Subgroups — C₂²×#384×C₁₀₉ #42152004

Order 27904.

5502 conjugacy classes of subgroups.

The subgroup lattice has 126 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄⋊C₄×C₁₀₉	C₂	1	6	yes	maximal
2	C₂³×C₄²×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#384×C₁₀₉	C₂	1	24	yes	maximal, direct product factor
4	C₂⁴×C₄×C₁₀₉	C₂²	1	4	yes	—
5	C₂⁴×C₄×C₁₀₉	C₂²	1	3	yes	—
6	C₂²×C₄⋊C₄×C₁₀₉	—	2	48	no	—
7	C₂²×C₄⋊C₄×C₁₀₉	C₂²	1	72	yes	—
8	C₂²×C₄²×C₁₀₉	C₄	1	16	yes	complemented
9	C₂²×C₄²×C₁₀₉	C₂²	1	12	yes	—
10	#384×C₁₀₉	C₂²	1	64	yes	direct product factor
11	C₂⁵×C₁₀₉	C₂³	1	1	yes	characteristic
12	C₂³×C₄×C₁₀₉	—	2	48	no	—
13	C₂³×C₄×C₁₀₉	C₂×C₄	1	24	yes	—
14	C₂³×C₄×C₁₀₉	D₄	1	24	yes	—
15	C₂³×C₄×C₁₀₉	Q₈	1	24	yes	—
16	C₂³×C₄×C₁₀₉	C₂³	1	24	yes	—
17	C₂³×C₄×C₁₀₉	C₂³	1	18	yes	—
18	C₂×C₄⋊C₄×C₁₀₉	—	2	288	no	—
19	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	96	yes	—
20	C₂×C₄²×C₁₀₉	C₂×C₄	1	96	yes	complemented
21	C₂×C₄²×C₁₀₉	C₂³	1	16	yes	—
22	C₂⁴×C₁₀₉	C₂²×C₄	1	4	yes	—
23	C₂⁴×C₁₀₉	C₂×D₄	1	12	yes	—
24	C₂⁴×C₁₀₉	C₂×Q₈	1	12	yes	—
25	C₂⁴×C₁₀₉	C₂⁴	1	3	yes	—
26	C₂²×C₄×C₁₀₉	—	2	144	no	—
27	C₂²×C₄×C₁₀₉	—	4	64	no	—
28	C₂²×C₄×C₁₀₉	C₄⋊C₄	1	192	yes	complemented
29	C₂²×C₄×C₁₀₉	C₂²×C₄	1	72	yes	—
30	C₂²×C₄×C₁₀₉	C₂×D₄	1	72	yes	—
31	C₂²×C₄×C₁₀₉	C₂×Q₈	1	72	yes	—
32	C₂²×C₄×C₁₀₉	C₂⁴	1	16	yes	—
33	C₂²×C₄×C₁₀₉	C₂⁴	1	12	yes	—
34	C₄⋊C₄×C₁₀₉	—	2	192	no	—
35	C₄²×C₁₀₉	C₂²×C₄	1	64	yes	complemented
36	C₂³×C₁₀₉	C₂×C₄⋊C₄	1	48	yes	—
37	C₂³×C₁₀₉	#127	1	16	yes	—
38	C₂³×C₁₀₉	C₄⋊Q₈	1	48	yes	—
39	C₂³×C₁₀₉	C₂³×C₄	1	6	yes	—
40	C₂³×C₁₀₉	C₂²×D₄	1	18	yes	—
41	C₂³×C₁₀₉	C₂²×Q₈	1	18	yes	—
42	C₂³×C₁₀₉	C₂⁵	1	1	yes	characteristic
43	C₂×C₄×C₁₀₉	—	2	48	no	—
44	C₂×C₄×C₁₀₉	—	4	96	no	—
45	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	288	yes	complemented
46	C₂×C₄×C₁₀₉	C₂³×C₄	1	24	yes	—
47	C₂×C₄×C₁₀₉	C₂²×D₄	1	24	yes	—
48	C₂×C₄×C₁₀₉	C₂²×Q₈	1	24	yes	—
49	C₂²×C₁₀₉	#384	1	64	yes	direct product factor
50	C₂²×C₁₀₉	C₂²×C₄⋊C₄	1	36	yes	—
51	C₂²×C₁₀₉	C₂×#127	1	12	yes	—
52	C₂²×C₁₀₉	C₂×C₄⋊Q₈	1	36	yes	—
53	C₂²×C₁₀₉	C₂⁴×C₄	1	1	yes	characteristic
54	C₂²×C₁₀₉	C₂³×D₄	1	3	yes	—
55	C₂²×C₁₀₉	C₂³×Q₈	1	3	yes	—
56	C₄×C₁₀₉	—	4	16	no	—
57	C₄×C₁₀₉	C₂²×C₄⋊C₄	1	48	yes	complemented
58	C₂²×#384	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₂×C₁₀₉	C₂×#384	1	24	yes	direct product factor
60	C₂×C₁₀₉	C₂³×C₄⋊C₄	1	3	yes	—
61	C₂×C₁₀₉	C₂²×#127	1	1	yes	characteristic
62	C₂×C₁₀₉	C₂²×C₄⋊Q₈	1	3	yes	—
63	C₂³×C₄⋊C₄	C₂×C₁₀₉	1	6	yes	—
64	C₂³×C₄²	C₂×C₁₀₉	1	1	yes	characteristic
65	C₂×#384	C₂×C₁₀₉	1	24	yes	direct product factor
66	C₁₀₉	C₂²×#384	1	1	yes	characteristic, direct product factor, Sylow
67	C₂⁴×C₄	C₂²×C₁₀₉	1	4	yes	—
68	C₂⁴×C₄	C₂²×C₁₀₉	1	3	yes	—
69	C₂²×C₄⋊C₄	—	2	48	no	—
70	C₂²×C₄⋊C₄	C₂²×C₁₀₉	1	72	yes	—
71	C₂²×C₄²	C₄×C₁₀₉	1	16	yes	complemented
72	C₂²×C₄²	C₂²×C₁₀₉	1	12	yes	—
73	#384	C₂²×C₁₀₉	1	64	yes	direct product factor
74	C₂⁵	C₂³×C₁₀₉	1	1	yes	characteristic
75	C₂³×C₄	—	2	48	no	—
76	C₂³×C₄	C₂×C₄×C₁₀₉	1	24	yes	—
77	C₂³×C₄	D₄×C₁₀₉	1	24	yes	—
78	C₂³×C₄	Q₈×C₁₀₉	1	24	yes	—
79	C₂³×C₄	C₂³×C₁₀₉	1	24	yes	—
80	C₂³×C₄	C₂³×C₁₀₉	1	18	yes	—
81	C₂×C₄⋊C₄	—	2	288	no	—
82	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	96	yes	—
83	C₂×C₄²	C₂×C₄×C₁₀₉	1	96	yes	complemented
84	C₂×C₄²	C₂³×C₁₀₉	1	16	yes	—
85	C₂⁴	C₂²×C₄×C₁₀₉	1	4	yes	—
86	C₂⁴	C₂×D₄×C₁₀₉	1	12	yes	—
87	C₂⁴	C₂×Q₈×C₁₀₉	1	12	yes	—
88	C₂⁴	C₂⁴×C₁₀₉	1	3	yes	—
89	C₂²×C₄	—	2	144	no	—
90	C₂²×C₄	—	4	64	no	—
91	C₂²×C₄	C₄⋊C₄×C₁₀₉	1	192	yes	complemented
92	C₂²×C₄	C₂²×C₄×C₁₀₉	1	72	yes	—
93	C₂²×C₄	C₂×D₄×C₁₀₉	1	72	yes	—
94	C₂²×C₄	C₂×Q₈×C₁₀₉	1	72	yes	—
95	C₂²×C₄	C₂⁴×C₁₀₉	1	16	yes	—
96	C₂²×C₄	C₂⁴×C₁₀₉	1	12	yes	—
97	C₄⋊C₄	—	2	192	no	—
98	C₄²	C₂²×C₄×C₁₀₉	1	64	yes	complemented
99	C₂³	C₂×C₄⋊C₄×C₁₀₉	1	48	yes	—
100	C₂³	#127×C₁₀₉	1	16	yes	—
101	C₂³	C₄⋊Q₈×C₁₀₉	1	48	yes	—
102	C₂³	C₂³×C₄×C₁₀₉	1	6	yes	—
103	C₂³	C₂²×D₄×C₁₀₉	1	18	yes	—
104	C₂³	C₂²×Q₈×C₁₀₉	1	18	yes	—
105	C₂³	C₂⁵×C₁₀₉	1	1	yes	characteristic
106	C₂×C₄	—	2	48	no	—
107	C₂×C₄	—	4	96	no	—
108	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	288	yes	complemented
109	C₂×C₄	C₂³×C₄×C₁₀₉	1	24	yes	—
110	C₂×C₄	C₂²×D₄×C₁₀₉	1	24	yes	—
111	C₂×C₄	C₂²×Q₈×C₁₀₉	1	24	yes	—
112	C₂²	#384×C₁₀₉	1	64	yes	direct product factor
113	C₂²	C₂²×C₄⋊C₄×C₁₀₉	1	36	yes	—
114	C₂²	C₂×#127×C₁₀₉	1	12	yes	—
115	C₂²	C₂×C₄⋊Q₈×C₁₀₉	1	36	yes	—
116	C₂²	C₂⁴×C₄×C₁₀₉	1	1	yes	characteristic
117	C₂²	C₂³×D₄×C₁₀₉	1	3	yes	—
118	C₂²	C₂³×Q₈×C₁₀₉	1	3	yes	—
119	C₄	—	4	16	no	—
120	C₄	C₂²×C₄⋊C₄×C₁₀₉	1	48	yes	complemented
121	C₂	C₂×#384×C₁₀₉	1	24	yes	direct product factor
122	C₂	C₂³×C₄⋊C₄×C₁₀₉	1	3	yes	—
123	C₂	C₂²×#127×C₁₀₉	1	1	yes	characteristic
124	C₂	C₂²×C₄⋊Q₈×C₁₀₉	1	3	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁵×C₁₀₉	C₂³
commutator	C₂²	C₂⁴×C₄×C₁₀₉
Frattini, cosocle	C₂³	C₂⁵×C₁₀₉
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×#384×C₁₀₉	C₂	direct (N × H)
8. C₂²×C₄²×C₁₀₉	C₄	semidirect
10. #384×C₁₀₉	C₂²	direct (N × H)
20. C₂×C₄²×C₁₀₉	C₂×C₄	semidirect
28. C₂²×C₄×C₁₀₉	C₄⋊C₄	semidirect
35. C₄²×C₁₀₉	C₂²×C₄	semidirect
45. C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	semidirect
49. C₂²×C₁₀₉	#384	direct (N × H)
57. C₄×C₁₀₉	C₂²×C₄⋊C₄	semidirect
58. C₂²×#384	C₁₀₉	direct (N × H)
59. C₂×C₁₀₉	C₂×#384	direct (N × H)
65. C₂×#384	C₂×C₁₀₉	direct (N × H)
66. C₁₀₉	C₂²×#384	direct (N × H)
71. C₂²×C₄²	C₄×C₁₀₉	semidirect
73. #384	C₂²×C₁₀₉	direct (N × H)
83. C₂×C₄²	C₂×C₄×C₁₀₉	semidirect
91. C₂²×C₄	C₄⋊C₄×C₁₀₉	semidirect
98. C₄²	C₂²×C₄×C₁₀₉	semidirect
108. C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	semidirect
112. C₂²	#384×C₁₀₉	direct (N × H)
120. C₄	C₂²×C₄⋊C₄×C₁₀₉	semidirect
121. C₂	C₂×#384×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×C₄⋊C₄×C₁₀₉	2
C₂³×C₄²×C₁₀₉	2
C₂×#384×C₁₀₉	2
C₂²×#384	109

Subgroups — C22×#384×C109 #42152004

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#384×C₁₀₉ #42152004