Subgroups — C₁₀₉×#3305 #28360559

Order 13952.

408 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#555×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
2	#550×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#486×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#482×C₁₀₉	C₂	1	2	yes	maximal, complemented
5	#477×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#466×C₁₀₉	C₂	1	2	yes	maximal, complemented
7	#459×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#449×C₁₀₉	C₂	1	2	yes	maximal, complemented
9	C₄×SD₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×D₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
11	#416×C₁₀₉	C₂	1	2	yes	maximal, complemented
12	C₂×#41×C₁₀₉	C₂²	1	2	yes	—
13	C₂×SD₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₂×D₈×C₁₀₉	C₂²	1	2	yes	—
15	C₂²×C₈×C₁₀₉	C₂²	1	2	yes	—
16	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
17	#127×C₁₀₉	—	2	1	no	—
18	#127×C₁₀₉	C₂²	1	1	yes	characteristic
19	#124×C₁₀₉	—	2	2	no	—
20	#121×C₁₀₉	—	2	1	no	—
21	#121×C₁₀₉	—	2	2	no	—
22	#121×C₁₀₉	C₂²	1	2	yes	—
23	#121×C₁₀₉	C₂²	1	2	yes	—
24	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
25	C₄×D₄×C₁₀₉	—	2	1	no	—
26	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
27	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
28	#117×C₁₀₉	—	2	1	no	—
29	#117×C₁₀₉	C₂²	1	2	yes	—
30	#107×C₁₀₉	C₂²	1	2	yes	—
31	#106×C₁₀₉	C₂²	1	1	yes	characteristic
32	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
33	#103×C₁₀₉	C₂²	1	1	yes	characteristic
34	#103×C₁₀₉	C₂²	1	2	yes	—
35	#102×C₁₀₉	C₂²	1	2	yes	—
36	#102×C₁₀₉	C₂²	1	2	yes	—
37	#102×C₁₀₉	C₂²	1	1	yes	characteristic
38	#102×C₁₀₉	C₂²	1	2	yes	—
39	C₂²⋊C₈×C₁₀₉	C₂²	1	2	yes	—
40	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
41	#41×C₁₀₉	—	2	4	no	—
42	#41×C₁₀₉	—	4	2	no	—
43	C₂×Q₈×C₁₀₉	—	2	1	no	—
44	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
45	C₂×D₄×C₁₀₉	—	2	2	no	—
46	C₂×D₄×C₁₀₉	—	2	2	no	—
47	C₂×D₄×C₁₀₉	—	2	2	no	—
48	C₂×D₄×C₁₀₉	—	2	1	no	—
49	C₂×D₄×C₁₀₉	—	2	1	no	—
50	C₂×D₄×C₁₀₉	D₄	1	1	yes	characteristic
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	C₂³	1	2	yes	—
56	SD₁₆×C₁₀₉	—	2	2	no	—
57	D₈×C₁₀₉	—	4	2	no	—
58	C₂×C₈×C₁₀₉	—	4	2	no	—
59	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
60	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
65	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
66	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
67	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
68	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
69	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
70	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
71	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
72	C₂²⋊C₄×C₁₀₉	D₄	1	2	yes	—
73	C₄²×C₁₀₉	—	2	2	no	—
74	C₄²×C₁₀₉	—	2	1	no	—
75	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
76	C₂³×C₁₀₉	—	2	1	no	—
77	C₂³×C₁₀₉	—	2	2	no	—
78	C₂³×C₁₀₉	C₂×D₄	1	2	yes	—
79	Q₈×C₁₀₉	—	2	1	no	—
80	Q₈×C₁₀₉	—	8	1	no	—
81	Q₈×C₁₀₉	#41	1	2	yes	—
82	D₄×C₁₀₉	—	2	2	no	—
83	D₄×C₁₀₉	—	2	1	no	—
84	D₄×C₁₀₉	—	2	2	no	—
85	D₄×C₁₀₉	—	2	1	no	—
86	D₄×C₁₀₉	—	2	2	no	—
87	D₄×C₁₀₉	—	4	4	no	—
88	D₄×C₁₀₉	—	4	2	no	—
89	D₄×C₁₀₉	—	8	1	no	—
90	D₄×C₁₀₉	—	8	2	no	—
91	D₄×C₁₀₉	—	8	2	no	—
92	C₂×C₄×C₁₀₉	—	2	1	no	—
93	C₂×C₄×C₁₀₉	—	2	2	no	—
94	C₂×C₄×C₁₀₉	—	2	2	no	—
95	C₂×C₄×C₁₀₉	—	2	1	no	—
96	C₂×C₄×C₁₀₉	—	2	1	no	—
97	C₂×C₄×C₁₀₉	—	2	2	no	—
98	C₂×C₄×C₁₀₉	—	4	4	no	—
99	C₂×C₄×C₁₀₉	—	4	2	no	—
100	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
101	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
104	C₈×C₁₀₉	—	2	1	no	—
105	C₈×C₁₀₉	—	2	1	no	—
106	C₈×C₁₀₉	—	2	2	no	—
107	C₂²×C₁₀₉	—	2	2	no	—
108	C₂²×C₁₀₉	—	2	2	no	—
109	C₂²×C₁₀₉	—	2	2	no	—
110	C₂²×C₁₀₉	—	4	2	no	—
111	C₂²×C₁₀₉	—	4	1	no	—
112	C₂²×C₁₀₉	—	4	1	no	—
113	C₂²×C₁₀₉	—	4	2	no	—
114	C₂²×C₁₀₉	—	4	2	no	—
115	C₂²×C₁₀₉	—	4	1	no	—
116	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
117	C₄×C₁₀₉	—	2	2	no	—
118	C₄×C₁₀₉	—	2	2	no	—
119	C₄×C₁₀₉	—	2	1	no	—
120	C₄×C₁₀₉	—	4	1	no	—
121	C₄×C₁₀₉	—	4	1	no	—
122	C₄×C₁₀₉	—	4	2	no	—
123	C₄×C₁₀₉	#135	1	2	yes	—
124	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
125	C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
126	C₂×C₁₀₉	—	4	2	no	—
127	C₂×C₁₀₉	—	8	2	no	—
128	C₂×C₁₀₉	—	8	1	no	—
129	C₂×C₁₀₉	#546	1	1	yes	characteristic
130	C₂×C₁₀₉	C₂×#135	1	1	yes	characteristic
131	C₂×C₁₀₉	#576	1	1	yes	characteristic
132	#3305	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
133	C₁₀₉	#3305	1	1	yes	characteristic, direct product factor, Sylow
134	#555	C₂×C₁₀₉	1	1	yes	characteristic, complemented
135	#550	C₂×C₁₀₉	1	1	yes	characteristic, complemented
136	#486	C₂×C₁₀₉	1	1	yes	characteristic, complemented
137	#482	C₂×C₁₀₉	1	2	yes	complemented
138	#477	C₂×C₁₀₉	1	1	yes	characteristic, complemented
139	#466	C₂×C₁₀₉	1	2	yes	complemented
140	#459	C₂×C₁₀₉	1	1	yes	characteristic, complemented
141	#449	C₂×C₁₀₉	1	2	yes	complemented
142	C₄×SD₁₆	C₂×C₁₀₉	1	1	yes	characteristic, complemented
143	C₈×D₄	C₂×C₁₀₉	1	1	yes	characteristic, complemented
144	#416	C₂×C₁₀₉	1	2	yes	complemented
145	C₂×#41	C₂²×C₁₀₉	1	2	yes	—
146	C₂×SD₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
147	C₂×D₈	C₂²×C₁₀₉	1	2	yes	—
148	C₂²×C₈	C₂²×C₁₀₉	1	2	yes	—
149	C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
150	#127	—	2	1	no	—
151	#127	C₂²×C₁₀₉	1	1	yes	characteristic
152	#124	—	2	2	no	—
153	#121	—	2	1	no	—
154	#121	—	2	2	no	—
155	#121	C₂²×C₁₀₉	1	2	yes	—
156	#121	C₂²×C₁₀₉	1	2	yes	—
157	C₄×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
158	C₄×D₄	—	2	1	no	—
159	C₄×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
160	C₄×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
161	#117	—	2	1	no	—
162	#117	C₂²×C₁₀₉	1	2	yes	—
163	#107	C₂²×C₁₀₉	1	2	yes	—
164	#106	C₂²×C₁₀₉	1	1	yes	characteristic
165	C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
166	#103	C₂²×C₁₀₉	1	1	yes	characteristic
167	#103	C₂²×C₁₀₉	1	2	yes	—
168	#102	C₂²×C₁₀₉	1	2	yes	—
169	#102	C₂²×C₁₀₉	1	2	yes	—
170	#102	C₂²×C₁₀₉	1	1	yes	characteristic
171	#102	C₂²×C₁₀₉	1	2	yes	—
172	C₂²⋊C₈	C₂²×C₁₀₉	1	2	yes	—
173	C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
174	#41	—	2	4	no	—
175	#41	—	4	2	no	—
176	C₂×Q₈	—	2	1	no	—
177	C₂×Q₈	C₂³×C₁₀₉	1	1	yes	characteristic
178	C₂×D₄	—	2	2	no	—
179	C₂×D₄	—	2	2	no	—
180	C₂×D₄	—	2	2	no	—
181	C₂×D₄	—	2	1	no	—
182	C₂×D₄	—	2	1	no	—
183	C₂×D₄	D₄×C₁₀₉	1	1	yes	characteristic
184	C₂×D₄	C₂³×C₁₀₉	1	2	yes	—
185	C₂×D₄	C₂³×C₁₀₉	1	1	yes	characteristic
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
189	SD₁₆	—	2	2	no	—
190	D₈	—	4	2	no	—
191	C₂×C₈	—	4	2	no	—
192	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
193	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
194	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
195	C₄⋊C₄	—	2	1	no	—
196	C₄⋊C₄	—	2	1	no	—
197	C₄⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
198	C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
199	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
200	C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
201	C₂²⋊C₄	—	2	2	no	—
202	C₂²⋊C₄	—	2	2	no	—
203	C₂²⋊C₄	—	2	2	no	—
204	C₂²⋊C₄	—	2	1	no	—
205	C₂²⋊C₄	D₄×C₁₀₉	1	2	yes	—
206	C₄²	—	2	2	no	—
207	C₄²	—	2	1	no	—
208	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
209	C₂³	—	2	1	no	—
210	C₂³	—	2	2	no	—
211	C₂³	C₂×D₄×C₁₀₉	1	2	yes	—
212	Q₈	—	2	1	no	—
213	Q₈	—	8	1	no	—
214	Q₈	#41×C₁₀₉	1	2	yes	—
215	D₄	—	2	2	no	—
216	D₄	—	2	1	no	—
217	D₄	—	2	2	no	—
218	D₄	—	2	1	no	—
219	D₄	—	2	2	no	—
220	D₄	—	4	4	no	—
221	D₄	—	4	2	no	—
222	D₄	—	8	1	no	—
223	D₄	—	8	2	no	—
224	D₄	—	8	2	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	4	4	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
234	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
235	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
236	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
237	C₈	—	2	1	no	—
238	C₈	—	2	1	no	—
239	C₈	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	2	2	no	—
243	C₂²	—	4	2	no	—
244	C₂²	—	4	1	no	—
245	C₂²	—	4	1	no	—
246	C₂²	—	4	2	no	—
247	C₂²	—	4	2	no	—
248	C₂²	—	4	1	no	—
249	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
250	C₄	—	2	2	no	—
251	C₄	—	2	2	no	—
252	C₄	—	2	1	no	—
253	C₄	—	4	1	no	—
254	C₄	—	4	1	no	—
255	C₄	—	4	2	no	—
256	C₄	#135×C₁₀₉	1	2	yes	—
257	C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
258	C₄	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
259	C₂	—	4	2	no	—
260	C₂	—	8	2	no	—
261	C₂	—	8	1	no	—
262	C₂	#546×C₁₀₉	1	1	yes	characteristic
263	C₂	C₂×#135×C₁₀₉	1	1	yes	characteristic
264	C₂	#576×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 ◁ C₄×D₄×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. #555×C₁₀₉	C₂	semidirect
2. #550×C₁₀₉	C₂	semidirect
3. #486×C₁₀₉	C₂	semidirect
4. #482×C₁₀₉	C₂	semidirect
5. #477×C₁₀₉	C₂	semidirect
6. #466×C₁₀₉	C₂	semidirect
7. #459×C₁₀₉	C₂	semidirect
8. #449×C₁₀₉	C₂	semidirect
9. C₄×SD₁₆×C₁₀₉	C₂	semidirect
10. C₈×D₄×C₁₀₉	C₂	semidirect
11. #416×C₁₀₉	C₂	semidirect
132. #3305	C₁₀₉	direct (N × H)
133. C₁₀₉	#3305	direct (N × H)
134. #555	C₂×C₁₀₉	semidirect
135. #550	C₂×C₁₀₉	semidirect
136. #486	C₂×C₁₀₉	semidirect
137. #482	C₂×C₁₀₉	semidirect
138. #477	C₂×C₁₀₉	semidirect
139. #466	C₂×C₁₀₉	semidirect
140. #459	C₂×C₁₀₉	semidirect
141. #449	C₂×C₁₀₉	semidirect
142. C₄×SD₁₆	C₂×C₁₀₉	semidirect
143. C₈×D₄	C₂×C₁₀₉	semidirect
144. #416	C₂×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#555×C₁₀₉	2
#550×C₁₀₉	2
#486×C₁₀₉	2
#482×C₁₀₉	2
#477×C₁₀₉	2
#466×C₁₀₉	2
#459×C₁₀₉	2
#449×C₁₀₉	2
C₄×SD₁₆×C₁₀₉	2
C₈×D₄×C₁₀₉	2
#416×C₁₀₉	2
#3305	109

Subgroups — C109×#3305 #28360559

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#3305 #28360559