Subgroups — C₁₀₉×#3399 #28360653

Order 13952.

418 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#559×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
2	#550×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#496×C₁₀₉	C₂	1	2	yes	maximal, complemented
4	#493×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#463×C₁₀₉	C₂	1	2	yes	maximal, complemented
6	#460×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
7	#459×C₁₀₉	C₂	1	2	yes	maximal, complemented
8	#459×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
9	#446×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
10	#442×C₁₀₉	C₂	1	2	yes	maximal, complemented
11	C₄×SD₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₁₀₉	—	2	1	no	—
13	C₂×SD₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₂×SD₁₆×C₁₀₉	C₂²	1	2	yes	—
15	C₂×D₈×C₁₀₉	C₂²	1	2	yes	—
16	C₂×D₈×C₁₀₉	C₂²	1	2	yes	—
17	C₂×D₈×C₁₀₉	C₂²	1	2	yes	—
18	#127×C₁₀₉	C₂²	1	2	yes	—
19	#127×C₁₀₉	C₂²	1	1	yes	characteristic
20	#127×C₁₀₉	C₂²	1	1	yes	characteristic
21	#124×C₁₀₉	—	2	1	no	—
22	#124×C₁₀₉	C₂²	1	2	yes	—
23	#123×C₁₀₉	—	2	2	no	—
24	#121×C₁₀₉	—	2	1	no	—
25	#121×C₁₀₉	—	2	2	no	—
26	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
27	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
28	C₄×D₄×C₁₀₉	—	2	1	no	—
29	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
30	C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
31	C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
32	#106×C₁₀₉	C₂²	1	1	yes	characteristic
33	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
34	C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
35	#103×C₁₀₉	C₂²	1	1	yes	characteristic
36	#102×C₁₀₉	C₂²	1	1	yes	characteristic
37	#102×C₁₀₉	C₂²	1	2	yes	—
38	#102×C₁₀₉	C₂²	1	2	yes	—
39	#97×C₁₀₉	C₂²	1	2	yes	—
40	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
41	#41×C₁₀₉	—	4	4	no	—
42	C₂×Q₈×C₁₀₉	D₄	1	1	yes	characteristic
43	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
44	C₂×D₄×C₁₀₉	—	2	1	no	—
45	C₂×D₄×C₁₀₉	—	2	2	no	—
46	C₂×D₄×C₁₀₉	—	2	1	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	C₂³	1	2	yes	—
51	C₂×D₄×C₁₀₉	C₂³	1	2	yes	—
52	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₁₀₉	—	2	2	no	—
54	C₂²×C₄×C₁₀₉	—	2	1	no	—
55	C₂²×C₄×C₁₀₉	—	2	2	no	—
56	SD₁₆×C₁₀₉	—	2	2	no	—
57	SD₁₆×C₁₀₉	—	4	2	no	—
58	D₈×C₁₀₉	—	2	4	no	—
59	D₈×C₁₀₉	—	4	2	no	—
60	D₈×C₁₀₉	—	4	2	no	—
61	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
62	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
63	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
64	C₄⋊C₄×C₁₀₉	—	2	1	no	—
65	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
66	C₄⋊C₄×C₁₀₉	D₄	1	2	yes	—
67	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
68	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
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	—	2	2	no	—
73	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
74	C₄²×C₁₀₉	—	2	1	no	—
75	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
76	C₄²×C₁₀₉	C₂³	1	2	yes	—
77	C₂³×C₁₀₉	—	2	2	no	—
78	C₂³×C₁₀₉	—	2	2	no	—
79	C₂³×C₁₀₉	—	2	1	no	—
80	Q₈×C₁₀₉	—	2	2	no	—
81	Q₈×C₁₀₉	—	2	1	no	—
82	Q₈×C₁₀₉	—	2	1	no	—
83	D₄×C₁₀₉	—	2	2	no	—
84	D₄×C₁₀₉	—	2	2	no	—
85	D₄×C₁₀₉	—	2	1	no	—
86	D₄×C₁₀₉	—	2	2	no	—
87	D₄×C₁₀₉	—	2	1	no	—
88	D₄×C₁₀₉	—	2	2	no	—
89	D₄×C₁₀₉	—	4	4	no	—
90	D₄×C₁₀₉	—	4	2	no	—
91	D₄×C₁₀₉	—	4	2	no	—
92	D₄×C₁₀₉	—	8	2	no	—
93	D₄×C₁₀₉	—	8	1	no	—
94	D₄×C₁₀₉	—	8	1	no	—
95	C₂×C₄×C₁₀₉	—	2	1	no	—
96	C₂×C₄×C₁₀₉	—	2	1	no	—
97	C₂×C₄×C₁₀₉	—	2	1	no	—
98	C₂×C₄×C₁₀₉	—	4	4	no	—
99	C₂×C₄×C₁₀₉	—	4	2	no	—
100	C₂×C₄×C₁₀₉	—	8	2	no	—
101	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
102	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
103	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
106	C₈×C₁₀₉	—	2	1	no	—
107	C₈×C₁₀₉	—	2	2	no	—
108	C₈×C₁₀₉	#41	1	2	yes	complemented
109	C₂²×C₁₀₉	—	4	2	no	—
110	C₂²×C₁₀₉	—	4	2	no	—
111	C₂²×C₁₀₉	—	4	1	no	—
112	C₂²×C₁₀₉	—	4	2	no	—
113	C₂²×C₁₀₉	—	4	1	no	—
114	C₂²×C₁₀₉	—	4	2	no	—
115	C₂²×C₁₀₉	—	4	2	no	—
116	C₂²×C₁₀₉	—	4	1	no	—
117	C₂²×C₁₀₉	—	4	2	no	—
118	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
119	C₄×C₁₀₉	—	2	2	no	—
120	C₄×C₁₀₉	—	2	2	no	—
121	C₄×C₁₀₉	—	2	1	no	—
122	C₄×C₁₀₉	—	4	1	no	—
123	C₄×C₁₀₉	—	4	1	no	—
124	C₄×C₁₀₉	—	4	1	no	—
125	C₄×C₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
126	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
127	C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
128	C₂×C₁₀₉	—	8	2	no	—
129	C₂×C₁₀₉	—	8	1	no	—
130	C₂×C₁₀₉	—	8	2	no	—
131	C₂×C₁₀₉	#550	1	1	yes	characteristic
132	C₂×C₁₀₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
133	C₂×C₁₀₉	#576	1	1	yes	characteristic
134	#3399	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
135	C₁₀₉	#3399	1	1	yes	characteristic, direct product factor, Sylow
136	#559	C₂×C₁₀₉	1	1	yes	characteristic, complemented
137	#550	C₂×C₁₀₉	1	1	yes	characteristic, complemented
138	#496	C₂×C₁₀₉	1	2	yes	complemented
139	#493	C₂×C₁₀₉	1	1	yes	characteristic, complemented
140	#463	C₂×C₁₀₉	1	2	yes	complemented
141	#460	C₂×C₁₀₉	1	1	yes	characteristic, complemented
142	#459	C₂×C₁₀₉	1	2	yes	complemented
143	#459	C₂×C₁₀₉	1	1	yes	characteristic, complemented
144	#446	C₂×C₁₀₉	1	1	yes	characteristic, complemented
145	#442	C₂×C₁₀₉	1	2	yes	complemented
146	C₄×SD₁₆	C₂×C₁₀₉	1	1	yes	characteristic, complemented
147	C₂×#41	—	2	1	no	—
148	C₂×SD₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
149	C₂×SD₁₆	C₂²×C₁₀₉	1	2	yes	—
150	C₂×D₈	C₂²×C₁₀₉	1	2	yes	—
151	C₂×D₈	C₂²×C₁₀₉	1	2	yes	—
152	C₂×D₈	C₂²×C₁₀₉	1	2	yes	—
153	#127	C₂²×C₁₀₉	1	2	yes	—
154	#127	C₂²×C₁₀₉	1	1	yes	characteristic
155	#127	C₂²×C₁₀₉	1	1	yes	characteristic
156	#124	—	2	1	no	—
157	#124	C₂²×C₁₀₉	1	2	yes	—
158	#123	—	2	2	no	—
159	#121	—	2	1	no	—
160	#121	—	2	2	no	—
161	C₄×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
162	C₄×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
163	C₄×D₄	—	2	1	no	—
164	C₄×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
165	C₄×D₄	C₂²×C₁₀₉	1	2	yes	—
166	C₄×D₄	C₂²×C₁₀₉	1	2	yes	—
167	#106	C₂²×C₁₀₉	1	1	yes	characteristic
168	C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
169	C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
170	#103	C₂²×C₁₀₉	1	1	yes	characteristic
171	#102	C₂²×C₁₀₉	1	1	yes	characteristic
172	#102	C₂²×C₁₀₉	1	2	yes	—
173	#102	C₂²×C₁₀₉	1	2	yes	—
174	#97	C₂²×C₁₀₉	1	2	yes	—
175	C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
176	#41	—	4	4	no	—
177	C₂×Q₈	D₄×C₁₀₉	1	1	yes	characteristic
178	C₂×Q₈	C₂³×C₁₀₉	1	1	yes	characteristic
179	C₂×D₄	—	2	1	no	—
180	C₂×D₄	—	2	2	no	—
181	C₂×D₄	—	2	1	no	—
182	C₂×D₄	—	2	2	no	—
183	C₂×D₄	—	2	1	no	—
184	C₂×D₄	—	2	1	no	—
185	C₂×D₄	C₂³×C₁₀₉	1	2	yes	—
186	C₂×D₄	C₂³×C₁₀₉	1	2	yes	—
187	C₂×D₄	C₂³×C₁₀₉	1	1	yes	characteristic
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	—	2	2	no	—
191	SD₁₆	—	2	2	no	—
192	SD₁₆	—	4	2	no	—
193	D₈	—	2	4	no	—
194	D₈	—	4	2	no	—
195	D₈	—	4	2	no	—
196	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
197	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
198	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
199	C₄⋊C₄	—	2	1	no	—
200	C₄⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
201	C₄⋊C₄	D₄×C₁₀₉	1	2	yes	—
202	C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
203	C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
204	C₂²⋊C₄	—	2	2	no	—
205	C₂²⋊C₄	—	2	2	no	—
206	C₂²⋊C₄	—	2	1	no	—
207	C₂²⋊C₄	—	2	2	no	—
208	C₂²⋊C₄	—	2	2	no	—
209	C₄²	—	2	1	no	—
210	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
211	C₄²	C₂³×C₁₀₉	1	2	yes	—
212	C₂³	—	2	2	no	—
213	C₂³	—	2	2	no	—
214	C₂³	—	2	1	no	—
215	Q₈	—	2	2	no	—
216	Q₈	—	2	1	no	—
217	Q₈	—	2	1	no	—
218	D₄	—	2	2	no	—
219	D₄	—	2	2	no	—
220	D₄	—	2	1	no	—
221	D₄	—	2	2	no	—
222	D₄	—	2	1	no	—
223	D₄	—	2	2	no	—
224	D₄	—	4	4	no	—
225	D₄	—	4	2	no	—
226	D₄	—	4	2	no	—
227	D₄	—	8	2	no	—
228	D₄	—	8	1	no	—
229	D₄	—	8	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	8	2	no	—
236	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
237	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
238	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
239	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
240	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
241	C₈	—	2	1	no	—
242	C₈	—	2	2	no	—
243	C₈	#41×C₁₀₉	1	2	yes	complemented
244	C₂²	—	4	2	no	—
245	C₂²	—	4	2	no	—
246	C₂²	—	4	1	no	—
247	C₂²	—	4	2	no	—
248	C₂²	—	4	1	no	—
249	C₂²	—	4	2	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	4	1	no	—
252	C₂²	—	4	2	no	—
253	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	2	1	no	—
257	C₄	—	4	1	no	—
258	C₄	—	4	1	no	—
259	C₄	—	4	1	no	—
260	C₄	AGL(1,ℤ/8)×C₁₀₉	1	2	yes	—
261	C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
262	C₄	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
263	C₂	—	8	2	no	—
264	C₂	—	8	1	no	—
265	C₂	—	8	2	no	—
266	C₂	#550×C₁₀₉	1	1	yes	characteristic
267	C₂	C₂×AGL(1,ℤ/8)×C₁₀₉	1	1	yes	characteristic
268	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×Q₈×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. #559×C₁₀₉	C₂	semidirect
2. #550×C₁₀₉	C₂	semidirect
3. #496×C₁₀₉	C₂	semidirect
4. #493×C₁₀₉	C₂	semidirect
5. #463×C₁₀₉	C₂	semidirect
6. #460×C₁₀₉	C₂	semidirect
7. #459×C₁₀₉	C₂	semidirect
8. #459×C₁₀₉	C₂	semidirect
9. #446×C₁₀₉	C₂	semidirect
10. #442×C₁₀₉	C₂	semidirect
11. C₄×SD₁₆×C₁₀₉	C₂	semidirect
108. C₈×C₁₀₉	#41	semidirect
134. #3399	C₁₀₉	direct (N × H)
135. C₁₀₉	#3399	direct (N × H)
136. #559	C₂×C₁₀₉	semidirect
137. #550	C₂×C₁₀₉	semidirect
138. #496	C₂×C₁₀₉	semidirect
139. #493	C₂×C₁₀₉	semidirect
140. #463	C₂×C₁₀₉	semidirect
141. #460	C₂×C₁₀₉	semidirect
142. #459	C₂×C₁₀₉	semidirect
143. #459	C₂×C₁₀₉	semidirect
144. #446	C₂×C₁₀₉	semidirect
145. #442	C₂×C₁₀₉	semidirect
146. C₄×SD₁₆	C₂×C₁₀₉	semidirect
243. C₈	#41×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#559×C₁₀₉	2
#550×C₁₀₉	2
#496×C₁₀₉	2
#493×C₁₀₉	2
#463×C₁₀₉	2
#460×C₁₀₉	2
#459×C₁₀₉	2
#446×C₁₀₉	2
#442×C₁₀₉	2
C₄×SD₁₆×C₁₀₉	2
#3399	109

Subgroups — C109×#3399 #28360653

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#3399 #28360653