Subgroups — #28376663

Order 13952.

1900 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₄×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295664	C₂	1	8	yes	maximal, complemented
3	D₄×C₂²⋊D₁₀₉	C₂	1	8	yes	maximal, complemented
4	#20295590	C₂	1	2	yes	maximal, complemented
5	#20295585	C₂	1	4	yes	maximal, complemented
6	#20295533	C₂	1	2	yes	maximal, complemented
7	C₂×#16812667	C₂	1	4	yes	maximal, complemented
8	#20295501	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×H(5,𝔽₂)×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂⁴×D₁₀₉	C₂²	1	2	yes	complemented
11	C₂²×D₄₃₆	C₂²	1	1	yes	characteristic, complemented
12	C₂²×C₂²⋊D₁₀₉	C₂²	1	2	yes	complemented
13	C₂²×C₂²⋊D₁₀₉	C₂²	1	8	yes	complemented
14	C₂×D₄×D₁₀₉	—	2	8	no	—
15	C₂×D₄×D₁₀₉	C₂²	1	8	yes	complemented
16	C₂²×C₄×D₁₀₉	C₂²	1	1	yes	characteristic, complemented
17	#16812695	C₂²	1	4	yes	complemented
18	#16812688	C₂²	1	4	yes	complemented
19	#16812682	C₂²	1	8	yes	complemented
20	#16812680	C₂²	1	4	yes	complemented
21	#16812675	C₂²	1	4	yes	complemented
22	#16812673	C₂²	1	8	yes	complemented
23	#16812671	C₂²	1	16	yes	complemented
24	#16812670	C₂²	1	4	yes	complemented
25	#16812668	C₂²	1	8	yes	complemented
26	#16812667	C₂²	1	4	yes	complemented
27	#16812667	C₂²	1	8	yes	complemented
28	#16812667	C₂²	1	16	yes	complemented
29	D₄×Dic₁₀₉	C₂²	1	8	yes	complemented
30	C₄×C₂²⋊D₁₀₉	C₂²	1	8	yes	complemented
31	#16812658	C₂²	1	2	yes	complemented
32	C₂×C₂²⋊Dic₁₀₉	C₂²	1	4	yes	complemented
33	C₂×#13182006	C₂²	1	8	yes	complemented
34	H(5,𝔽₂)×C₁₀₉	—	2	8	no	—
35	C₂×#41×C₁₀₉	C₂²	1	2	yes	complemented
36	C₂×#41×C₁₀₉	C₂²	1	4	yes	complemented
37	C₂²×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
38	C₂²×D₄×C₁₀₉	C₂²	1	4	yes	complemented
39	C₂²×D₄×C₁₀₉	C₂²	1	4	yes	complemented
40	C₂³×D₁₀₉	—	2	8	no	—
41	C₂³×D₁₀₉	—	4	8	no	—
42	C₂³×D₁₀₉	C₂³	1	1	yes	characteristic
43	C₂³×D₁₀₉	C₂³	1	8	yes	—
44	C₂³×D₁₀₉	C₂³	1	2	yes	—
45	C₂×D₄₃₆	—	2	4	no	—
46	C₂×D₄₃₆	C₂³	1	4	yes	—
47	C₂×C₂²⋊D₁₀₉	—	2	8	no	—
48	C₂×C₂²⋊D₁₀₉	—	4	16	no	—
49	C₂×C₂²⋊D₁₀₉	C₂³	1	8	yes	—
50	C₂×C₂²⋊D₁₀₉	C₂³	1	8	yes	—
51	C₂×C₂²⋊D₁₀₉	C₂³	1	16	yes	complemented
52	D₄×D₁₀₉	—	2	32	no	—
53	C₂²×Dic₁₀₉	C₂³	1	1	yes	characteristic
54	C₂²×Dic₁₀₉	C₂³	1	8	yes	complemented
55	C₂×C₄×D₁₀₉	—	2	4	no	—
56	C₂×C₄×D₁₀₉	C₂³	1	4	yes	—
57	C₄⋊Dic₁₀₉	C₂³	1	4	yes	complemented
58	#13182008	C₂³	1	8	yes	complemented
59	C₂²⋊Dic₁₀₉	C₂³	1	4	yes	—
60	C₂²⋊Dic₁₀₉	C₂³	1	8	yes	complemented
61	C₂²⋊Dic₁₀₉	C₂³	1	8	yes	complemented
62	#13182006	C₂³	1	8	yes	—
63	#13182006	C₂³	1	16	yes	complemented
64	C₄×Dic₁₀₉	C₂³	1	4	yes	complemented
65	C₂⁴×C₁₀₉	C₂³	1	2	yes	—
66	C₂⁴×C₁₀₉	C₂³	1	4	yes	complemented
67	#41×C₁₀₉	—	2	16	no	—
68	#41×C₁₀₉	—	2	8	no	—
69	C₂×Q₈×C₁₀₉	C₂³	1	2	yes	complemented
70	C₂×D₄×C₁₀₉	—	2	16	no	—
71	C₂×D₄×C₁₀₉	—	2	16	no	—
72	C₂×D₄×C₁₀₉	D₄	1	8	yes	complemented
73	C₂×D₄×C₁₀₉	C₂³	1	4	yes	—
74	C₂×D₄×C₁₀₉	C₂³	1	4	yes	—
75	C₂×D₄×C₁₀₉	C₂³	1	8	yes	complemented
76	C₂×D₄×C₁₀₉	C₂³	1	2	yes	complemented
77	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
78	C₂²×C₄×C₁₀₉	C₂³	1	4	yes	complemented
79	C₂²×C₄×C₁₀₉	C₂³	1	4	yes	complemented
80	C₂³×C₁₀₉	—	2	8	no	—
81	C₂³×C₁₀₉	—	4	4	no	—
82	C₂³×C₁₀₉	—	4	8	no	—
83	C₂³×C₁₀₉	C₂×D₄	1	8	yes	complemented
84	C₂³×C₁₀₉	C₂⁴	1	4	yes	—
85	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
86	C₂³×C₁₀₉	C₂⁴	1	4	yes	—
87	C₂²×D₁₀₉	—	2	2	no	—
88	C₂²×D₁₀₉	—	2	4	no	—
89	C₂²×D₁₀₉	—	2	16	no	—
90	C₂²×D₁₀₉	—	2	4	no	—
91	C₂²×D₁₀₉	—	2	16	no	—
92	C₂²×D₁₀₉	—	4	8	no	—
93	C₂²×D₁₀₉	—	4	4	no	—
94	C₂²×D₁₀₉	—	4	32	no	—
95	C₂²×D₁₀₉	C₂⁴	1	4	yes	—
96	C₂²×D₁₀₉	C₂⁴	1	2	yes	—
97	Q₈×C₁₀₉	—	2	4	no	—
98	D₄×C₁₀₉	—	2	8	no	—
99	D₄×C₁₀₉	—	2	16	no	—
100	D₄×C₁₀₉	—	2	4	no	—
101	D₄×C₁₀₉	C₂×D₄	1	16	yes	complemented
102	C₂×C₄×C₁₀₉	—	2	8	no	—
103	C₂×C₄×C₁₀₉	—	2	8	no	—
104	C₂×C₄×C₁₀₉	C₂×D₄	1	4	yes	complemented
105	C₂×C₄×C₁₀₉	C₂⁴	1	4	yes	—
106	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
107	C₂²⋊D₁₀₉	—	2	16	no	—
108	C₂²⋊D₁₀₉	—	2	16	no	—
109	C₂²⋊D₁₀₉	—	4	16	no	—
110	C₂×Dic₁₀₉	—	2	2	no	—
111	C₂×Dic₁₀₉	—	4	8	no	—
112	C₂×Dic₁₀₉	C₂⁴	1	8	yes	—
113	C₂×Dic₁₀₉	C₂⁴	1	2	yes	—
114	D₄₃₆	—	2	8	no	—
115	C₄×D₁₀₉	—	2	8	no	—
116	C₂²×C₁₀₉	—	2	8	no	—
117	C₂²×C₁₀₉	—	2	2	no	—
118	C₂²×C₁₀₉	—	2	4	no	—
119	C₂²×C₁₀₉	—	2	8	no	—
120	C₂²×C₁₀₉	—	4	8	no	—
121	C₂²×C₁₀₉	—	8	8	no	—
122	C₂²×C₁₀₉	C₂²×D₄	1	2	yes	—
123	C₂²×C₁₀₉	C₂²×D₄	1	8	yes	—
124	C₂²×C₁₀₉	C₂⁵	1	1	yes	characteristic
125	C₂×D₁₀₉	—	2	8	no	—
126	C₂×D₁₀₉	—	2	2	no	—
127	C₂×D₁₀₉	—	2	2	no	—
128	C₂×D₁₀₉	—	2	4	no	—
129	C₂×D₁₀₉	—	2	8	no	—
130	C₂×D₁₀₉	—	4	32	no	—
131	C₂×D₁₀₉	—	4	8	no	—
132	C₂×D₁₀₉	H(5,𝔽₂)	1	4	yes	complemented
133	C₄×C₁₀₉	—	2	4	no	—
134	C₄×C₁₀₉	C₂²×D₄	1	4	yes	—
135	Dic₁₀₉	—	2	2	no	—
136	Dic₁₀₉	—	2	8	no	—
137	C₂×C₁₀₉	—	2	2	no	—
138	C₂×C₁₀₉	—	2	8	no	—
139	C₂×C₁₀₉	—	4	4	no	—
140	C₂×C₁₀₉	C₂³×D₄	1	1	yes	characteristic
141	C₂×C₁₀₉	C₂×H(5,𝔽₂)	1	2	yes	—
142	D₁₀₉	—	2	4	no	—
143	D₁₀₉	—	4	4	no	—
144	#3484	—	109	1	no	maximal, Sylow
145	C₁₀₉	#3484	1	1	yes	characteristic, complemented, Sylow
146	C₂×H(5,𝔽₂)	D₁₀₉	1	1	yes	characteristic, complemented
147	C₂³×D₄	—	109	1	no	—
148	#546	—	109	8	no	—
149	D₄²	—	109	8	no	—
150	#535	—	109	2	no	—
151	#534	—	109	4	no	—
152	#525	—	109	2	no	—
153	C₂×#120	—	109	4	no	—
154	#518	—	109	1	no	—
155	C₂⁵	—	109	2	no	—
156	H(5,𝔽₂)	—	2	8	no	—
157	C₂×#41	C₂×D₁₀₉	1	2	yes	complemented
158	C₂×#41	C₂×D₁₀₉	1	4	yes	complemented
159	C₂²×D₄	—	109	8	no	—
160	C₂²×D₄	—	109	2	no	—
161	C₂²×D₄	—	109	8	no	—
162	C₂²×D₄	—	109	1	no	—
163	C₂²×D₄	—	218	8	no	—
164	C₂²×D₄	C₂×D₁₀₉	1	1	yes	characteristic
165	C₂²×D₄	C₂×D₁₀₉	1	4	yes	complemented
166	C₂²×D₄	C₂×D₁₀₉	1	4	yes	complemented
167	C₂³×C₄	—	109	1	no	—
168	#127	—	109	4	no	—
169	#124	—	109	4	no	—
170	#123	—	109	4	no	—
171	#123	—	109	8	no	—
172	C₂²⋊Q₈	—	109	4	no	—
173	#121	—	109	4	no	—
174	#121	—	109	16	no	—
175	#121	—	109	8	no	—
176	#120	—	109	8	no	—
177	#120	—	109	4	no	—
178	#120	—	109	8	no	—
179	#120	—	109	16	no	—
180	C₄×D₄	—	109	8	no	—
181	C₄×D₄	—	109	8	no	—
182	#117	—	109	2	no	—
183	C₂×C₂²⋊C₄	—	109	4	no	—
184	C₂×C₂²⋊C₄	—	109	8	no	—
185	C₂⁴	—	109	2	no	—
186	C₂⁴	—	109	8	no	—
187	C₂⁴	—	109	1	no	—
188	C₂⁴	—	436	8	no	—
189	C₂⁴	—	218	8	no	—
190	C₂⁴	C₂²×D₁₀₉	1	2	yes	—
191	C₂⁴	C₂²×D₁₀₉	1	4	yes	complemented
192	#41	—	2	16	no	—
193	#41	—	2	8	no	—
194	C₂×Q₈	C₂²×D₁₀₉	1	2	yes	complemented
195	C₂×D₄	—	2	16	no	—
196	C₂×D₄	—	2	16	no	—
197	C₂×D₄	—	109	16	no	—
198	C₂×D₄	—	109	8	no	—
199	C₂×D₄	—	109	8	no	—
200	C₂×D₄	—	109	4	no	—
201	C₂×D₄	—	436	16	no	—
202	C₂×D₄	—	218	8	no	—
203	C₂×D₄	—	218	32	no	—
204	C₂×D₄	—	218	4	no	—
205	C₂×D₄	C₂²⋊D₁₀₉	1	8	yes	complemented
206	C₂×D₄	C₂²×D₁₀₉	1	4	yes	—
207	C₂×D₄	C₂²×D₁₀₉	1	4	yes	—
208	C₂×D₄	C₂²×D₁₀₉	1	8	yes	complemented
209	C₂×D₄	C₂²×D₁₀₉	1	2	yes	complemented
210	C₂²×C₄	—	109	8	no	—
211	C₂²×C₄	—	109	1	no	—
212	C₂²×C₄	—	109	4	no	—
213	C₂²×C₄	—	218	4	no	—
214	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
215	C₂²×C₄	C₂²×D₁₀₉	1	4	yes	complemented
216	C₂²×C₄	C₂²×D₁₀₉	1	4	yes	complemented
217	C₄⋊C₄	—	109	8	no	—
218	C₄⋊C₄	—	109	4	no	—
219	C₂²⋊C₄	—	109	4	no	—
220	C₂²⋊C₄	—	109	8	no	—
221	C₂²⋊C₄	—	109	8	no	—
222	C₂²⋊C₄	—	109	8	no	—
223	C₂²⋊C₄	—	109	16	no	—
224	C₄²	—	109	4	no	—
225	C₂³	—	2	8	no	—
226	C₂³	—	4	4	no	—
227	C₂³	—	4	8	no	—
228	C₂³	—	109	2	no	—
229	C₂³	—	109	4	no	—
230	C₂³	—	436	8	no	—
231	C₂³	—	436	4	no	—
232	C₂³	—	436	32	no	—
233	C₂³	—	218	16	no	—
234	C₂³	—	218	4	no	—
235	C₂³	—	218	4	no	—
236	C₂³	—	218	16	no	—
237	C₂³	—	218	2	no	—
238	C₂³	C₂×C₂²⋊D₁₀₉	1	8	yes	complemented
239	C₂³	C₂³×D₁₀₉	1	4	yes	—
240	C₂³	C₂³×D₁₀₉	1	1	yes	characteristic
241	C₂³	C₂³×D₁₀₉	1	4	yes	—
242	Q₈	—	2	4	no	—
243	D₄	—	2	8	no	—
244	D₄	—	2	16	no	—
245	D₄	—	2	4	no	—
246	D₄	—	436	16	no	—
247	D₄	—	218	16	no	—
248	D₄	—	218	16	no	—
249	D₄	—	218	8	no	—
250	D₄	C₂×C₂²⋊D₁₀₉	1	16	yes	complemented
251	C₂×C₄	—	2	8	no	—
252	C₂×C₄	—	2	8	no	—
253	C₂×C₄	—	109	8	no	—
254	C₂×C₄	—	109	2	no	—
255	C₂×C₄	—	436	8	no	—
256	C₂×C₄	—	218	2	no	—
257	C₂×C₄	—	218	8	no	—
258	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	4	yes	complemented
259	C₂×C₄	C₂³×D₁₀₉	1	4	yes	—
260	C₂×C₄	C₂³×D₁₀₉	1	2	yes	—
261	C₂²	—	2	8	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	4	no	—
264	C₂²	—	2	8	no	—
265	C₂²	—	4	8	no	—
266	C₂²	—	8	8	no	—
267	C₂²	—	109	4	no	—
268	C₂²	—	436	8	no	—
269	C₂²	—	436	32	no	—
270	C₂²	—	218	8	no	—
271	C₂²	—	218	4	no	—
272	C₂²	—	218	8	no	—
273	C₂²	—	218	2	no	—
274	C₂²	—	218	2	no	—
275	C₂²	C₂²×C₂²⋊D₁₀₉	1	8	yes	—
276	C₂²	C₂²×C₂²⋊D₁₀₉	1	2	yes	—
277	C₂²	C₂⁴×D₁₀₉	1	1	yes	characteristic
278	C₄	—	2	4	no	—
279	C₄	—	218	2	no	—
280	C₄	—	218	8	no	—
281	C₄	C₂²×C₂²⋊D₁₀₉	1	4	yes	—
282	C₂	—	2	2	no	—
283	C₂	—	2	8	no	—
284	C₂	—	4	4	no	—
285	C₂	—	436	4	no	—
286	C₂	—	218	4	no	—
287	C₂	C₂³×C₂²⋊D₁₀₉	1	1	yes	characteristic
288	C₂	H(5,𝔽₂)×D₁₀₉	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₀₉
commutator, socle, cosocle	C₂²×C₁₀₉	C₂⁵
Fitting	C₂×H(5,𝔽₂)×C₁₀₉	C₂
hypercenter	C₂×H(5,𝔽₂)	D₁₀₉
hypocenter	C₁₀₉	#3484
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. C₂²×D₄×D₁₀₉	C₂	semidirect
2. #20295664	C₂	semidirect
3. D₄×C₂²⋊D₁₀₉	C₂	semidirect
4. #20295590	C₂	semidirect
5. #20295585	C₂	semidirect
6. #20295533	C₂	semidirect
7. C₂×#16812667	C₂	semidirect
8. #20295501	C₂	semidirect
9. C₂×H(5,𝔽₂)×C₁₀₉	C₂	semidirect
10. C₂⁴×D₁₀₉	C₂²	semidirect
11. C₂²×D₄₃₆	C₂²	semidirect
12. C₂²×C₂²⋊D₁₀₉	C₂²	semidirect
13. C₂²×C₂²⋊D₁₀₉	C₂²	semidirect
15. C₂×D₄×D₁₀₉	C₂²	semidirect
16. C₂²×C₄×D₁₀₉	C₂²	semidirect
17. #16812695	C₂²	semidirect
18. #16812688	C₂²	semidirect
19. #16812682	C₂²	semidirect
20. #16812680	C₂²	semidirect
21. #16812675	C₂²	semidirect
22. #16812673	C₂²	semidirect
23. #16812671	C₂²	semidirect
24. #16812670	C₂²	semidirect
25. #16812668	C₂²	semidirect
26. #16812667	C₂²	semidirect
27. #16812667	C₂²	semidirect
28. #16812667	C₂²	semidirect
29. D₄×Dic₁₀₉	C₂²	semidirect
30. C₄×C₂²⋊D₁₀₉	C₂²	semidirect
31. #16812658	C₂²	semidirect
32. C₂×C₂²⋊Dic₁₀₉	C₂²	semidirect
33. C₂×#13182006	C₂²	semidirect
35. C₂×#41×C₁₀₉	C₂²	semidirect
36. C₂×#41×C₁₀₉	C₂²	semidirect
38. C₂²×D₄×C₁₀₉	C₂²	semidirect
39. C₂²×D₄×C₁₀₉	C₂²	semidirect
51. C₂×C₂²⋊D₁₀₉	C₂³	semidirect
54. C₂²×Dic₁₀₉	C₂³	semidirect
57. C₄⋊Dic₁₀₉	C₂³	semidirect
58. #13182008	C₂³	semidirect
60. C₂²⋊Dic₁₀₉	C₂³	semidirect
61. C₂²⋊Dic₁₀₉	C₂³	semidirect
63. #13182006	C₂³	semidirect
64. C₄×Dic₁₀₉	C₂³	semidirect
66. C₂⁴×C₁₀₉	C₂³	semidirect
69. C₂×Q₈×C₁₀₉	C₂³	semidirect
72. C₂×D₄×C₁₀₉	D₄	semidirect
75. C₂×D₄×C₁₀₉	C₂³	semidirect
76. C₂×D₄×C₁₀₉	C₂³	semidirect
78. C₂²×C₄×C₁₀₉	C₂³	semidirect
79. C₂²×C₄×C₁₀₉	C₂³	semidirect
83. C₂³×C₁₀₉	C₂×D₄	semidirect
101. D₄×C₁₀₉	C₂×D₄	semidirect
104. C₂×C₄×C₁₀₉	C₂×D₄	semidirect
132. C₂×D₁₀₉	H(5,𝔽₂)	semidirect
145. C₁₀₉	#3484	semidirect
146. C₂×H(5,𝔽₂)	D₁₀₉	semidirect
157. C₂×#41	C₂×D₁₀₉	semidirect
158. C₂×#41	C₂×D₁₀₉	semidirect
165. C₂²×D₄	C₂×D₁₀₉	semidirect
166. C₂²×D₄	C₂×D₁₀₉	semidirect
191. C₂⁴	C₂²×D₁₀₉	semidirect
194. C₂×Q₈	C₂²×D₁₀₉	semidirect
205. C₂×D₄	C₂²⋊D₁₀₉	semidirect
208. C₂×D₄	C₂²×D₁₀₉	semidirect
209. C₂×D₄	C₂²×D₁₀₉	semidirect
215. C₂²×C₄	C₂²×D₁₀₉	semidirect
216. C₂²×C₄	C₂²×D₁₀₉	semidirect
238. C₂³	C₂×C₂²⋊D₁₀₉	semidirect
250. D₄	C₂×C₂²⋊D₁₀₉	semidirect
258. C₂×C₄	C₂×C₂²⋊D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₄×D₁₀₉	2
#20295664	2
D₄×C₂²⋊D₁₀₉	2
#20295590	2
#20295585	2
#20295533	2
C₂×#16812667	2
#20295501	2
C₂×H(5,𝔽₂)×C₁₀₉	2
#3484	109