Subgroups — C₁₀₉×#3228 #28360482

Order 13952.

370 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#533×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
2	#532×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	#500×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#498×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#486×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#486×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
7	#482×C₁₀₉	C₂	1	4	yes	maximal, complemented
8	#478×C₁₀₉	C₂	1	4	yes	maximal, complemented
9	#431×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#41×C₁₀₉	—	2	1	no	—
11	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
12	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
14	#127×C₁₀₉	C₂²	1	1	yes	characteristic
15	#125×C₁₀₉	C₂²	1	2	yes	—
16	#124×C₁₀₉	—	2	1	no	—
17	C₂²⋊Q₈×C₁₀₉	—	2	1	no	—
18	C₂²⋊Q₈×C₁₀₉	—	2	1	no	—
19	#121×C₁₀₉	—	2	1	no	—
20	#121×C₁₀₉	C₂²	1	2	yes	—
21	C₄×Q₈×C₁₀₉	—	2	1	no	—
22	C₄×Q₈×C₁₀₉	—	2	1	no	—
23	C₄×D₄×C₁₀₉	—	2	1	no	—
24	C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
25	#117×C₁₀₉	C₂²	1	2	yes	—
26	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
27	#107×C₁₀₉	C₂²	1	4	yes	—
28	#106×C₁₀₉	C₂²	1	4	yes	—
29	C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
30	#102×C₁₀₉	C₂²	1	4	yes	—
31	#102×C₁₀₉	C₂²	1	4	yes	—
32	C₂²⋊C₈×C₁₀₉	C₂²	1	2	yes	—
33	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
34	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
35	#41×C₁₀₉	—	4	4	no	—
36	C₂×Q₈×C₁₀₉	—	2	1	no	—
37	C₂×Q₈×C₁₀₉	—	2	1	no	—
38	C₂×Q₈×C₁₀₉	—	2	1	no	—
39	C₂×D₄×C₁₀₉	—	2	2	no	—
40	C₂×D₄×C₁₀₉	—	2	1	no	—
41	C₂×D₄×C₁₀₉	—	2	1	no	—
42	C₂×D₄×C₁₀₉	C₂³	1	2	yes	—
43	C₂²×C₄×C₁₀₉	—	2	2	no	—
44	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
47	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
48	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
49	C₄⋊C₄×C₁₀₉	—	2	1	no	—
50	C₄⋊C₄×C₁₀₉	—	2	2	no	—
51	C₄⋊C₄×C₁₀₉	—	2	2	no	—
52	C₄⋊C₄×C₁₀₉	—	2	1	no	—
53	C₄⋊C₄×C₁₀₉	—	2	1	no	—
54	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
55	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
56	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
57	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
58	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
59	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
60	C₄²×C₁₀₉	—	2	2	no	—
61	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
62	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
63	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
64	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
65	C₂³×C₁₀₉	—	2	2	no	—
66	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
67	Q₈×C₁₀₉	—	4	2	no	—
68	Q₈×C₁₀₉	—	4	2	no	—
69	Q₈×C₁₀₉	—	4	2	no	—
70	D₄×C₁₀₉	—	2	4	no	—
71	D₄×C₁₀₉	—	4	4	no	—
72	D₄×C₁₀₉	—	4	2	no	—
73	D₄×C₁₀₉	—	4	2	no	—
74	C₂×C₄×C₁₀₉	—	2	2	no	—
75	C₂×C₄×C₁₀₉	—	2	2	no	—
76	C₂×C₄×C₁₀₉	—	2	2	no	—
77	C₂×C₄×C₁₀₉	—	2	2	no	—
78	C₂×C₄×C₁₀₉	—	2	2	no	—
79	C₂×C₄×C₁₀₉	—	2	2	no	—
80	C₂×C₄×C₁₀₉	—	4	4	no	—
81	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
83	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
84	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
85	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
86	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
87	C₈×C₁₀₉	—	2	2	no	—
88	C₈×C₁₀₉	—	2	2	no	—
89	C₂²×C₁₀₉	—	2	1	no	—
90	C₂²×C₁₀₉	—	2	1	no	—
91	C₂²×C₁₀₉	—	2	1	no	—
92	C₂²×C₁₀₉	—	4	2	no	—
93	C₂²×C₁₀₉	—	4	2	no	—
94	C₂²×C₁₀₉	—	4	2	no	—
95	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
96	C₄×C₁₀₉	—	2	1	no	—
97	C₄×C₁₀₉	—	2	1	no	—
98	C₄×C₁₀₉	—	2	1	no	—
99	C₄×C₁₀₉	—	4	2	no	—
100	C₄×C₁₀₉	—	4	2	no	—
101	C₄×C₁₀₉	—	4	2	no	—
102	C₄×C₁₀₉	#135	1	2	yes	—
103	C₄×C₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
104	C₄×C₁₀₉	2₋¹⁺⁴	1	2	yes	—
105	C₂×C₁₀₉	—	4	1	no	—
106	C₂×C₁₀₉	—	8	2	no	—
107	C₂×C₁₀₉	#536	1	1	yes	characteristic
108	C₂×C₁₀₉	C₂×#135	1	1	yes	characteristic
109	C₂×C₁₀₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
110	#3228	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₁₀₉	#3228	1	1	yes	characteristic, direct product factor, Sylow
112	#533	C₂×C₁₀₉	1	1	yes	characteristic, complemented
113	#532	C₂×C₁₀₉	1	1	yes	characteristic
114	#500	C₂×C₁₀₉	1	1	yes	characteristic, complemented
115	#498	C₂×C₁₀₉	1	1	yes	characteristic, complemented
116	#486	C₂×C₁₀₉	1	1	yes	characteristic, complemented
117	#486	C₂×C₁₀₉	1	1	yes	characteristic, complemented
118	#482	C₂×C₁₀₉	1	4	yes	complemented
119	#478	C₂×C₁₀₉	1	4	yes	complemented
120	#431	C₂×C₁₀₉	1	1	yes	characteristic, complemented
121	C₂×#41	—	2	1	no	—
122	C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
123	C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
124	C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
125	#127	C₂²×C₁₀₉	1	1	yes	characteristic
126	#125	C₂²×C₁₀₉	1	2	yes	—
127	#124	—	2	1	no	—
128	C₂²⋊Q₈	—	2	1	no	—
129	C₂²⋊Q₈	—	2	1	no	—
130	#121	—	2	1	no	—
131	#121	C₂²×C₁₀₉	1	2	yes	—
132	C₄×Q₈	—	2	1	no	—
133	C₄×Q₈	—	2	1	no	—
134	C₄×D₄	—	2	1	no	—
135	C₄×D₄	C₂²×C₁₀₉	1	2	yes	—
136	#117	C₂²×C₁₀₉	1	2	yes	—
137	C₂×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
138	#107	C₂²×C₁₀₉	1	4	yes	—
139	#106	C₂²×C₁₀₉	1	4	yes	—
140	C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
141	#102	C₂²×C₁₀₉	1	4	yes	—
142	#102	C₂²×C₁₀₉	1	4	yes	—
143	C₂²⋊C₈	C₂²×C₁₀₉	1	2	yes	—
144	C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
145	C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
146	#41	—	4	4	no	—
147	C₂×Q₈	—	2	1	no	—
148	C₂×Q₈	—	2	1	no	—
149	C₂×Q₈	—	2	1	no	—
150	C₂×D₄	—	2	2	no	—
151	C₂×D₄	—	2	1	no	—
152	C₂×D₄	—	2	1	no	—
153	C₂×D₄	C₂³×C₁₀₉	1	2	yes	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	D₄×C₁₀₉	1	1	yes	characteristic
156	C₂²×C₄	D₄×C₁₀₉	1	1	yes	characteristic
157	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
158	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
159	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
160	C₄⋊C₄	—	2	1	no	—
161	C₄⋊C₄	—	2	2	no	—
162	C₄⋊C₄	—	2	2	no	—
163	C₄⋊C₄	—	2	1	no	—
164	C₄⋊C₄	—	2	1	no	—
165	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
166	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
167	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
168	C₂²⋊C₄	—	2	2	no	—
169	C₂²⋊C₄	—	2	2	no	—
170	C₂²⋊C₄	—	2	2	no	—
171	C₄²	—	2	2	no	—
172	C₄²	D₄×C₁₀₉	1	1	yes	characteristic
173	C₄²	D₄×C₁₀₉	1	1	yes	characteristic
174	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
175	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
176	C₂³	—	2	2	no	—
177	C₂³	C₂×D₄×C₁₀₉	1	1	yes	characteristic
178	Q₈	—	4	2	no	—
179	Q₈	—	4	2	no	—
180	Q₈	—	4	2	no	—
181	D₄	—	2	4	no	—
182	D₄	—	4	4	no	—
183	D₄	—	4	2	no	—
184	D₄	—	4	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	4	4	no	—
192	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
193	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
194	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
195	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
196	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
197	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
198	C₈	—	2	2	no	—
199	C₈	—	2	2	no	—
200	C₂²	—	2	1	no	—
201	C₂²	—	2	1	no	—
202	C₂²	—	2	1	no	—
203	C₂²	—	4	2	no	—
204	C₂²	—	4	2	no	—
205	C₂²	—	4	2	no	—
206	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
207	C₄	—	2	1	no	—
208	C₄	—	2	1	no	—
209	C₄	—	2	1	no	—
210	C₄	—	4	2	no	—
211	C₄	—	4	2	no	—
212	C₄	—	4	2	no	—
213	C₄	#135×C₁₀₉	1	2	yes	—
214	C₄	AGL(1,ℤ/8)×C₁₀₉	1	2	yes	—
215	C₄	2₋¹⁺⁴×C₁₀₉	1	2	yes	—
216	C₂	—	4	1	no	—
217	C₂	—	8	2	no	—
218	C₂	#536×C₁₀₉	1	1	yes	characteristic
219	C₂	C₂×#135×C₁₀₉	1	1	yes	characteristic
220	C₂	C₂×AGL(1,ℤ/8)×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₀₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₀₉
prime radical	#532×C₁₀₉	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. #533×C₁₀₉	C₂	semidirect
3. #500×C₁₀₉	C₂	semidirect
4. #498×C₁₀₉	C₂	semidirect
5. #486×C₁₀₉	C₂	semidirect
6. #486×C₁₀₉	C₂	semidirect
7. #482×C₁₀₉	C₂	semidirect
8. #478×C₁₀₉	C₂	semidirect
9. #431×C₁₀₉	C₂	semidirect
110. #3228	C₁₀₉	direct (N × H)
111. C₁₀₉	#3228	direct (N × H)
112. #533	C₂×C₁₀₉	semidirect
114. #500	C₂×C₁₀₉	semidirect
115. #498	C₂×C₁₀₉	semidirect
116. #486	C₂×C₁₀₉	semidirect
117. #486	C₂×C₁₀₉	semidirect
118. #482	C₂×C₁₀₉	semidirect
119. #478	C₂×C₁₀₉	semidirect
120. #431	C₂×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#533×C₁₀₉	2
#532×C₁₀₉	2
#500×C₁₀₉	2
#498×C₁₀₉	2
#486×C₁₀₉	2
#482×C₁₀₉	2
#478×C₁₀₉	2
#431×C₁₀₉	2
#3228	109

Subgroups — C109×#3228 #28360482

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#3228 #28360482