Subgroups — C₂²×#121×C₁₀₉ #28360686

Order 13952.

2528 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, socle	C₂⁴×C₁₀₉	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁵×C₁₀₉
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×D₄×C₁₀₉	C₂	semidirect
2. C₂³×D₄×C₁₀₉	C₂	semidirect
3. C₂⁴×C₄×C₁₀₉	C₂	semidirect
4. C₂×#121×C₁₀₉	C₂	direct (N × H)
5. C₂²×C₄⋊C₄×C₁₀₉	C₂	semidirect
6. C₂²×C₂²⋊C₄×C₁₀₉	C₂	semidirect
11. C₂²×D₄×C₁₀₉	C₂²	semidirect
12. C₂²×D₄×C₁₀₉	C₂²	semidirect
17. C₂³×C₄×C₁₀₉	C₂²	semidirect
18. #121×C₁₀₉	C₂²	direct (N × H)
19. C₂×C₄⋊C₄×C₁₀₉	C₂²	semidirect
20. C₂×C₂²⋊C₄×C₁₀₉	C₂²	semidirect
26. C₂⁴×C₁₀₉	D₄	semidirect
32. C₂×D₄×C₁₀₉	C₂³	semidirect
33. C₂×D₄×C₁₀₉	C₂³	semidirect
37. C₂²×C₄×C₁₀₉	D₄	semidirect
41. C₂²×C₄×C₁₀₉	C₂³	semidirect
42. C₄⋊C₄×C₁₀₉	C₂³	semidirect
43. C₂²⋊C₄×C₁₀₉	C₂³	semidirect
53. C₂³×C₁₀₉	C₂×D₄	semidirect
63. C₂×C₄×C₁₀₉	C₂×D₄	semidirect
74. C₂²×C₁₀₉	#121	direct (N × H)
77. C₂²×C₁₀₉	C₂²×D₄	semidirect
82. C₄×C₁₀₉	C₂²×D₄	semidirect
85. C₂×C₁₀₉	C₂×#121	direct (N × H)
89. C₂²×#121	C₁₀₉	direct (N × H)
90. C₁₀₉	C₂²×#121	direct (N × H)
91. C₂³×D₄	C₂×C₁₀₉	semidirect
92. C₂³×D₄	C₂×C₁₀₉	semidirect
93. C₂⁴×C₄	C₂×C₁₀₉	semidirect
94. C₂×#121	C₂×C₁₀₉	direct (N × H)
95. C₂²×C₄⋊C₄	C₂×C₁₀₉	semidirect
96. C₂²×C₂²⋊C₄	C₂×C₁₀₉	semidirect
101. C₂²×D₄	C₂²×C₁₀₉	semidirect
102. C₂²×D₄	C₂²×C₁₀₉	semidirect
107. C₂³×C₄	C₂²×C₁₀₉	semidirect
108. #121	C₂²×C₁₀₉	direct (N × H)
109. C₂×C₄⋊C₄	C₂²×C₁₀₉	semidirect
110. C₂×C₂²⋊C₄	C₂²×C₁₀₉	semidirect
116. C₂⁴	D₄×C₁₀₉	semidirect
122. C₂×D₄	C₂³×C₁₀₉	semidirect
123. C₂×D₄	C₂³×C₁₀₉	semidirect
127. C₂²×C₄	D₄×C₁₀₉	semidirect
131. C₂²×C₄	C₂³×C₁₀₉	semidirect
132. C₄⋊C₄	C₂³×C₁₀₉	semidirect
133. C₂²⋊C₄	C₂³×C₁₀₉	semidirect
143. C₂³	C₂×D₄×C₁₀₉	semidirect
153. C₂×C₄	C₂×D₄×C₁₀₉	semidirect
164. C₂²	#121×C₁₀₉	direct (N × H)
167. C₂²	C₂²×D₄×C₁₀₉	semidirect
172. C₄	C₂²×D₄×C₁₀₉	semidirect
175. C₂	C₂×#121×C₁₀₉	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×D₄×C₁₀₉	2
C₂⁴×C₄×C₁₀₉	2
C₂×#121×C₁₀₉	2
C₂²×C₄⋊C₄×C₁₀₉	2
C₂²×C₂²⋊C₄×C₁₀₉	2
C₂²×#121	109

Subgroups — C22×#121×C109 #28360686

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#121×C₁₀₉ #28360686