Subgroups — C₁₀₉×#3232 #28360486

Order 13952.

334 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#529×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#125×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#499×C₁₀₉	C₂	1	2	yes	maximal, complemented
4	#488×C₁₀₉	C₂	1	2	yes	maximal, complemented
5	#484×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#483×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
7	#481×C₁₀₉	C₂	1	1	yes	maximal, characteristic
8	#480×C₁₀₉	C₂	1	1	yes	maximal, characteristic
9	#479×C₁₀₉	C₂	1	2	yes	maximal, complemented
10	#478×C₁₀₉	C₂	1	2	yes	maximal, complemented
11	#431×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#126×C₁₀₉	—	2	1	no	—
13	#125×C₁₀₉	—	2	1	no	—
14	#125×C₁₀₉	—	2	1	no	—
15	#125×C₁₀₉	C₂²	1	1	yes	characteristic
16	#125×C₁₀₉	C₂²	1	2	yes	—
17	#125×C₁₀₉	C₂²	1	2	yes	—
18	#124×C₁₀₉	C₂²	1	1	yes	characteristic
19	#123×C₁₀₉	—	2	1	no	—
20	C₂²⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
21	#121×C₁₀₉	C₂²	1	1	yes	characteristic
22	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₁₀₉	—	2	1	no	—
24	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
25	#117×C₁₀₉	—	2	1	no	—
26	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
27	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
28	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
30	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
31	#107×C₁₀₉	C₂²	1	2	yes	—
32	#107×C₁₀₉	C₂²	1	2	yes	—
33	#106×C₁₀₉	C₂²	1	2	yes	—
34	#106×C₁₀₉	C₂²	1	2	yes	—
35	C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
36	#103×C₁₀₉	C₂²	1	2	yes	—
37	#103×C₁₀₉	C₂²	1	2	yes	—
38	#102×C₁₀₉	C₂²	1	2	yes	—
39	#102×C₁₀₉	C₂²	1	2	yes	—
40	C₂²⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
41	C₂²⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
42	C₄×C₈×C₁₀₉	C₂²	1	2	yes	—
43	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
44	C₂×D₄×C₁₀₉	—	2	1	no	—
45	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₁₀₉	—	2	1	no	—
47	C₂²×C₄×C₁₀₉	—	2	1	no	—
48	C₂²×C₄×C₁₀₉	—	2	1	no	—
49	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
52	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
53	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
54	C₄⋊C₄×C₁₀₉	—	2	2	no	—
55	C₄⋊C₄×C₁₀₉	—	2	2	no	—
56	C₄⋊C₄×C₁₀₉	—	2	2	no	—
57	C₄⋊C₄×C₁₀₉	—	2	2	no	—
58	C₄⋊C₄×C₁₀₉	—	2	1	no	—
59	C₄⋊C₄×C₁₀₉	—	2	1	no	—
60	C₄⋊C₄×C₁₀₉	—	2	1	no	—
61	C₄⋊C₄×C₁₀₉	—	2	1	no	—
62	C₄⋊C₄×C₁₀₉	—	2	1	no	—
63	C₄⋊C₄×C₁₀₉	—	2	1	no	—
64	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
65	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
66	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
67	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
68	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
69	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
70	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
71	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
72	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
73	C₄²×C₁₀₉	—	2	1	no	—
74	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
75	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
76	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
77	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
78	C₂³×C₁₀₉	—	2	1	no	—
79	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
80	Q₈×C₁₀₉	—	2	2	no	—
81	D₄×C₁₀₉	—	2	2	no	—
82	D₄×C₁₀₉	—	4	2	no	—
83	C₂×C₄×C₁₀₉	—	2	2	no	—
84	C₂×C₄×C₁₀₉	—	2	2	no	—
85	C₂×C₄×C₁₀₉	—	2	1	no	—
86	C₂×C₄×C₁₀₉	—	2	1	no	—
87	C₂×C₄×C₁₀₉	—	2	1	no	—
88	C₂×C₄×C₁₀₉	—	2	2	no	—
89	C₂×C₄×C₁₀₉	—	2	2	no	—
90	C₂×C₄×C₁₀₉	—	4	1	no	—
91	C₂×C₄×C₁₀₉	—	4	2	no	—
92	C₂×C₄×C₁₀₉	—	4	2	no	—
93	C₂×C₄×C₁₀₉	—	4	2	no	—
94	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
100	C₈×C₁₀₉	—	2	2	no	—
101	C₈×C₁₀₉	—	2	2	no	—
102	C₂²×C₁₀₉	—	2	1	no	—
103	C₂²×C₁₀₉	—	2	1	no	—
104	C₂²×C₁₀₉	—	4	1	no	—
105	C₂²×C₁₀₉	—	4	1	no	—
106	C₂²×C₁₀₉	—	4	1	no	—
107	C₂²×C₁₀₉	#135	1	2	yes	—
108	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
109	C₄×C₁₀₉	—	2	1	no	—
110	C₄×C₁₀₉	—	2	1	no	—
111	C₄×C₁₀₉	—	2	1	no	—
112	C₄×C₁₀₉	—	2	1	no	—
113	C₄×C₁₀₉	—	2	1	no	—
114	C₄×C₁₀₉	—	4	1	no	—
115	C₄×C₁₀₉	—	4	2	no	—
116	C₄×C₁₀₉	—	4	1	no	—
117	C₄×C₁₀₉	—	4	1	no	—
118	C₄×C₁₀₉	—	4	2	no	—
119	C₄×C₁₀₉	2₋¹⁺⁴	1	2	yes	—
120	C₂×C₁₀₉	—	2	2	no	—
121	C₂×C₁₀₉	—	8	1	no	—
122	C₂×C₁₀₉	#536	1	1	yes	characteristic
123	C₂×C₁₀₉	C₂×#135	1	1	yes	characteristic
124	C₂×C₁₀₉	#575	1	1	yes	characteristic
125	#3232	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
126	C₁₀₉	#3232	1	1	yes	characteristic, direct product factor, Sylow
127	#529	C₂×C₁₀₉	1	1	yes	characteristic
128	C₂×#125	C₂×C₁₀₉	1	1	yes	characteristic, complemented
129	#499	C₂×C₁₀₉	1	2	yes	complemented
130	#488	C₂×C₁₀₉	1	2	yes	complemented
131	#484	C₂×C₁₀₉	1	1	yes	characteristic, complemented
132	#483	C₂×C₁₀₉	1	1	yes	characteristic, complemented
133	#481	C₂×C₁₀₉	1	1	yes	characteristic
134	#480	C₂×C₁₀₉	1	1	yes	characteristic
135	#479	C₂×C₁₀₉	1	2	yes	complemented
136	#478	C₂×C₁₀₉	1	2	yes	complemented
137	#431	C₂×C₁₀₉	1	1	yes	characteristic, complemented
138	#126	—	2	1	no	—
139	#125	—	2	1	no	—
140	#125	—	2	1	no	—
141	#125	C₂²×C₁₀₉	1	2	yes	—
142	#125	C₂²×C₁₀₉	1	2	yes	—
143	#125	C₂²×C₁₀₉	1	1	yes	characteristic
144	#124	C₂²×C₁₀₉	1	1	yes	characteristic
145	#123	—	2	1	no	—
146	C₂²⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
147	#121	C₂²×C₁₀₉	1	1	yes	characteristic
148	C₄×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
149	C₄×D₄	—	2	1	no	—
150	C₄×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
151	#117	—	2	1	no	—
152	C₂×C₄⋊C₄	—	2	1	no	—
153	C₂×C₄⋊C₄	—	2	1	no	—
154	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
155	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
156	C₂×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
157	#107	C₂²×C₁₀₉	1	2	yes	—
158	#107	C₂²×C₁₀₉	1	2	yes	—
159	#106	C₂²×C₁₀₉	1	2	yes	—
160	#106	C₂²×C₁₀₉	1	2	yes	—
161	C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
162	#103	C₂²×C₁₀₉	1	2	yes	—
163	#103	C₂²×C₁₀₉	1	2	yes	—
164	#102	C₂²×C₁₀₉	1	2	yes	—
165	#102	C₂²×C₁₀₉	1	2	yes	—
166	C₂²⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
167	C₂²⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
168	C₄×C₈	C₂²×C₁₀₉	1	2	yes	—
169	C₂×Q₈	C₂³×C₁₀₉	1	1	yes	characteristic
170	C₂×D₄	—	2	1	no	—
171	C₂×D₄	C₂³×C₁₀₉	1	1	yes	characteristic
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	—	2	1	no	—
174	C₂²×C₄	—	2	1	no	—
175	C₂²×C₄	D₄×C₁₀₉	1	1	yes	characteristic
176	C₂²×C₄	D₄×C₁₀₉	1	1	yes	characteristic
177	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
178	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
179	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	—	2	2	no	—
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	—	2	2	no	—
184	C₄⋊C₄	—	2	1	no	—
185	C₄⋊C₄	—	2	1	no	—
186	C₄⋊C₄	—	2	1	no	—
187	C₄⋊C₄	—	2	1	no	—
188	C₄⋊C₄	—	2	1	no	—
189	C₄⋊C₄	—	2	1	no	—
190	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
191	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
192	C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
193	C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
194	C₂²⋊C₄	—	2	1	no	—
195	C₂²⋊C₄	—	2	1	no	—
196	C₂²⋊C₄	—	2	1	no	—
197	C₂²⋊C₄	—	2	1	no	—
198	C₂²⋊C₄	—	2	1	no	—
199	C₄²	—	2	1	no	—
200	C₄²	D₄×C₁₀₉	1	1	yes	characteristic
201	C₄²	D₄×C₁₀₉	1	1	yes	characteristic
202	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
203	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
204	C₂³	—	2	1	no	—
205	C₂³	C₂×D₄×C₁₀₉	1	1	yes	characteristic
206	Q₈	—	2	2	no	—
207	D₄	—	2	2	no	—
208	D₄	—	4	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	4	1	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
221	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
222	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
223	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
224	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
225	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
226	C₈	—	2	2	no	—
227	C₈	—	2	2	no	—
228	C₂²	—	2	1	no	—
229	C₂²	—	2	1	no	—
230	C₂²	—	4	1	no	—
231	C₂²	—	4	1	no	—
232	C₂²	—	4	1	no	—
233	C₂²	#135×C₁₀₉	1	2	yes	—
234	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
235	C₄	—	2	1	no	—
236	C₄	—	2	1	no	—
237	C₄	—	2	1	no	—
238	C₄	—	2	1	no	—
239	C₄	—	2	1	no	—
240	C₄	—	4	1	no	—
241	C₄	—	4	2	no	—
242	C₄	—	4	1	no	—
243	C₄	—	4	1	no	—
244	C₄	—	4	2	no	—
245	C₄	2₋¹⁺⁴×C₁₀₉	1	2	yes	—
246	C₂	—	2	2	no	—
247	C₂	—	8	1	no	—
248	C₂	#536×C₁₀₉	1	1	yes	characteristic
249	C₂	C₂×#135×C₁₀₉	1	1	yes	characteristic
250	C₂	#575×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₀₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₀₉
prime radical	#121×C₁₀₉	C₂²
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
2. C₂×#125×C₁₀₉	C₂	semidirect
3. #499×C₁₀₉	C₂	semidirect
4. #488×C₁₀₉	C₂	semidirect
5. #484×C₁₀₉	C₂	semidirect
6. #483×C₁₀₉	C₂	semidirect
9. #479×C₁₀₉	C₂	semidirect
10. #478×C₁₀₉	C₂	semidirect
11. #431×C₁₀₉	C₂	semidirect
125. #3232	C₁₀₉	direct (N × H)
126. C₁₀₉	#3232	direct (N × H)
128. C₂×#125	C₂×C₁₀₉	semidirect
129. #499	C₂×C₁₀₉	semidirect
130. #488	C₂×C₁₀₉	semidirect
131. #484	C₂×C₁₀₉	semidirect
132. #483	C₂×C₁₀₉	semidirect
135. #479	C₂×C₁₀₉	semidirect
136. #478	C₂×C₁₀₉	semidirect
137. #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
#529×C₁₀₉	2
C₂×#125×C₁₀₉	2
#499×C₁₀₉	2
#488×C₁₀₉	2
#484×C₁₀₉	2
#483×C₁₀₉	2
#481×C₁₀₉	2
#480×C₁₀₉	2
#479×C₁₀₉	2
#478×C₁₀₉	2
#431×C₁₀₉	2
#3232	109

Subgroups — C109×#3232 #28360486

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#3232 #28360486