Subgroups — C₁₀₉×#3105 #28360359

Order 13952.

454 conjugacy classes of subgroups.

The subgroup lattice has 282 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, complemented
2	C₂×C₄×D₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#484×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#481×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#478×C₁₀₉	C₂	1	2	yes	maximal, complemented
6	#463×C₁₀₉	C₂	1	2	yes	maximal, complemented
7	#450×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#447×C₁₀₉	C₂	1	1	yes	maximal, characteristic
9	C₄×SD₁₆×C₁₀₉	C₂	1	2	yes	maximal, complemented
10	C₄×D₈×C₁₀₉	C₂	1	2	yes	maximal, complemented
11	#431×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₁₀₉	—	2	1	no	—
14	C₂×SD₁₆×C₁₀₉	C₂²	1	2	yes	—
15	C₂×D₈×C₁₀₉	C₂²	1	2	yes	—
16	#126×C₁₀₉	—	2	1	no	—
17	#125×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
18	#124×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
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, complemented
23	C₄×D₄×C₁₀₉	—	2	1	no	—
24	C₄×D₄×C₁₀₉	—	2	1	no	—
25	C₄×D₄×C₁₀₉	—	2	1	no	—
26	C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
27	C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
28	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
29	#117×C₁₀₉	—	2	1	no	—
30	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
31	C₂×C₂²⋊C₄×C₁₀₉	—	2	1	no	—
32	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
33	#107×C₁₀₉	C₂²	1	2	yes	complemented
34	#106×C₁₀₉	C₂²	1	2	yes	complemented
35	C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	complemented
36	#103×C₁₀₉	C₂²	1	2	yes	complemented
37	#102×C₁₀₉	C₂²	1	2	yes	—
38	#102×C₁₀₉	C₂²	1	2	yes	—
39	#102×C₁₀₉	C₂²	1	2	yes	complemented
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	complemented
43	C₂⁴×C₁₀₉	—	2	1	no	—
44	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
45	C₂×D₄×C₁₀₉	—	2	4	no	—
46	C₂×D₄×C₁₀₉	—	2	1	no	—
47	C₂×D₄×C₁₀₉	—	2	1	no	—
48	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
49	C₂×D₄×C₁₀₉	C₂³	1	2	yes	—
50	C₂²×C₄×C₁₀₉	—	2	1	no	—
51	C₂²×C₄×C₁₀₉	—	2	2	no	—
52	C₂²×C₄×C₁₀₉	—	2	1	no	—
53	C₂²×C₄×C₁₀₉	—	2	2	no	—
54	C₂²×C₄×C₁₀₉	—	4	4	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	SD₁₆×C₁₀₉	—	2	4	no	—
59	D₈×C₁₀₉	—	2	4	no	—
60	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
61	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
62	C₄⋊C₄×C₁₀₉	—	2	1	no	—
63	C₄⋊C₄×C₁₀₉	—	2	1	no	—
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	C₂³	1	1	yes	characteristic
68	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
69	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
70	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
71	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
72	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
73	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
74	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
75	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
76	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
77	C₄²×C₁₀₉	—	2	1	no	—
78	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
79	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
80	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
81	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
82	C₂³×C₁₀₉	—	2	4	no	—
83	C₂³×C₁₀₉	—	2	2	no	—
84	C₂³×C₁₀₉	—	2	1	no	—
85	C₂³×C₁₀₉	—	4	2	no	—
86	C₂³×C₁₀₉	—	4	2	no	—
87	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
88	Q₈×C₁₀₉	—	2	2	no	—
89	D₄×C₁₀₉	—	2	4	no	—
90	D₄×C₁₀₉	—	2	2	no	—
91	D₄×C₁₀₉	—	2	2	no	—
92	D₄×C₁₀₉	—	4	2	no	—
93	D₄×C₁₀₉	#41	1	4	yes	—
94	C₂×C₄×C₁₀₉	—	2	2	no	—
95	C₂×C₄×C₁₀₉	—	2	2	no	—
96	C₂×C₄×C₁₀₉	—	2	2	no	—
97	C₂×C₄×C₁₀₉	—	2	1	no	—
98	C₂×C₄×C₁₀₉	—	2	1	no	—
99	C₂×C₄×C₁₀₉	—	2	1	no	—
100	C₂×C₄×C₁₀₉	—	2	2	no	—
101	C₂×C₄×C₁₀₉	—	4	4	no	—
102	C₂×C₄×C₁₀₉	—	4	2	no	—
103	C₂×C₄×C₁₀₉	—	4	2	no	—
104	C₂×C₄×C₁₀₉	—	4	4	no	—
105	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
111	C₈×C₁₀₉	—	2	2	no	—
112	C₈×C₁₀₉	—	2	2	no	—
113	C₂²×C₁₀₉	—	2	1	no	—
114	C₂²×C₁₀₉	—	2	1	no	—
115	C₂²×C₁₀₉	—	2	4	no	—
116	C₂²×C₁₀₉	—	4	8	no	—
117	C₂²×C₁₀₉	—	4	2	no	—
118	C₂²×C₁₀₉	—	4	2	no	—
119	C₂²×C₁₀₉	—	4	1	no	—
120	C₂²×C₁₀₉	—	4	1	no	—
121	C₂²×C₁₀₉	—	4	1	no	—
122	C₂²×C₁₀₉	#135	1	2	yes	—
123	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
124	C₄×C₁₀₉	—	2	1	no	—
125	C₄×C₁₀₉	—	2	1	no	—
126	C₄×C₁₀₉	—	2	1	no	—
127	C₄×C₁₀₉	—	2	1	no	—
128	C₄×C₁₀₉	—	4	1	no	—
129	C₄×C₁₀₉	—	4	1	no	—
130	C₄×C₁₀₉	—	4	1	no	—
131	C₄×C₁₀₉	—	4	2	no	—
132	C₄×C₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
133	C₄×C₁₀₉	C₂×#41	1	2	yes	—
134	C₂×C₁₀₉	—	2	2	no	—
135	C₂×C₁₀₉	—	4	4	no	—
136	C₂×C₁₀₉	—	8	1	no	—
137	C₂×C₁₀₉	#525	1	1	yes	characteristic
138	C₂×C₁₀₉	C₂×#135	1	1	yes	characteristic
139	C₂×C₁₀₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
140	#3105	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
141	C₁₀₉	#3105	1	1	yes	characteristic, direct product factor, Sylow
142	#529	C₂×C₁₀₉	1	1	yes	characteristic, complemented
143	C₂×C₄×D₄	C₂×C₁₀₉	1	1	yes	characteristic, complemented
144	#484	C₂×C₁₀₉	1	1	yes	characteristic, complemented
145	#481	C₂×C₁₀₉	1	1	yes	characteristic, complemented
146	#478	C₂×C₁₀₉	1	2	yes	complemented
147	#463	C₂×C₁₀₉	1	2	yes	complemented
148	#450	C₂×C₁₀₉	1	1	yes	characteristic, complemented
149	#447	C₂×C₁₀₉	1	1	yes	characteristic
150	C₄×SD₁₆	C₂×C₁₀₉	1	2	yes	complemented
151	C₄×D₈	C₂×C₁₀₉	1	2	yes	complemented
152	#431	C₂×C₁₀₉	1	1	yes	characteristic, complemented
153	C₂²×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
154	C₂³×C₄	—	2	1	no	—
155	C₂×SD₁₆	C₂²×C₁₀₉	1	2	yes	—
156	C₂×D₈	C₂²×C₁₀₉	1	2	yes	—
157	#126	—	2	1	no	—
158	#125	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
159	#124	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
160	#123	—	2	1	no	—
161	C₂²⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
162	#121	C₂²×C₁₀₉	1	1	yes	characteristic
163	C₄×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
164	C₄×D₄	—	2	1	no	—
165	C₄×D₄	—	2	1	no	—
166	C₄×D₄	—	2	1	no	—
167	C₄×D₄	C₂²×C₁₀₉	1	2	yes	—
168	C₄×D₄	C₂²×C₁₀₉	1	2	yes	—
169	C₄×D₄	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
170	#117	—	2	1	no	—
171	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
172	C₂×C₂²⋊C₄	—	2	1	no	—
173	C₂×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
174	#107	C₂²×C₁₀₉	1	2	yes	complemented
175	#106	C₂²×C₁₀₉	1	2	yes	complemented
176	C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	complemented
177	#103	C₂²×C₁₀₉	1	2	yes	complemented
178	#102	C₂²×C₁₀₉	1	2	yes	—
179	#102	C₂²×C₁₀₉	1	2	yes	—
180	#102	C₂²×C₁₀₉	1	2	yes	complemented
181	C₂²⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
182	C₂²⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
183	C₄×C₈	C₂²×C₁₀₉	1	2	yes	complemented
184	C₂⁴	—	2	1	no	—
185	C₂×Q₈	C₂³×C₁₀₉	1	1	yes	characteristic
186	C₂×D₄	—	2	4	no	—
187	C₂×D₄	—	2	1	no	—
188	C₂×D₄	—	2	1	no	—
189	C₂×D₄	C₂³×C₁₀₉	1	1	yes	characteristic
190	C₂×D₄	C₂³×C₁₀₉	1	2	yes	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	4	4	no	—
196	C₂²×C₄	D₄×C₁₀₉	1	1	yes	characteristic
197	C₂²×C₄	D₄×C₁₀₉	1	1	yes	characteristic
198	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
199	SD₁₆	—	2	4	no	—
200	D₈	—	2	4	no	—
201	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
202	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
203	C₄⋊C₄	—	2	1	no	—
204	C₄⋊C₄	—	2	1	no	—
205	C₄⋊C₄	—	2	1	no	—
206	C₄⋊C₄	—	2	1	no	—
207	C₄⋊C₄	—	2	1	no	—
208	C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
209	C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
210	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
211	C₂²⋊C₄	—	2	1	no	—
212	C₂²⋊C₄	—	2	2	no	—
213	C₂²⋊C₄	—	2	2	no	—
214	C₂²⋊C₄	—	2	1	no	—
215	C₂²⋊C₄	—	2	1	no	—
216	C₂²⋊C₄	—	2	1	no	—
217	C₂²⋊C₄	—	2	1	no	—
218	C₄²	—	2	1	no	—
219	C₄²	D₄×C₁₀₉	1	1	yes	characteristic
220	C₄²	D₄×C₁₀₉	1	1	yes	characteristic
221	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
222	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
223	C₂³	—	2	4	no	—
224	C₂³	—	2	2	no	—
225	C₂³	—	2	1	no	—
226	C₂³	—	4	2	no	—
227	C₂³	—	4	2	no	—
228	C₂³	C₂×D₄×C₁₀₉	1	1	yes	characteristic
229	Q₈	—	2	2	no	—
230	D₄	—	2	4	no	—
231	D₄	—	2	2	no	—
232	D₄	—	2	2	no	—
233	D₄	—	4	2	no	—
234	D₄	#41×C₁₀₉	1	4	yes	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
247	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
248	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
249	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
250	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
251	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
252	C₈	—	2	2	no	—
253	C₈	—	2	2	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	2	4	no	—
257	C₂²	—	4	8	no	—
258	C₂²	—	4	2	no	—
259	C₂²	—	4	2	no	—
260	C₂²	—	4	1	no	—
261	C₂²	—	4	1	no	—
262	C₂²	—	4	1	no	—
263	C₂²	#135×C₁₀₉	1	2	yes	—
264	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	2	1	no	—
268	C₄	—	2	1	no	—
269	C₄	—	4	1	no	—
270	C₄	—	4	1	no	—
271	C₄	—	4	2	no	—
272	C₄	—	4	1	no	—
273	C₄	AGL(1,ℤ/8)×C₁₀₉	1	2	yes	—
274	C₄	C₂×#41×C₁₀₉	1	2	yes	—
275	C₂	—	2	2	no	—
276	C₂	—	4	4	no	—
277	C₂	—	8	1	no	—
278	C₂	#525×C₁₀₉	1	1	yes	characteristic
279	C₂	C₂×#135×C₁₀₉	1	1	yes	characteristic
280	C₂	C₂×AGL(1,ℤ/8)×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₀₉	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₀₉
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	#447×C₁₀₉	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. #529×C₁₀₉	C₂	semidirect
2. C₂×C₄×D₄×C₁₀₉	C₂	semidirect
3. #484×C₁₀₉	C₂	semidirect
4. #481×C₁₀₉	C₂	semidirect
5. #478×C₁₀₉	C₂	semidirect
6. #463×C₁₀₉	C₂	semidirect
7. #450×C₁₀₉	C₂	semidirect
9. C₄×SD₁₆×C₁₀₉	C₂	semidirect
10. C₄×D₈×C₁₀₉	C₂	semidirect
11. #431×C₁₀₉	C₂	semidirect
17. #125×C₁₀₉	C₂²	semidirect
18. #124×C₁₀₉	C₂²	semidirect
22. C₄×Q₈×C₁₀₉	C₂²	semidirect
28. C₄×D₄×C₁₀₉	C₂²	semidirect
33. #107×C₁₀₉	C₂²	semidirect
34. #106×C₁₀₉	C₂²	semidirect
35. C₄⋊C₈×C₁₀₉	C₂²	semidirect
36. #103×C₁₀₉	C₂²	semidirect
39. #102×C₁₀₉	C₂²	semidirect
42. C₄×C₈×C₁₀₉	C₂²	semidirect
140. #3105	C₁₀₉	direct (N × H)
141. C₁₀₉	#3105	direct (N × H)
142. #529	C₂×C₁₀₉	semidirect
143. C₂×C₄×D₄	C₂×C₁₀₉	semidirect
144. #484	C₂×C₁₀₉	semidirect
145. #481	C₂×C₁₀₉	semidirect
146. #478	C₂×C₁₀₉	semidirect
147. #463	C₂×C₁₀₉	semidirect
148. #450	C₂×C₁₀₉	semidirect
150. C₄×SD₁₆	C₂×C₁₀₉	semidirect
151. C₄×D₈	C₂×C₁₀₉	semidirect
152. #431	C₂×C₁₀₉	semidirect
158. #125	C₂²×C₁₀₉	semidirect
159. #124	C₂²×C₁₀₉	semidirect
163. C₄×Q₈	C₂²×C₁₀₉	semidirect
169. C₄×D₄	C₂²×C₁₀₉	semidirect
174. #107	C₂²×C₁₀₉	semidirect
175. #106	C₂²×C₁₀₉	semidirect
176. C₄⋊C₈	C₂²×C₁₀₉	semidirect
177. #103	C₂²×C₁₀₉	semidirect
180. #102	C₂²×C₁₀₉	semidirect
183. C₄×C₈	C₂²×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×C₄×D₄×C₁₀₉	2
#484×C₁₀₉	2
#481×C₁₀₉	2
#478×C₁₀₉	2
#463×C₁₀₉	2
#450×C₁₀₉	2
#447×C₁₀₉	2
C₄×SD₁₆×C₁₀₉	2
C₄×D₈×C₁₀₉	2
#431×C₁₀₉	2
#3105	109

Subgroups — C109×#3105 #28360359

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#3105 #28360359