Subgroups — C₂×#117×D₁₀₉ #28376194

Order 13952.

1880 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

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

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₀₉
commutator	C₂×C₁₀₉	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₁₀₉
Fitting	C₂×#117×C₁₀₉	C₂
socle	C₂³×C₁₀₉	C₂⁴
prime radical	C₂⁴×D₁₀₉	C₂²
cosocle	C₂²×C₁₀₉	C₂⁵
hypercenter	C₂×#117	D₁₀₉
hypocenter	C₁₀₉	C₂²×#117
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
2. C₂×#16812658	C₂	semidirect
3. C₂×#16812657	C₂	semidirect
4. C₂×#16812656	C₂	semidirect
5. C₂×#16812655	C₂	semidirect
6. #117×D₁₀₉	C₂	direct (N × H)
7. C₂×C₄⋊C₄×D₁₀₉	C₂	semidirect
9. C₂×C₄²×D₁₀₉	C₂	semidirect
10. C₂×#117×C₁₀₉	C₂	semidirect
13. C₂²×C₄×D₁₀₉	C₄	semidirect
20. #16812658	C₂²	semidirect
21. #16812657	C₂²	semidirect
22. #16812656	C₂²	semidirect
23. #16812655	C₂²	semidirect
24. C₂×C₄⋊Dic₁₀₉	C₂²	semidirect
25. C₂×#13182008	C₂²	semidirect
26. C₄⋊C₄×D₁₀₉	C₂²	semidirect
30. C₂×C₄×Dic₁₀₉	C₂²	semidirect
31. C₂×C₄×Dic₁₀₉	C₂²	semidirect
32. C₄²×D₁₀₉	C₂²	semidirect
34. #117×C₁₀₉	C₂²	semidirect
35. C₂×C₄⋊C₄×C₁₀₉	C₂²	semidirect
37. C₂×C₄²×C₁₀₉	C₂²	semidirect
43. C₂²×Dic₁₀₉	C₂×C₄	semidirect
51. C₂×C₄×D₁₀₉	C₂×C₄	semidirect
52. C₂×C₄×D₁₀₉	C₂×C₄	semidirect
53. C₂×C₄×D₁₀₉	C₂×C₄	semidirect
59. C₄⋊Dic₁₀₉	C₂³	semidirect
60. #13182008	C₂³	semidirect
63. C₄×Dic₁₀₉	C₂³	semidirect
64. C₄×Dic₁₀₉	C₂³	semidirect
66. C₂²×C₄×C₁₀₉	C₂×C₄	semidirect
71. C₄⋊C₄×C₁₀₉	C₂³	semidirect
73. C₄²×C₁₀₉	C₂³	semidirect
90. C₂×C₄×C₁₀₉	C₂²×C₄	semidirect
97. C₂×Dic₁₀₉	C₂²×C₄	semidirect
104. C₄×D₁₀₉	C₂²×C₄	semidirect
105. C₄×D₁₀₉	C₂²×C₄	semidirect
115. C₂×D₁₀₉	#117	direct (N × H)
126. C₂×C₁₀₉	C₂×#117	semidirect
130. D₁₀₉	C₂×#117	direct (N × H)
132. C₁₀₉	C₂²×#117	semidirect
139. C₂×#117	D₁₀₉	direct (N × H)
157. #117	C₂×D₁₀₉	direct (N × H)
161. C₂×C₄⋊C₄	C₂×D₁₀₉	semidirect
169. C₂×C₄²	C₂×D₁₀₉	semidirect
192. C₂²×C₄	C₄×D₁₀₉	semidirect
199. C₄⋊C₄	C₂²×D₁₀₉	semidirect
205. C₄²	C₂²×D₁₀₉	semidirect
234. C₂×C₄	C₂×C₄×D₁₀₉	semidirect
260. C₂	#117×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
C₂³×C₄×D₁₀₉	2
C₂×#16812658	2
C₂×#16812657	2
C₂×#16812656	2
C₂×#16812655	2
#117×D₁₀₉	2
C₂×C₄⋊C₄×D₁₀₉	2
C₂×C₂²⋊C₄×D₁₀₉	2
C₂×C₄²×D₁₀₉	2
C₂×#117×C₁₀₉	2
C₂²×#117	109

Subgroups — C2×#117×D109 #28376194

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#117×D₁₀₉ #28376194