Subgroups — C₁₀₉×#3333 #28360587

Order 13952.

398 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#554×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
2	#550×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	#486×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#480×C₁₀₉	C₂	1	2	yes	maximal, complemented
5	#477×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#468×C₁₀₉	C₂	1	2	yes	maximal, complemented
7	#460×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#452×C₁₀₉	C₂	1	2	yes	maximal, complemented
9	C₄×Q₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
10	#436×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
11	#417×C₁₀₉	C₂	1	2	yes	maximal, complemented
12	C₂×#41×C₁₀₉	C₂²	1	2	yes	—
13	C₂×Q₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₂×SD₁₆×C₁₀₉	C₂²	1	2	yes	—
15	C₂×M₁₆×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	#125×C₁₀₉	—	2	1	no	—
20	C₂²⋊Q₈×C₁₀₉	—	2	2	no	—
21	C₂²⋊Q₈×C₁₀₉	C₂²	1	2	yes	—
22	#121×C₁₀₉	—	2	2	no	—
23	#121×C₁₀₉	C₂²	1	2	yes	—
24	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
25	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
26	C₄×D₄×C₁₀₉	—	2	1	no	—
27	C₄×D₄×C₁₀₉	—	2	1	no	—
28	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
30	#107×C₁₀₉	C₂²	1	1	yes	characteristic
31	#107×C₁₀₉	C₂²	1	2	yes	—
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	#103×C₁₀₉	C₂²	1	1	yes	characteristic
36	#102×C₁₀₉	C₂²	1	2	yes	—
37	#102×C₁₀₉	C₂²	1	2	yes	—
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×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
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	C₂²×C₄×C₁₀₉	—	2	2	no	—
53	C₂²×C₄×C₁₀₉	—	2	2	no	—
54	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
55	Q₁₆×C₁₀₉	—	2	2	no	—
56	SD₁₆×C₁₀₉	—	4	2	no	—
57	M₁₆×C₁₀₉	—	4	2	no	—
58	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
59	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
60	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
61	C₄⋊C₄×C₁₀₉	—	2	2	no	—
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	2	no	—
66	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
67	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
68	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
69	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
70	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
71	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
72	C₂²⋊C₄×C₁₀₉	D₄	1	2	yes	—
73	C₄²×C₁₀₉	—	2	1	no	—
74	C₄²×C₁₀₉	—	2	1	no	—
75	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
76	C₂³×C₁₀₉	—	2	2	no	—
77	C₂³×C₁₀₉	C₂×D₄	1	2	yes	—
78	Q₈×C₁₀₉	—	2	1	no	—
79	Q₈×C₁₀₉	—	2	1	no	—
80	Q₈×C₁₀₉	—	2	1	no	—
81	Q₈×C₁₀₉	—	8	1	no	—
82	Q₈×C₁₀₉	#41	1	2	yes	—
83	D₄×C₁₀₉	—	2	2	no	—
84	D₄×C₁₀₉	—	2	2	no	—
85	D₄×C₁₀₉	—	2	2	no	—
86	D₄×C₁₀₉	—	4	4	no	—
87	D₄×C₁₀₉	—	4	2	no	—
88	D₄×C₁₀₉	—	8	1	no	—
89	D₄×C₁₀₉	—	8	2	no	—
90	C₂×C₄×C₁₀₉	—	2	1	no	—
91	C₂×C₄×C₁₀₉	—	2	2	no	—
92	C₂×C₄×C₁₀₉	—	2	2	no	—
93	C₂×C₄×C₁₀₉	—	2	1	no	—
94	C₂×C₄×C₁₀₉	—	2	1	no	—
95	C₂×C₄×C₁₀₉	—	2	1	no	—
96	C₂×C₄×C₁₀₉	—	2	2	no	—
97	C₂×C₄×C₁₀₉	—	4	4	no	—
98	C₂×C₄×C₁₀₉	—	8	2	no	—
99	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
101	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
103	C₈×C₁₀₉	—	2	1	no	—
104	C₈×C₁₀₉	—	2	1	no	—
105	C₈×C₁₀₉	—	2	2	no	—
106	C₂²×C₁₀₉	—	2	2	no	—
107	C₂²×C₁₀₉	—	2	2	no	—
108	C₂²×C₁₀₉	—	2	2	no	—
109	C₂²×C₁₀₉	—	4	2	no	—
110	C₂²×C₁₀₉	—	4	2	no	—
111	C₂²×C₁₀₉	—	4	2	no	—
112	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
113	C₄×C₁₀₉	—	2	2	no	—
114	C₄×C₁₀₉	—	2	2	no	—
115	C₄×C₁₀₉	—	2	1	no	—
116	C₄×C₁₀₉	—	4	1	no	—
117	C₄×C₁₀₉	—	4	1	no	—
118	C₄×C₁₀₉	—	4	2	no	—
119	C₄×C₁₀₉	—	4	1	no	—
120	C₄×C₁₀₉	#137	1	2	yes	—
121	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
122	C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
123	C₂×C₁₀₉	—	4	2	no	—
124	C₂×C₁₀₉	—	8	2	no	—
125	C₂×C₁₀₉	#546	1	1	yes	characteristic
126	C₂×C₁₀₉	C₂×#137	1	1	yes	characteristic
127	C₂×C₁₀₉	#576	1	1	yes	characteristic
128	#3333	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
129	C₁₀₉	#3333	1	1	yes	characteristic, direct product factor, Sylow
130	#554	C₂×C₁₀₉	1	1	yes	characteristic, complemented
131	#550	C₂×C₁₀₉	1	1	yes	characteristic
132	#486	C₂×C₁₀₉	1	1	yes	characteristic, complemented
133	#480	C₂×C₁₀₉	1	2	yes	complemented
134	#477	C₂×C₁₀₉	1	1	yes	characteristic, complemented
135	#468	C₂×C₁₀₉	1	2	yes	complemented
136	#460	C₂×C₁₀₉	1	1	yes	characteristic, complemented
137	#452	C₂×C₁₀₉	1	2	yes	complemented
138	C₄×Q₁₆	C₂×C₁₀₉	1	1	yes	characteristic, complemented
139	#436	C₂×C₁₀₉	1	1	yes	characteristic, complemented
140	#417	C₂×C₁₀₉	1	2	yes	complemented
141	C₂×#41	C₂²×C₁₀₉	1	2	yes	—
142	C₂×Q₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
143	C₂×SD₁₆	C₂²×C₁₀₉	1	2	yes	—
144	C₂×M₁₆	C₂²×C₁₀₉	1	2	yes	—
145	C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
146	#127	—	2	1	no	—
147	#127	C₂²×C₁₀₉	1	1	yes	characteristic
148	#125	—	2	1	no	—
149	C₂²⋊Q₈	—	2	2	no	—
150	C₂²⋊Q₈	C₂²×C₁₀₉	1	2	yes	—
151	#121	—	2	2	no	—
152	#121	C₂²×C₁₀₉	1	2	yes	—
153	C₄×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
154	C₄×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
155	C₄×D₄	—	2	1	no	—
156	C₄×D₄	—	2	1	no	—
157	C₄×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
158	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
159	#107	C₂²×C₁₀₉	1	1	yes	characteristic
160	#107	C₂²×C₁₀₉	1	2	yes	—
161	C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
162	#103	C₂²×C₁₀₉	1	1	yes	characteristic
163	#103	C₂²×C₁₀₉	1	2	yes	—
164	#103	C₂²×C₁₀₉	1	1	yes	characteristic
165	#102	C₂²×C₁₀₉	1	2	yes	—
166	#102	C₂²×C₁₀₉	1	2	yes	—
167	#102	C₂²×C₁₀₉	1	2	yes	—
168	C₂²⋊C₈	C₂²×C₁₀₉	1	2	yes	—
169	C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
170	#41	—	2	4	no	—
171	#41	—	4	2	no	—
172	C₂×Q₈	—	2	1	no	—
173	C₂×Q₈	C₂³×C₁₀₉	1	1	yes	characteristic
174	C₂×Q₈	C₂³×C₁₀₉	1	1	yes	characteristic
175	C₂×D₄	—	2	2	no	—
176	C₂×D₄	—	2	2	no	—
177	C₂×D₄	—	2	1	no	—
178	C₂×D₄	—	2	1	no	—
179	C₂×D₄	D₄×C₁₀₉	1	1	yes	characteristic
180	C₂×D₄	C₂³×C₁₀₉	1	2	yes	—
181	C₂²×C₄	—	2	2	no	—
182	C₂²×C₄	—	2	2	no	—
183	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
184	Q₁₆	—	2	2	no	—
185	SD₁₆	—	4	2	no	—
186	M₁₆	—	4	2	no	—
187	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
188	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
189	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
190	C₄⋊C₄	—	2	2	no	—
191	C₄⋊C₄	—	2	1	no	—
192	C₄⋊C₄	—	2	1	no	—
193	C₄⋊C₄	—	2	1	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
196	C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
197	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
198	C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
199	C₂²⋊C₄	—	2	2	no	—
200	C₂²⋊C₄	—	2	2	no	—
201	C₂²⋊C₄	D₄×C₁₀₉	1	2	yes	—
202	C₄²	—	2	1	no	—
203	C₄²	—	2	1	no	—
204	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
205	C₂³	—	2	2	no	—
206	C₂³	C₂×D₄×C₁₀₉	1	2	yes	—
207	Q₈	—	2	1	no	—
208	Q₈	—	2	1	no	—
209	Q₈	—	2	1	no	—
210	Q₈	—	8	1	no	—
211	Q₈	#41×C₁₀₉	1	2	yes	—
212	D₄	—	2	2	no	—
213	D₄	—	2	2	no	—
214	D₄	—	2	2	no	—
215	D₄	—	4	4	no	—
216	D₄	—	4	2	no	—
217	D₄	—	8	1	no	—
218	D₄	—	8	2	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	4	4	no	—
227	C₂×C₄	—	8	2	no	—
228	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
229	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
230	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
231	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
232	C₈	—	2	1	no	—
233	C₈	—	2	1	no	—
234	C₈	—	2	2	no	—
235	C₂²	—	2	2	no	—
236	C₂²	—	2	2	no	—
237	C₂²	—	2	2	no	—
238	C₂²	—	4	2	no	—
239	C₂²	—	4	2	no	—
240	C₂²	—	4	2	no	—
241	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
242	C₄	—	2	2	no	—
243	C₄	—	2	2	no	—
244	C₄	—	2	1	no	—
245	C₄	—	4	1	no	—
246	C₄	—	4	1	no	—
247	C₄	—	4	2	no	—
248	C₄	—	4	1	no	—
249	C₄	#137×C₁₀₉	1	2	yes	—
250	C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
251	C₄	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
252	C₂	—	4	2	no	—
253	C₂	—	8	2	no	—
254	C₂	#546×C₁₀₉	1	1	yes	characteristic
255	C₂	C₂×#137×C₁₀₉	1	1	yes	characteristic
256	C₂	#576×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₀₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₀₉
prime radical	#550×C₁₀₉	C₂
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. #554×C₁₀₉	C₂	semidirect
3. #486×C₁₀₉	C₂	semidirect
4. #480×C₁₀₉	C₂	semidirect
5. #477×C₁₀₉	C₂	semidirect
6. #468×C₁₀₉	C₂	semidirect
7. #460×C₁₀₉	C₂	semidirect
8. #452×C₁₀₉	C₂	semidirect
9. C₄×Q₁₆×C₁₀₉	C₂	semidirect
10. #436×C₁₀₉	C₂	semidirect
11. #417×C₁₀₉	C₂	semidirect
128. #3333	C₁₀₉	direct (N × H)
129. C₁₀₉	#3333	direct (N × H)
130. #554	C₂×C₁₀₉	semidirect
132. #486	C₂×C₁₀₉	semidirect
133. #480	C₂×C₁₀₉	semidirect
134. #477	C₂×C₁₀₉	semidirect
135. #468	C₂×C₁₀₉	semidirect
136. #460	C₂×C₁₀₉	semidirect
137. #452	C₂×C₁₀₉	semidirect
138. C₄×Q₁₆	C₂×C₁₀₉	semidirect
139. #436	C₂×C₁₀₉	semidirect
140. #417	C₂×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#554×C₁₀₉	2
#550×C₁₀₉	2
#486×C₁₀₉	2
#480×C₁₀₉	2
#477×C₁₀₉	2
#468×C₁₀₉	2
#460×C₁₀₉	2
#452×C₁₀₉	2
C₄×Q₁₆×C₁₀₉	2
#436×C₁₀₉	2
#417×C₁₀₉	2
#3333	109

Subgroups — C109×#3333 #28360587

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#3333 #28360587