Subgroups — C₂²×#41×D₁₀₉ #28377828

Order 13952.

6520 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13182030	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#13182029	C₂	1	3	yes	maximal, complemented
3	C₂²×#13182028	C₂	1	3	yes	maximal, complemented
4	C₂×#41×D₁₀₉	C₂	1	48	yes	maximal, direct product factor
5	C₂²×Q₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₄×D₁₀₉	C₂	1	3	yes	maximal, complemented
7	C₂³×C₄×D₁₀₉	C₂	1	3	yes	maximal, complemented
8	C₂²×#41×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁴×D₁₀₉	C₂²	1	3	yes	—
10	C₂×#13182030	C₂²	1	24	yes	complemented
11	C₂×#13182029	C₂²	1	72	yes	complemented
12	C₂×#13182028	C₂²	1	72	yes	complemented
13	#41×D₁₀₉	C₂²	1	256	yes	direct product factor
14	C₂²×Dic₂₁₈	C₂²	1	3	yes	complemented
15	C₂×Q₈×D₁₀₉	C₂²	1	24	yes	complemented
16	C₂²×D₄₃₆	C₂²	1	3	yes	complemented
17	C₂²×C₂²⋊D₁₀₉	C₂²	1	6	yes	complemented
18	C₂×D₄×D₁₀₉	C₂²	1	72	yes	complemented
19	C₂³×Dic₁₀₉	C₂²	1	3	yes	complemented
20	C₂²×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
21	C₂²×C₄×D₁₀₉	C₂²	1	3	yes	—
22	C₂²×C₄×D₁₀₉	C₂²	1	3	yes	complemented
23	C₂²×C₄×D₁₀₉	C₂²	1	3	yes	complemented
24	C₂²×C₄×D₁₀₉	C₂²	1	72	yes	complemented
25	C₂×#41×C₁₀₉	C₂²	1	24	yes	complemented
26	C₂²×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
27	C₂²×D₄×C₁₀₉	C₂²	1	3	yes	complemented
28	C₂³×C₄×C₁₀₉	C₂²	1	3	yes	complemented
29	C₂³×D₁₀₉	—	2	24	no	—
30	C₂³×D₁₀₉	C₂³	1	36	yes	—
31	C₂³×D₁₀₉	C₂³	1	3	yes	—
32	C₂³×D₁₀₉	C₂³	1	1	yes	characteristic
33	#13182030	C₂³	1	64	yes	complemented
34	#13182029	C₂³	1	192	yes	complemented
35	#13182028	C₂³	1	192	yes	complemented
36	C₂×Dic₂₁₈	C₂³	1	36	yes	complemented
37	Q₈×D₁₀₉	C₂³	1	64	yes	complemented
38	C₂×D₄₃₆	C₂³	1	36	yes	complemented
39	C₂×C₂²⋊D₁₀₉	C₂³	1	72	yes	complemented
40	D₄×D₁₀₉	C₂³	1	192	yes	complemented
41	C₂²×Dic₁₀₉	C₂³	1	1	yes	characteristic
42	C₂²×Dic₁₀₉	C₂³	1	3	yes	—
43	C₂²×Dic₁₀₉	C₂³	1	36	yes	complemented
44	C₂×C₄×D₁₀₉	C₂³	1	12	yes	—
45	C₂×C₄×D₁₀₉	C₂³	1	36	yes	—
46	C₂×C₄×D₁₀₉	C₂³	1	192	yes	complemented
47	C₂×C₄×D₁₀₉	C₂³	1	36	yes	complemented
48	C₂×C₄×D₁₀₉	C₂³	1	36	yes	complemented
49	C₂⁴×C₁₀₉	C₂³	1	3	yes	—
50	#41×C₁₀₉	C₂³	1	64	yes	complemented
51	C₂×Q₈×C₁₀₉	C₂³	1	12	yes	complemented
52	C₂×D₄×C₁₀₉	C₂³	1	36	yes	complemented
53	C₂²×C₄×C₁₀₉	C₂³	1	3	yes	—
54	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₁₀₉	C₂³	1	36	yes	complemented
56	C₂³×C₁₀₉	—	2	12	no	—
57	C₂³×C₁₀₉	C₂⁴	1	18	yes	—
58	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
59	C₂²×D₁₀₉	—	2	12	no	—
60	C₂²×D₁₀₉	—	2	144	no	—
61	C₂²×D₁₀₉	#41	1	8	yes	direct product factor
62	C₂²×D₁₀₉	C₂⁴	1	48	yes	—
63	C₂²×D₁₀₉	C₂⁴	1	18	yes	—
64	C₂²×D₁₀₉	C₂⁴	1	6	yes	—
65	Q₈×C₁₀₉	C₂⁴	1	16	yes	complemented
66	D₄×C₁₀₉	C₂⁴	1	48	yes	complemented
67	C₂×C₄×C₁₀₉	C₂⁴	1	6	yes	—
68	C₂×C₄×C₁₀₉	C₂⁴	1	18	yes	—
69	C₂×C₄×C₁₀₉	C₂⁴	1	48	yes	complemented
70	C₂²⋊D₁₀₉	C₂⁴	1	96	yes	complemented
71	C₂×Dic₁₀₉	C₂⁴	1	6	yes	—
72	C₂×Dic₁₀₉	C₂⁴	1	18	yes	—
73	C₂×Dic₁₀₉	C₂⁴	1	48	yes	complemented
74	D₄₃₆	C₂⁴	1	48	yes	complemented
75	C₄×D₁₀₉	C₂⁴	1	16	yes	—
76	C₄×D₁₀₉	C₂⁴	1	48	yes	—
77	C₄×D₁₀₉	C₂⁴	1	48	yes	complemented
78	C₄×D₁₀₉	C₂⁴	1	48	yes	complemented
79	Dic₂₁₈	C₂⁴	1	48	yes	complemented
80	C₂²×C₁₀₉	—	2	36	no	—
81	C₂²×C₁₀₉	C₂×#41	1	4	yes	complemented
82	C₂²×C₁₀₉	C₂⁵	1	12	yes	—
83	C₂²×C₁₀₉	C₂⁵	1	3	yes	—
84	C₂×D₁₀₉	—	2	36	no	—
85	C₂×D₁₀₉	—	2	96	no	—
86	C₂×D₁₀₉	C₂×#41	1	24	yes	direct product factor
87	C₂×D₁₀₉	C₂⁵	1	12	yes	—
88	C₂×D₁₀₉	C₂⁵	1	4	yes	—
89	C₄×C₁₀₉	C₂⁵	1	12	yes	—
90	C₄×C₁₀₉	C₂⁵	1	4	yes	—
91	Dic₁₀₉	C₂⁵	1	12	yes	—
92	Dic₁₀₉	C₂⁵	1	4	yes	—
93	C₂×C₁₀₉	—	2	12	no	—
94	C₂×C₁₀₉	C₂²×#41	1	6	yes	complemented
95	C₂×C₁₀₉	C₂⁶	1	1	yes	characteristic
96	D₁₀₉	—	2	12	no	—
97	D₁₀₉	C₂²×#41	1	8	yes	direct product factor
98	C₂³×#41	—	109	1	no	maximal, Sylow
99	C₁₀₉	C₂³×#41	1	1	yes	characteristic, complemented, Sylow
100	C₂²×#41	—	109	48	no	—
101	C₂²×#41	—	109	1	no	—
102	C₂²×#41	—	109	3	no	—
103	C₂²×#41	—	109	3	no	—
104	C₂²×#41	D₁₀₉	1	1	yes	characteristic, direct product factor
105	C₂³×Q₈	—	109	1	no	—
106	C₂³×D₄	—	109	3	no	—
107	C₂⁴×C₄	—	109	3	no	—
108	C₂⁵	—	109	3	no	—
109	C₂×#41	—	109	256	no	—
110	C₂×#41	—	109	24	no	—
111	C₂×#41	—	109	72	no	—
112	C₂×#41	—	109	72	no	—
113	C₂×#41	C₂×D₁₀₉	1	24	yes	direct product factor
114	C₂²×Q₈	—	109	24	no	—
115	C₂²×Q₈	—	109	3	no	—
116	C₂²×Q₈	C₂×D₁₀₉	1	1	yes	characteristic, complemented
117	C₂²×D₄	—	109	6	no	—
118	C₂²×D₄	—	109	72	no	—
119	C₂²×D₄	—	109	3	no	—
120	C₂²×D₄	C₂×D₁₀₉	1	3	yes	complemented
121	C₂³×C₄	—	109	3	no	—
122	C₂³×C₄	—	109	3	no	—
123	C₂³×C₄	—	109	3	no	—
124	C₂³×C₄	—	109	1	no	—
125	C₂³×C₄	—	109	3	no	—
126	C₂³×C₄	—	109	72	no	—
127	C₂³×C₄	C₂×D₁₀₉	1	3	yes	complemented
128	C₂⁴	—	109	1	no	—
129	C₂⁴	—	109	36	no	—
130	C₂⁴	—	109	3	no	—
131	C₂⁴	—	218	24	no	—
132	C₂⁴	C₂²×D₁₀₉	1	3	yes	—
133	#41	—	109	64	no	—
134	#41	—	109	192	no	—
135	#41	—	109	192	no	—
136	#41	C₂²×D₁₀₉	1	64	yes	direct product factor
137	C₂×Q₈	—	109	64	no	—
138	C₂×Q₈	—	109	36	no	—
139	C₂×Q₈	C₂²×D₁₀₉	1	12	yes	complemented
140	C₂×D₄	—	109	72	no	—
141	C₂×D₄	—	109	192	no	—
142	C₂×D₄	—	109	36	no	—
143	C₂×D₄	C₂²×D₁₀₉	1	36	yes	complemented
144	C₂²×C₄	—	109	1	no	—
145	C₂²×C₄	—	109	3	no	—
146	C₂²×C₄	—	109	36	no	—
147	C₂²×C₄	—	109	12	no	—
148	C₂²×C₄	—	109	36	no	—
149	C₂²×C₄	—	109	36	no	—
150	C₂²×C₄	—	109	36	no	—
151	C₂²×C₄	—	109	192	no	—
152	C₂²×C₄	C₂²×D₁₀₉	1	3	yes	—
153	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
154	C₂²×C₄	C₂²×D₁₀₉	1	36	yes	complemented
155	C₂³	—	2	12	no	—
156	C₂³	—	109	8	no	—
157	C₂³	—	109	6	no	—
158	C₂³	—	109	48	no	—
159	C₂³	—	109	18	no	—
160	C₂³	—	218	144	no	—
161	C₂³	—	218	12	no	—
162	C₂³	C₂³×D₁₀₉	1	18	yes	—
163	C₂³	C₂³×D₁₀₉	1	1	yes	characteristic
164	Q₈	—	109	48	no	—
165	Q₈	C₂³×D₁₀₉	1	16	yes	complemented
166	D₄	—	109	96	no	—
167	D₄	—	109	48	no	—
168	D₄	C₂³×D₁₀₉	1	48	yes	complemented
169	C₂×C₄	—	109	6	no	—
170	C₂×C₄	—	109	18	no	—
171	C₂×C₄	—	109	48	no	—
172	C₂×C₄	—	109	16	no	—
173	C₂×C₄	—	109	48	no	—
174	C₂×C₄	—	109	48	no	—
175	C₂×C₄	—	109	48	no	—
176	C₂×C₄	C₂³×D₁₀₉	1	6	yes	—
177	C₂×C₄	C₂³×D₁₀₉	1	18	yes	—
178	C₂×C₄	C₂³×D₁₀₉	1	48	yes	complemented
179	C₂²	—	2	36	no	—
180	C₂²	—	109	24	no	—
181	C₂²	—	109	4	no	—
182	C₂²	—	109	12	no	—
183	C₂²	—	218	96	no	—
184	C₂²	—	218	36	no	—
185	C₂²	#41×D₁₀₉	1	4	yes	direct product factor
186	C₂²	C₂⁴×D₁₀₉	1	12	yes	—
187	C₂²	C₂⁴×D₁₀₉	1	3	yes	—
188	C₄	—	109	4	no	—
189	C₄	—	109	12	no	—
190	C₄	C₂⁴×D₁₀₉	1	4	yes	—
191	C₄	C₂⁴×D₁₀₉	1	12	yes	—
192	C₂	—	2	12	no	—
193	C₂	—	109	8	no	—
194	C₂	—	218	12	no	—
195	C₂	C₂×#41×D₁₀₉	1	6	yes	direct product factor
196	C₂	C₂⁵×D₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₀₉
commutator, cosocle	C₂×C₁₀₉	C₂⁶
Frattini	C₂	C₂⁵×D₁₀₉
Fitting	C₂²×#41×C₁₀₉	C₂
socle	C₂³×C₁₀₉	C₂⁴
hypercenter	C₂²×#41	D₁₀₉
hypocenter	C₁₀₉	C₂³×#41
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#41

Complements

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

N	H ≅ G/N	type
1. C₂²×#13182030	C₂	semidirect
2. C₂²×#13182029	C₂	semidirect
3. C₂²×#13182028	C₂	semidirect
4. C₂×#41×D₁₀₉	C₂	direct (N × H)
5. C₂²×Q₈×D₁₀₉	C₂	semidirect
6. C₂²×D₄×D₁₀₉	C₂	semidirect
7. C₂³×C₄×D₁₀₉	C₂	semidirect
8. C₂²×#41×C₁₀₉	C₂	semidirect
10. C₂×#13182030	C₂²	semidirect
11. C₂×#13182029	C₂²	semidirect
12. C₂×#13182028	C₂²	semidirect
13. #41×D₁₀₉	C₂²	direct (N × H)
14. C₂²×Dic₂₁₈	C₂²	semidirect
15. C₂×Q₈×D₁₀₉	C₂²	semidirect
16. C₂²×D₄₃₆	C₂²	semidirect
17. C₂²×C₂²⋊D₁₀₉	C₂²	semidirect
18. C₂×D₄×D₁₀₉	C₂²	semidirect
19. C₂³×Dic₁₀₉	C₂²	semidirect
22. C₂²×C₄×D₁₀₉	C₂²	semidirect
23. C₂²×C₄×D₁₀₉	C₂²	semidirect
24. C₂²×C₄×D₁₀₉	C₂²	semidirect
25. C₂×#41×C₁₀₉	C₂²	semidirect
26. C₂²×Q₈×C₁₀₉	C₂²	semidirect
27. C₂²×D₄×C₁₀₉	C₂²	semidirect
28. C₂³×C₄×C₁₀₉	C₂²	semidirect
33. #13182030	C₂³	semidirect
34. #13182029	C₂³	semidirect
35. #13182028	C₂³	semidirect
36. C₂×Dic₂₁₈	C₂³	semidirect
37. Q₈×D₁₀₉	C₂³	semidirect
38. C₂×D₄₃₆	C₂³	semidirect
39. C₂×C₂²⋊D₁₀₉	C₂³	semidirect
40. D₄×D₁₀₉	C₂³	semidirect
43. C₂²×Dic₁₀₉	C₂³	semidirect
46. C₂×C₄×D₁₀₉	C₂³	semidirect
47. C₂×C₄×D₁₀₉	C₂³	semidirect
48. C₂×C₄×D₁₀₉	C₂³	semidirect
50. #41×C₁₀₉	C₂³	semidirect
51. C₂×Q₈×C₁₀₉	C₂³	semidirect
52. C₂×D₄×C₁₀₉	C₂³	semidirect
55. C₂²×C₄×C₁₀₉	C₂³	semidirect
61. C₂²×D₁₀₉	#41	direct (N × H)
65. Q₈×C₁₀₉	C₂⁴	semidirect
66. D₄×C₁₀₉	C₂⁴	semidirect
69. C₂×C₄×C₁₀₉	C₂⁴	semidirect
70. C₂²⋊D₁₀₉	C₂⁴	semidirect
73. C₂×Dic₁₀₉	C₂⁴	semidirect
74. D₄₃₆	C₂⁴	semidirect
77. C₄×D₁₀₉	C₂⁴	semidirect
78. C₄×D₁₀₉	C₂⁴	semidirect
79. Dic₂₁₈	C₂⁴	semidirect
81. C₂²×C₁₀₉	C₂×#41	semidirect
86. C₂×D₁₀₉	C₂×#41	direct (N × H)
94. C₂×C₁₀₉	C₂²×#41	semidirect
97. D₁₀₉	C₂²×#41	direct (N × H)
99. C₁₀₉	C₂³×#41	semidirect
104. C₂²×#41	D₁₀₉	direct (N × H)
113. C₂×#41	C₂×D₁₀₉	direct (N × H)
116. C₂²×Q₈	C₂×D₁₀₉	semidirect
120. C₂²×D₄	C₂×D₁₀₉	semidirect
127. C₂³×C₄	C₂×D₁₀₉	semidirect
136. #41	C₂²×D₁₀₉	direct (N × H)
139. C₂×Q₈	C₂²×D₁₀₉	semidirect
143. C₂×D₄	C₂²×D₁₀₉	semidirect
154. C₂²×C₄	C₂²×D₁₀₉	semidirect
165. Q₈	C₂³×D₁₀₉	semidirect
168. D₄	C₂³×D₁₀₉	semidirect
178. C₂×C₄	C₂³×D₁₀₉	semidirect
185. C₂²	#41×D₁₀₉	direct (N × H)
195. C₂	C₂×#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
C₂²×#13182030	2
C₂²×#13182029	2
C₂²×#13182028	2
C₂×#41×D₁₀₉	2
C₂²×Q₈×D₁₀₉	2
C₂²×D₄×D₁₀₉	2
C₂³×C₄×D₁₀₉	2
C₂²×#41×C₁₀₉	2
C₂³×#41	109

Subgroups — C22×#41×D109 #28377828

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#41×D₁₀₉ #28377828