Subgroups — C₁₀₉×#3164 #28360418

Order 13952.

340 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#544×C₁₀₉	C₂	1	2	yes	maximal
2	#501×C₁₀₉	C₂	1	4	yes	maximal, complemented
3	#500×C₁₀₉	C₂	1	2	yes	maximal, complemented
4	#499×C₁₀₉	C₂	1	2	yes	maximal, complemented
5	#428×C₁₀₉	C₂	1	2	yes	maximal
6	#427×C₁₀₉	C₂	1	1	yes	maximal, characteristic
7	C₂×#107×C₁₀₉	C₂	1	1	yes	maximal, characteristic
8	#405×C₁₀₉	C₂	1	1	yes	maximal, characteristic
9	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₄⋊Q₈×C₁₀₉	—	2	2	no	—
12	C₄⋊Q₈×C₁₀₉	C₂²	1	4	yes	—
13	#125×C₁₀₉	—	2	2	no	—
14	#125×C₁₀₉	C₂²	1	4	yes	—
15	C₂²⋊Q₈×C₁₀₉	—	2	4	no	—
16	#117×C₁₀₉	C₂²	1	2	yes	—
17	#117×C₁₀₉	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
19	#107×C₁₀₉	C₂²	1	1	yes	characteristic
20	#107×C₁₀₉	C₂²	1	2	yes	—
21	#107×C₁₀₉	C₂²	1	1	yes	characteristic
22	#107×C₁₀₉	C₂²	1	4	yes	—
23	#107×C₁₀₉	C₂²	1	2	yes	—
24	#106×C₁₀₉	C₂²	1	4	yes	—
25	#106×C₁₀₉	C₂²	1	2	yes	—
26	#97×C₁₀₉	C₂²	1	2	yes	—
27	C₄×C₈×C₁₀₉	C₂²	1	2	yes	—
28	C₂×Q₈×C₁₀₉	—	2	4	no	—
29	C₂²×C₄×C₁₀₉	—	2	2	no	—
30	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
31	M₁₆×C₁₀₉	Q₈	1	4	yes	complemented
32	C₂×C₈×C₁₀₉	Q₈	1	4	yes	complemented
33	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
34	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
35	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
36	C₄⋊C₄×C₁₀₉	—	2	2	no	—
37	C₄⋊C₄×C₁₀₉	—	2	4	no	—
38	C₄⋊C₄×C₁₀₉	—	2	4	no	—
39	C₄⋊C₄×C₁₀₉	—	2	4	no	—
40	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
41	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
42	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
43	C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
44	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
45	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
46	C₂²⋊C₄×C₁₀₉	D₄	1	2	yes	—
47	C₄²×C₁₀₉	—	2	2	no	—
48	C₄²×C₁₀₉	C₂³	1	2	yes	—
49	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
50	Q₈×C₁₀₉	—	8	4	no	—
51	C₂×C₄×C₁₀₉	—	2	2	no	—
52	C₂×C₄×C₁₀₉	—	2	2	no	—
53	C₂×C₄×C₁₀₉	—	2	4	no	—
54	C₂×C₄×C₁₀₉	—	8	2	no	—
55	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
59	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
60	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
61	C₈×C₁₀₉	C₂×Q₈	1	2	yes	—
62	C₈×C₁₀₉	C₂×Q₈	1	2	yes	—
63	C₈×C₁₀₉	C₂×Q₈	1	4	yes	—
64	C₂²×C₁₀₉	—	2	1	no	—
65	C₂²×C₁₀₉	—	2	1	no	—
66	C₂²×C₁₀₉	C₄⋊Q₈	1	2	yes	—
67	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
68	C₄×C₁₀₉	—	2	2	no	—
69	C₄×C₁₀₉	—	2	2	no	—
70	C₄×C₁₀₉	—	4	4	no	—
71	C₄×C₁₀₉	—	4	2	no	—
72	C₄×C₁₀₉	—	4	2	no	—
73	C₄×C₁₀₉	C₄⋊Q₈	1	2	yes	—
74	C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
75	C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
76	C₂×C₁₀₉	—	2	2	no	—
77	C₂×C₁₀₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
78	C₂×C₁₀₉	#576	1	1	yes	characteristic
79	C₂×C₁₀₉	#578	1	1	yes	characteristic
80	#3164	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₁₀₉	#3164	1	1	yes	characteristic, direct product factor, Sylow
82	#544	C₂×C₁₀₉	1	2	yes	—
83	#501	C₂×C₁₀₉	1	4	yes	complemented
84	#500	C₂×C₁₀₉	1	2	yes	complemented
85	#499	C₂×C₁₀₉	1	2	yes	complemented
86	#428	C₂×C₁₀₉	1	2	yes	—
87	#427	C₂×C₁₀₉	1	1	yes	characteristic
88	C₂×#107	C₂×C₁₀₉	1	1	yes	characteristic
89	#405	C₂×C₁₀₉	1	1	yes	characteristic
90	C₂×M₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
91	C₂²×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
92	C₄⋊Q₈	—	2	2	no	—
93	C₄⋊Q₈	C₂²×C₁₀₉	1	4	yes	—
94	#125	—	2	2	no	—
95	#125	C₂²×C₁₀₉	1	4	yes	—
96	C₂²⋊Q₈	—	2	4	no	—
97	#117	C₂²×C₁₀₉	1	2	yes	—
98	#117	C₂²×C₁₀₉	1	1	yes	characteristic
99	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
100	#107	C₂²×C₁₀₉	1	2	yes	—
101	#107	C₂²×C₁₀₉	1	2	yes	—
102	#107	C₂²×C₁₀₉	1	1	yes	characteristic
103	#107	C₂²×C₁₀₉	1	1	yes	characteristic
104	#107	C₂²×C₁₀₉	1	4	yes	—
105	#106	C₂²×C₁₀₉	1	2	yes	—
106	#106	C₂²×C₁₀₉	1	4	yes	—
107	#97	C₂²×C₁₀₉	1	2	yes	—
108	C₄×C₈	C₂²×C₁₀₉	1	2	yes	—
109	C₂×Q₈	—	2	4	no	—
110	C₂²×C₄	—	2	2	no	—
111	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
112	M₁₆	Q₈×C₁₀₉	1	4	yes	complemented
113	C₂×C₈	Q₈×C₁₀₉	1	4	yes	complemented
114	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
115	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
116	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
117	C₄⋊C₄	—	2	2	no	—
118	C₄⋊C₄	—	2	4	no	—
119	C₄⋊C₄	—	2	4	no	—
120	C₄⋊C₄	—	2	4	no	—
121	C₄⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
122	C₄⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
123	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
124	C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
125	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
126	C₂²⋊C₄	—	2	2	no	—
127	C₂²⋊C₄	D₄×C₁₀₉	1	2	yes	—
128	C₄²	—	2	2	no	—
129	C₄²	C₂³×C₁₀₉	1	2	yes	—
130	C₂³	C₂×D₄×C₁₀₉	1	1	yes	characteristic
131	Q₈	—	8	4	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	4	no	—
135	C₂×C₄	—	8	2	no	—
136	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
137	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
138	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
139	C₂×C₄	C₂×Q₈×C₁₀₉	1	2	yes	—
140	C₂×C₄	C₂×Q₈×C₁₀₉	1	2	yes	—
141	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
142	C₈	C₂×Q₈×C₁₀₉	1	2	yes	—
143	C₈	C₂×Q₈×C₁₀₉	1	2	yes	—
144	C₈	C₂×Q₈×C₁₀₉	1	4	yes	—
145	C₂²	—	2	1	no	—
146	C₂²	—	2	1	no	—
147	C₂²	C₄⋊Q₈×C₁₀₉	1	2	yes	—
148	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	—	2	2	no	—
151	C₄	—	4	4	no	—
152	C₄	—	4	2	no	—
153	C₄	—	4	2	no	—
154	C₄	C₄⋊Q₈×C₁₀₉	1	2	yes	—
155	C₄	C₂²×Q₈×C₁₀₉	1	1	yes	characteristic
156	C₄	C₂²×Q₈×C₁₀₉	1	1	yes	characteristic
157	C₂	—	2	2	no	—
158	C₂	C₂×C₄⋊Q₈×C₁₀₉	1	1	yes	characteristic
159	C₂	#576×C₁₀₉	1	1	yes	characteristic
160	C₂	#578×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₀₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₀₉
prime radical	C₂³×C₁₀₉	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ #117×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. #501×C₁₀₉	C₂	semidirect
3. #500×C₁₀₉	C₂	semidirect
4. #499×C₁₀₉	C₂	semidirect
31. M₁₆×C₁₀₉	Q₈	semidirect
32. C₂×C₈×C₁₀₉	Q₈	semidirect
80. #3164	C₁₀₉	direct (N × H)
81. C₁₀₉	#3164	direct (N × H)
83. #501	C₂×C₁₀₉	semidirect
84. #500	C₂×C₁₀₉	semidirect
85. #499	C₂×C₁₀₉	semidirect
112. M₁₆	Q₈×C₁₀₉	semidirect
113. C₂×C₈	Q₈×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#544×C₁₀₉	2
#501×C₁₀₉	2
#500×C₁₀₉	2
#499×C₁₀₉	2
#428×C₁₀₉	2
#427×C₁₀₉	2
C₂×#107×C₁₀₉	2
#405×C₁₀₉	2
#3164	109

Subgroups — C109×#3164 #28360418

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#3164 #28360418