Subgroups — C₁₀₉×#3390 #28360644

Order 13952.

320 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#557×C₁₀₉	C₂	1	2	yes	maximal
2	#501×C₁₀₉	C₂	1	2	yes	maximal
3	#499×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	#479×C₁₀₉	C₂	1	2	yes	maximal
5	#479×C₁₀₉	C₂	1	2	yes	maximal
6	#477×C₁₀₉	C₂	1	2	yes	maximal
7	#446×C₁₀₉	C₂	1	1	yes	maximal, characteristic
8	#441×C₁₀₉	C₂	1	2	yes	maximal
9	C₄×Q₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
10	C₂×Q₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₄⋊Q₈×C₁₀₉	—	2	2	no	—
12	C₄⋊Q₈×C₁₀₉	C₂²	1	2	yes	—
13	#125×C₁₀₉	—	2	2	no	—
14	#125×C₁₀₉	—	2	2	no	—
15	#125×C₁₀₉	C₂²	1	2	yes	—
16	#125×C₁₀₉	C₂²	1	2	yes	—
17	C₄×Q₈×C₁₀₉	—	2	2	no	—
18	C₄×Q₈×C₁₀₉	C₂²	1	2	yes	—
19	C₄×Q₈×C₁₀₉	C₂²	1	2	yes	—
20	C₄×Q₈×C₁₀₉	C₂²	1	2	yes	—
21	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
22	#107×C₁₀₉	C₂²	1	1	yes	characteristic
23	#107×C₁₀₉	C₂²	1	1	yes	characteristic
24	#107×C₁₀₉	C₂²	1	1	yes	characteristic
25	#107×C₁₀₉	C₂²	1	2	yes	—
26	#106×C₁₀₉	C₂²	1	2	yes	—
27	#106×C₁₀₉	C₂²	1	2	yes	—
28	C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
29	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
30	#103×C₁₀₉	C₂²	1	2	yes	—
31	#103×C₁₀₉	C₂²	1	2	yes	—
32	#103×C₁₀₉	C₂²	1	2	yes	—
33	#97×C₁₀₉	C₂²	1	2	yes	—
34	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
35	C₂×Q₈×C₁₀₉	—	2	2	no	—
36	C₂×Q₈×C₁₀₉	D₄	1	1	yes	characteristic
37	C₂×Q₈×C₁₀₉	C₂³	1	2	yes	—
38	Q₁₆×C₁₀₉	Q₈	1	4	yes	complemented
39	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
40	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
41	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
42	C₄⋊C₄×C₁₀₉	—	2	2	no	—
43	C₄⋊C₄×C₁₀₉	—	2	2	no	—
44	C₄⋊C₄×C₁₀₉	—	2	2	no	—
45	C₄⋊C₄×C₁₀₉	—	2	2	no	—
46	C₄⋊C₄×C₁₀₉	—	2	2	no	—
47	C₄⋊C₄×C₁₀₉	—	2	2	no	—
48	C₄⋊C₄×C₁₀₉	—	2	2	no	—
49	C₄⋊C₄×C₁₀₉	—	2	2	no	—
50	C₄⋊C₄×C₁₀₉	D₄	1	2	yes	—
51	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
52	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
53	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
54	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
55	C₄²×C₁₀₉	—	2	2	no	—
56	C₄²×C₁₀₉	—	2	2	no	—
57	C₄²×C₁₀₉	—	2	2	no	—
58	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
59	C₄²×C₁₀₉	C₂³	1	2	yes	—
60	Q₈×C₁₀₉	—	2	2	no	—
61	Q₈×C₁₀₉	—	4	1	no	—
62	Q₈×C₁₀₉	—	4	4	no	—
63	Q₈×C₁₀₉	C₂×Q₈	1	4	yes	—
64	C₂×C₄×C₁₀₉	—	2	2	no	—
65	C₂×C₄×C₁₀₉	—	2	2	no	—
66	C₂×C₄×C₁₀₉	—	2	2	no	—
67	C₂×C₄×C₁₀₉	—	2	2	no	—
68	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
70	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
71	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
73	C₈×C₁₀₉	—	2	1	no	—
74	C₈×C₁₀₉	—	2	2	no	—
75	C₈×C₁₀₉	C₂×Q₈	1	2	yes	—
76	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
77	C₄×C₁₀₉	—	2	4	no	—
78	C₄×C₁₀₉	—	2	1	no	—
79	C₄×C₁₀₉	—	2	2	no	—
80	C₄×C₁₀₉	—	2	1	no	—
81	C₄×C₁₀₉	—	2	2	no	—
82	C₄×C₁₀₉	—	4	2	no	—
83	C₄×C₁₀₉	—	4	2	no	—
84	C₄×C₁₀₉	—	4	2	no	—
85	C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
86	C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
87	C₂×C₁₀₉	D₄×Q₈	1	1	yes	characteristic
88	C₂×C₁₀₉	#575	1	1	yes	characteristic
89	C₂×C₁₀₉	#578	1	1	yes	characteristic
90	#3390	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₁₀₉	#3390	1	1	yes	characteristic, direct product factor, Sylow
92	#557	C₂×C₁₀₉	1	2	yes	—
93	#501	C₂×C₁₀₉	1	2	yes	—
94	#499	C₂×C₁₀₉	1	1	yes	characteristic
95	#479	C₂×C₁₀₉	1	2	yes	—
96	#479	C₂×C₁₀₉	1	2	yes	—
97	#477	C₂×C₁₀₉	1	2	yes	—
98	#446	C₂×C₁₀₉	1	1	yes	characteristic
99	#441	C₂×C₁₀₉	1	2	yes	—
100	C₄×Q₁₆	C₂×C₁₀₉	1	1	yes	characteristic
101	C₂×Q₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
102	C₄⋊Q₈	—	2	2	no	—
103	C₄⋊Q₈	C₂²×C₁₀₉	1	2	yes	—
104	#125	—	2	2	no	—
105	#125	—	2	2	no	—
106	#125	C₂²×C₁₀₉	1	2	yes	—
107	#125	C₂²×C₁₀₉	1	2	yes	—
108	C₄×Q₈	—	2	2	no	—
109	C₄×Q₈	C₂²×C₁₀₉	1	2	yes	—
110	C₄×Q₈	C₂²×C₁₀₉	1	2	yes	—
111	C₄×Q₈	C₂²×C₁₀₉	1	2	yes	—
112	C₄×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
113	#107	C₂²×C₁₀₉	1	1	yes	characteristic
114	#107	C₂²×C₁₀₉	1	1	yes	characteristic
115	#107	C₂²×C₁₀₉	1	2	yes	—
116	#107	C₂²×C₁₀₉	1	1	yes	characteristic
117	#106	C₂²×C₁₀₉	1	2	yes	—
118	#106	C₂²×C₁₀₉	1	2	yes	—
119	C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
120	C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
121	#103	C₂²×C₁₀₉	1	2	yes	—
122	#103	C₂²×C₁₀₉	1	2	yes	—
123	#103	C₂²×C₁₀₉	1	2	yes	—
124	#97	C₂²×C₁₀₉	1	2	yes	—
125	C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
126	C₂×Q₈	—	2	2	no	—
127	C₂×Q₈	D₄×C₁₀₉	1	1	yes	characteristic
128	C₂×Q₈	C₂³×C₁₀₉	1	2	yes	—
129	Q₁₆	Q₈×C₁₀₉	1	4	yes	complemented
130	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
131	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
132	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
133	C₄⋊C₄	—	2	2	no	—
134	C₄⋊C₄	—	2	2	no	—
135	C₄⋊C₄	—	2	2	no	—
136	C₄⋊C₄	—	2	2	no	—
137	C₄⋊C₄	—	2	2	no	—
138	C₄⋊C₄	—	2	2	no	—
139	C₄⋊C₄	—	2	2	no	—
140	C₄⋊C₄	—	2	2	no	—
141	C₄⋊C₄	D₄×C₁₀₉	1	2	yes	—
142	C₄⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
143	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
144	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
145	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
146	C₄²	—	2	2	no	—
147	C₄²	—	2	2	no	—
148	C₄²	—	2	2	no	—
149	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
150	C₄²	C₂³×C₁₀₉	1	2	yes	—
151	Q₈	—	2	2	no	—
152	Q₈	—	4	1	no	—
153	Q₈	—	4	4	no	—
154	Q₈	C₂×Q₈×C₁₀₉	1	4	yes	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
160	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
161	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
162	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
163	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
164	C₈	—	2	1	no	—
165	C₈	—	2	2	no	—
166	C₈	C₂×Q₈×C₁₀₉	1	2	yes	—
167	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
168	C₄	—	2	4	no	—
169	C₄	—	2	2	no	—
170	C₄	—	2	1	no	—
171	C₄	—	2	1	no	—
172	C₄	—	2	2	no	—
173	C₄	—	4	2	no	—
174	C₄	—	4	2	no	—
175	C₄	—	4	2	no	—
176	C₄	C₂²×Q₈×C₁₀₉	1	1	yes	characteristic
177	C₄	2₋¹⁺⁴×C₁₀₉	1	1	yes	characteristic
178	C₂	D₄×Q₈×C₁₀₉	1	1	yes	characteristic
179	C₂	#575×C₁₀₉	1	1	yes	characteristic
180	C₂	#578×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₁₀₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₀₉
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₄×Q₈×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
38. Q₁₆×C₁₀₉	Q₈	semidirect
90. #3390	C₁₀₉	direct (N × H)
91. C₁₀₉	#3390	direct (N × H)
129. Q₁₆	Q₈×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#557×C₁₀₉	2
#501×C₁₀₉	2
#499×C₁₀₉	2
#479×C₁₀₉	2
#477×C₁₀₉	2
#446×C₁₀₉	2
#441×C₁₀₉	2
C₄×Q₁₆×C₁₀₉	2
#3390	109

Subgroups — C109×#3390 #28360644

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#3390 #28360644