4	no	—
102	C₂×C₄×C₁₀₉	—	2	2	no	—
103	C₂×C₄×C₁₀₉	—	2	8	no	—
104	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
105	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
106	C₂×C₄×C₁₀₉	C₂⁴	1	4	yes	—
107	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
108	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
109	C₂²⋊D₁₀₉	—	2	16	no	—
110	C₂²⋊D₁₀₉	—	2	8	no	—
111	C₂×Dic₁₀₉	—	4	4	no	—
112	C₂×Dic₁₀₉	C₂⁴	1	4	yes	—
113	C₂×Dic₁₀₉	C₂⁴	1	4	yes	—
114	C₂×Dic₁₀₉	C₂⁴	1	4	yes	—
115	D₄₃₆	#41	1	8	yes	complemented
116	C₄×D₁₀₉	—	2	8	no	—
117	C₄×D₁₀₉	—	2	8	no	—
118	C₄×D₁₀₉	—	2	8	no	—
119	C₄×D₁₀₉	—	4	8	no	—
120	Dic₂₁₈	—	2	4	no	—
121	C₂²×C₁₀₉	—	2	2	no	—
122	C₂²×C₁₀₉	—	2	8	no	—
123	C₂²×C₁₀₉	—	2	4	no	—
124	C₂²×C₁₀₉	—	2	1	no	—
125	C₂²×C₁₀₉	C₂⁵	1	1	yes	characteristic
126	C₂×D₁₀₉	—	2	4	no	—
127	C₂×D₁₀₉	—	2	4	no	—
128	C₂×D₁₀₉	—	4	16	no	—
129	C₂×D₁₀₉	C₂×#41	1	8	yes	—
130	C₄×C₁₀₉	—	2	2	no	—
131	C₄×C₁₀₉	—	2	4	no	—
132	C₄×C₁₀₉	—	2	1	no	—
133	C₄×C₁₀₉	C₂×#41	1	4	yes	—
134	C₄×C₁₀₉	H(5,𝔽₂)	1	2	yes	—
135	Dic₁₀₉	—	2	4	no	—
136	Dic₁₀₉	—	2	4	no	—
137	Dic₁₀₉	—	2	4	no	—
138	C₂×C₁₀₉	—	4	4	no	—
139	C₂×C₁₀₉	—	4	1	no	—
140	C₂×C₁₀₉	C₂²×#41	1	2	yes	—
141	C₂×C₁₀₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
142	D₁₀₉	—	2	8	no	—
143	#3481	—	109	1	no	maximal, Sylow
144	C₁₀₉	#3481	1	1	yes	characteristic, complemented, Sylow
145	#559	—	109	1	no	—
146	#554	—	109	2	no	—
147	#553	—	109	4	no	—
148	#551	—	109	4	no	—
149	#546	—	109	4	no	—
150	D₄²	—	109	1	no	—
151	#532	D₁₀₉	1	1	yes	characteristic, complemented
152	#529	—	109	4	no	—
153	#525	—	109	4	no	—
154	C₄×#41	—	109	2	no	—
155	C₂×C₄×D₄	—	109	4	no	—
156	C₂×#41	—	109	2	no	—
157	C₂×#41	C₂×D₁₀₉	1	2	yes	complemented
158	C₂²×D₄	—	109	4	no	—
159	C₂³×C₄	—	109	4	no	—
160	C₄⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic, complemented
161	#127	C₂×D₁₀₉	1	1	yes	characteristic
162	#126	—	109	8	no	—
163	#125	—	109	4	no	—
164	#124	—	109	4	no	—
165	#124	C₂×D₁₀₉	1	2	yes	complemented
166	#123	—	109	4	no	—
167	#123	—	109	4	no	—
168	#123	—	109	8	no	—
169	C₂²⋊Q₈	—	109	8	no	—
170	C₂²⋊Q₈	—	109	4	no	—
171	#121	—	109	8	no	—
172	#121	—	109	4	no	—
173	#121	C₂×D₁₀₉	1	4	yes	complemented
174	#120	—	109	4	no	—
175	#120	—	109	4	no	—
176	C₄×Q₈	—	109	2	no	—
177	C₄×Q₈	—	109	2	no	—
178	C₄×D₄	—	109	4	no	—
179	C₄×D₄	—	109	8	no	—
180	C₄×D₄	—	109	4	no	—
181	C₄×D₄	—	109	4	no	—
182	C₄×D₄	—	109	4	no	—
183	C₄×D₄	—	109	8	no	—
184	C₄×D₄	—	109	2	no	—
185	C₄×D₄	—	109	2	no	—
186	C₄×D₄	C₂×D₁₀₉	1	4	yes	complemented
187	#117	—	109	2	no	—
188	#117	—	109	4	no	—
189	#117	—	109	4	no	—
190	C₂×C₄⋊C₄	—	109	4	no	—
191	C₂×C₂²⋊C₄	—	109	8	no	—
192	C₂×C₄²	—	109	4	no	—
193	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic, complemented
194	C₂⁴	—	109	4	no	—
195	#41	—	4	4	no	—
196	#41	—	218	8	no	—
197	C₂×Q₈	—	109	2	no	—
198	C₂×Q₈	C₂²×D₁₀₉	1	2	yes	—
199	C₂×D₄	—	109	4	no	—
200	C₂×D₄	—	109	8	no	—
201	C₂×D₄	—	109	2	no	—
202	C₂×D₄	—	218	16	no	—
203	C₂×D₄	C₂²×D₁₀₉	1	4	yes	—
204	C₂×D₄	C₂²×D₁₀₉	1	2	yes	—
205	C₂×D₄	C₂²×D₁₀₉	1	4	yes	—
206	C₂²×C₄	—	109	4	no	—
207	C₂²×C₄	—	109	8	no	—
208	C₂²×C₄	—	109	4	no	—
209	C₂²×C₄	—	109	4	no	—
210	C₂²×C₄	—	109	4	no	—
211	C₂²×C₄	—	436	8	no	—
212	C₂²×C₄	C₂²×D₁₀₉	1	2	yes	—
213	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
214	C₂²×C₄	C₂²×D₁₀₉	1	4	yes	—
215	C₄⋊C₄	—	109	8	no	—
216	C₄⋊C₄	—	109	4	no	—
217	C₄⋊C₄	—	109	4	no	—
218	C₄⋊C₄	—	109	2	no	—
219	C₄⋊C₄	—	109	2	no	—
220	C₄⋊C₄	—	109	8	no	—
221	C₄⋊C₄	C₂²×D₁₀₉	1	4	yes	—
222	C₂²⋊C₄	—	109	8	no	—
223	C₂²⋊C₄	—	109	4	no	—
224	C₂²⋊C₄	—	109	4	no	—
225	C₂²⋊C₄	—	109	4	no	—
226	C₂²⋊C₄	—	109	4	no	—
227	C₂²⋊C₄	—	109	8	no	—
228	C₂²⋊C₄	—	109	4	no	—
229	C₂²⋊C₄	C₂²×D₁₀₉	1	8	yes	—
230	C₄²	—	109	4	no	—
231	C₄²	—	109	4	no	—
232	C₄²	—	109	4	no	—
233	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
234	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
235	C₄²	C₂²×D₁₀₉	1	2	yes	—
236	C₂³	—	109	4	no	—
237	C₂³	—	436	4	no	—
238	C₂³	—	218	16	no	—
239	C₂³	C₂³×D₁₀₉	1	4	yes	—
240	C₂³	C₂³×D₁₀₉	1	1	yes	characteristic
241	Q₈	—	4	2	no	—
242	Q₈	—	218	4	no	—
243	D₄	—	2	8	no	—
244	D₄	—	4	4	no	—
245	D₄	—	4	2	no	—
246	D₄	—	109	8	no	—
247	D₄	—	218	16	no	—
248	D₄	—	218	8	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	8	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	109	4	no	—
253	C₂×C₄	—	109	4	no	—
254	C₂×C₄	—	109	4	no	—
255	C₂×C₄	—	436	8	no	—
256	C₂×C₄	—	436	4	no	—
257	C₂×C₄	—	218	8	no	—
258	C₂×C₄	—	218	8	no	—
259	C₂×C₄	—	218	8	no	—
260	C₂×C₄	C₂³×D₁₀₉	1	2	yes	—
261	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
262	C₂×C₄	C₂³×D₁₀₉	1	4	yes	—
263	C₂×C₄	C₂³×D₁₀₉	1	2	yes	—
264	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
265	C₂²	—	2	2	no	—
266	C₂²	—	2	8	no	—
267	C₂²	—	2	4	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	109	8	no	—
270	C₂²	—	436	16	no	—
271	C₂²	—	218	4	no	—
272	C₂²	—	218	4	no	—
273	C₂²	C₂⁴×D₁₀₉	1	1	yes	characteristic
274	C₄	—	2	2	no	—
275	C₄	—	2	4	no	—
276	C₄	—	2	1	no	—
277	C₄	—	218	4	no	—
278	C₄	—	218	4	no	—
279	C₄	—	218	4	no	—
280	C₄	#41×D₁₀₉	1	4	yes	—
281	C₄	#16812744	1	2	yes	—
282	C₂	—	4	4	no	—
283	C₂	—	4	1	no	—
284	C₂	—	218	8	no	—
285	C₂	C₂×#41×D₁₀₉	1	2	yes	—
286	C₂	C₂×#16812744	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂³×D₁₀₉
commutator, socle, cosocle	C₂²×C₁₀₉	C₂⁵
Frattini	C₂²	C₂⁴×D₁₀₉
Fitting	#532×C₁₀₉	C₂
hypercenter	#532	D₁₀₉
hypocenter	C₁₀₉	#3481
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ #532

Complements

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

N	H ≅ G/N	type
1. #20295762	C₂	semidirect
2. #20295737	C₂	semidirect
3. #20295723	C₂	semidirect
4. #20295706	C₂	semidirect
5. #20295673	C₂	semidirect
6. #20295661	C₂	semidirect
7. #20295558	C₂	semidirect
8. #20295533	C₂	semidirect
9. C₄×#13182029	C₂	semidirect
10. C₄×D₄×D₁₀₉	C₂	semidirect
11. #532×C₁₀₉	C₂	semidirect
14. C₂²×C₄×D₁₀₉	C₂²	semidirect
15. #16812692	C₂²	semidirect
16. #16812690	C₂²	semidirect
17. #16812686	C₂²	semidirect
18. #16812681	C₂²	semidirect
19. #16812680	C₂²	semidirect
20. #16812680	C₂²	semidirect
21. #16812679	C₂²	semidirect
22. #16812675	C₂²	semidirect
24. #16812670	C₂²	semidirect
26. #16812667	C₂²	semidirect
27. C₄×Dic₂₁₈	C₂²	semidirect
28. C₄×Dic₂₁₈	C₂²	semidirect
31. D₄×Dic₁₀₉	C₂²	semidirect
33. C₄×C₂²⋊D₁₀₉	C₂²	semidirect
34. C₄×C₂²⋊D₁₀₉	C₂²	semidirect
35. C₄×C₂²⋊D₁₀₉	C₂²	semidirect
36. #16812659	C₂²	semidirect
37. #16812658	C₂²	semidirect
38. #16812658	C₂²	semidirect
39. #16812656	C₂²	semidirect
40. C₄⋊C₄×D₁₀₉	C₂²	semidirect
41. C₂²⋊C₄×D₁₀₉	C₂²	semidirect
42. C₄²×D₁₀₉	C₂²	semidirect
43. C₂×#41×C₁₀₉	C₂²	semidirect
44. C₄⋊Q₈×C₁₀₉	C₂²	semidirect
46. #124×C₁₀₉	C₂²	semidirect
47. #121×C₁₀₉	C₂²	semidirect
48. C₄×D₄×C₁₀₉	C₂²	semidirect
49. C₂×C₄²×C₁₀₉	C₂²	semidirect
115. D₄₃₆	#41	semidirect
144. C₁₀₉	#3481	semidirect
151. #532	D₁₀₉	semidirect
157. C₂×#41	C₂×D₁₀₉	semidirect
160. C₄⋊Q₈	C₂×D₁₀₉	semidirect
165. #124	C₂×D₁₀₉	semidirect
173. #121	C₂×D₁₀₉	semidirect
186. C₄×D₄	C₂×D₁₀₉	semidirect
193. C₂×C₄²	C₂×D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295762	2
#20295737	2
#20295723	2
#20295706	2
#20295673	2
#20295661	2
#20295558	2
#20295533	2
C₄×#13182029	2
C₄×D₄×D₁₀₉	2
#532×C₁₀₉	2
#3481	109