Subgroups — C₁₀₉×#3389 #28360643

Order 13952.

348 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#554×C₁₀₉	C₂	1	2	yes	maximal, complemented
2	#501×C₁₀₉	C₂	1	2	yes	maximal, complemented
3	#499×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#478×C₁₀₉	C₂	1	2	yes	maximal, complemented
5	#478×C₁₀₉	C₂	1	2	yes	maximal, complemented
6	#474×C₁₀₉	C₂	1	2	yes	maximal, complemented
7	#446×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#442×C₁₀₉	C₂	1	2	yes	maximal
9	C₄×D₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
10	C₂×D₈×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₄⋊Q₈×C₁₀₉	C₂²	1	2	yes	—
12	#125×C₁₀₉	C₂²	1	2	yes	—
13	#125×C₁₀₉	C₂²	1	2	yes	—
14	C₂²⋊Q₈×C₁₀₉	—	2	2	no	—
15	C₂²⋊Q₈×C₁₀₉	—	2	2	no	—
16	C₂²⋊Q₈×C₁₀₉	—	2	2	no	—
17	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
18	C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
19	C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
20	C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
21	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
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	#102×C₁₀₉	C₂²	1	2	yes	—
31	#102×C₁₀₉	C₂²	1	2	yes	—
32	#102×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×D₄×C₁₀₉	C₂³	1	2	yes	—
38	C₂²×C₄×C₁₀₉	—	2	2	no	—
39	C₂²×C₄×C₁₀₉	—	2	2	no	—
40	C₂²×C₄×C₁₀₉	—	2	2	no	—
41	D₈×C₁₀₉	Q₈	1	4	yes	complemented
42	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
43	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
44	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
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	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
56	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
57	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
58	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
59	C₄²×C₁₀₉	C₂³	1	2	yes	—
60	C₂³×C₁₀₉	—	2	2	no	—
61	Q₈×C₁₀₉	—	4	1	no	—
62	Q₈×C₁₀₉	—	8	2	no	—
63	D₄×C₁₀₉	—	2	2	no	—
64	D₄×C₁₀₉	C₂×Q₈	1	4	yes	—
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	—	4	4	no	—
69	C₂×C₄×C₁₀₉	—	8	2	no	—
70	C₂×C₄×C₁₀₉	—	8	2	no	—
71	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
73	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
74	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
76	C₈×C₁₀₉	—	2	1	no	—
77	C₈×C₁₀₉	—	2	2	no	—
78	C₈×C₁₀₉	C₂×Q₈	1	2	yes	—
79	C₂²×C₁₀₉	—	2	4	no	—
80	C₂²×C₁₀₉	—	4	2	no	—
81	C₂²×C₁₀₉	—	4	2	no	—
82	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
83	C₄×C₁₀₉	—	2	2	no	—
84	C₄×C₁₀₉	—	2	1	no	—
85	C₄×C₁₀₉	—	2	1	no	—
86	C₄×C₁₀₉	—	2	2	no	—
87	C₄×C₁₀₉	—	4	2	no	—
88	C₄×C₁₀₉	—	4	2	no	—
89	C₄×C₁₀₉	—	4	2	no	—
90	C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
91	C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
92	C₂×C₁₀₉	—	4	4	no	—
93	C₂×C₁₀₉	D₄×Q₈	1	1	yes	characteristic
94	C₂×C₁₀₉	#575	1	1	yes	characteristic
95	C₂×C₁₀₉	#576	1	1	yes	characteristic
96	#3389	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₁₀₉	#3389	1	1	yes	characteristic, direct product factor, Sylow
98	#554	C₂×C₁₀₉	1	2	yes	complemented
99	#501	C₂×C₁₀₉	1	2	yes	complemented
100	#499	C₂×C₁₀₉	1	1	yes	characteristic, complemented
101	#478	C₂×C₁₀₉	1	2	yes	complemented
102	#478	C₂×C₁₀₉	1	2	yes	complemented
103	#474	C₂×C₁₀₉	1	2	yes	complemented
104	#446	C₂×C₁₀₉	1	1	yes	characteristic, complemented
105	#442	C₂×C₁₀₉	1	2	yes	—
106	C₄×D₈	C₂×C₁₀₉	1	1	yes	characteristic
107	C₂×D₈	C₂²×C₁₀₉	1	1	yes	characteristic
108	C₄⋊Q₈	C₂²×C₁₀₉	1	2	yes	—
109	#125	C₂²×C₁₀₉	1	2	yes	—
110	#125	C₂²×C₁₀₉	1	2	yes	—
111	C₂²⋊Q₈	—	2	2	no	—
112	C₂²⋊Q₈	—	2	2	no	—
113	C₂²⋊Q₈	—	2	2	no	—
114	C₄×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
115	C₄×D₄	C₂²×C₁₀₉	1	2	yes	—
116	C₄×D₄	C₂²×C₁₀₉	1	2	yes	—
117	C₄×D₄	C₂²×C₁₀₉	1	2	yes	—
118	C₂×C₄⋊C₄	—	2	2	no	—
119	#107	C₂²×C₁₀₉	1	1	yes	characteristic
120	#107	C₂²×C₁₀₉	1	1	yes	characteristic
121	#107	C₂²×C₁₀₉	1	2	yes	—
122	#107	C₂²×C₁₀₉	1	1	yes	characteristic
123	#106	C₂²×C₁₀₉	1	2	yes	—
124	#106	C₂²×C₁₀₉	1	2	yes	—
125	C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
126	C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
127	#102	C₂²×C₁₀₉	1	2	yes	—
128	#102	C₂²×C₁₀₉	1	2	yes	—
129	#102	C₂²×C₁₀₉	1	2	yes	—
130	#97	C₂²×C₁₀₉	1	2	yes	—
131	C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
132	C₂×Q₈	—	2	2	no	—
133	C₂×Q₈	D₄×C₁₀₉	1	1	yes	characteristic
134	C₂×D₄	C₂³×C₁₀₉	1	2	yes	—
135	C₂²×C₄	—	2	2	no	—
136	C₂²×C₄	—	2	2	no	—
137	C₂²×C₄	—	2	2	no	—
138	D₈	Q₈×C₁₀₉	1	4	yes	complemented
139	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
140	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
141	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
142	C₄⋊C₄	—	2	2	no	—
143	C₄⋊C₄	—	2	2	no	—
144	C₄⋊C₄	—	2	2	no	—
145	C₄⋊C₄	—	2	2	no	—
146	C₄⋊C₄	—	2	2	no	—
147	C₄⋊C₄	D₄×C₁₀₉	1	2	yes	—
148	C₄⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
149	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
150	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
151	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
152	C₂²⋊C₄	—	2	2	no	—
153	C₂²⋊C₄	—	2	2	no	—
154	C₂²⋊C₄	—	2	2	no	—
155	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
156	C₄²	C₂³×C₁₀₉	1	2	yes	—
157	C₂³	—	2	2	no	—
158	Q₈	—	4	1	no	—
159	Q₈	—	8	2	no	—
160	D₄	—	2	2	no	—
161	D₄	C₂×Q₈×C₁₀₉	1	4	yes	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	4	4	no	—
166	C₂×C₄	—	8	2	no	—
167	C₂×C₄	—	8	2	no	—
168	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
169	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
170	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
171	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
172	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
173	C₈	—	2	1	no	—
174	C₈	—	2	2	no	—
175	C₈	C₂×Q₈×C₁₀₉	1	2	yes	—
176	C₂²	—	2	4	no	—
177	C₂²	—	4	2	no	—
178	C₂²	—	4	2	no	—
179	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
180	C₄	—	2	2	no	—
181	C₄	—	2	1	no	—
182	C₄	—	2	1	no	—
183	C₄	—	2	2	no	—
184	C₄	—	4	2	no	—
185	C₄	—	4	2	no	—
186	C₄	—	4	2	no	—
187	C₄	C₂²×Q₈×C₁₀₉	1	1	yes	characteristic
188	C₄	2₋¹⁺⁴×C₁₀₉	1	1	yes	characteristic
189	C₂	—	4	4	no	—
190	C₂	D₄×Q₈×C₁₀₉	1	1	yes	characteristic
191	C₂	#575×C₁₀₉	1	1	yes	characteristic
192	C₂	#576×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₀₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₀₉
prime radical	C₂×D₈×C₁₀₉	C₂²
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
1. #554×C₁₀₉	C₂	semidirect
2. #501×C₁₀₉	C₂	semidirect
3. #499×C₁₀₉	C₂	semidirect
4. #478×C₁₀₉	C₂	semidirect
5. #478×C₁₀₉	C₂	semidirect
6. #474×C₁₀₉	C₂	semidirect
7. #446×C₁₀₉	C₂	semidirect
41. D₈×C₁₀₉	Q₈	semidirect
96. #3389	C₁₀₉	direct (N × H)
97. C₁₀₉	#3389	direct (N × H)
98. #554	C₂×C₁₀₉	semidirect
99. #501	C₂×C₁₀₉	semidirect
100. #499	C₂×C₁₀₉	semidirect
101. #478	C₂×C₁₀₉	semidirect
102. #478	C₂×C₁₀₉	semidirect
103. #474	C₂×C₁₀₉	semidirect
104. #446	C₂×C₁₀₉	semidirect
138. D₈	Q₈×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#554×C₁₀₉	2
#501×C₁₀₉	2
#499×C₁₀₉	2
#478×C₁₀₉	2
#474×C₁₀₉	2
#446×C₁₀₉	2
#442×C₁₀₉	2
C₄×D₈×C₁₀₉	2
#3389	109

Subgroups — C109×#3389 #28360643

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#3389 #28360643