Subgroups — C₄×#41×D₁₀₉ #28376209

Order 13952.

1728 conjugacy classes of subgroups.

The subgroup lattice has 282 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	1	yes	maximal, characteristic
2	#41×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#13182030	C₂	1	2	yes	maximal, complemented
4	C₄×#13182029	C₂	1	3	yes	maximal, complemented
5	#20295494	C₂	1	3	yes	maximal, complemented
6	C₄×#13182028	C₂	1	6	yes	maximal, complemented
7	C₄×Q₈×D₁₀₉	C₂	1	2	yes	maximal, complemented
8	C₄×D₄×D₁₀₉	C₂	1	6	yes	maximal, complemented
9	#117×D₁₀₉	C₂	1	3	yes	maximal, complemented
10	C₂×C₄²×D₁₀₉	C₂	1	3	yes	maximal, complemented
11	C₄×#41×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13182030	C₂²	1	1	yes	characteristic
13	C₂×#13182029	C₂²	1	3	yes	—
14	C₂×#13182028	C₂²	1	3	yes	—
15	#41×D₁₀₉	C₄	1	16	yes	direct product factor
16	C₂×Q₈×D₁₀₉	C₂²	1	1	yes	characteristic
17	C₂×D₄×D₁₀₉	C₂²	1	3	yes	—
18	C₂²×C₄×D₁₀₉	C₂²	1	6	yes	—
19	C₂²×C₄×D₁₀₉	C₂²	1	3	yes	—
20	Q₈×Dic₁₀₉	C₂²	1	2	yes	complemented
21	C₄×Dic₂₁₈	C₂²	1	6	yes	complemented
22	#16812664	C₂²	1	6	yes	complemented
23	C₄×D₄₃₆	C₂²	1	6	yes	complemented
24	D₄×Dic₁₀₉	C₂²	1	6	yes	complemented
25	#16812661	C₂²	1	6	yes	complemented
26	C₄×C₂²⋊D₁₀₉	C₂²	1	12	yes	complemented
27	#16812659	C₂²	1	12	yes	complemented
28	#16812658	C₂²	1	3	yes	complemented
29	#16812657	C₂²	1	6	yes	complemented
30	#16812656	C₂²	1	6	yes	complemented
31	#16812655	C₂²	1	6	yes	complemented
32	C₄⋊C₄×D₁₀₉	C₂²	1	6	yes	—
33	C₂²⋊C₄×D₁₀₉	C₂²	1	6	yes	—
34	C₂×C₄×Dic₁₀₉	C₂²	1	6	yes	complemented
35	C₂×C₄×Dic₁₀₉	C₂²	1	3	yes	complemented
36	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
37	C₄²×D₁₀₉	C₂²	1	3	yes	—
38	C₄²×D₁₀₉	C₂²	1	6	yes	—
39	C₄²×D₁₀₉	C₂²	1	6	yes	complemented
40	C₄²×D₁₀₉	C₂²	1	3	yes	complemented
41	C₄²×D₁₀₉	C₂²	1	3	yes	complemented
42	C₂×#41×C₁₀₉	C₂²	1	1	yes	characteristic
43	C₄×Q₈×C₁₀₉	C₂²	1	2	yes	complemented
44	C₄×D₄×C₁₀₉	C₂²	1	6	yes	complemented
45	#117×C₁₀₉	C₂²	1	3	yes	complemented
46	C₂×C₄²×C₁₀₉	C₂²	1	3	yes	complemented
47	C₂³×D₁₀₉	C₂³	1	3	yes	—
48	#13182030	C₂×C₄	1	8	yes	complemented
49	#13182029	C₂×C₄	1	24	yes	complemented
50	#13182028	C₂×C₄	1	24	yes	complemented
51	C₂×Dic₂₁₈	C₂³	1	3	yes	—
52	Q₈×D₁₀₉	C₂×C₄	1	8	yes	complemented
53	C₂×D₄₃₆	C₂³	1	3	yes	—
54	C₂×C₂²⋊D₁₀₉	C₂³	1	6	yes	—
55	D₄×D₁₀₉	C₂×C₄	1	24	yes	complemented
56	C₂²×Dic₁₀₉	C₂³	1	6	yes	—
57	C₂²×Dic₁₀₉	C₂³	1	3	yes	—
58	C₂×C₄×D₁₀₉	—	2	24	no	—
59	C₂×C₄×D₁₀₉	C₂×C₄	1	24	yes	complemented
60	C₂×C₄×D₁₀₉	C₂³	1	6	yes	—
61	C₂×C₄×D₁₀₉	C₂³	1	6	yes	—
62	C₂×C₄×D₁₀₉	C₂³	1	6	yes	—
63	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
64	C₂×C₄×D₁₀₉	C₂³	1	3	yes	—
65	C₂×C₄×D₁₀₉	C₂³	1	3	yes	—
66	C₂×C₄×D₁₀₉	C₂³	1	3	yes	—
67	C₂×C₄×D₁₀₉	C₂³	1	2	yes	—
68	C₄⋊Dic₁₀₉	C₂³	1	6	yes	—
69	#13182008	C₂³	1	12	yes	—
70	C₂²⋊Dic₁₀₉	C₂³	1	6	yes	—
71	#13182006	C₂³	1	12	yes	—
72	C₄×Dic₁₀₉	C₂³	1	2	yes	—
73	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
74	C₄×Dic₁₀₉	C₂³	1	6	yes	—
75	C₄×Dic₁₀₉	C₂³	1	3	yes	—
76	C₄×Dic₁₀₉	C₂³	1	6	yes	—
77	C₄×Dic₁₀₉	C₂³	1	6	yes	—
78	C₄×Dic₁₀₉	C₂³	1	6	yes	—
79	#41×C₁₀₉	C₂×C₄	1	8	yes	complemented
80	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
81	C₂×D₄×C₁₀₉	C₂³	1	3	yes	—
82	C₂²×C₄×C₁₀₉	C₂³	1	6	yes	—
83	C₂²×C₄×C₁₀₉	C₂³	1	3	yes	—
84	C₄⋊C₄×C₁₀₉	C₂³	1	6	yes	—
85	C₂²⋊C₄×C₁₀₉	C₂³	1	6	yes	—
86	C₄²×C₁₀₉	C₂³	1	3	yes	—
87	C₄²×C₁₀₉	C₂³	1	6	yes	—
88	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
89	C₂³×C₁₀₉	C₂⁴	1	3	yes	—
90	C₂²×D₁₀₉	—	2	12	no	—
91	C₂²×D₁₀₉	C₂²×C₄	1	12	yes	—
92	C₂²×D₁₀₉	C₂⁴	1	3	yes	—
93	C₂²×D₁₀₉	C₂⁴	1	1	yes	characteristic
94	Q₈×C₁₀₉	C₂²×C₄	1	4	yes	complemented
95	D₄×C₁₀₉	C₂²×C₄	1	12	yes	complemented
96	C₂×C₄×C₁₀₉	—	2	12	no	—
97	C₂×C₄×C₁₀₉	C₂²×C₄	1	12	yes	complemented
98	C₂×C₄×C₁₀₉	C₂⁴	1	6	yes	—
99	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
100	C₂×C₄×C₁₀₉	C₂⁴	1	3	yes	—
101	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
102	C₂²⋊D₁₀₉	C₂²×C₄	1	24	yes	complemented
103	C₂×Dic₁₀₉	—	2	12	no	—
104	C₂×Dic₁₀₉	C₂²×C₄	1	12	yes	complemented
105	C₂×Dic₁₀₉	C₂⁴	1	6	yes	—
106	C₂×Dic₁₀₉	C₂⁴	1	1	yes	characteristic
107	C₂×Dic₁₀₉	C₂⁴	1	3	yes	—
108	C₂×Dic₁₀₉	C₂⁴	1	2	yes	—
109	D₄₃₆	C₂²×C₄	1	12	yes	complemented
110	C₄×D₁₀₉	—	2	12	no	—
111	C₄×D₁₀₉	—	2	12	no	—
112	C₄×D₁₀₉	—	2	12	no	—
113	C₄×D₁₀₉	C₂²×C₄	1	4	yes	—
114	C₄×D₁₀₉	C₂²×C₄	1	12	yes	—
115	C₄×D₁₀₉	C₂²×C₄	1	12	yes	complemented
116	C₄×D₁₀₉	C₂²×C₄	1	12	yes	complemented
117	C₄×D₁₀₉	#41	1	8	yes	direct product factor
118	Dic₂₁₈	C₂²×C₄	1	12	yes	complemented
119	C₂²×C₁₀₉	—	2	6	no	—
120	C₂²×C₁₀₉	C₂³×C₄	1	6	yes	—
121	C₂²×C₁₀₉	C₂⁵	1	1	yes	characteristic
122	C₂×D₁₀₉	—	2	6	no	—
123	C₂×D₁₀₉	—	2	24	no	—
124	C₂×D₁₀₉	C₂³×C₄	1	6	yes	—
125	C₂×D₁₀₉	C₂³×C₄	1	2	yes	—
126	C₂×D₁₀₉	C₂×#41	1	4	yes	—
127	C₄×C₁₀₉	—	2	6	no	—
128	C₄×C₁₀₉	C₂³×C₄	1	6	yes	—
129	C₄×C₁₀₉	C₂³×C₄	1	2	yes	—
130	C₄×C₁₀₉	C₂×#41	1	4	yes	complemented
131	Dic₁₀₉	—	2	6	no	—
132	Dic₁₀₉	C₂³×C₄	1	6	yes	—
133	Dic₁₀₉	C₂³×C₄	1	2	yes	—
134	Dic₁₀₉	C₂×#41	1	4	yes	complemented
135	C₂×C₁₀₉	—	2	6	no	—
136	C₂×C₁₀₉	C₂⁴×C₄	1	1	yes	characteristic
137	C₂×C₁₀₉	C₂²×#41	1	2	yes	—
138	D₁₀₉	—	2	6	no	—
139	D₁₀₉	C₄×#41	1	4	yes	direct product factor
140	C₂×C₄×#41	—	109	1	no	maximal, Sylow
141	C₁₀₉	C₂×C₄×#41	1	1	yes	characteristic, complemented, Sylow
142	C₂²×#41	—	109	1	no	—
143	C₄×#41	—	109	1	no	—
144	C₄×#41	—	109	2	no	—
145	C₄×#41	—	109	6	no	—
146	C₄×#41	—	109	3	no	—
147	C₄×#41	—	109	3	no	—
148	C₄×#41	D₁₀₉	1	1	yes	characteristic, direct product factor
149	C₂×C₄×Q₈	—	109	2	no	—
150	C₂×C₄×D₄	—	109	6	no	—
151	C₂×#117	—	109	3	no	—
152	C₂²×C₄²	—	109	3	no	—
153	C₂×#41	—	109	16	no	—
154	C₂×#41	—	109	1	no	—
155	C₂×#41	—	109	3	no	—
156	C₂×#41	—	109	3	no	—
157	C₂×#41	C₂×D₁₀₉	1	1	yes	characteristic
158	C₂²×Q₈	—	109	1	no	—
159	C₂²×D₄	—	109	3	no	—
160	C₂³×C₄	—	109	3	no	—
161	C₂³×C₄	—	109	6	no	—
162	C₄×Q₈	—	109	2	no	—
163	C₄×Q₈	—	109	6	no	—
164	C₄×Q₈	—	109	6	no	—
165	C₄×Q₈	C₂×D₁₀₉	1	2	yes	complemented
166	C₄×D₄	—	109	6	no	—
167	C₄×D₄	—	109	12	no	—
168	C₄×D₄	—	109	6	no	—
169	C₄×D₄	—	109	12	no	—
170	C₄×D₄	—	109	6	no	—
171	C₄×D₄	C₂×D₁₀₉	1	6	yes	complemented
172	#117	—	109	3	no	—
173	#117	—	109	6	no	—
174	#117	—	109	6	no	—
175	#117	—	109	6	no	—
176	#117	C₂×D₁₀₉	1	3	yes	complemented
177	C₂×C₄⋊C₄	—	109	6	no	—
178	C₂×C₂²⋊C₄	—	109	6	no	—
179	C₂×C₄²	—	109	6	no	—
180	C₂×C₄²	—	109	3	no	—
181	C₂×C₄²	—	109	3	no	—
182	C₂×C₄²	—	109	6	no	—
183	C₂×C₄²	—	109	6	no	—
184	C₂×C₄²	—	109	3	no	—
185	C₂×C₄²	—	109	3	no	—
186	C₂×C₄²	—	109	1	no	—
187	C₂×C₄²	C₂×D₁₀₉	1	3	yes	complemented
188	C₂⁴	—	109	3	no	—
189	#41	—	109	8	no	—
190	#41	—	109	24	no	—
191	#41	—	109	24	no	—
192	#41	C₄×D₁₀₉	1	8	yes	direct product factor
193	C₂×Q₈	—	109	8	no	—
194	C₂×Q₈	—	109	3	no	—
195	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
196	C₂×D₄	—	109	6	no	—
197	C₂×D₄	—	109	24	no	—
198	C₂×D₄	—	109	3	no	—
199	C₂×D₄	C₂²×D₁₀₉	1	3	yes	—
200	C₂²×C₄	—	109	3	no	—
201	C₂²×C₄	—	109	24	no	—
202	C₂²×C₄	—	109	1	no	—
203	C₂²×C₄	—	109	2	no	—
204	C₂²×C₄	—	109	3	no	—
205	C₂²×C₄	—	109	3	no	—
206	C₂²×C₄	—	109	3	no	—
207	C₂²×C₄	—	109	6	no	—
208	C₂²×C₄	—	109	6	no	—
209	C₂²×C₄	—	109	6	no	—
210	C₂²×C₄	—	109	6	no	—
211	C₂²×C₄	—	218	24	no	—
212	C₂²×C₄	C₂²×D₁₀₉	1	6	yes	—
213	C₂²×C₄	C₂²×D₁₀₉	1	3	yes	—
214	C₄⋊C₄	—	109	12	no	—
215	C₄⋊C₄	—	109	6	no	—
216	C₄⋊C₄	C₂²×D₁₀₉	1	6	yes	—
217	C₂²⋊C₄	—	109	6	no	—
218	C₂²⋊C₄	—	109	12	no	—
219	C₂²⋊C₄	C₂²×D₁₀₉	1	6	yes	—
220	C₄²	—	109	1	no	—
221	C₄²	—	109	2	no	—
222	C₄²	—	109	6	no	—
223	C₄²	—	109	3	no	—
224	C₄²	—	109	6	no	—
225	C₄²	—	109	6	no	—
226	C₄²	—	109	6	no	—
227	C₄²	C₂²×D₁₀₉	1	3	yes	—
228	C₄²	C₂²×D₁₀₉	1	6	yes	—
229	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
230	C₂³	—	109	1	no	—
231	C₂³	—	109	12	no	—
232	C₂³	—	109	3	no	—
233	C₂³	—	218	12	no	—
234	C₂³	C₂³×D₁₀₉	1	3	yes	—
235	Q₈	—	109	12	no	—
236	Q₈	C₂×C₄×D₁₀₉	1	4	yes	complemented
237	D₄	—	109	24	no	—
238	D₄	—	109	12	no	—
239	D₄	C₂×C₄×D₁₀₉	1	12	yes	complemented
240	C₂×C₄	—	2	12	no	—
241	C₂×C₄	—	109	12	no	—
242	C₂×C₄	—	109	1	no	—
243	C₂×C₄	—	109	2	no	—
244	C₂×C₄	—	109	3	no	—
245	C₂×C₄	—	109	4	no	—
246	C₂×C₄	—	109	12	no	—
247	C₂×C₄	—	109	12	no	—
248	C₂×C₄	—	109	12	no	—
249	C₂×C₄	—	109	8	no	—
250	C₂×C₄	—	109	6	no	—
251	C₂×C₄	—	218	12	no	—
252	C₂×C₄	—	218	12	no	—
253	C₂×C₄	—	218	12	no	—
254	C₂×C₄	—	218	12	no	—
255	C₂×C₄	C₂×C₄×D₁₀₉	1	12	yes	complemented
256	C₂×C₄	C₂³×D₁₀₉	1	6	yes	—
257	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
258	C₂×C₄	C₂³×D₁₀₉	1	3	yes	—
259	C₂×C₄	C₂³×D₁₀₉	1	2	yes	—
260	C₂²	—	2	6	no	—
261	C₂²	—	109	2	no	—
262	C₂²	—	109	4	no	—
263	C₂²	—	109	6	no	—
264	C₂²	—	218	24	no	—
265	C₂²	—	218	6	no	—
266	C₂²	C₂²×C₄×D₁₀₉	1	6	yes	—
267	C₂²	C₂⁴×D₁₀₉	1	1	yes	characteristic
268	C₄	—	2	6	no	—
269	C₄	—	109	2	no	—
270	C₄	—	109	6	no	—
271	C₄	—	109	4	no	—
272	C₄	—	218	6	no	—
273	C₄	C₂²×C₄×D₁₀₉	1	2	yes	—
274	C₄	C₂²×C₄×D₁₀₉	1	6	yes	—
275	C₄	#41×D₁₀₉	1	4	yes	direct product factor
276	C₂	—	2	6	no	—
277	C₂	—	109	4	no	—
278	C₂	—	218	6	no	—
279	C₂	C₂³×C₄×D₁₀₉	1	1	yes	characteristic
280	C₂	C₂×#41×D₁₀₉	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂²×D₁₀₉
commutator	C₂×C₁₀₉	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₁₀₉
Fitting	C₄×#41×C₁₀₉	C₂
socle, cosocle	C₂²×C₁₀₉	C₂⁵
prime radical	C₂×#41×D₁₀₉	C₂
hypercenter	C₄×#41	D₁₀₉
hypocenter	C₁₀₉	C₂×C₄×#41
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
2. #41×Dic₁₀₉	C₂	semidirect
3. C₄×#13182030	C₂	semidirect
4. C₄×#13182029	C₂	semidirect
5. #20295494	C₂	semidirect
6. C₄×#13182028	C₂	semidirect
7. C₄×Q₈×D₁₀₉	C₂	semidirect
8. C₄×D₄×D₁₀₉	C₂	semidirect
9. #117×D₁₀₉	C₂	semidirect
10. C₂×C₄²×D₁₀₉	C₂	semidirect
11. C₄×#41×C₁₀₉	C₂	semidirect
15. #41×D₁₀₉	C₄	direct (N × H)
20. Q₈×Dic₁₀₉	C₂²	semidirect
21. C₄×Dic₂₁₈	C₂²	semidirect
22. #16812664	C₂²	semidirect
23. C₄×D₄₃₆	C₂²	semidirect
24. D₄×Dic₁₀₉	C₂²	semidirect
25. #16812661	C₂²	semidirect
26. C₄×C₂²⋊D₁₀₉	C₂²	semidirect
27. #16812659	C₂²	semidirect
28. #16812658	C₂²	semidirect
29. #16812657	C₂²	semidirect
30. #16812656	C₂²	semidirect
31. #16812655	C₂²	semidirect
34. C₂×C₄×Dic₁₀₉	C₂²	semidirect
35. C₂×C₄×Dic₁₀₉	C₂²	semidirect
39. C₄²×D₁₀₉	C₂²	semidirect
40. C₄²×D₁₀₉	C₂²	semidirect
41. C₄²×D₁₀₉	C₂²	semidirect
43. C₄×Q₈×C₁₀₉	C₂²	semidirect
44. C₄×D₄×C₁₀₉	C₂²	semidirect
45. #117×C₁₀₉	C₂²	semidirect
46. C₂×C₄²×C₁₀₉	C₂²	semidirect
48. #13182030	C₂×C₄	semidirect
49. #13182029	C₂×C₄	semidirect
50. #13182028	C₂×C₄	semidirect
52. Q₈×D₁₀₉	C₂×C₄	semidirect
55. D₄×D₁₀₉	C₂×C₄	semidirect
59. C₂×C₄×D₁₀₉	C₂×C₄	semidirect
79. #41×C₁₀₉	C₂×C₄	semidirect
94. Q₈×C₁₀₉	C₂²×C₄	semidirect
95. D₄×C₁₀₉	C₂²×C₄	semidirect
97. C₂×C₄×C₁₀₉	C₂²×C₄	semidirect
102. C₂²⋊D₁₀₉	C₂²×C₄	semidirect
104. C₂×Dic₁₀₉	C₂²×C₄	semidirect
109. D₄₃₆	C₂²×C₄	semidirect
115. C₄×D₁₀₉	C₂²×C₄	semidirect
116. C₄×D₁₀₉	C₂²×C₄	semidirect
117. C₄×D₁₀₉	#41	direct (N × H)
118. Dic₂₁₈	C₂²×C₄	semidirect
130. C₄×C₁₀₉	C₂×#41	semidirect
134. Dic₁₀₉	C₂×#41	semidirect
139. D₁₀₉	C₄×#41	direct (N × H)
141. C₁₀₉	C₂×C₄×#41	semidirect
148. C₄×#41	D₁₀₉	direct (N × H)
165. C₄×Q₈	C₂×D₁₀₉	semidirect
171. C₄×D₄	C₂×D₁₀₉	semidirect
176. #117	C₂×D₁₀₉	semidirect
187. C₂×C₄²	C₂×D₁₀₉	semidirect
192. #41	C₄×D₁₀₉	direct (N × H)
236. Q₈	C₂×C₄×D₁₀₉	semidirect
239. D₄	C₂×C₄×D₁₀₉	semidirect
255. C₂×C₄	C₂×C₄×D₁₀₉	semidirect
275. C₄	#41×D₁₀₉	direct (N × H)

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
#41×Dic₁₀₉	2
C₄×#13182030	2
C₄×#13182029	2
#20295494	2
C₄×#13182028	2
C₄×Q₈×D₁₀₉	2
C₄×D₄×D₁₀₉	2
#117×D₁₀₉	2
C₂×C₄²×D₁₀₉	2
C₄×#41×C₁₀₉	2
C₂×C₄×#41	109

Subgroups — C4×#41×D109 #28376209

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#41×D₁₀₉ #28376209