Subgroups — C₁₀₉×#3210 #28360464

Order 13952.

394 conjugacy classes of subgroups.

The subgroup lattice has 226 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	#494×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#492×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#487×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#486×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
7	#464×C₁₀₉	C₂	1	4	yes	maximal, complemented
8	#452×C₁₀₉	C₂	1	4	yes	maximal, complemented
9	#431×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#41×C₁₀₉	C₂²	1	2	yes	—
11	C₂×Q₁₆×C₁₀₉	C₂²	1	4	yes	—
12	C₂×SD₁₆×C₁₀₉	C₂²	1	4	yes	—
13	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
16	#127×C₁₀₉	C₂²	1	1	yes	characteristic
17	#125×C₁₀₉	—	2	1	no	—
18	#124×C₁₀₉	C₂²	1	2	yes	—
19	C₂²⋊Q₈×C₁₀₉	—	2	1	no	—
20	C₂²⋊Q₈×C₁₀₉	C₂²	1	2	yes	—
21	#121×C₁₀₉	—	2	1	no	—
22	#121×C₁₀₉	—	2	1	no	—
23	C₄×Q₈×C₁₀₉	—	2	1	no	—
24	C₄×Q₈×C₁₀₉	C₂²	1	2	yes	—
25	C₄×D₄×C₁₀₉	—	2	1	no	—
26	C₄×D₄×C₁₀₉	—	2	1	no	—
27	#117×C₁₀₉	—	2	1	no	—
28	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
29	C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
30	#103×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	C₂³	1	2	yes	—
38	C₂×Q₈×C₁₀₉	C₂³	1	2	yes	—
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	Q₁₆×C₁₀₉	—	4	4	no	—
48	SD₁₆×C₁₀₉	—	4	4	no	—
49	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
50	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
51	C₄⋊C₄×C₁₀₉	—	2	1	no	—
52	C₄⋊C₄×C₁₀₉	—	2	1	no	—
53	C₄⋊C₄×C₁₀₉	—	2	1	no	—
54	C₄⋊C₄×C₁₀₉	—	2	1	no	—
55	C₄⋊C₄×C₁₀₉	—	2	1	no	—
56	C₄⋊C₄×C₁₀₉	—	2	2	no	—
57	C₄⋊C₄×C₁₀₉	—	2	2	no	—
58	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
59	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
60	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
61	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
62	C₄²×C₁₀₉	—	2	2	no	—
63	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
64	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
65	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
66	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
67	C₂³×C₁₀₉	—	2	2	no	—
68	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
69	Q₈×C₁₀₉	—	2	4	no	—
70	Q₈×C₁₀₉	—	2	4	no	—
71	Q₈×C₁₀₉	—	4	2	no	—
72	D₄×C₁₀₉	—	2	4	no	—
73	D₄×C₁₀₉	—	4	4	no	—
74	D₄×C₁₀₉	—	4	2	no	—
75	D₄×C₁₀₉	—	4	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	—	2	2	no	—
81	C₂×C₄×C₁₀₉	—	2	2	no	—
82	C₂×C₄×C₁₀₉	—	4	4	no	—
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	C₂×D₄	1	1	yes	characteristic
87	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
88	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
89	C₈×C₁₀₉	—	2	2	no	—
90	C₈×C₁₀₉	—	2	2	no	—
91	C₂²×C₁₀₉	—	2	1	no	—
92	C₂²×C₁₀₉	—	2	1	no	—
93	C₂²×C₁₀₉	—	2	1	no	—
94	C₂²×C₁₀₉	—	4	2	no	—
95	C₂²×C₁₀₉	—	4	2	no	—
96	C₂²×C₁₀₉	—	4	2	no	—
97	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
98	C₄×C₁₀₉	—	2	1	no	—
99	C₄×C₁₀₉	—	2	1	no	—
100	C₄×C₁₀₉	—	2	1	no	—
101	C₄×C₁₀₉	—	4	2	no	—
102	C₄×C₁₀₉	—	4	2	no	—
103	C₄×C₁₀₉	—	4	2	no	—
104	C₄×C₁₀₉	#135	1	2	yes	—
105	C₄×C₁₀₉	#137	1	2	yes	—
106	C₄×C₁₀₉	H(5,𝔽₂)	1	2	yes	—
107	C₂×C₁₀₉	—	4	1	no	—
108	C₂×C₁₀₉	—	8	2	no	—
109	C₂×C₁₀₉	#535	1	1	yes	characteristic
110	C₂×C₁₀₉	C₂×#135	1	1	yes	characteristic
111	C₂×C₁₀₉	C₂×#137	1	1	yes	characteristic
112	#3210	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₁₀₉	#3210	1	1	yes	characteristic, direct product factor, Sylow
114	#533	C₂×C₁₀₉	1	1	yes	characteristic, complemented
115	#532	C₂×C₁₀₉	1	1	yes	characteristic
116	#494	C₂×C₁₀₉	1	1	yes	characteristic, complemented
117	#492	C₂×C₁₀₉	1	1	yes	characteristic, complemented
118	#487	C₂×C₁₀₉	1	1	yes	characteristic, complemented
119	#486	C₂×C₁₀₉	1	1	yes	characteristic, complemented
120	#464	C₂×C₁₀₉	1	4	yes	complemented
121	#452	C₂×C₁₀₉	1	4	yes	complemented
122	#431	C₂×C₁₀₉	1	1	yes	characteristic, complemented
123	C₂×#41	C₂²×C₁₀₉	1	2	yes	—
124	C₂×Q₁₆	C₂²×C₁₀₉	1	4	yes	—
125	C₂×SD₁₆	C₂²×C₁₀₉	1	4	yes	—
126	C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
127	C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
128	C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
129	#127	C₂²×C₁₀₉	1	1	yes	characteristic
130	#125	—	2	1	no	—
131	#124	C₂²×C₁₀₉	1	2	yes	—
132	C₂²⋊Q₈	—	2	1	no	—
133	C₂²⋊Q₈	C₂²×C₁₀₉	1	2	yes	—
134	#121	—	2	1	no	—
135	#121	—	2	1	no	—
136	C₄×Q₈	—	2	1	no	—
137	C₄×Q₈	C₂²×C₁₀₉	1	2	yes	—
138	C₄×D₄	—	2	1	no	—
139	C₄×D₄	—	2	1	no	—
140	#117	—	2	1	no	—
141	C₂×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
142	C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
143	#103	C₂²×C₁₀₉	1	4	yes	—
144	#102	C₂²×C₁₀₉	1	4	yes	—
145	C₂²⋊C₈	C₂²×C₁₀₉	1	2	yes	—
146	C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
147	C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
148	#41	—	4	4	no	—
149	C₂×Q₈	—	2	1	no	—
150	C₂×Q₈	C₂³×C₁₀₉	1	2	yes	—
151	C₂×Q₈	C₂³×C₁₀₉	1	2	yes	—
152	C₂×D₄	—	2	2	no	—
153	C₂×D₄	—	2	1	no	—
154	C₂×D₄	—	2	1	no	—
155	C₂×D₄	C₂³×C₁₀₉	1	2	yes	—
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	D₄×C₁₀₉	1	1	yes	characteristic
158	C₂²×C₄	D₄×C₁₀₉	1	1	yes	characteristic
159	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
160	Q₁₆	—	4	4	no	—
161	SD₁₆	—	4	4	no	—
162	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
163	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
164	C₄⋊C₄	—	2	1	no	—
165	C₄⋊C₄	—	2	1	no	—
166	C₄⋊C₄	—	2	1	no	—
167	C₄⋊C₄	—	2	1	no	—
168	C₄⋊C₄	—	2	1	no	—
169	C₄⋊C₄	—	2	2	no	—
170	C₄⋊C₄	—	2	2	no	—
171	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
172	C₂²⋊C₄	—	2	2	no	—
173	C₂²⋊C₄	—	2	2	no	—
174	C₂²⋊C₄	—	2	2	no	—
175	C₄²	—	2	2	no	—
176	C₄²	D₄×C₁₀₉	1	1	yes	characteristic
177	C₄²	D₄×C₁₀₉	1	1	yes	characteristic
178	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
179	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
180	C₂³	—	2	2	no	—
181	C₂³	C₂×D₄×C₁₀₉	1	1	yes	characteristic
182	Q₈	—	2	4	no	—
183	Q₈	—	2	4	no	—
184	Q₈	—	4	2	no	—
185	D₄	—	2	4	no	—
186	D₄	—	4	4	no	—
187	D₄	—	4	2	no	—
188	D₄	—	4	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	4	4	no	—
196	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
197	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
198	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
199	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
200	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
201	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
202	C₈	—	2	2	no	—
203	C₈	—	2	2	no	—
204	C₂²	—	2	1	no	—
205	C₂²	—	2	1	no	—
206	C₂²	—	2	1	no	—
207	C₂²	—	4	2	no	—
208	C₂²	—	4	2	no	—
209	C₂²	—	4	2	no	—
210	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	4	2	no	—
215	C₄	—	4	2	no	—
216	C₄	—	4	2	no	—
217	C₄	#135×C₁₀₉	1	2	yes	—
218	C₄	#137×C₁₀₉	1	2	yes	—
219	C₄	H(5,𝔽₂)×C₁₀₉	1	2	yes	—
220	C₂	—	4	1	no	—
221	C₂	—	8	2	no	—
222	C₂	#535×C₁₀₉	1	1	yes	characteristic
223	C₂	C₂×#135×C₁₀₉	1	1	yes	characteristic
224	C₂	C₂×#137×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. #494×C₁₀₉	C₂	semidirect
4. #492×C₁₀₉	C₂	semidirect
5. #487×C₁₀₉	C₂	semidirect
6. #486×C₁₀₉	C₂	semidirect
7. #464×C₁₀₉	C₂	semidirect
8. #452×C₁₀₉	C₂	semidirect
9. #431×C₁₀₉	C₂	semidirect
112. #3210	C₁₀₉	direct (N × H)
113. C₁₀₉	#3210	direct (N × H)
114. #533	C₂×C₁₀₉	semidirect
116. #494	C₂×C₁₀₉	semidirect
117. #492	C₂×C₁₀₉	semidirect
118. #487	C₂×C₁₀₉	semidirect
119. #486	C₂×C₁₀₉	semidirect
120. #464	C₂×C₁₀₉	semidirect
121. #452	C₂×C₁₀₉	semidirect
122. #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
#494×C₁₀₉	2
#492×C₁₀₉	2
#487×C₁₀₉	2
#486×C₁₀₉	2
#464×C₁₀₉	2
#452×C₁₀₉	2
#431×C₁₀₉	2
#3210	109

Subgroups — C109×#3210 #28360464

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#3210 #28360464