Subgroups — C₂×#124×D₁₀₉ #28376277

Order 13952.

1880 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×D₄×D₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#16812688	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16812687	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16812686	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16812685	C₂	1	4	yes	maximal, complemented
7	#124×D₁₀₉	C₂	1	16	yes	maximal, direct product factor
8	C₂×C₂²⋊C₄×D₁₀₉	C₂	1	4	yes	maximal, complemented
9	C₂×C₄²×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#124×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂⁴×D₁₀₉	C₂²	1	2	yes	—
12	C₂²×Dic₂₁₈	C₂²	1	2	yes	complemented
13	C₂²×Dic₂₁₈	C₂²	1	1	yes	characteristic, complemented
14	C₂×Q₈×D₁₀₉	—	2	8	no	—
15	C₂×Q₈×D₁₀₉	C₂²	1	8	yes	complemented
16	C₂²×D₄₃₆	C₂²	1	1	yes	characteristic
17	C₂²×C₂²⋊D₁₀₉	C₂²	1	2	yes	—
18	C₂×D₄×D₁₀₉	—	2	8	no	—
19	C₂×D₄×D₁₀₉	C₂²	1	8	yes	—
20	C₂²×C₄×D₁₀₉	C₂²	1	2	yes	—
21	C₂²×C₄×D₁₀₉	C₂²	1	2	yes	—
22	C₂²×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
23	#16812688	C₂²	1	8	yes	complemented
24	#16812687	C₂²	1	8	yes	complemented
25	#16812686	C₂²	1	8	yes	complemented
26	#16812685	C₂²	1	32	yes	complemented
27	C₂×C₂²⋊Dic₁₀₉	C₂²	1	4	yes	complemented
28	C₂×#13182006	C₂²	1	4	yes	complemented
29	C₂×#13182006	C₂²	1	4	yes	complemented
30	C₂²⋊C₄×D₁₀₉	C₂²	1	32	yes	complemented
31	C₂×C₄×Dic₁₀₉	C₂²	1	2	yes	complemented
32	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic, complemented
33	C₄²×D₁₀₉	C₂²	1	8	yes	complemented
34	C₂²×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
35	C₂²×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
36	#124×C₁₀₉	C₂²	1	8	yes	complemented
37	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	4	yes	complemented
38	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
39	C₂³×D₁₀₉	—	2	8	no	—
40	C₂³×D₁₀₉	—	2	16	no	—
41	C₂³×D₁₀₉	C₂³	1	8	yes	—
42	C₂³×D₁₀₉	C₂³	1	2	yes	—
43	C₂³×D₁₀₉	C₂³	1	1	yes	characteristic
44	C₂×Dic₂₁₈	—	2	4	no	—
45	C₂×Dic₂₁₈	—	2	8	no	—
46	C₂×Dic₂₁₈	C₂³	1	8	yes	complemented
47	C₂×Dic₂₁₈	C₂³	1	4	yes	complemented
48	Q₈×D₁₀₉	—	2	32	no	—
49	C₂×D₄₃₆	—	2	4	no	—
50	C₂×D₄₃₆	C₂³	1	4	yes	—
51	C₂×C₂²⋊D₁₀₉	—	2	8	no	—
52	C₂×C₂²⋊D₁₀₉	C₂³	1	8	yes	—
53	D₄×D₁₀₉	—	2	32	no	—
54	C₂²×Dic₁₀₉	C₂³	1	2	yes	—
55	C₂²×Dic₁₀₉	C₂³	1	2	yes	—
56	C₂²×Dic₁₀₉	C₂³	1	1	yes	characteristic
57	C₂×C₄×D₁₀₉	—	2	8	no	—
58	C₂×C₄×D₁₀₉	—	2	8	no	—
59	C₂×C₄×D₁₀₉	D₄	1	8	yes	—
60	C₂×C₄×D₁₀₉	C₂³	1	8	yes	—
61	C₂×C₄×D₁₀₉	C₂³	1	8	yes	—
62	C₂×C₄×D₁₀₉	C₂³	1	4	yes	—
63	C₂²⋊Dic₁₀₉	C₂³	1	16	yes	complemented
64	#13182006	C₂³	1	16	yes	complemented
65	#13182006	C₂³	1	16	yes	complemented
66	C₄×Dic₁₀₉	C₂³	1	8	yes	complemented
67	C₄×Dic₁₀₉	C₂³	1	4	yes	complemented
68	C₂⁴×C₁₀₉	C₂³	1	2	yes	—
69	C₂×Q₈×C₁₀₉	—	2	4	no	—
70	C₂×Q₈×C₁₀₉	C₂³	1	4	yes	complemented
71	C₂×D₄×C₁₀₉	—	2	4	no	—
72	C₂×D₄×C₁₀₉	C₂³	1	4	yes	—
73	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
74	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
75	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
76	C₂²⋊C₄×C₁₀₉	C₂³	1	16	yes	complemented
77	C₄²×C₁₀₉	C₂³	1	4	yes	complemented
78	C₂³×C₁₀₉	—	2	4	no	—
79	C₂³×C₁₀₉	—	2	8	no	—
80	C₂³×C₁₀₉	C₂⁴	1	4	yes	—
81	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
82	C₂²×D₁₀₉	—	2	4	no	—
83	C₂²×D₁₀₉	—	2	8	no	—
84	C₂²×D₁₀₉	—	2	16	no	—
85	C₂²×D₁₀₉	—	2	32	no	—
86	C₂²×D₁₀₉	—	4	32	no	—
87	C₂²×D₁₀₉	C₂×D₄	1	4	yes	—
88	C₂²×D₁₀₉	#41	1	8	yes	—
89	C₂²×D₁₀₉	C₂⁴	1	4	yes	—
90	C₂²×D₁₀₉	C₂⁴	1	2	yes	—
91	Q₈×C₁₀₉	—	2	8	no	—
92	D₄×C₁₀₉	—	2	8	no	—
93	C₂×C₄×C₁₀₉	—	2	4	no	—
94	C₂×C₄×C₁₀₉	—	2	4	no	—
95	C₂×C₄×C₁₀₉	C₂×D₄	1	4	yes	—
96	C₂×C₄×C₁₀₉	C₂⁴	1	4	yes	—
97	C₂×C₄×C₁₀₉	C₂⁴	1	4	yes	—
98	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
99	C₂²⋊D₁₀₉	—	2	16	no	—
100	C₂×Dic₁₀₉	—	2	4	no	—
101	C₂×Dic₁₀₉	—	2	4	no	—
102	C₂×Dic₁₀₉	C₂×D₄	1	4	yes	—
103	C₂×Dic₁₀₉	C₂⁴	1	4	yes	—
104	C₂×Dic₁₀₉	C₂⁴	1	4	yes	—
105	C₂×Dic₁₀₉	C₂⁴	1	2	yes	—
106	D₄₃₆	—	2	8	no	—
107	C₄×D₁₀₉	—	2	16	no	—
108	C₄×D₁₀₉	—	2	16	no	—
109	C₄×D₁₀₉	C₂×D₄	1	16	yes	—
110	Dic₂₁₈	—	2	8	no	—
111	Dic₂₁₈	—	2	16	no	—
112	C₂²×C₁₀₉	—	2	4	no	—
113	C₂²×C₁₀₉	—	2	8	no	—
114	C₂²×C₁₀₉	—	4	8	no	—
115	C₂²×C₁₀₉	C₂²×D₄	1	2	yes	—
116	C₂²×C₁₀₉	C₂×#41	1	4	yes	—
117	C₂²×C₁₀₉	C₂⁵	1	1	yes	characteristic
118	C₂×D₁₀₉	—	2	4	no	—
119	C₂×D₁₀₉	—	2	8	no	—
120	C₂×D₁₀₉	—	4	8	no	—
121	C₂×D₁₀₉	—	4	32	no	—
122	C₂×D₁₀₉	#124	1	16	yes	direct product factor
123	C₂×D₁₀₉	C₂²×D₄	1	4	yes	—
124	C₂×D₁₀₉	C₂×#41	1	8	yes	—
125	C₄×C₁₀₉	—	2	4	no	—
126	C₄×C₁₀₉	—	2	4	no	—
127	C₄×C₁₀₉	C₂²×D₄	1	4	yes	—
128	Dic₁₀₉	—	2	4	no	—
129	Dic₁₀₉	—	2	4	no	—
130	Dic₁₀₉	C₂²×D₄	1	4	yes	—
131	C₂×C₁₀₉	—	4	4	no	—
132	C₂×C₁₀₉	C₂×#124	1	4	yes	complemented
133	C₂×C₁₀₉	C₂³×D₄	1	1	yes	characteristic
134	C₂×C₁₀₉	C₂²×#41	1	2	yes	—
135	D₁₀₉	—	4	4	no	—
136	D₁₀₉	C₂×#124	1	8	yes	direct product factor
137	C₂²×#124	—	109	1	no	maximal, Sylow
138	C₁₀₉	C₂²×#124	1	1	yes	characteristic, complemented, Sylow
139	C₂³×Q₈	—	109	1	no	—
140	C₂³×D₄	—	109	1	no	—
141	C₂×#124	—	109	1	no	—
142	C₂×#124	—	109	1	no	—
143	C₂×#124	—	109	16	no	—
144	C₂×#124	—	109	4	no	—
145	C₂×#124	—	109	1	no	—
146	C₂×#124	D₁₀₉	1	1	yes	characteristic, direct product factor
147	C₂²×C₂²⋊C₄	—	109	4	no	—
148	C₂²×C₄²	—	109	1	no	—
149	C₂⁵	—	109	2	no	—
150	C₂²×Q₈	—	109	8	no	—
151	C₂²×Q₈	—	109	2	no	—
152	C₂²×Q₈	—	109	1	no	—
153	C₂²×Q₈	—	218	8	no	—
154	C₂²×Q₈	C₂×D₁₀₉	1	1	yes	characteristic, complemented
155	C₂²×D₄	—	109	2	no	—
156	C₂²×D₄	—	109	8	no	—
157	C₂²×D₄	—	109	1	no	—
158	C₂²×D₄	—	218	8	no	—
159	C₂²×D₄	C₂×D₁₀₉	1	1	yes	characteristic
160	C₂³×C₄	—	109	1	no	—
161	C₂³×C₄	—	109	2	no	—
162	C₂³×C₄	—	109	2	no	—
163	#124	—	109	8	no	—
164	#124	—	109	8	no	—
165	#124	—	109	32	no	—
166	#124	—	109	8	no	—
167	#124	C₂×D₁₀₉	1	8	yes	direct product factor
168	C₂×C₂²⋊C₄	—	109	4	no	—
169	C₂×C₂²⋊C₄	—	109	4	no	—
170	C₂×C₂²⋊C₄	—	109	4	no	—
171	C₂×C₂²⋊C₄	—	109	32	no	—
172	C₂×C₂²⋊C₄	C₂×D₁₀₉	1	4	yes	complemented
173	C₂×C₄²	—	109	2	no	—
174	C₂×C₄²	—	109	1	no	—
175	C₂×C₄²	—	109	8	no	—
176	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic, complemented
177	C₂⁴	—	109	1	no	—
178	C₂⁴	—	109	8	no	—
179	C₂⁴	—	109	2	no	—
180	C₂⁴	—	218	8	no	—
181	C₂⁴	—	218	16	no	—
182	C₂⁴	C₂²×D₁₀₉	1	2	yes	—
183	C₂×Q₈	—	2	4	no	—
184	C₂×Q₈	—	109	8	no	—
185	C₂×Q₈	—	109	4	no	—
186	C₂×Q₈	—	218	8	no	—
187	C₂×Q₈	—	218	32	no	—
188	C₂×Q₈	—	218	4	no	—
189	C₂×Q₈	C₂²×D₁₀₉	1	4	yes	complemented
190	C₂×D₄	—	2	4	no	—
191	C₂×D₄	—	109	8	no	—
192	C₂×D₄	—	109	4	no	—
193	C₂×D₄	—	218	32	no	—
194	C₂×D₄	—	218	8	no	—
195	C₂×D₄	—	218	4	no	—
196	C₂×D₄	C₂²×D₁₀₉	1	4	yes	—
197	C₂²×C₄	—	109	1	no	—
198	C₂²×C₄	—	109	8	no	—
199	C₂²×C₄	—	109	4	no	—
200	C₂²×C₄	—	109	2	no	—
201	C₂²×C₄	—	109	2	no	—
202	C₂²×C₄	—	109	8	no	—
203	C₂²×C₄	—	109	8	no	—
204	C₂²×C₄	—	218	8	no	—
205	C₂²×C₄	—	218	8	no	—
206	C₂²×C₄	C₂²×D₁₀₉	1	2	yes	—
207	C₂²×C₄	C₂²×D₁₀₉	1	2	yes	—
208	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
209	C₂²⋊C₄	—	109	16	no	—
210	C₂²⋊C₄	—	109	16	no	—
211	C₂²⋊C₄	—	109	16	no	—
212	C₂²⋊C₄	C₂²×D₁₀₉	1	16	yes	complemented
213	C₄²	—	109	8	no	—
214	C₄²	—	109	4	no	—
215	C₄²	C₂²×D₁₀₉	1	4	yes	complemented
216	C₂³	—	2	4	no	—
217	C₂³	—	2	8	no	—
218	C₂³	—	109	4	no	—
219	C₂³	—	109	8	no	—
220	C₂³	—	109	2	no	—
221	C₂³	—	109	4	no	—
222	C₂³	—	436	32	no	—
223	C₂³	—	218	16	no	—
224	C₂³	—	218	32	no	—
225	C₂³	—	218	4	no	—
226	C₂³	—	218	8	no	—
227	C₂³	C₂³×D₁₀₉	1	4	yes	—
228	C₂³	C₂³×D₁₀₉	1	1	yes	characteristic
229	Q₈	—	2	8	no	—
230	Q₈	—	218	16	no	—
231	Q₈	—	218	8	no	—
232	D₄	—	2	8	no	—
233	D₄	—	218	16	no	—
234	D₄	—	218	8	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	109	4	no	—
238	C₂×C₄	—	109	2	no	—
239	C₂×C₄	—	109	16	no	—
240	C₂×C₄	—	109	4	no	—
241	C₂×C₄	—	109	4	no	—
242	C₂×C₄	—	218	16	no	—
243	C₂×C₄	—	218	16	no	—
244	C₂×C₄	—	218	4	no	—
245	C₂×C₄	—	218	4	no	—
246	C₂×C₄	D₄×D₁₀₉	1	4	yes	—
247	C₂×C₄	C₂³×D₁₀₉	1	4	yes	—
248	C₂×C₄	C₂³×D₁₀₉	1	4	yes	—
249	C₂×C₄	C₂³×D₁₀₉	1	2	yes	—
250	C₂²	—	2	4	no	—
251	C₂²	—	2	8	no	—
252	C₂²	—	4	8	no	—
253	C₂²	—	109	16	no	—
254	C₂²	—	109	4	no	—
255	C₂²	—	109	8	no	—
256	C₂²	—	436	32	no	—
257	C₂²	—	436	8	no	—
258	C₂²	—	218	4	no	—
259	C₂²	—	218	8	no	—
260	C₂²	C₂×D₄×D₁₀₉	1	2	yes	—
261	C₂²	#41×D₁₀₉	1	4	yes	—
262	C₂²	C₂⁴×D₁₀₉	1	1	yes	characteristic
263	C₄	—	2	4	no	—
264	C₄	—	2	4	no	—
265	C₄	—	109	4	no	—
266	C₄	—	218	4	no	—
267	C₄	—	218	4	no	—
268	C₄	C₂×D₄×D₁₀₉	1	4	yes	—
269	C₂	—	4	4	no	—
270	C₂	—	109	8	no	—
271	C₂	—	436	4	no	—
272	C₂	#124×D₁₀₉	1	4	yes	direct product factor
273	C₂	C₂²×D₄×D₁₀₉	1	1	yes	characteristic
274	C₂	C₂×#41×D₁₀₉	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂³×D₁₀₉
commutator, cosocle	C₂²×C₁₀₉	C₂⁵
Frattini	C₂²	C₂⁴×D₁₀₉
Fitting	C₂×#124×C₁₀₉	C₂
socle	C₂³×C₁₀₉	C₂⁴
prime radical	C₂²×D₄×D₁₀₉	C₂
hypercenter	C₂×#124	D₁₀₉
hypocenter	C₁₀₉	C₂²×#124
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂×#124

Complements

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

N	H ≅ G/N	type
1. C₂²×Q₈×D₁₀₉	C₂	semidirect
3. C₂×#16812688	C₂	semidirect
4. C₂×#16812687	C₂	semidirect
5. C₂×#16812686	C₂	semidirect
6. C₂×#16812685	C₂	semidirect
7. #124×D₁₀₉	C₂	direct (N × H)
8. C₂×C₂²⋊C₄×D₁₀₉	C₂	semidirect
9. C₂×C₄²×D₁₀₉	C₂	semidirect
10. C₂×#124×C₁₀₉	C₂	semidirect
12. C₂²×Dic₂₁₈	C₂²	semidirect
13. C₂²×Dic₂₁₈	C₂²	semidirect
15. C₂×Q₈×D₁₀₉	C₂²	semidirect
23. #16812688	C₂²	semidirect
24. #16812687	C₂²	semidirect
25. #16812686	C₂²	semidirect
26. #16812685	C₂²	semidirect
27. C₂×C₂²⋊Dic₁₀₉	C₂²	semidirect
28. C₂×#13182006	C₂²	semidirect
29. C₂×#13182006	C₂²	semidirect
30. C₂²⋊C₄×D₁₀₉	C₂²	semidirect
31. C₂×C₄×Dic₁₀₉	C₂²	semidirect
32. C₂×C₄×Dic₁₀₉	C₂²	semidirect
33. C₄²×D₁₀₉	C₂²	semidirect
34. C₂²×Q₈×C₁₀₉	C₂²	semidirect
36. #124×C₁₀₉	C₂²	semidirect
37. C₂×C₂²⋊C₄×C₁₀₉	C₂²	semidirect
38. C₂×C₄²×C₁₀₉	C₂²	semidirect
46. C₂×Dic₂₁₈	C₂³	semidirect
47. C₂×Dic₂₁₈	C₂³	semidirect
63. C₂²⋊Dic₁₀₉	C₂³	semidirect
64. #13182006	C₂³	semidirect
65. #13182006	C₂³	semidirect
66. C₄×Dic₁₀₉	C₂³	semidirect
67. C₄×Dic₁₀₉	C₂³	semidirect
70. C₂×Q₈×C₁₀₉	C₂³	semidirect
76. C₂²⋊C₄×C₁₀₉	C₂³	semidirect
77. C₄²×C₁₀₉	C₂³	semidirect
122. C₂×D₁₀₉	#124	direct (N × H)
132. C₂×C₁₀₉	C₂×#124	semidirect
136. D₁₀₉	C₂×#124	direct (N × H)
138. C₁₀₉	C₂²×#124	semidirect
146. C₂×#124	D₁₀₉	direct (N × H)
154. C₂²×Q₈	C₂×D₁₀₉	semidirect
167. #124	C₂×D₁₀₉	direct (N × H)
172. C₂×C₂²⋊C₄	C₂×D₁₀₉	semidirect
176. C₂×C₄²	C₂×D₁₀₉	semidirect
189. C₂×Q₈	C₂²×D₁₀₉	semidirect
212. C₂²⋊C₄	C₂²×D₁₀₉	semidirect
215. C₄²	C₂²×D₁₀₉	semidirect
272. C₂	#124×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
C₂²×Q₈×D₁₀₉	2
C₂²×D₄×D₁₀₉	2
C₂×#16812688	2
C₂×#16812687	2
C₂×#16812686	2
C₂×#16812685	2
#124×D₁₀₉	2
C₂×C₂²⋊C₄×D₁₀₉	2
C₂×C₄²×D₁₀₉	2
C₂×#124×C₁₀₉	2
C₂²×#124	109

Subgroups — C2×#124×D109 #28376277

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#124×D₁₀₉ #28376277