Subgroups — #550×D₁₀₉ #28376508

Order 13952.

1736 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41×D₁₀₉	C₂	1	2	yes	maximal, complemented
2	#20295698	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295697	C₂	1	3	yes	maximal, complemented
4	#20295696	C₂	1	3	yes	maximal, complemented
5	#20295695	C₂	1	2	yes	maximal, complemented
6	#20295694	C₂	1	6	yes	maximal, complemented
7	#127×D₁₀₉	C₂	1	3	yes	maximal, complemented
8	#121×D₁₀₉	C₂	1	6	yes	maximal, complemented
9	C₄×Q₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×D₄×D₁₀₉	C₂	1	3	yes	maximal, complemented
11	#550×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13182030	C₂²	1	2	yes	complemented
13	C₂×#13182029	C₂²	1	6	yes	complemented
14	C₂×#13182028	C₂²	1	6	yes	complemented
15	#41×D₁₀₉	—	2	16	no	—
16	C₂×Q₈×D₁₀₉	C₂²	1	1	yes	characteristic
17	C₂×D₄×D₁₀₉	C₂²	1	3	yes	—
18	C₂×D₄×D₁₀₉	C₂²	1	6	yes	—
19	C₂×D₄×D₁₀₉	C₂²	1	6	yes	—
20	C₂²×C₄×D₁₀₉	C₂²	1	6	yes	—
21	#16812696	C₂²	1	3	yes	complemented
22	#16812695	C₂²	1	6	yes	complemented
23	#16812695	C₂²	1	6	yes	complemented
24	#16812695	C₂²	1	6	yes	complemented
25	#16812674	C₂²	1	6	yes	complemented
26	#16812673	C₂²	1	6	yes	complemented
27	#16812672	C₂²	1	12	yes	complemented
28	#16812671	C₂²	1	12	yes	complemented
29	#16812670	C₂²	1	6	yes	complemented
30	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic, complemented
31	C₄×Dic₂₁₈	C₂²	1	3	yes	complemented
32	#16812664	C₂²	1	3	yes	complemented
33	C₄×D₄₃₆	C₂²	1	3	yes	complemented
34	D₄×Dic₁₀₉	C₂²	1	3	yes	complemented
35	#16812661	C₂²	1	3	yes	complemented
36	C₄×C₂²⋊D₁₀₉	C₂²	1	6	yes	complemented
37	#16812659	C₂²	1	6	yes	complemented
38	C₄⋊C₄×D₁₀₉	C₂²	1	1	yes	characteristic
39	C₄⋊C₄×D₁₀₉	C₂²	1	3	yes	—
40	C₂²⋊C₄×D₁₀₉	C₂²	1	6	yes	—
41	C₄²×D₁₀₉	C₂²	1	3	yes	—
42	C₂×#41×C₁₀₉	C₂²	1	2	yes	complemented
43	#127×C₁₀₉	C₂²	1	3	yes	complemented
44	#121×C₁₀₉	C₂²	1	6	yes	complemented
45	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
46	C₄×D₄×C₁₀₉	C₂²	1	3	yes	complemented
47	C₂³×D₁₀₉	C₂³	1	6	yes	—
48	#13182030	—	2	8	no	—
49	#13182029	—	2	24	no	—
50	#13182028	—	2	24	no	—
51	C₂×Dic₂₁₈	C₂³	1	3	yes	—
52	Q₈×D₁₀₉	D₄	1	8	yes	complemented
53	C₂×D₄₃₆	C₂³	1	3	yes	—
54	C₂×D₄₃₆	C₂³	1	6	yes	—
55	C₂×D₄₃₆	C₂³	1	6	yes	—
56	C₂×C₂²⋊D₁₀₉	C₂³	1	6	yes	—
57	C₂×C₂²⋊D₁₀₉	C₂³	1	12	yes	—
58	C₂×C₂²⋊D₁₀₉	C₂³	1	12	yes	—
59	D₄×D₁₀₉	—	2	24	no	—
60	D₄×D₁₀₉	—	2	12	no	—
61	D₄×D₁₀₉	—	4	12	no	—
62	C₂²×Dic₁₀₉	C₂³	1	6	yes	—
63	C₂×C₄×D₁₀₉	—	2	24	no	—
64	C₂×C₄×D₁₀₉	C₂³	1	3	yes	—
65	C₂×C₄×D₁₀₉	C₂³	1	2	yes	—
66	C₂×C₄×D₁₀₉	C₂³	1	6	yes	—
67	C₂×C₄×D₁₀₉	C₂³	1	6	yes	—
68	C₂×C₄×D₁₀₉	C₂³	1	3	yes	—
69	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
70	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
71	C₄⋊Dic₁₀₉	C₂³	1	3	yes	—
72	#13182008	C₂³	1	2	yes	—
73	#13182008	C₂³	1	6	yes	—
74	C₂²⋊Dic₁₀₉	C₂³	1	6	yes	—
75	#13182006	C₂³	1	6	yes	—
76	#13182006	C₂³	1	6	yes	—
77	C₄×Dic₁₀₉	C₂³	1	3	yes	—
78	C₄×Dic₁₀₉	C₂³	1	3	yes	—
79	C₄×Dic₁₀₉	C₂³	1	3	yes	—
80	#41×C₁₀₉	—	2	8	no	—
81	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
82	C₂×D₄×C₁₀₉	C₂³	1	3	yes	—
83	C₂×D₄×C₁₀₉	C₂³	1	6	yes	—
84	C₂×D₄×C₁₀₉	C₂³	1	6	yes	—
85	C₂²×C₄×C₁₀₉	C₂³	1	6	yes	—
86	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
87	C₄⋊C₄×C₁₀₉	C₂³	1	3	yes	—
88	C₂²⋊C₄×C₁₀₉	C₂³	1	6	yes	—
89	C₄²×C₁₀₉	C₂³	1	3	yes	—
90	C₂³×C₁₀₉	C₂⁴	1	6	yes	—
91	C₂²×D₁₀₉	—	2	12	no	—
92	C₂²×D₁₀₉	—	2	12	no	—
93	C₂²×D₁₀₉	—	2	12	no	—
94	C₂²×D₁₀₉	C₂⁴	1	6	yes	—
95	C₂²×D₁₀₉	C₂⁴	1	1	yes	characteristic
96	Q₈×C₁₀₉	C₂×D₄	1	4	yes	complemented
97	D₄×C₁₀₉	—	2	12	no	—
98	D₄×C₁₀₉	—	2	6	no	—
99	D₄×C₁₀₉	—	4	6	no	—
100	C₂×C₄×C₁₀₉	—	2	12	no	—
101	C₂×C₄×C₁₀₉	C₂⁴	1	3	yes	—
102	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
103	C₂×C₄×C₁₀₉	C₂⁴	1	3	yes	—
104	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
105	C₂²⋊D₁₀₉	—	2	24	no	—
106	C₂²⋊D₁₀₉	—	2	12	no	—
107	C₂²⋊D₁₀₉	—	4	12	no	—
108	C₂×Dic₁₀₉	—	2	12	no	—
109	C₂×Dic₁₀₉	C₂⁴	1	3	yes	—
110	C₂×Dic₁₀₉	C₂⁴	1	2	yes	—
111	C₂×Dic₁₀₉	C₂⁴	1	3	yes	—
112	C₂×Dic₁₀₉	C₂⁴	1	1	yes	characteristic
113	D₄₃₆	—	2	12	no	—
114	D₄₃₆	—	2	6	no	—
115	D₄₃₆	—	4	6	no	—
116	C₄×D₁₀₉	—	2	4	no	—
117	C₄×D₁₀₉	—	2	6	no	—
118	C₄×D₁₀₉	—	2	12	no	—
119	C₄×D₁₀₉	—	2	12	no	—
120	C₄×D₁₀₉	C₂×D₄	1	12	yes	—
121	C₄×D₁₀₉	#41	1	4	yes	complemented
122	Dic₂₁₈	C₂×D₄	1	12	yes	complemented
123	C₂²×C₁₀₉	—	2	6	no	—
124	C₂²×C₁₀₉	—	2	6	no	—
125	C₂²×C₁₀₉	—	2	6	no	—
126	C₂²×C₁₀₉	C₂⁵	1	1	yes	characteristic
127	C₂×D₁₀₉	—	2	6	no	—
128	C₂×D₁₀₉	—	2	6	no	—
129	C₂×D₁₀₉	—	2	6	no	—
130	C₂×D₁₀₉	—	4	24	no	—
131	C₂×D₁₀₉	C₂²×D₄	1	2	yes	—
132	C₂×D₁₀₉	C₂×#41	1	2	yes	—
133	C₂×D₁₀₉	H(5,𝔽₂)	1	2	yes	—
134	C₄×C₁₀₉	—	2	2	no	—
135	C₄×C₁₀₉	—	2	3	no	—
136	C₄×C₁₀₉	C₂²×D₄	1	6	yes	—
137	C₄×C₁₀₉	C₂×#41	1	2	yes	complemented
138	Dic₁₀₉	—	2	2	no	—
139	Dic₁₀₉	—	2	3	no	—
140	Dic₁₀₉	C₂²×D₄	1	6	yes	—
141	Dic₁₀₉	C₂×#41	1	2	yes	complemented
142	C₂×C₁₀₉	—	4	6	no	—
143	C₂×C₁₀₉	C₂³×D₄	1	1	yes	characteristic
144	C₂×C₁₀₉	C₂²×#41	1	1	yes	characteristic
145	C₂×C₁₀₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
146	D₁₀₉	—	4	6	no	—
147	D₁₀₉	#550	1	4	yes	direct product factor
148	C₂×#550	—	109	1	no	maximal, Sylow
149	C₁₀₉	C₂×#550	1	1	yes	characteristic, complemented, Sylow
150	C₂²×#41	—	109	2	no	—
151	#550	—	109	2	no	—
152	#550	—	109	3	no	—
153	#550	—	109	6	no	—
154	#550	—	109	1	no	—
155	#550	—	109	3	no	—
156	#550	D₁₀₉	1	1	yes	characteristic, direct product factor
157	C₂×#127	—	109	3	no	—
158	C₂×#121	—	109	6	no	—
159	C₂×C₄×Q₈	—	109	1	no	—
160	C₂×C₄×D₄	—	109	3	no	—
161	C₂×#41	—	109	2	no	—
162	C₂×#41	—	109	6	no	—
163	C₂×#41	—	109	6	no	—
164	C₂×#41	—	218	16	no	—
165	C₂×#41	C₂×D₁₀₉	1	2	yes	complemented
166	C₂²×Q₈	—	109	1	no	—
167	C₂²×D₄	—	109	6	no	—
168	C₂²×D₄	—	109	3	no	—
169	C₂²×D₄	—	109	6	no	—
170	C₂³×C₄	—	109	6	no	—
171	#127	—	109	6	no	—
172	#127	—	109	6	no	—
173	#127	—	109	6	no	—
174	#127	—	109	3	no	—
175	#127	C₂×D₁₀₉	1	3	yes	complemented
176	#121	—	109	6	no	—
177	#121	—	109	12	no	—
178	#121	—	109	6	no	—
179	#121	—	109	6	no	—
180	#121	—	109	12	no	—
181	#121	C₂×D₁₀₉	1	6	yes	complemented
182	C₄×Q₈	—	109	1	no	—
183	C₄×Q₈	—	109	3	no	—
184	C₄×Q₈	—	109	3	no	—
185	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic, complemented
186	C₄×D₄	—	109	3	no	—
187	C₄×D₄	—	109	6	no	—
188	C₄×D₄	—	109	3	no	—
189	C₄×D₄	—	109	6	no	—
190	C₄×D₄	—	109	3	no	—
191	C₄×D₄	C₂×D₁₀₉	1	3	yes	complemented
192	C₂×C₄⋊C₄	—	109	1	no	—
193	C₂×C₄⋊C₄	—	109	3	no	—
194	C₂×C₂²⋊C₄	—	109	6	no	—
195	C₂×C₄²	—	109	3	no	—
196	C₂⁴	—	109	6	no	—
197	#41	—	2	8	no	—
198	#41	—	218	24	no	—
199	#41	—	218	24	no	—
200	#41	—	218	8	no	—
201	C₂×Q₈	—	109	8	no	—
202	C₂×Q₈	—	109	3	no	—
203	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
204	C₂×D₄	—	109	12	no	—
205	C₂×D₄	—	109	6	no	—
206	C₂×D₄	—	109	12	no	—
207	C₂×D₄	—	109	6	no	—
208	C₂×D₄	—	109	3	no	—
209	C₂×D₄	—	109	6	no	—
210	C₂×D₄	—	436	12	no	—
211	C₂×D₄	—	218	12	no	—
212	C₂×D₄	—	218	24	no	—
213	C₂×D₄	C₂²×D₁₀₉	1	3	yes	—
214	C₂×D₄	C₂²×D₁₀₉	1	6	yes	—
215	C₂×D₄	C₂²×D₁₀₉	1	6	yes	—
216	C₂²×C₄	—	109	1	no	—
217	C₂²×C₄	—	109	3	no	—
218	C₂²×C₄	—	109	6	no	—
219	C₂²×C₄	—	109	3	no	—
220	C₂²×C₄	—	109	2	no	—
221	C₂²×C₄	—	109	6	no	—
222	C₂²×C₄	—	109	6	no	—
223	C₂²×C₄	—	218	24	no	—
224	C₂²×C₄	C₂²×D₁₀₉	1	6	yes	—
225	C₄⋊C₄	—	109	2	no	—
226	C₄⋊C₄	—	109	6	no	—
227	C₄⋊C₄	—	109	1	no	—
228	C₄⋊C₄	—	109	3	no	—
229	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
230	C₄⋊C₄	C₂²×D₁₀₉	1	3	yes	—
231	C₂²⋊C₄	—	109	6	no	—
232	C₂²⋊C₄	—	109	6	no	—
233	C₂²⋊C₄	—	109	6	no	—
234	C₂²⋊C₄	C₂²×D₁₀₉	1	6	yes	—
235	C₄²	—	109	3	no	—
236	C₄²	—	109	3	no	—
237	C₄²	—	109	3	no	—
238	C₄²	C₂²×D₁₀₉	1	3	yes	—
239	C₂³	—	109	1	no	—
240	C₂³	—	109	6	no	—
241	C₂³	—	218	12	no	—
242	C₂³	—	218	12	no	—
243	C₂³	—	218	12	no	—
244	C₂³	C₂³×D₁₀₉	1	6	yes	—
245	Q₈	—	109	12	no	—
246	Q₈	D₄×D₁₀₉	1	4	yes	complemented
247	D₄	—	2	12	no	—
248	D₄	—	2	6	no	—
249	D₄	—	4	6	no	—
250	D₄	—	436	6	no	—
251	D₄	—	436	12	no	—
252	D₄	—	218	24	no	—
253	D₄	—	218	12	no	—
254	D₄	—	218	12	no	—
255	D₄	—	218	6	no	—
256	C₂×C₄	—	2	12	no	—
257	C₂×C₄	—	109	1	no	—
258	C₂×C₄	—	109	3	no	—
259	C₂×C₄	—	109	12	no	—
260	C₂×C₄	—	109	4	no	—
261	C₂×C₄	—	109	3	no	—
262	C₂×C₄	—	109	2	no	—
263	C₂×C₄	—	218	12	no	—
264	C₂×C₄	—	218	4	no	—
265	C₂×C₄	—	218	6	no	—
266	C₂×C₄	—	218	12	no	—
267	C₂×C₄	—	218	12	no	—
268	C₂×C₄	C₂³×D₁₀₉	1	3	yes	—
269	C₂×C₄	C₂³×D₁₀₉	1	2	yes	—
270	C₂×C₄	C₂³×D₁₀₉	1	3	yes	—
271	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
272	C₂²	—	2	6	no	—
273	C₂²	—	2	6	no	—
274	C₂²	—	2	6	no	—
275	C₂²	—	109	2	no	—
276	C₂²	—	109	2	no	—
277	C₂²	—	109	2	no	—
278	C₂²	—	436	24	no	—
279	C₂²	—	218	6	no	—
280	C₂²	—	218	6	no	—
281	C₂²	—	218	6	no	—
282	C₂²	C₂⁴×D₁₀₉	1	1	yes	characteristic
283	C₄	—	2	2	no	—
284	C₄	—	2	3	no	—
285	C₄	—	109	6	no	—
286	C₄	—	109	2	no	—
287	C₄	—	218	2	no	—
288	C₄	—	218	3	no	—
289	C₄	C₂×D₄×D₁₀₉	1	6	yes	—
290	C₄	#41×D₁₀₉	1	2	yes	complemented
291	C₂	—	4	6	no	—
292	C₂	—	109	4	no	—
293	C₂	—	436	6	no	—
294	C₂	C₂²×D₄×D₁₀₉	1	1	yes	characteristic
295	C₂	C₂×#41×D₁₀₉	1	1	yes	characteristic
296	C₂	H(5,𝔽₂)×D₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₀₉
commutator, socle, cosocle	C₂²×C₁₀₉	C₂⁵
Fitting	#550×C₁₀₉	C₂
hypercenter	#550	D₁₀₉
hypocenter	C₁₀₉	C₂×#550
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ #550

Complements

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

N	H ≅ G/N	type
1. C₂×#41×D₁₀₉	C₂	semidirect
2. #20295698	C₂	semidirect
3. #20295697	C₂	semidirect
4. #20295696	C₂	semidirect
5. #20295695	C₂	semidirect
6. #20295694	C₂	semidirect
7. #127×D₁₀₉	C₂	semidirect
8. #121×D₁₀₉	C₂	semidirect
9. C₄×Q₈×D₁₀₉	C₂	semidirect
10. C₄×D₄×D₁₀₉	C₂	semidirect
11. #550×C₁₀₉	C₂	semidirect
12. C₂×#13182030	C₂²	semidirect
13. C₂×#13182029	C₂²	semidirect
14. C₂×#13182028	C₂²	semidirect
21. #16812696	C₂²	semidirect
22. #16812695	C₂²	semidirect
23. #16812695	C₂²	semidirect
24. #16812695	C₂²	semidirect
25. #16812674	C₂²	semidirect
26. #16812673	C₂²	semidirect
27. #16812672	C₂²	semidirect
28. #16812671	C₂²	semidirect
29. #16812670	C₂²	semidirect
30. Q₈×Dic₁₀₉	C₂²	semidirect
31. C₄×Dic₂₁₈	C₂²	semidirect
32. #16812664	C₂²	semidirect
33. C₄×D₄₃₆	C₂²	semidirect
34. D₄×Dic₁₀₉	C₂²	semidirect
35. #16812661	C₂²	semidirect
36. C₄×C₂²⋊D₁₀₉	C₂²	semidirect
37. #16812659	C₂²	semidirect
42. C₂×#41×C₁₀₉	C₂²	semidirect
43. #127×C₁₀₉	C₂²	semidirect
44. #121×C₁₀₉	C₂²	semidirect
45. C₄×Q₈×C₁₀₉	C₂²	semidirect
46. C₄×D₄×C₁₀₉	C₂²	semidirect
52. Q₈×D₁₀₉	D₄	semidirect
96. Q₈×C₁₀₉	C₂×D₄	semidirect
121. C₄×D₁₀₉	#41	semidirect
122. Dic₂₁₈	C₂×D₄	semidirect
137. C₄×C₁₀₉	C₂×#41	semidirect
141. Dic₁₀₉	C₂×#41	semidirect
147. D₁₀₉	#550	direct (N × H)
149. C₁₀₉	C₂×#550	semidirect
156. #550	D₁₀₉	direct (N × H)
165. C₂×#41	C₂×D₁₀₉	semidirect
175. #127	C₂×D₁₀₉	semidirect
181. #121	C₂×D₁₀₉	semidirect
185. C₄×Q₈	C₂×D₁₀₉	semidirect
191. C₄×D₄	C₂×D₁₀₉	semidirect
246. Q₈	D₄×D₁₀₉	semidirect
290. C₄	#41×D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#41×D₁₀₉	2
#20295698	2
#20295697	2
#20295696	2
#20295695	2
#20295694	2
#127×D₁₀₉	2
#121×D₁₀₉	2
C₄×Q₈×D₁₀₉	2
C₄×D₄×D₁₀₉	2
#550×C₁₀₉	2
C₂×#550	109

Subgroups — #550×D109 #28376508

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #550×D₁₀₉ #28376508