Subgroups — C₁₀₉×#2298 #28359552

Order 13952.

524 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#117×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	#388×C₁₀₉	C₂	1	4	yes	maximal
3	#387×C₁₀₉	C₂	1	4	yes	maximal, complemented
4	C₄×C₄⋊C₄×C₁₀₉	C₂	1	2	yes	maximal, complemented
5	C₄×C₂²⋊C₄×C₁₀₉	C₂	1	2	yes	maximal
6	#376×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄³×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
9	#117×C₁₀₉	C₄	1	8	yes	complemented
10	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	4	yes	—
11	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
12	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	4	yes	—
13	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₀₉	C₂²	1	4	yes	—
15	C₂×C₄²×C₁₀₉	C₂²	1	2	yes	—
16	C₂×C₄²×C₁₀₉	C₂²	1	2	yes	—
17	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
18	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
19	#95×C₁₀₉	C₂²	1	4	yes	—
20	#95×C₁₀₉	C₂²	1	4	yes	—
21	#95×C₁₀₉	C₂²	1	4	yes	—
22	C₂⁴×C₁₀₉	C₂³	1	1	yes	characteristic
23	C₂²×C₄×C₁₀₉	—	2	4	no	—
24	C₂²×C₄×C₁₀₉	C₂×C₄	1	4	yes	—
25	C₂²×C₄×C₁₀₉	C₂³	1	4	yes	—
26	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
27	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
28	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
29	C₂²×C₄×C₁₀₉	C₂³	1	4	yes	—
30	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
31	C₄⋊C₄×C₁₀₉	—	2	8	no	—
32	C₄⋊C₄×C₁₀₉	C₂×C₄	1	8	yes	—
33	C₂²⋊C₄×C₁₀₉	—	2	8	no	—
34	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	8	yes	—
35	C₄²×C₁₀₉	—	2	8	no	—
36	C₄²×C₁₀₉	—	2	4	no	—
37	C₄²×C₁₀₉	—	2	4	no	—
38	C₄²×C₁₀₉	C₂×C₄	1	4	yes	—
39	C₄²×C₁₀₉	C₂×C₄	1	4	yes	—
40	C₂³×C₁₀₉	—	2	4	no	—
41	C₂³×C₁₀₉	—	2	2	no	—
42	C₂³×C₁₀₉	C₂²×C₄	1	2	yes	—
43	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
44	C₂×C₄×C₁₀₉	—	2	4	no	—
45	C₂×C₄×C₁₀₉	—	2	4	no	—
46	C₂×C₄×C₁₀₉	—	2	4	no	—
47	C₂×C₄×C₁₀₉	—	2	4	no	—
48	C₂×C₄×C₁₀₉	—	2	2	no	—
49	C₂×C₄×C₁₀₉	—	2	8	no	—
50	C₂×C₄×C₁₀₉	—	2	4	no	—
51	C₂×C₄×C₁₀₉	—	2	4	no	—
52	C₂×C₄×C₁₀₉	—	2	4	no	—
53	C₂×C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
54	C₂×C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
55	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
56	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₁₀₉	#41	1	8	yes	—
58	C₂×C₄×C₁₀₉	#41	1	4	yes	—
59	C₂²×C₁₀₉	—	2	2	no	—
60	C₂²×C₁₀₉	—	2	4	no	—
61	C₂²×C₁₀₉	—	4	4	no	—
62	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
63	C₂²×C₁₀₉	C₂×#41	1	2	yes	—
64	C₂²×C₁₀₉	C₂×#41	1	4	yes	—
65	C₄×C₁₀₉	—	2	2	no	—
66	C₄×C₁₀₉	—	2	4	no	—
67	C₄×C₁₀₉	—	2	4	no	—
68	C₄×C₁₀₉	—	2	8	no	—
69	C₄×C₁₀₉	—	4	4	no	—
70	C₄×C₁₀₉	#117	1	4	yes	—
71	C₂×C₁₀₉	—	4	2	no	—
72	C₂×C₁₀₉	C₂×#117	1	1	yes	characteristic
73	C₂×C₁₀₉	C₄×#41	1	2	yes	—
74	C₂×C₁₀₉	#529	1	4	yes	—
75	#2298	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₁₀₉	#2298	1	1	yes	characteristic, direct product factor, Sylow
77	C₂×#117	C₂×C₁₀₉	1	1	yes	characteristic
78	#388	C₂×C₁₀₉	1	4	yes	—
79	#387	C₂×C₁₀₉	1	4	yes	complemented
80	C₄×C₄⋊C₄	C₂×C₁₀₉	1	2	yes	complemented
81	C₄×C₂²⋊C₄	C₂×C₁₀₉	1	2	yes	—
82	#376	C₂×C₁₀₉	1	1	yes	characteristic, complemented
83	C₄³	C₂×C₁₀₉	1	1	yes	characteristic, complemented
84	C₂³×C₄	C₂²×C₁₀₉	1	1	yes	characteristic
85	#117	C₄×C₁₀₉	1	8	yes	complemented
86	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	4	yes	—
87	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
88	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	4	yes	—
89	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	2	yes	—
90	C₂×C₄²	C₂²×C₁₀₉	1	2	yes	—
91	C₂×C₄²	C₂²×C₁₀₉	1	4	yes	—
92	C₂×C₄²	C₂²×C₁₀₉	1	2	yes	—
93	C₂×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
94	C₂×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
95	#95	C₂²×C₁₀₉	1	4	yes	—
96	#95	C₂²×C₁₀₉	1	4	yes	—
97	#95	C₂²×C₁₀₉	1	4	yes	—
98	C₂⁴	C₂³×C₁₀₉	1	1	yes	characteristic
99	C₂²×C₄	—	2	4	no	—
100	C₂²×C₄	C₂×C₄×C₁₀₉	1	4	yes	—
101	C₂²×C₄	C₂³×C₁₀₉	1	4	yes	—
102	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
103	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
104	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
105	C₂²×C₄	C₂³×C₁₀₉	1	4	yes	—
106	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
107	C₄⋊C₄	—	2	8	no	—
108	C₄⋊C₄	C₂×C₄×C₁₀₉	1	8	yes	—
109	C₂²⋊C₄	—	2	8	no	—
110	C₂²⋊C₄	C₂×C₄×C₁₀₉	1	8	yes	—
111	C₄²	—	2	8	no	—
112	C₄²	—	2	4	no	—
113	C₄²	—	2	4	no	—
114	C₄²	C₂×C₄×C₁₀₉	1	4	yes	—
115	C₄²	C₂×C₄×C₁₀₉	1	4	yes	—
116	C₂³	—	2	4	no	—
117	C₂³	—	2	2	no	—
118	C₂³	C₂²×C₄×C₁₀₉	1	2	yes	—
119	C₂³	C₂⁴×C₁₀₉	1	1	yes	characteristic
120	C₂×C₄	—	2	4	no	—
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	—	2	4	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	8	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	C₂²×C₄×C₁₀₉	1	4	yes	—
130	C₂×C₄	C₂²×C₄×C₁₀₉	1	4	yes	—
131	C₂×C₄	C₂²×C₄×C₁₀₉	1	2	yes	—
132	C₂×C₄	C₂²×C₄×C₁₀₉	1	2	yes	—
133	C₂×C₄	#41×C₁₀₉	1	8	yes	—
134	C₂×C₄	#41×C₁₀₉	1	4	yes	—
135	C₂²	—	2	2	no	—
136	C₂²	—	2	4	no	—
137	C₂²	—	4	4	no	—
138	C₂²	C₂³×C₄×C₁₀₉	1	1	yes	characteristic
139	C₂²	C₂×#41×C₁₀₉	1	2	yes	—
140	C₂²	C₂×#41×C₁₀₉	1	4	yes	—
141	C₄	—	2	2	no	—
142	C₄	—	2	4	no	—
143	C₄	—	2	4	no	—
144	C₄	—	2	8	no	—
145	C₄	—	4	4	no	—
146	C₄	#117×C₁₀₉	1	4	yes	—
147	C₂	—	4	2	no	—
148	C₂	C₂×#117×C₁₀₉	1	1	yes	characteristic
149	C₂	C₄×#41×C₁₀₉	1	2	yes	—
150	C₂	#529×C₁₀₉	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₀₉	C₂³
commutator	C₂²	C₂³×C₄×C₁₀₉
Frattini, cosocle	C₂³	C₂⁴×C₁₀₉
socle	C₂³×C₁₀₉	C₂⁴
prime radical	C₂⁴×C₁₀₉	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. #387×C₁₀₉	C₂	semidirect
4. C₄×C₄⋊C₄×C₁₀₉	C₂	semidirect
6. #376×C₁₀₉	C₂	semidirect
7. C₄³×C₁₀₉	C₂	semidirect
9. #117×C₁₀₉	C₄	semidirect
75. #2298	C₁₀₉	direct (N × H)
76. C₁₀₉	#2298	direct (N × H)
79. #387	C₂×C₁₀₉	semidirect
80. C₄×C₄⋊C₄	C₂×C₁₀₉	semidirect
82. #376	C₂×C₁₀₉	semidirect
83. C₄³	C₂×C₁₀₉	semidirect
85. #117	C₄×C₁₀₉	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#117×C₁₀₉	2
#388×C₁₀₉	2
#387×C₁₀₉	2
C₄×C₄⋊C₄×C₁₀₉	2
C₄×C₂²⋊C₄×C₁₀₉	2
#376×C₁₀₉	2
C₄³×C₁₀₉	2
#2298	109

Subgroups — C109×#2298 #28359552

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#2298 #28359552