Subgroups — #28377802

Order 13952.

1424 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295863	C₂	1	2	yes	maximal, complemented
2	#20295839	C₂	1	3	yes	maximal, complemented
3	#20295835	C₂	1	6	yes	maximal, complemented
4	#20295834	C₂	1	2	yes	maximal, complemented
5	#20295831	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295828	C₂	1	3	yes	maximal, complemented
7	#20295825	C₂	1	6	yes	maximal, complemented
8	C₂×#16812722	C₂	1	3	yes	maximal, complemented
9	#135×D₁₀₉	C₂	1	3	yes	maximal, complemented
10	#131×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
11	#578×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#16812747	—	2	2	no	—
13	#16812747	C₂²	1	6	yes	complemented
14	#16812745	—	2	6	no	—
15	#16812745	C₂²	1	2	yes	complemented
16	#16812745	C₂²	1	6	yes	complemented
17	C₂×#13182030	—	2	2	no	—
18	C₂×#13182030	C₂²	1	6	yes	complemented
19	#41×D₁₀₉	—	2	6	no	—
20	#41×D₁₀₉	C₂²	1	1	yes	characteristic
21	#41×D₁₀₉	C₂²	1	6	yes	complemented
22	#16812731	C₂²	1	12	yes	complemented
23	#16812730	C₂²	1	6	yes	complemented
24	#16812727	C₂²	1	6	yes	complemented
25	#16812726	C₂²	1	12	yes	complemented
26	#16812724	C₂²	1	6	yes	complemented
27	#16812722	C₂²	1	3	yes	complemented
28	#16812722	C₂²	1	6	yes	complemented
29	#16812722	C₂²	1	6	yes	complemented
30	#16812722	C₂²	1	6	yes	complemented
31	#16812721	C₂²	1	3	yes	complemented
32	#16812720	C₂²	1	6	yes	complemented
33	#16812719	C₂²	1	6	yes	complemented
34	#16812718	C₂²	1	1	yes	characteristic, complemented
35	Q₁₆×D₁₀₉	C₂²	1	3	yes	complemented
36	C₂×#13182016	C₂²	1	6	yes	complemented
37	SD₁₆×D₁₀₉	C₂²	1	6	yes	complemented
38	C₂×D₈₇₂	C₂²	1	3	yes	complemented
39	D₈×D₁₀₉	C₂²	1	3	yes	complemented
40	#16812707	C₂²	1	3	yes	complemented
41	#16812706	C₂²	1	1	yes	characteristic, complemented
42	#16812705	C₂²	1	3	yes	complemented
43	M₁₆×D₁₀₉	C₂²	1	3	yes	complemented
44	C₂×C₈×D₁₀₉	C₂²	1	3	yes	complemented
45	2₋¹⁺⁴×C₁₀₉	C₂²	1	2	yes	complemented
46	#137×C₁₀₉	C₂²	1	6	yes	complemented
47	#135×C₁₀₉	C₂²	1	3	yes	complemented
48	C₂×Q₁₆×C₁₀₉	C₂²	1	3	yes	complemented
49	#131×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
50	#13182030	—	2	2	no	—
51	#13182030	—	2	6	no	—
52	#13182030	—	2	6	no	—
53	#13182030	—	2	6	no	—
54	#13182030	D₄	1	1	yes	characteristic
55	#13182030	C₂³	1	6	yes	—
56	#13182030	C₂³	1	12	yes	—
57	#13182029	—	2	12	no	—
58	#13182029	—	2	6	no	—
59	#13182029	—	2	6	no	—
60	#13182029	C₂³	1	3	yes	—
61	#13182029	C₂³	1	6	yes	—
62	#13182028	—	2	6	no	—
63	#13182028	D₄	1	3	yes	—
64	#13182028	C₂³	1	2	yes	—
65	Q₈×D₁₀₉	—	2	6	no	—
66	Q₈×D₁₀₉	D₄	1	1	yes	characteristic
67	Q₈×D₁₀₉	C₂³	1	6	yes	—
68	C₂×D₄₃₆	—	2	6	no	—
69	C₂×D₄₃₆	—	2	6	no	—
70	C₂×D₄₃₆	C₂³	1	6	yes	—
71	D₄×D₁₀₉	—	2	6	no	—
72	D₄×D₁₀₉	—	2	6	no	—
73	D₄×D₁₀₉	—	2	12	no	—
74	D₄×D₁₀₉	D₄	1	3	yes	—
75	D₄×D₁₀₉	C₂³	1	6	yes	—
76	C₂×C₄×D₁₀₉	—	2	6	no	—
77	C₂×C₄×D₁₀₉	—	2	6	no	—
78	C₂×C₄×D₁₀₉	C₂³	1	3	yes	—
79	Dic₄₃₆	C₂³	1	1	yes	characteristic
80	#13182019	C₂³	1	6	yes	—
81	#13182018	C₂³	1	6	yes	—
82	#13182017	C₂³	1	2	yes	—
83	#13182016	C₂³	1	6	yes	—
84	#13182016	C₂³	1	12	yes	—
85	D₈₇₂	C₂³	1	3	yes	—
86	D₈₇₂	C₂³	1	6	yes	—
87	#13182014	C₂³	1	6	yes	—
88	#13182013	C₂³	1	3	yes	—
89	#13182012	C₂³	1	6	yes	—
90	C₂×#11724784	C₂³	1	3	yes	—
91	C₈×D₁₀₉	C₂³	1	1	yes	characteristic
92	C₈×D₁₀₉	C₂³	1	3	yes	—
93	C₈×D₁₀₉	C₂³	1	3	yes	—
94	C₈×D₁₀₉	C₂³	1	3	yes	—
95	#41×C₁₀₉	—	2	6	no	—
96	#41×C₁₀₉	C₂³	1	6	yes	—
97	#41×C₁₀₉	C₂³	1	1	yes	characteristic
98	C₂×Q₈×C₁₀₉	—	2	2	no	—
99	C₂×Q₈×C₁₀₉	C₂³	1	6	yes	—
100	Q₁₆×C₁₀₉	C₂³	1	3	yes	—
101	Q₁₆×C₁₀₉	C₂³	1	6	yes	—
102	SD₁₆×C₁₀₉	C₂³	1	6	yes	—
103	D₈×C₁₀₉	C₂³	1	1	yes	characteristic
104	M₁₆×C₁₀₉	C₂³	1	3	yes	—
105	C₂×C₈×C₁₀₉	C₂³	1	3	yes	—
106	C₂²×D₁₀₉	—	2	6	no	—
107	C₂²×D₁₀₉	—	2	6	no	—
108	C₂²×D₁₀₉	C₂×D₄	1	3	yes	—
109	Q₈×C₁₀₉	—	2	6	no	—
110	Q₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
111	Q₈×C₁₀₉	C₂⁴	1	6	yes	—
112	D₄×C₁₀₉	—	2	6	no	—
113	D₄×C₁₀₉	C₂×D₄	1	3	yes	—
114	D₄×C₁₀₉	C₂⁴	1	2	yes	—
115	C₂×C₄×C₁₀₉	—	2	6	no	—
116	C₂×C₄×C₁₀₉	—	2	6	no	—
117	C₂×C₄×C₁₀₉	C₂⁴	1	3	yes	—
118	C₂²⋊D₁₀₉	—	2	6	no	—
119	C₂²⋊D₁₀₉	—	2	2	no	—
120	C₂²⋊D₁₀₉	—	2	6	no	—
121	C₂²⋊D₁₀₉	C₂×D₄	1	6	yes	—
122	C₂×Dic₁₀₉	—	2	2	no	—
123	C₂×Dic₁₀₉	C₂×D₄	1	3	yes	—
124	D₄₃₆	—	2	12	no	—
125	D₄₃₆	—	2	6	no	—
126	D₄₃₆	—	2	6	no	—
127	D₄₃₆	C₂×D₄	1	3	yes	—
128	D₄₃₆	C₂⁴	1	6	yes	—
129	C₄×D₁₀₉	—	2	6	no	—
130	C₄×D₁₀₉	—	2	6	no	—
131	C₄×D₁₀₉	—	2	6	no	—
132	C₄×D₁₀₉	—	2	12	no	—
133	C₄×D₁₀₉	C₂×D₄	1	3	yes	—
134	C₄×D₁₀₉	C₂×D₄	1	3	yes	—
135	C₄×D₁₀₉	C₂⁴	1	1	yes	characteristic
136	C₄×D₁₀₉	C₂⁴	1	3	yes	—
137	Dic₂₁₈	—	2	6	no	—
138	Dic₂₁₈	C₂×D₄	1	3	yes	—
139	Dic₂₁₈	C₂⁴	1	2	yes	—
140	C₈×C₁₀₉	C₂⁴	1	1	yes	characteristic
141	C₈×C₁₀₉	C₂⁴	1	3	yes	—
142	#11724784	C₂⁴	1	1	yes	characteristic
143	#11724784	C₂⁴	1	3	yes	—
144	C₂²×C₁₀₉	—	2	2	no	—
145	C₂²×C₁₀₉	C₂²×D₄	1	3	yes	—
146	C₂×D₁₀₉	—	2	6	no	—
147	C₂×D₁₀₉	—	4	3	no	—
148	C₂×D₁₀₉	—	8	6	no	—
149	C₂×D₁₀₉	—	8	6	no	—
150	C₂×D₁₀₉	C₂²×D₄	1	3	yes	—
151	C₂×D₁₀₉	C₂²×D₄	1	1	yes	characteristic
152	C₄×C₁₀₉	—	2	6	no	—
153	C₄×C₁₀₉	C₂²×D₄	1	3	yes	—
154	C₄×C₁₀₉	C₂⁵	1	1	yes	characteristic
155	Dic₁₀₉	—	2	2	no	—
156	Dic₁₀₉	C₂²×D₄	1	3	yes	—
157	Dic₁₀₉	C₂²×D₄	1	1	yes	characteristic
158	C₂×C₁₀₉	—	2	3	no	—
159	C₂×C₁₀₉	—	4	2	no	—
160	C₂×C₁₀₉	C₂³×D₄	1	1	yes	characteristic
161	D₁₀₉	—	2	1	no	—
162	D₁₀₉	—	2	3	no	—
163	D₁₀₉	—	4	6	no	—
164	#3584	—	109	1	no	maximal, Sylow
165	C₁₀₉	#3584	1	1	yes	characteristic, complemented, Sylow
166	#585	—	109	2	no	—
167	#578	—	109	3	no	—
168	#578	D₁₀₉	1	1	yes	characteristic, complemented
169	#577	—	109	2	no	—
170	#577	—	109	6	no	—
171	#576	—	109	3	no	—
172	#576	—	109	1	no	—
173	#575	—	109	6	no	—
174	C₂×#135	—	109	3	no	—
175	C₂×#135	—	109	3	no	—
176	C₂×#131	—	109	1	no	—
177	2₋¹⁺⁴	—	109	6	no	—
178	2₋¹⁺⁴	—	218	2	no	—
179	2₋¹⁺⁴	C₂×D₁₀₉	1	2	yes	complemented
180	H(5,𝔽₂)	—	109	6	no	—
181	H(5,𝔽₂)	—	109	2	no	—
182	H(5,𝔽₂)	—	218	6	no	—
183	C₂×#41	—	109	1	no	—
184	C₂×#41	—	109	6	no	—
185	C₂×#41	—	109	6	no	—
186	C₂×#41	—	218	2	no	—
187	C₂×#41	—	218	6	no	—
188	#137	—	109	6	no	—
189	#137	—	109	12	no	—
190	#137	C₂×D₁₀₉	1	6	yes	complemented
191	AGL(1,ℤ/8)	—	109	6	no	—
192	AGL(1,ℤ/8)	—	109	12	no	—
193	AGL(1,ℤ/8)	—	109	6	no	—
194	#135	—	109	6	no	—
195	#135	—	109	3	no	—
196	#135	—	109	6	no	—
197	#135	—	109	6	no	—
198	#135	—	109	6	no	—
199	#135	—	109	6	no	—
200	#135	—	109	1	no	—
201	#135	—	109	3	no	—
202	#135	C₂×D₁₀₉	1	3	yes	complemented
203	C₂×Q₁₆	—	109	3	no	—
204	C₂×Q₁₆	C₂×D₁₀₉	1	3	yes	complemented
205	C₂×SD₁₆	—	109	6	no	—
206	C₂×SD₁₆	—	109	6	no	—
207	C₂×D₈	—	109	3	no	—
208	C₂×D₈	—	109	3	no	—
209	#131	—	109	1	no	—
210	#131	—	109	3	no	—
211	#131	—	109	3	no	—
212	#131	C₂×D₁₀₉	1	1	yes	characteristic, complemented
213	C₂×M₁₆	—	109	3	no	—
214	C₂²×C₈	—	109	3	no	—
215	#41	—	2	6	no	—
216	#41	—	109	1	no	—
217	#41	—	109	3	no	—
218	#41	—	109	3	no	—
219	#41	—	109	6	no	—
220	#41	—	109	12	no	—
221	#41	—	109	2	no	—
222	#41	—	109	6	no	—
223	#41	—	218	6	no	—
224	#41	—	218	12	no	—
225	#41	—	218	6	no	—
226	#41	—	218	6	no	—
227	#41	—	218	6	no	—
228	#41	—	218	6	no	—
229	#41	—	218	6	no	—
230	#41	—	218	2	no	—
231	#41	C₂²×D₁₀₉	1	6	yes	—
232	#41	C₂²×D₁₀₉	1	1	yes	characteristic
233	C₂×Q₈	—	2	2	no	—
234	C₂×Q₈	—	109	1	no	—
235	C₂×Q₈	—	109	6	no	—
236	C₂×Q₈	—	218	6	no	—
237	C₂×Q₈	C₂²×D₁₀₉	1	6	yes	—
238	C₂×D₄	—	109	3	no	—
239	C₂×D₄	—	109	6	no	—
240	C₂×D₄	—	109	6	no	—
241	C₂×D₄	—	218	6	no	—
242	C₂×D₄	—	218	6	no	—
243	C₂×D₄	—	218	12	no	—
244	C₂×D₄	—	218	6	no	—
245	C₂×D₄	—	218	6	no	—
246	C₂²×C₄	—	109	3	no	—
247	C₂²×C₄	—	218	6	no	—
248	C₂²×C₄	—	218	6	no	—
249	Q₁₆	—	109	6	no	—
250	Q₁₆	—	109	1	no	—
251	Q₁₆	C₂²×D₁₀₉	1	3	yes	—
252	Q₁₆	C₂²×D₁₀₉	1	6	yes	—
253	SD₁₆	—	109	6	no	—
254	SD₁₆	—	109	12	no	—
255	SD₁₆	—	109	2	no	—
256	SD₁₆	—	109	6	no	—
257	SD₁₆	C₂²×D₁₀₉	1	6	yes	—
258	D₈	—	109	6	no	—
259	D₈	—	109	3	no	—
260	D₈	—	109	6	no	—
261	D₈	C₂²×D₁₀₉	1	1	yes	characteristic
262	M₁₆	—	109	3	no	—
263	M₁₆	—	109	6	no	—
264	M₁₆	C₂²×D₁₀₉	1	3	yes	—
265	C₂×C₈	—	109	3	no	—
266	C₂×C₈	—	109	1	no	—
267	C₂×C₈	—	109	3	no	—
268	C₂×C₈	—	109	3	no	—
269	C₂×C₈	—	109	3	no	—
270	C₂×C₈	C₂²×D₁₀₉	1	3	yes	—
271	C₂³	—	109	3	no	—
272	C₂³	—	218	6	no	—
273	C₂³	—	218	6	no	—
274	Q₈	—	2	6	no	—
275	Q₈	—	109	3	no	—
276	Q₈	—	109	2	no	—
277	Q₈	—	218	6	no	—
278	Q₈	D₄×D₁₀₉	1	1	yes	characteristic
279	Q₈	C₂³×D₁₀₉	1	6	yes	—
280	D₄	—	2	6	no	—
281	D₄	—	109	6	no	—
282	D₄	—	109	3	no	—
283	D₄	—	109	6	no	—
284	D₄	—	218	6	no	—
285	D₄	—	218	12	no	—
286	D₄	—	218	6	no	—
287	D₄	—	218	2	no	—
288	D₄	—	218	6	no	—
289	D₄	—	218	6	no	—
290	D₄	D₄×D₁₀₉	1	3	yes	—
291	D₄	C₂³×D₁₀₉	1	2	yes	—
292	C₂×C₄	—	2	6	no	—
293	C₂×C₄	—	2	6	no	—
294	C₂×C₄	—	109	3	no	—
295	C₂×C₄	—	109	3	no	—
296	C₂×C₄	—	109	3	no	—
297	C₂×C₄	—	109	1	no	—
298	C₂×C₄	—	109	3	no	—
299	C₂×C₄	—	218	6	no	—
300	C₂×C₄	—	218	12	no	—
301	C₂×C₄	—	218	6	no	—
302	C₂×C₄	—	218	2	no	—
303	C₂×C₄	—	218	6	no	—
304	C₂×C₄	C₂³×D₁₀₉	1	3	yes	—
305	C₈	—	109	1	no	—
306	C₈	—	109	3	no	—
307	C₈	C₂³×D₁₀₉	1	1	yes	characteristic
308	C₈	C₂³×D₁₀₉	1	3	yes	—
309	C₂²	—	2	2	no	—
310	C₂²	—	872	6	no	—
311	C₂²	—	872	6	no	—
312	C₂²	—	109	1	no	—
313	C₂²	—	109	3	no	—
314	C₂²	—	436	3	no	—
315	C₂²	—	218	6	no	—
316	C₂²	C₂×D₄×D₁₀₉	1	3	yes	—
317	C₄	—	2	6	no	—
318	C₄	—	109	1	no	—
319	C₄	—	109	3	no	—
320	C₄	—	218	2	no	—
321	C₄	C₂×D₄×D₁₀₉	1	3	yes	—
322	C₄	C₂⁴×D₁₀₉	1	1	yes	characteristic
323	C₂	—	2	3	no	—
324	C₂	—	4	2	no	—
325	C₂	—	436	6	no	—
326	C₂	—	218	3	no	—
327	C₂	—	218	1	no	—
328	C₂	C₂²×D₄×D₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₄×D₁₀₉
commutator, cosocle	C₄×C₁₀₉	C₂⁵
Frattini	C₄	C₂⁴×D₁₀₉
Fitting	#578×C₁₀₉	C₂
socle	C₂×C₁₀₉	C₂³×D₄
hypercenter	#578	D₁₀₉
hypocenter	C₁₀₉	#3584
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ #41 ◁ #578

Complements

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

N	H ≅ G/N	type
1. #20295863	C₂	semidirect
2. #20295839	C₂	semidirect
3. #20295835	C₂	semidirect
4. #20295834	C₂	semidirect
5. #20295831	C₂	semidirect
6. #20295828	C₂	semidirect
7. #20295825	C₂	semidirect
8. C₂×#16812722	C₂	semidirect
9. #135×D₁₀₉	C₂	semidirect
10. #131×D₁₀₉	C₂	semidirect
11. #578×C₁₀₉	C₂	semidirect
13. #16812747	C₂²	semidirect
15. #16812745	C₂²	semidirect
16. #16812745	C₂²	semidirect
18. C₂×#13182030	C₂²	semidirect
21. #41×D₁₀₉	C₂²	semidirect
22. #16812731	C₂²	semidirect
23. #16812730	C₂²	semidirect
24. #16812727	C₂²	semidirect
25. #16812726	C₂²	semidirect
26. #16812724	C₂²	semidirect
27. #16812722	C₂²	semidirect
28. #16812722	C₂²	semidirect
29. #16812722	C₂²	semidirect
30. #16812722	C₂²	semidirect
31. #16812721	C₂²	semidirect
32. #16812720	C₂²	semidirect
33. #16812719	C₂²	semidirect
34. #16812718	C₂²	semidirect
35. Q₁₆×D₁₀₉	C₂²	semidirect
36. C₂×#13182016	C₂²	semidirect
37. SD₁₆×D₁₀₉	C₂²	semidirect
38. C₂×D₈₇₂	C₂²	semidirect
39. D₈×D₁₀₉	C₂²	semidirect
40. #16812707	C₂²	semidirect
41. #16812706	C₂²	semidirect
42. #16812705	C₂²	semidirect
43. M₁₆×D₁₀₉	C₂²	semidirect
44. C₂×C₈×D₁₀₉	C₂²	semidirect
45. 2₋¹⁺⁴×C₁₀₉	C₂²	semidirect
46. #137×C₁₀₉	C₂²	semidirect
47. #135×C₁₀₉	C₂²	semidirect
48. C₂×Q₁₆×C₁₀₉	C₂²	semidirect
49. #131×C₁₀₉	C₂²	semidirect
165. C₁₀₉	#3584	semidirect
168. #578	D₁₀₉	semidirect
179. 2₋¹⁺⁴	C₂×D₁₀₉	semidirect
190. #137	C₂×D₁₀₉	semidirect
202. #135	C₂×D₁₀₉	semidirect
204. C₂×Q₁₆	C₂×D₁₀₉	semidirect
212. #131	C₂×D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295863	2
#20295839	2
#20295835	2
#20295834	2
#20295831	2
#20295828	2
#20295825	2
C₂×#16812722	2
#135×D₁₀₉	2
#131×D₁₀₉	2
#578×C₁₀₉	2
#3584	109