Subgroups — C₁₀₉×#3212 #28360466

Order 13952.

400 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#532×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
2	#529×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#495×C₁₀₉	C₂	1	2	yes	maximal, complemented
4	#488×C₁₀₉	C₂	1	2	yes	maximal, complemented
5	#462×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#461×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
7	#460×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#459×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
9	#452×C₁₀₉	C₂	1	2	yes	maximal, complemented
10	#449×C₁₀₉	C₂	1	2	yes	maximal, complemented
11	#431×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₁₀₉	C₂²	1	2	yes	—
13	C₂×Q₁₆×C₁₀₉	C₂²	1	2	yes	—
14	C₂×SD₁₆×C₁₀₉	C₂²	1	2	yes	—
15	C₂×SD₁₆×C₁₀₉	C₂²	1	2	yes	—
16	C₂×D₈×C₁₀₉	C₂²	1	2	yes	—
17	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
18	#127×C₁₀₉	C₂²	1	1	yes	characteristic
19	#126×C₁₀₉	—	2	1	no	—
20	#125×C₁₀₉	C₂²	1	1	yes	characteristic
21	#124×C₁₀₉	C₂²	1	1	yes	characteristic
22	#124×C₁₀₉	C₂²	1	2	yes	—
23	#123×C₁₀₉	—	2	1	no	—
24	C₂²⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
25	#121×C₁₀₉	—	2	1	no	—
26	#121×C₁₀₉	—	2	1	no	—
27	#121×C₁₀₉	C₂²	1	1	yes	characteristic
28	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
29	C₄×D₄×C₁₀₉	—	2	1	no	—
30	C₄×D₄×C₁₀₉	—	2	1	no	—
31	C₄×D₄×C₁₀₉	—	2	1	no	—
32	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
33	#117×C₁₀₉	—	2	1	no	—
34	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
35	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
36	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
37	#103×C₁₀₉	C₂²	1	2	yes	—
38	#103×C₁₀₉	C₂²	1	2	yes	—
39	#102×C₁₀₉	C₂²	1	2	yes	—
40	#102×C₁₀₉	C₂²	1	2	yes	—
41	C₂²⋊C₈×C₁₀₉	C₂²	1	2	yes	—
42	C₄×C₈×C₁₀₉	C₂²	1	2	yes	—
43	#41×C₁₀₉	—	4	2	no	—
44	#41×C₁₀₉	—	4	2	no	—
45	C₂×Q₈×C₁₀₉	C₂³	1	2	yes	—
46	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
47	C₂×D₄×C₁₀₉	—	2	1	no	—
48	C₂×D₄×C₁₀₉	—	2	2	no	—
49	C₂×D₄×C₁₀₉	—	2	1	no	—
50	C₂×D₄×C₁₀₉	—	2	1	no	—
51	C₂×D₄×C₁₀₉	C₂³	1	2	yes	—
52	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₁₀₉	—	2	1	no	—
54	C₂²×C₄×C₁₀₉	—	2	2	no	—
55	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
58	Q₁₆×C₁₀₉	—	4	2	no	—
59	SD₁₆×C₁₀₉	—	4	2	no	—
60	SD₁₆×C₁₀₉	—	4	2	no	—
61	D₈×C₁₀₉	—	4	2	no	—
62	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
63	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
64	C₄⋊C₄×C₁₀₉	—	2	1	no	—
65	C₄⋊C₄×C₁₀₉	—	2	1	no	—
66	C₄⋊C₄×C₁₀₉	—	2	1	no	—
67	C₄⋊C₄×C₁₀₉	—	2	1	no	—
68	C₄⋊C₄×C₁₀₉	—	2	1	no	—
69	C₄⋊C₄×C₁₀₉	—	2	1	no	—
70	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
71	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
72	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
73	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
74	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
75	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
76	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
77	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
78	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
79	C₄²×C₁₀₉	—	2	1	no	—
80	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
81	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
82	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
83	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
84	C₂³×C₁₀₉	—	2	1	no	—
85	C₂³×C₁₀₉	—	2	2	no	—
86	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
87	Q₈×C₁₀₉	—	2	1	no	—
88	Q₈×C₁₀₉	—	2	2	no	—
89	Q₈×C₁₀₉	—	2	1	no	—
90	Q₈×C₁₀₉	—	2	2	no	—
91	D₄×C₁₀₉	—	2	1	no	—
92	D₄×C₁₀₉	—	2	2	no	—
93	D₄×C₁₀₉	—	2	1	no	—
94	D₄×C₁₀₉	—	2	2	no	—
95	D₄×C₁₀₉	—	4	2	no	—
96	D₄×C₁₀₉	—	4	2	no	—
97	D₄×C₁₀₉	—	4	4	no	—
98	D₄×C₁₀₉	—	4	2	no	—
99	C₂×C₄×C₁₀₉	—	2	2	no	—
100	C₂×C₄×C₁₀₉	—	2	1	no	—
101	C₂×C₄×C₁₀₉	—	2	1	no	—
102	C₂×C₄×C₁₀₉	—	2	1	no	—
103	C₂×C₄×C₁₀₉	—	2	1	no	—
104	C₂×C₄×C₁₀₉	—	2	1	no	—
105	C₂×C₄×C₁₀₉	—	2	2	no	—
106	C₂×C₄×C₁₀₉	—	4	1	no	—
107	C₂×C₄×C₁₀₉	—	4	4	no	—
108	C₂×C₄×C₁₀₉	—	4	2	no	—
109	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
115	C₈×C₁₀₉	—	2	2	no	—
116	C₈×C₁₀₉	—	2	2	no	—
117	C₂²×C₁₀₉	—	2	1	no	—
118	C₂²×C₁₀₉	—	2	1	no	—
119	C₂²×C₁₀₉	—	2	1	no	—
120	C₂²×C₁₀₉	—	4	2	no	—
121	C₂²×C₁₀₉	—	4	1	no	—
122	C₂²×C₁₀₉	—	4	1	no	—
123	C₂²×C₁₀₉	—	4	2	no	—
124	C₂²×C₁₀₉	—	4	2	no	—
125	C₂²×C₁₀₉	—	4	1	no	—
126	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
127	C₄×C₁₀₉	—	2	1	no	—
128	C₄×C₁₀₉	—	2	1	no	—
129	C₄×C₁₀₉	—	2	1	no	—
130	C₄×C₁₀₉	—	2	1	no	—
131	C₄×C₁₀₉	—	4	2	no	—
132	C₄×C₁₀₉	—	4	1	no	—
133	C₄×C₁₀₉	—	4	1	no	—
134	C₄×C₁₀₉	—	4	1	no	—
135	C₄×C₁₀₉	#135	1	2	yes	—
136	C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
137	C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
138	C₂×C₁₀₉	—	4	1	no	—
139	C₂×C₁₀₉	—	8	2	no	—
140	C₂×C₁₀₉	—	8	1	no	—
141	C₂×C₁₀₉	#535	1	1	yes	characteristic
142	C₂×C₁₀₉	C₂×#135	1	1	yes	characteristic
143	C₂×C₁₀₉	#575	1	1	yes	characteristic
144	#3212	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
145	C₁₀₉	#3212	1	1	yes	characteristic, direct product factor, Sylow
146	#532	C₂×C₁₀₉	1	1	yes	characteristic, complemented
147	#529	C₂×C₁₀₉	1	1	yes	characteristic, complemented
148	#495	C₂×C₁₀₉	1	2	yes	complemented
149	#488	C₂×C₁₀₉	1	2	yes	complemented
150	#462	C₂×C₁₀₉	1	1	yes	characteristic, complemented
151	#461	C₂×C₁₀₉	1	1	yes	characteristic, complemented
152	#460	C₂×C₁₀₉	1	1	yes	characteristic, complemented
153	#459	C₂×C₁₀₉	1	1	yes	characteristic, complemented
154	#452	C₂×C₁₀₉	1	2	yes	complemented
155	#449	C₂×C₁₀₉	1	2	yes	complemented
156	#431	C₂×C₁₀₉	1	1	yes	characteristic, complemented
157	C₂×#41	C₂²×C₁₀₉	1	2	yes	—
158	C₂×Q₁₆	C₂²×C₁₀₉	1	2	yes	—
159	C₂×SD₁₆	C₂²×C₁₀₉	1	2	yes	—
160	C₂×SD₁₆	C₂²×C₁₀₉	1	2	yes	—
161	C₂×D₈	C₂²×C₁₀₉	1	2	yes	—
162	C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
163	#127	C₂²×C₁₀₉	1	1	yes	characteristic
164	#126	—	2	1	no	—
165	#125	C₂²×C₁₀₉	1	1	yes	characteristic
166	#124	C₂²×C₁₀₉	1	1	yes	characteristic
167	#124	C₂²×C₁₀₉	1	2	yes	—
168	#123	—	2	1	no	—
169	C₂²⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
170	#121	—	2	1	no	—
171	#121	—	2	1	no	—
172	#121	C₂²×C₁₀₉	1	1	yes	characteristic
173	C₄×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
174	C₄×D₄	—	2	1	no	—
175	C₄×D₄	—	2	1	no	—
176	C₄×D₄	—	2	1	no	—
177	C₄×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
178	#117	—	2	1	no	—
179	C₂×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
180	C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
181	C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
182	#103	C₂²×C₁₀₉	1	2	yes	—
183	#103	C₂²×C₁₀₉	1	2	yes	—
184	#102	C₂²×C₁₀₉	1	2	yes	—
185	#102	C₂²×C₁₀₉	1	2	yes	—
186	C₂²⋊C₈	C₂²×C₁₀₉	1	2	yes	—
187	C₄×C₈	C₂²×C₁₀₉	1	2	yes	—
188	#41	—	4	2	no	—
189	#41	—	4	2	no	—
190	C₂×Q₈	C₂³×C₁₀₉	1	2	yes	—
191	C₂×Q₈	C₂³×C₁₀₉	1	1	yes	characteristic
192	C₂×D₄	—	2	1	no	—
193	C₂×D₄	—	2	2	no	—
194	C₂×D₄	—	2	1	no	—
195	C₂×D₄	—	2	1	no	—
196	C₂×D₄	C₂³×C₁₀₉	1	2	yes	—
197	C₂×D₄	C₂³×C₁₀₉	1	1	yes	characteristic
198	C₂²×C₄	—	2	1	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	D₄×C₁₀₉	1	1	yes	characteristic
201	C₂²×C₄	D₄×C₁₀₉	1	1	yes	characteristic
202	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
203	Q₁₆	—	4	2	no	—
204	SD₁₆	—	4	2	no	—
205	SD₁₆	—	4	2	no	—
206	D₈	—	4	2	no	—
207	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
208	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
209	C₄⋊C₄	—	2	1	no	—
210	C₄⋊C₄	—	2	1	no	—
211	C₄⋊C₄	—	2	1	no	—
212	C₄⋊C₄	—	2	1	no	—
213	C₄⋊C₄	—	2	1	no	—
214	C₄⋊C₄	—	2	1	no	—
215	C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
216	C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
217	C₂²⋊C₄	—	2	1	no	—
218	C₂²⋊C₄	—	2	1	no	—
219	C₂²⋊C₄	—	2	1	no	—
220	C₂²⋊C₄	—	2	2	no	—
221	C₂²⋊C₄	—	2	2	no	—
222	C₂²⋊C₄	—	2	1	no	—
223	C₂²⋊C₄	—	2	1	no	—
224	C₄²	—	2	1	no	—
225	C₄²	D₄×C₁₀₉	1	1	yes	characteristic
226	C₄²	D₄×C₁₀₉	1	1	yes	characteristic
227	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
228	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
229	C₂³	—	2	1	no	—
230	C₂³	—	2	2	no	—
231	C₂³	C₂×D₄×C₁₀₉	1	1	yes	characteristic
232	Q₈	—	2	1	no	—
233	Q₈	—	2	2	no	—
234	Q₈	—	2	1	no	—
235	Q₈	—	2	2	no	—
236	D₄	—	2	1	no	—
237	D₄	—	2	2	no	—
238	D₄	—	2	1	no	—
239	D₄	—	2	2	no	—
240	D₄	—	4	2	no	—
241	D₄	—	4	4	no	—
242	D₄	—	4	2	no	—
243	D₄	—	4	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
255	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
256	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
257	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
258	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
259	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
260	C₈	—	2	2	no	—
261	C₈	—	2	2	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	4	2	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	4	1	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	4	2	no	—
270	C₂²	—	4	1	no	—
271	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	4	2	no	—
277	C₄	—	4	1	no	—
278	C₄	—	4	1	no	—
279	C₄	—	4	1	no	—
280	C₄	#135×C₁₀₉	1	2	yes	—
281	C₄	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
282	C₄	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
283	C₂	—	4	1	no	—
284	C₂	—	8	2	no	—
285	C₂	—	8	1	no	—
286	C₂	#535×C₁₀₉	1	1	yes	characteristic
287	C₂	C₂×#135×C₁₀₉	1	1	yes	characteristic
288	C₂	#575×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₀₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₀₉
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. #532×C₁₀₉	C₂	semidirect
2. #529×C₁₀₉	C₂	semidirect
3. #495×C₁₀₉	C₂	semidirect
4. #488×C₁₀₉	C₂	semidirect
5. #462×C₁₀₉	C₂	semidirect
6. #461×C₁₀₉	C₂	semidirect
7. #460×C₁₀₉	C₂	semidirect
8. #459×C₁₀₉	C₂	semidirect
9. #452×C₁₀₉	C₂	semidirect
10. #449×C₁₀₉	C₂	semidirect
11. #431×C₁₀₉	C₂	semidirect
144. #3212	C₁₀₉	direct (N × H)
145. C₁₀₉	#3212	direct (N × H)
146. #532	C₂×C₁₀₉	semidirect
147. #529	C₂×C₁₀₉	semidirect
148. #495	C₂×C₁₀₉	semidirect
149. #488	C₂×C₁₀₉	semidirect
150. #462	C₂×C₁₀₉	semidirect
151. #461	C₂×C₁₀₉	semidirect
152. #460	C₂×C₁₀₉	semidirect
153. #459	C₂×C₁₀₉	semidirect
154. #452	C₂×C₁₀₉	semidirect
155. #449	C₂×C₁₀₉	semidirect
156. #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
#532×C₁₀₉	2
#529×C₁₀₉	2
#495×C₁₀₉	2
#488×C₁₀₉	2
#462×C₁₀₉	2
#461×C₁₀₉	2
#460×C₁₀₉	2
#459×C₁₀₉	2
#452×C₁₀₉	2
#449×C₁₀₉	2
#431×C₁₀₉	2
#3212	109

Subgroups — C109×#3212 #28360466

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#3212 #28360466