Subgroups — #28377803

Order 13952.

1424 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295862	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295841	C₂	1	4	yes	maximal, complemented
3	#20295837	C₂	1	4	yes	maximal, complemented
4	#20295834	C₂	1	4	yes	maximal, complemented
5	#20295829	C₂	1	4	yes	maximal, complemented
6	#20295824	C₂	1	6	yes	maximal, complemented
7	C₂×#16812719	C₂	1	6	yes	maximal, complemented
8	C₂×#16812706	C₂	1	1	yes	maximal, characteristic, complemented
9	#585×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
10	#16812747	—	2	1	no	—
11	#16812746	C₂²	1	4	yes	complemented
12	#16812745	C₂²	1	4	yes	complemented
13	#16812744	—	2	3	no	—
14	C₂×#13182029	C₂²	1	6	yes	complemented
15	#41×D₁₀₉	—	2	4	no	—
16	#16812730	C₂²	1	12	yes	complemented
17	#16812729	C₂²	1	12	yes	complemented
18	#16812725	C₂²	1	12	yes	complemented
19	#16812724	C₂²	1	12	yes	complemented
20	#16812719	C₂²	1	4	yes	complemented
21	#16812719	C₂²	1	12	yes	complemented
22	#16812719	C₂²	1	12	yes	complemented
23	#16812719	C₂²	1	12	yes	complemented
24	C₂×#13182019	C₂²	1	6	yes	complemented
25	C₂×#13182017	C₂²	1	6	yes	complemented
26	C₂×#13182016	C₂²	1	6	yes	complemented
27	C₂×#13182014	C₂²	1	6	yes	complemented
28	#16812706	C₂²	1	4	yes	complemented
29	#16812706	C₂²	1	4	yes	complemented
30	C₂×#13182013	C₂²	1	3	yes	complemented
31	C₂²×#11724784	C₂²	1	3	yes	complemented
32	2₋¹⁺⁴×C₁₀₉	—	2	1	no	—
33	2₋¹⁺⁴×C₁₀₉	C₂²	1	4	yes	complemented
34	H(5,𝔽₂)×C₁₀₉	—	2	3	no	—
35	H(5,𝔽₂)×C₁₀₉	C₂²	1	4	yes	complemented
36	C₂×#41×C₁₀₉	—	2	4	no	—
37	C₂×#41×C₁₀₉	C₂²	1	1	yes	characteristic
38	C₂×#41×C₁₀₉	C₂²	1	6	yes	complemented
39	#13182030	—	2	4	no	—
40	#13182029	—	2	6	no	—
41	#13182029	—	2	6	no	—
42	#13182029	C₂³	1	4	yes	—
43	#13182029	C₂³	1	12	yes	—
44	#13182028	—	2	12	no	—
45	C₂×Dic₂₁₈	C₂³	1	6	yes	—
46	Q₈×D₁₀₉	—	2	4	no	—
47	C₂×D₄₃₆	C₂³	1	6	yes	—
48	C₂×C₂²⋊D₁₀₉	—	2	6	no	—
49	D₄×D₁₀₉	—	2	12	no	—
50	C₂×C₄×D₁₀₉	—	2	6	no	—
51	#13182019	C₂³	1	4	yes	—
52	#13182019	C₂³	1	12	yes	—
53	#13182017	C₂³	1	4	yes	—
54	#13182017	C₂³	1	12	yes	—
55	#13182016	C₂³	1	4	yes	—
56	#13182016	C₂³	1	12	yes	—
57	#13182014	C₂³	1	4	yes	—
58	#13182014	C₂³	1	12	yes	—
59	#13182013	C₂³	1	12	yes	—
60	C₂×#11724784	C₂³	1	1	yes	characteristic
61	C₂×#11724784	C₂³	1	3	yes	—
62	C₂×#11724784	C₂³	1	6	yes	—
63	C₂×#11724784	C₂³	1	6	yes	—
64	#41×C₁₀₉	—	2	4	no	—
65	#41×C₁₀₉	—	2	12	no	—
66	#41×C₁₀₉	—	2	6	no	—
67	#41×C₁₀₉	—	2	6	no	—
68	#41×C₁₀₉	D₄	1	4	yes	—
69	#41×C₁₀₉	C₂³	1	4	yes	—
70	#41×C₁₀₉	C₂³	1	12	yes	—
71	#41×C₁₀₉	C₂³	1	4	yes	—
72	C₂×Q₈×C₁₀₉	—	2	4	no	—
73	C₂×Q₈×C₁₀₉	D₄	1	1	yes	characteristic
74	C₂×Q₈×C₁₀₉	C₂³	1	6	yes	—
75	C₂×D₄×C₁₀₉	—	2	6	no	—
76	C₂×D₄×C₁₀₉	—	2	12	no	—
77	C₂×D₄×C₁₀₉	D₄	1	3	yes	—
78	C₂×D₄×C₁₀₉	C₂³	1	6	yes	—
79	C₂²×C₄×C₁₀₉	—	2	6	no	—
80	C₂²×C₄×C₁₀₉	C₂³	1	3	yes	—
81	C₂³×C₁₀₉	—	2	6	no	—
82	C₂³×C₁₀₉	C₂×D₄	1	3	yes	—
83	C₂²×D₁₀₉	—	2	6	no	—
84	Q₈×C₁₀₉	—	2	6	no	—
85	Q₈×C₁₀₉	C₂×D₄	1	4	yes	—
86	Q₈×C₁₀₉	C₂⁴	1	4	yes	—
87	D₄×C₁₀₉	—	2	12	no	—
88	D₄×C₁₀₉	—	2	4	no	—
89	D₄×C₁₀₉	—	2	6	no	—
90	D₄×C₁₀₉	C₂×D₄	1	12	yes	—
91	D₄×C₁₀₉	C₂⁴	1	4	yes	—
92	C₂×C₄×C₁₀₉	—	2	6	no	—
93	C₂×C₄×C₁₀₉	—	2	4	no	—
94	C₂×C₄×C₁₀₉	—	2	12	no	—
95	C₂×C₄×C₁₀₉	C₂×D₄	1	6	yes	—
96	C₂×C₄×C₁₀₉	C₂×D₄	1	3	yes	—
97	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
98	C₂×C₄×C₁₀₉	C₂⁴	1	6	yes	—
99	C₂²⋊D₁₀₉	—	2	12	no	—
100	C₂²⋊D₁₀₉	—	2	4	no	—
101	C₂×Dic₁₀₉	—	2	6	no	—
102	D₄₃₆	—	2	6	no	—
103	D₄₃₆	C₂⁴	1	4	yes	—
104	C₄×D₁₀₉	—	2	4	no	—
105	C₄×D₁₀₉	—	2	12	no	—
106	Dic₂₁₈	—	2	6	no	—
107	Dic₂₁₈	C₂⁴	1	4	yes	—
108	#11724784	C₂⁴	1	1	yes	characteristic
109	#11724784	C₂⁴	1	1	yes	characteristic
110	#11724784	C₂⁴	1	6	yes	—
111	C₂²×C₁₀₉	—	2	4	no	—
112	C₂²×C₁₀₉	—	4	3	no	—
113	C₂²×C₁₀₉	—	8	6	no	—
114	C₂²×C₁₀₉	C₂²×D₄	1	6	yes	—
115	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
116	C₂×D₁₀₉	—	2	4	no	—
117	C₂×D₁₀₉	—	8	6	no	—
118	C₄×C₁₀₉	—	2	4	no	—
119	C₄×C₁₀₉	C₂²×D₄	1	6	yes	—
120	C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
121	C₄×C₁₀₉	C₂⁵	1	1	yes	characteristic
122	Dic₁₀₉	—	2	4	no	—
123	C₂×C₁₀₉	—	2	6	no	—
124	C₂×C₁₀₉	—	2	1	no	—
125	C₂×C₁₀₉	—	4	4	no	—
126	C₂×C₁₀₉	C₂³×D₄	1	1	yes	characteristic
127	D₁₀₉	—	4	4	no	—
128	#3584	—	109	1	no	maximal, Sylow
129	C₁₀₉	#3584	1	1	yes	characteristic, complemented, Sylow
130	#585	—	109	1	no	—
131	#585	D₁₀₉	1	1	yes	characteristic, complemented
132	#578	—	109	4	no	—
133	#577	—	109	4	no	—
134	#577	—	109	4	no	—
135	#576	—	109	4	no	—
136	#575	—	109	6	no	—
137	C₂×#135	—	109	6	no	—
138	C₂×#131	—	109	1	no	—
139	2₋¹⁺⁴	—	2	1	no	—
140	2₋¹⁺⁴	—	109	4	no	—
141	2₋¹⁺⁴	—	218	1	no	—
142	2₋¹⁺⁴	C₂×D₁₀₉	1	4	yes	complemented
143	H(5,𝔽₂)	—	2	3	no	—
144	H(5,𝔽₂)	—	109	4	no	—
145	H(5,𝔽₂)	—	218	3	no	—
146	H(5,𝔽₂)	C₂×D₁₀₉	1	4	yes	complemented
147	C₂×#41	—	2	4	no	—
148	C₂×#41	—	109	6	no	—
149	C₂×#41	—	218	4	no	—
150	C₂×#41	C₂×D₁₀₉	1	1	yes	characteristic
151	C₂×#41	C₂×D₁₀₉	1	6	yes	complemented
152	#137	—	109	12	no	—
153	#137	—	109	12	no	—
154	AGL(1,ℤ/8)	—	109	12	no	—
155	AGL(1,ℤ/8)	—	109	12	no	—
156	#135	—	109	4	no	—
157	#135	—	109	12	no	—
158	#135	—	109	12	no	—
159	#135	—	109	12	no	—
160	C₂×Q₁₆	—	109	6	no	—
161	C₂×SD₁₆	—	109	6	no	—
162	C₂×SD₁₆	—	109	6	no	—
163	C₂×D₈	—	109	6	no	—
164	#131	—	109	4	no	—
165	#131	—	109	4	no	—
166	C₂×M₁₆	—	109	3	no	—
167	C₂²×C₈	—	109	3	no	—
168	#41	—	2	4	no	—
169	#41	—	2	12	no	—
170	#41	—	2	6	no	—
171	#41	—	2	6	no	—
172	#41	—	109	4	no	—
173	#41	—	109	12	no	—
174	#41	—	218	4	no	—
175	#41	—	218	12	no	—
176	#41	—	218	6	no	—
177	#41	—	218	6	no	—
178	#41	C₂²⋊D₁₀₉	1	4	yes	—
179	#41	C₂²×D₁₀₉	1	4	yes	—
180	#41	C₂²×D₁₀₉	1	12	yes	—
181	#41	C₂²×D₁₀₉	1	4	yes	—
182	C₂×Q₈	—	2	4	no	—
183	C₂×Q₈	—	109	6	no	—
184	C₂×Q₈	—	218	4	no	—
185	C₂×Q₈	C₂²⋊D₁₀₉	1	1	yes	characteristic
186	C₂×Q₈	C₂²×D₁₀₉	1	6	yes	—
187	C₂×D₄	—	2	6	no	—
188	C₂×D₄	—	2	12	no	—
189	C₂×D₄	—	109	6	no	—
190	C₂×D₄	—	218	6	no	—
191	C₂×D₄	—	218	12	no	—
192	C₂×D₄	C₂²⋊D₁₀₉	1	3	yes	—
193	C₂×D₄	C₂²×D₁₀₉	1	6	yes	—
194	C₂²×C₄	—	2	6	no	—
195	C₂²×C₄	—	218	6	no	—
196	C₂²×C₄	C₂²×D₁₀₉	1	3	yes	—
197	Q₁₆	—	109	4	no	—
198	Q₁₆	—	109	12	no	—
199	SD₁₆	—	109	4	no	—
200	SD₁₆	—	109	12	no	—
201	SD₁₆	—	109	4	no	—
202	SD₁₆	—	109	12	no	—
203	D₈	—	109	4	no	—
204	D₈	—	109	12	no	—
205	M₁₆	—	109	12	no	—
206	C₂×C₈	—	109	1	no	—
207	C₂×C₈	—	109	3	no	—
208	C₂×C₈	—	109	6	no	—
209	C₂×C₈	—	109	6	no	—
210	C₂³	—	2	6	no	—
211	C₂³	—	218	6	no	—
212	C₂³	C₂×C₂²⋊D₁₀₉	1	3	yes	—
213	Q₈	—	2	6	no	—
214	Q₈	—	109	4	no	—
215	Q₈	—	218	6	no	—
216	Q₈	C₂×C₂²⋊D₁₀₉	1	4	yes	—
217	Q₈	C₂³×D₁₀₉	1	4	yes	—
218	D₄	—	2	12	no	—
219	D₄	—	2	4	no	—
220	D₄	—	2	6	no	—
221	D₄	—	109	4	no	—
222	D₄	—	218	12	no	—
223	D₄	—	218	4	no	—
224	D₄	—	218	6	no	—
225	D₄	C₂×C₂²⋊D₁₀₉	1	12	yes	—
226	D₄	C₂³×D₁₀₉	1	4	yes	—
227	C₂×C₄	—	2	6	no	—
228	C₂×C₄	—	2	12	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	218	6	no	—
231	C₂×C₄	—	218	12	no	—
232	C₂×C₄	—	218	4	no	—
233	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	3	yes	—
234	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	6	yes	—
235	C₂×C₄	C₂³×D₁₀₉	1	6	yes	—
236	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
237	C₈	—	109	1	no	—
238	C₈	—	109	1	no	—
239	C₈	—	109	6	no	—
240	C₂²	—	2	4	no	—
241	C₂²	—	4	3	no	—
242	C₂²	—	8	6	no	—
243	C₂²	—	872	6	no	—
244	C₂²	—	218	4	no	—
245	C₂²	C₂²×C₂²⋊D₁₀₉	1	6	yes	—
246	C₂²	C₂²×C₂²⋊D₁₀₉	1	1	yes	characteristic
247	C₄	—	2	4	no	—
248	C₄	—	218	4	no	—
249	C₄	C₂²×C₂²⋊D₁₀₉	1	6	yes	—
250	C₄	C₂²×C₂²⋊D₁₀₉	1	1	yes	characteristic
251	C₄	C₂⁴×D₁₀₉	1	1	yes	characteristic
252	C₂	—	2	1	no	—
253	C₂	—	2	6	no	—
254	C₂	—	4	4	no	—
255	C₂	—	436	4	no	—
256	C₂	C₂³×C₂²⋊D₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×C₂²⋊D₁₀₉
commutator, cosocle	C₄×C₁₀₉	C₂⁵
Frattini	C₄	C₂⁴×D₁₀₉
Fitting	#585×C₁₀₉	C₂
socle	C₂×C₁₀₉	C₂³×D₄
hypercenter	#585	D₁₀₉
hypocenter	C₁₀₉	#3584
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄ ◁ C₂×#41 ◁ #585

Complements

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

N	H ≅ G/N	type
1. #20295862	C₂	semidirect
2. #20295841	C₂	semidirect
3. #20295837	C₂	semidirect
4. #20295834	C₂	semidirect
5. #20295829	C₂	semidirect
6. #20295824	C₂	semidirect
7. C₂×#16812719	C₂	semidirect
8. C₂×#16812706	C₂	semidirect
9. #585×C₁₀₉	C₂	semidirect
11. #16812746	C₂²	semidirect
12. #16812745	C₂²	semidirect
14. C₂×#13182029	C₂²	semidirect
16. #16812730	C₂²	semidirect
17. #16812729	C₂²	semidirect
18. #16812725	C₂²	semidirect
19. #16812724	C₂²	semidirect
20. #16812719	C₂²	semidirect
21. #16812719	C₂²	semidirect
22. #16812719	C₂²	semidirect
23. #16812719	C₂²	semidirect
24. C₂×#13182019	C₂²	semidirect
25. C₂×#13182017	C₂²	semidirect
26. C₂×#13182016	C₂²	semidirect
27. C₂×#13182014	C₂²	semidirect
28. #16812706	C₂²	semidirect
29. #16812706	C₂²	semidirect
30. C₂×#13182013	C₂²	semidirect
31. C₂²×#11724784	C₂²	semidirect
33. 2₋¹⁺⁴×C₁₀₉	C₂²	semidirect
35. H(5,𝔽₂)×C₁₀₉	C₂²	semidirect
38. C₂×#41×C₁₀₉	C₂²	semidirect
129. C₁₀₉	#3584	semidirect
131. #585	D₁₀₉	semidirect
142. 2₋¹⁺⁴	C₂×D₁₀₉	semidirect
146. H(5,𝔽₂)	C₂×D₁₀₉	semidirect
151. C₂×#41	C₂×D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295862	2
#20295841	2
#20295837	2
#20295834	2
#20295829	2
#20295824	2
C₂×#16812719	2
C₂×#16812706	2
#585×C₁₀₉	2
#3584	109