Subgroups — C₁₀₉×#3353 #28360607

Order 13952.

374 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#556×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
2	#547×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	#500×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#484×C₁₀₉	C₂	1	2	yes	maximal, complemented
5	#482×C₁₀₉	C₂	1	2	yes	maximal, complemented
6	#475×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
7	#474×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#468×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	#428×C₁₀₉	C₂	1	2	yes	maximal, complemented
12	C₂×#41×C₁₀₉	—	2	1	no	—
13	C₂×SD₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×C₁₀₉	C₂²	1	2	yes	—
15	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
17	#126×C₁₀₉	—	2	2	no	—
18	#124×C₁₀₉	—	2	1	no	—
19	#123×C₁₀₉	—	2	2	no	—
20	C₂²⋊Q₈×C₁₀₉	—	2	1	no	—
21	C₂²⋊Q₈×C₁₀₉	C₂²	1	2	yes	—
22	#121×C₁₀₉	C₂²	1	2	yes	—
23	C₄×Q₈×C₁₀₉	—	2	1	no	—
24	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
25	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
26	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
27	#117×C₁₀₉	C₂²	1	2	yes	—
28	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
29	#107×C₁₀₉	C₂²	1	2	yes	—
30	#107×C₁₀₉	C₂²	1	2	yes	—
31	#107×C₁₀₉	C₂²	1	2	yes	—
32	#106×C₁₀₉	C₂²	1	1	yes	characteristic
33	#106×C₁₀₉	C₂²	1	2	yes	—
34	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
35	#103×C₁₀₉	C₂²	1	1	yes	characteristic
36	#103×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	—	4	4	no	—
42	C₂×Q₈×C₁₀₉	—	2	1	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	D₄	1	1	yes	characteristic
47	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₁₀₉	—	2	2	no	—
49	C₂²×C₄×C₁₀₉	—	2	1	no	—
50	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
51	SD₁₆×C₁₀₉	—	2	2	no	—
52	M₁₆×C₁₀₉	—	2	4	no	—
53	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
54	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
55	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
56	C₄⋊C₄×C₁₀₉	—	2	2	no	—
57	C₄⋊C₄×C₁₀₉	—	2	2	no	—
58	C₄⋊C₄×C₁₀₉	—	2	1	no	—
59	C₄⋊C₄×C₁₀₉	—	2	1	no	—
60	C₄⋊C₄×C₁₀₉	—	2	1	no	—
61	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
62	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
63	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
64	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
65	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
66	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
67	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
68	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
69	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
70	C₂²⋊C₄×C₁₀₉	D₄	1	2	yes	—
71	C₄²×C₁₀₉	—	2	2	no	—
72	C₄²×C₁₀₉	—	2	1	no	—
73	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
74	C₂³×C₁₀₉	—	2	1	no	—
75	C₂³×C₁₀₉	C₂×D₄	1	2	yes	—
76	Q₈×C₁₀₉	—	2	1	no	—
77	Q₈×C₁₀₉	—	2	1	no	—
78	Q₈×C₁₀₉	—	4	2	no	—
79	Q₈×C₁₀₉	—	4	2	no	—
80	D₄×C₁₀₉	—	2	2	no	—
81	D₄×C₁₀₉	—	2	1	no	—
82	D₄×C₁₀₉	—	2	1	no	—
83	D₄×C₁₀₉	—	4	4	no	—
84	C₂×C₄×C₁₀₉	—	2	2	no	—
85	C₂×C₄×C₁₀₉	—	2	2	no	—
86	C₂×C₄×C₁₀₉	—	2	1	no	—
87	C₂×C₄×C₁₀₉	—	2	1	no	—
88	C₂×C₄×C₁₀₉	—	2	2	no	—
89	C₂×C₄×C₁₀₉	—	2	1	no	—
90	C₂×C₄×C₁₀₉	—	2	2	no	—
91	C₂×C₄×C₁₀₉	—	4	4	no	—
92	C₂×C₄×C₁₀₉	—	4	2	no	—
93	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
94	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
97	C₈×C₁₀₉	—	2	2	no	—
98	C₈×C₁₀₉	—	2	1	no	—
99	C₈×C₁₀₉	#41	1	2	yes	complemented
100	C₂²×C₁₀₉	—	2	2	no	—
101	C₂²×C₁₀₉	—	2	2	no	—
102	C₂²×C₁₀₉	—	2	2	no	—
103	C₂²×C₁₀₉	—	4	1	no	—
104	C₂²×C₁₀₉	—	4	1	no	—
105	C₂²×C₁₀₉	—	4	1	no	—
106	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
107	C₄×C₁₀₉	—	2	2	no	—
108	C₄×C₁₀₉	—	2	1	no	—
109	C₄×C₁₀₉	—	4	1	no	—
110	C₄×C₁₀₉	—	4	1	no	—
111	C₄×C₁₀₉	—	4	2	no	—
112	C₄×C₁₀₉	—	4	2	no	—
113	C₄×C₁₀₉	—	4	1	no	—
114	C₄×C₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
115	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
116	C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
117	C₂×C₁₀₉	—	4	2	no	—
118	C₂×C₁₀₉	—	8	1	no	—
119	C₂×C₁₀₉	#547	1	1	yes	characteristic
120	C₂×C₁₀₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
121	C₂×C₁₀₉	#578	1	1	yes	characteristic
122	#3353	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
123	C₁₀₉	#3353	1	1	yes	characteristic, direct product factor, Sylow
124	#556	C₂×C₁₀₉	1	1	yes	characteristic, complemented
125	#547	C₂×C₁₀₉	1	1	yes	characteristic
126	#500	C₂×C₁₀₉	1	1	yes	characteristic, complemented
127	#484	C₂×C₁₀₉	1	2	yes	complemented
128	#482	C₂×C₁₀₉	1	2	yes	complemented
129	#475	C₂×C₁₀₉	1	1	yes	characteristic, complemented
130	#474	C₂×C₁₀₉	1	1	yes	characteristic, complemented
131	#468	C₂×C₁₀₉	1	2	yes	complemented
132	C₄×SD₁₆	C₂×C₁₀₉	1	1	yes	characteristic, complemented
133	#436	C₂×C₁₀₉	1	1	yes	characteristic, complemented
134	#428	C₂×C₁₀₉	1	2	yes	complemented
135	C₂×#41	—	2	1	no	—
136	C₂×SD₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
137	C₂×M₁₆	C₂²×C₁₀₉	1	2	yes	—
138	C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
139	C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
140	#126	—	2	2	no	—
141	#124	—	2	1	no	—
142	#123	—	2	2	no	—
143	C₂²⋊Q₈	—	2	1	no	—
144	C₂²⋊Q₈	C₂²×C₁₀₉	1	2	yes	—
145	#121	C₂²×C₁₀₉	1	2	yes	—
146	C₄×Q₈	—	2	1	no	—
147	C₄×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
148	C₄×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
149	C₄×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
150	#117	C₂²×C₁₀₉	1	2	yes	—
151	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
152	#107	C₂²×C₁₀₉	1	2	yes	—
153	#107	C₂²×C₁₀₉	1	2	yes	—
154	#107	C₂²×C₁₀₉	1	2	yes	—
155	#106	C₂²×C₁₀₉	1	1	yes	characteristic
156	#106	C₂²×C₁₀₉	1	2	yes	—
157	C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
158	#103	C₂²×C₁₀₉	1	1	yes	characteristic
159	#103	C₂²×C₁₀₉	1	2	yes	—
160	#102	C₂²×C₁₀₉	1	1	yes	characteristic
161	#102	C₂²×C₁₀₉	1	2	yes	—
162	C₂²⋊C₈	C₂²×C₁₀₉	1	2	yes	—
163	C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
164	#41	—	4	4	no	—
165	C₂×Q₈	—	2	1	no	—
166	C₂×Q₈	—	2	1	no	—
167	C₂×Q₈	C₂³×C₁₀₉	1	1	yes	characteristic
168	C₂×D₄	—	2	2	no	—
169	C₂×D₄	D₄×C₁₀₉	1	1	yes	characteristic
170	C₂×D₄	C₂³×C₁₀₉	1	1	yes	characteristic
171	C₂²×C₄	—	2	2	no	—
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
174	SD₁₆	—	2	2	no	—
175	M₁₆	—	2	4	no	—
176	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
177	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
178	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	—	2	1	no	—
182	C₄⋊C₄	—	2	1	no	—
183	C₄⋊C₄	—	2	1	no	—
184	C₄⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
185	C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
186	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
187	C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
188	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
189	C₂²⋊C₄	—	2	2	no	—
190	C₂²⋊C₄	—	2	2	no	—
191	C₂²⋊C₄	—	2	1	no	—
192	C₂²⋊C₄	—	2	2	no	—
193	C₂²⋊C₄	D₄×C₁₀₉	1	2	yes	—
194	C₄²	—	2	2	no	—
195	C₄²	—	2	1	no	—
196	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
197	C₂³	—	2	1	no	—
198	C₂³	C₂×D₄×C₁₀₉	1	2	yes	—
199	Q₈	—	2	1	no	—
200	Q₈	—	2	1	no	—
201	Q₈	—	4	2	no	—
202	Q₈	—	4	2	no	—
203	D₄	—	2	2	no	—
204	D₄	—	2	1	no	—
205	D₄	—	2	1	no	—
206	D₄	—	4	4	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	4	4	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
217	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
218	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
219	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
220	C₈	—	2	2	no	—
221	C₈	—	2	1	no	—
222	C₈	#41×C₁₀₉	1	2	yes	complemented
223	C₂²	—	2	2	no	—
224	C₂²	—	2	2	no	—
225	C₂²	—	2	2	no	—
226	C₂²	—	4	1	no	—
227	C₂²	—	4	1	no	—
228	C₂²	—	4	1	no	—
229	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
230	C₄	—	2	2	no	—
231	C₄	—	2	1	no	—
232	C₄	—	4	1	no	—
233	C₄	—	4	1	no	—
234	C₄	—	4	2	no	—
235	C₄	—	4	2	no	—
236	C₄	—	4	1	no	—
237	C₄	AGL(1,ℤ/8)×C₁₀₉	1	2	yes	—
238	C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
239	C₄	2₋¹⁺⁴×C₁₀₉	1	1	yes	characteristic
240	C₂	—	4	2	no	—
241	C₂	—	8	1	no	—
242	C₂	#547×C₁₀₉	1	1	yes	characteristic
243	C₂	C₂×AGL(1,ℤ/8)×C₁₀₉	1	1	yes	characteristic
244	C₂	#578×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₀₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₀₉
prime radical	#547×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. #556×C₁₀₉	C₂	semidirect
3. #500×C₁₀₉	C₂	semidirect
4. #484×C₁₀₉	C₂	semidirect
5. #482×C₁₀₉	C₂	semidirect
6. #475×C₁₀₉	C₂	semidirect
7. #474×C₁₀₉	C₂	semidirect
8. #468×C₁₀₉	C₂	semidirect
9. C₄×SD₁₆×C₁₀₉	C₂	semidirect
10. #436×C₁₀₉	C₂	semidirect
11. #428×C₁₀₉	C₂	semidirect
99. C₈×C₁₀₉	#41	semidirect
122. #3353	C₁₀₉	direct (N × H)
123. C₁₀₉	#3353	direct (N × H)
124. #556	C₂×C₁₀₉	semidirect
126. #500	C₂×C₁₀₉	semidirect
127. #484	C₂×C₁₀₉	semidirect
128. #482	C₂×C₁₀₉	semidirect
129. #475	C₂×C₁₀₉	semidirect
130. #474	C₂×C₁₀₉	semidirect
131. #468	C₂×C₁₀₉	semidirect
132. C₄×SD₁₆	C₂×C₁₀₉	semidirect
133. #436	C₂×C₁₀₉	semidirect
134. #428	C₂×C₁₀₉	semidirect
222. 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
#556×C₁₀₉	2
#547×C₁₀₉	2
#500×C₁₀₉	2
#484×C₁₀₉	2
#482×C₁₀₉	2
#475×C₁₀₉	2
#474×C₁₀₉	2
#468×C₁₀₉	2
C₄×SD₁₆×C₁₀₉	2
#436×C₁₀₉	2
#428×C₁₀₉	2
#3353	109

Subgroups — C109×#3353 #28360607

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#3353 #28360607