Subgroups — C₁₀₉×#3332 #28360586

Order 13952.

430 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	#550×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
2	#547×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#486×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#481×C₁₀₉	C₂	1	2	yes	maximal, complemented
5	#475×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#469×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	#436×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
11	#417×C₁₀₉	C₂	1	2	yes	maximal, complemented
12	C₂×#41×C₁₀₉	—	2	1	no	—
13	C₂×#41×C₁₀₉	C₂²	1	2	yes	—
14	C₂×SD₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂×D₈×C₁₀₉	C₂²	1	2	yes	—
16	C₂×M₁₆×C₁₀₉	C₂²	1	2	yes	—
17	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
18	#127×C₁₀₉	—	2	1	no	—
19	#127×C₁₀₉	C₂²	1	1	yes	characteristic
20	#123×C₁₀₉	—	2	2	no	—
21	C₂²⋊Q₈×C₁₀₉	—	2	1	no	—
22	#121×C₁₀₉	—	2	2	no	—
23	#121×C₁₀₉	C₂²	1	2	yes	—
24	#121×C₁₀₉	C₂²	1	2	yes	—
25	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
26	C₄×D₄×C₁₀₉	—	2	1	no	—
27	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
28	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
30	#106×C₁₀₉	C₂²	1	1	yes	characteristic
31	#106×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	#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	#41×C₁₀₉	—	4	4	no	—
44	C₂×Q₈×C₁₀₉	—	2	1	no	—
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	2	no	—
49	C₂×D₄×C₁₀₉	—	2	1	no	—
50	C₂×D₄×C₁₀₉	—	2	1	no	—
51	C₂×D₄×C₁₀₉	D₄	1	1	yes	characteristic
52	C₂×D₄×C₁₀₉	C₂³	1	2	yes	—
53	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₁₀₉	—	2	2	no	—
55	C₂²×C₄×C₁₀₉	—	2	2	no	—
56	C₂²×C₄×C₁₀₉	—	2	1	no	—
57	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
58	SD₁₆×C₁₀₉	—	2	2	no	—
59	D₈×C₁₀₉	—	4	2	no	—
60	M₁₆×C₁₀₉	—	4	2	no	—
61	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
62	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
63	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
64	C₄⋊C₄×C₁₀₉	—	2	2	no	—
65	C₄⋊C₄×C₁₀₉	—	2	1	no	—
66	C₄⋊C₄×C₁₀₉	—	2	1	no	—
67	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
68	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
69	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
70	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
71	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
72	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
73	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
74	C₂²⋊C₄×C₁₀₉	D₄	1	2	yes	—
75	C₄²×C₁₀₉	—	2	1	no	—
76	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
77	C₂³×C₁₀₉	—	2	1	no	—
78	C₂³×C₁₀₉	—	2	2	no	—
79	C₂³×C₁₀₉	C₂×D₄	1	2	yes	—
80	Q₈×C₁₀₉	—	2	1	no	—
81	Q₈×C₁₀₉	—	4	2	no	—
82	Q₈×C₁₀₉	#41	1	2	yes	complemented
83	D₄×C₁₀₉	—	2	2	no	—
84	D₄×C₁₀₉	—	2	1	no	—
85	D₄×C₁₀₉	—	2	2	no	—
86	D₄×C₁₀₉	—	2	1	no	—
87	D₄×C₁₀₉	—	2	2	no	—
88	D₄×C₁₀₉	—	4	4	no	—
89	D₄×C₁₀₉	—	4	4	no	—
90	D₄×C₁₀₉	—	4	2	no	—
91	D₄×C₁₀₉	—	8	1	no	—
92	D₄×C₁₀₉	—	8	2	no	—
93	C₂×C₄×C₁₀₉	—	2	1	no	—
94	C₂×C₄×C₁₀₉	—	2	2	no	—
95	C₂×C₄×C₁₀₉	—	2	2	no	—
96	C₂×C₄×C₁₀₉	—	2	1	no	—
97	C₂×C₄×C₁₀₉	—	2	1	no	—
98	C₂×C₄×C₁₀₉	—	2	2	no	—
99	C₂×C₄×C₁₀₉	—	4	4	no	—
100	C₂×C₄×C₁₀₉	—	4	4	no	—
101	C₂×C₄×C₁₀₉	—	4	2	no	—
102	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
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	1	no	—
108	C₈×C₁₀₉	—	2	2	no	—
109	C₂²×C₁₀₉	—	2	2	no	—
110	C₂²×C₁₀₉	—	2	2	no	—
111	C₂²×C₁₀₉	—	2	2	no	—
112	C₂²×C₁₀₉	—	4	2	no	—
113	C₂²×C₁₀₉	—	4	1	no	—
114	C₂²×C₁₀₉	—	4	1	no	—
115	C₂²×C₁₀₉	—	4	2	no	—
116	C₂²×C₁₀₉	—	4	2	no	—
117	C₂²×C₁₀₉	—	4	1	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	2	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	—	4	2	no	—
129	C₂×C₁₀₉	—	8	2	no	—
130	C₂×C₁₀₉	—	8	1	no	—
131	C₂×C₁₀₉	#546	1	1	yes	characteristic
132	C₂×C₁₀₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
133	C₂×C₁₀₉	#577	1	1	yes	characteristic
134	#3332	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
135	C₁₀₉	#3332	1	1	yes	characteristic, direct product factor, Sylow
136	#550	C₂×C₁₀₉	1	1	yes	characteristic, complemented
137	#547	C₂×C₁₀₉	1	1	yes	characteristic, complemented
138	#486	C₂×C₁₀₉	1	1	yes	characteristic, complemented
139	#481	C₂×C₁₀₉	1	2	yes	complemented
140	#475	C₂×C₁₀₉	1	1	yes	characteristic, complemented
141	#469	C₂×C₁₀₉	1	2	yes	complemented
142	#459	C₂×C₁₀₉	1	1	yes	characteristic, complemented
143	#449	C₂×C₁₀₉	1	2	yes	complemented
144	C₄×SD₁₆	C₂×C₁₀₉	1	1	yes	characteristic, complemented
145	#436	C₂×C₁₀₉	1	1	yes	characteristic, complemented
146	#417	C₂×C₁₀₉	1	2	yes	complemented
147	C₂×#41	—	2	1	no	—
148	C₂×#41	C₂²×C₁₀₉	1	2	yes	—
149	C₂×SD₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
150	C₂×D₈	C₂²×C₁₀₉	1	2	yes	—
151	C₂×M₁₆	C₂²×C₁₀₉	1	2	yes	—
152	C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
153	#127	—	2	1	no	—
154	#127	C₂²×C₁₀₉	1	1	yes	characteristic
155	#123	—	2	2	no	—
156	C₂²⋊Q₈	—	2	1	no	—
157	#121	—	2	2	no	—
158	#121	C₂²×C₁₀₉	1	2	yes	—
159	#121	C₂²×C₁₀₉	1	2	yes	—
160	C₄×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
161	C₄×D₄	—	2	1	no	—
162	C₄×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
163	C₄×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
164	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
165	#106	C₂²×C₁₀₉	1	1	yes	characteristic
166	#106	C₂²×C₁₀₉	1	2	yes	—
167	C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
168	#103	C₂²×C₁₀₉	1	1	yes	characteristic
169	#103	C₂²×C₁₀₉	1	2	yes	—
170	#102	C₂²×C₁₀₉	1	2	yes	—
171	#102	C₂²×C₁₀₉	1	2	yes	—
172	#102	C₂²×C₁₀₉	1	1	yes	characteristic
173	#102	C₂²×C₁₀₉	1	2	yes	—
174	C₂²⋊C₈	C₂²×C₁₀₉	1	2	yes	—
175	C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
176	#41	—	2	4	no	—
177	#41	—	4	2	no	—
178	#41	—	4	4	no	—
179	C₂×Q₈	—	2	1	no	—
180	C₂×Q₈	C₂³×C₁₀₉	1	1	yes	characteristic
181	C₂×D₄	—	2	2	no	—
182	C₂×D₄	—	2	2	no	—
183	C₂×D₄	—	2	2	no	—
184	C₂×D₄	—	2	1	no	—
185	C₂×D₄	—	2	1	no	—
186	C₂×D₄	D₄×C₁₀₉	1	1	yes	characteristic
187	C₂×D₄	C₂³×C₁₀₉	1	2	yes	—
188	C₂×D₄	C₂³×C₁₀₉	1	1	yes	characteristic
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
193	SD₁₆	—	2	2	no	—
194	D₈	—	4	2	no	—
195	M₁₆	—	4	2	no	—
196	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
197	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
198	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
199	C₄⋊C₄	—	2	2	no	—
200	C₄⋊C₄	—	2	1	no	—
201	C₄⋊C₄	—	2	1	no	—
202	C₄⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
203	C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
204	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
205	C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
206	C₂²⋊C₄	—	2	2	no	—
207	C₂²⋊C₄	—	2	1	no	—
208	C₂²⋊C₄	—	2	2	no	—
209	C₂²⋊C₄	D₄×C₁₀₉	1	2	yes	—
210	C₄²	—	2	1	no	—
211	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
212	C₂³	—	2	1	no	—
213	C₂³	—	2	2	no	—
214	C₂³	C₂×D₄×C₁₀₉	1	2	yes	—
215	Q₈	—	2	1	no	—
216	Q₈	—	4	2	no	—
217	Q₈	#41×C₁₀₉	1	2	yes	complemented
218	D₄	—	2	2	no	—
219	D₄	—	2	1	no	—
220	D₄	—	2	2	no	—
221	D₄	—	2	1	no	—
222	D₄	—	2	2	no	—
223	D₄	—	4	4	no	—
224	D₄	—	4	4	no	—
225	D₄	—	4	2	no	—
226	D₄	—	8	1	no	—
227	D₄	—	8	2	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	4	4	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
238	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
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	1	no	—
243	C₈	—	2	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	2	2	no	—
247	C₂²	—	4	2	no	—
248	C₂²	—	4	1	no	—
249	C₂²	—	4	1	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	4	2	no	—
252	C₂²	—	4	1	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	2	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	—	4	2	no	—
264	C₂	—	8	2	no	—
265	C₂	—	8	1	no	—
266	C₂	#546×C₁₀₉	1	1	yes	characteristic
267	C₂	C₂×AGL(1,ℤ/8)×C₁₀₉	1	1	yes	characteristic
268	C₂	#577×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. #550×C₁₀₉	C₂	semidirect
2. #547×C₁₀₉	C₂	semidirect
3. #486×C₁₀₉	C₂	semidirect
4. #481×C₁₀₉	C₂	semidirect
5. #475×C₁₀₉	C₂	semidirect
6. #469×C₁₀₉	C₂	semidirect
7. #459×C₁₀₉	C₂	semidirect
8. #449×C₁₀₉	C₂	semidirect
9. C₄×SD₁₆×C₁₀₉	C₂	semidirect
10. #436×C₁₀₉	C₂	semidirect
11. #417×C₁₀₉	C₂	semidirect
82. Q₈×C₁₀₉	#41	semidirect
134. #3332	C₁₀₉	direct (N × H)
135. C₁₀₉	#3332	direct (N × H)
136. #550	C₂×C₁₀₉	semidirect
137. #547	C₂×C₁₀₉	semidirect
138. #486	C₂×C₁₀₉	semidirect
139. #481	C₂×C₁₀₉	semidirect
140. #475	C₂×C₁₀₉	semidirect
141. #469	C₂×C₁₀₉	semidirect
142. #459	C₂×C₁₀₉	semidirect
143. #449	C₂×C₁₀₉	semidirect
144. C₄×SD₁₆	C₂×C₁₀₉	semidirect
145. #436	C₂×C₁₀₉	semidirect
146. #417	C₂×C₁₀₉	semidirect
217. Q₈	#41×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#550×C₁₀₉	2
#547×C₁₀₉	2
#486×C₁₀₉	2
#481×C₁₀₉	2
#475×C₁₀₉	2
#469×C₁₀₉	2
#459×C₁₀₉	2
#449×C₁₀₉	2
C₄×SD₁₆×C₁₀₉	2
#436×C₁₀₉	2
#417×C₁₀₉	2
#3332	109

Subgroups — C109×#3332 #28360586

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#3332 #28360586