Subgroups — C₁₀₉×#3372 #28360626

Order 13952.

340 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#557×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
2	#554×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	#491×C₁₀₉	C₂	1	2	yes	maximal, complemented
4	#487×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#479×C₁₀₉	C₂	1	2	yes	maximal, complemented
6	#477×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
7	#461×C₁₀₉	C₂	1	2	yes	maximal
8	#461×C₁₀₉	C₂	1	1	yes	maximal, characteristic
9	#446×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
10	#440×C₁₀₉	C₂	1	2	yes	maximal
11	C₄×SD₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
12	C₂×SD₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₂×SD₁₆×C₁₀₉	C₂²	1	2	yes	—
14	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₁₀₉	C₂²	1	2	yes	—
16	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
17	#125×C₁₀₉	—	2	1	no	—
18	#125×C₁₀₉	—	2	2	no	—
19	#125×C₁₀₉	C₂²	1	2	yes	—
20	C₂²⋊Q₈×C₁₀₉	—	2	1	no	—
21	C₂²⋊Q₈×C₁₀₉	—	2	2	no	—
22	C₄×Q₈×C₁₀₉	—	2	1	no	—
23	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
24	C₄×Q₈×C₁₀₉	C₂²	1	2	yes	—
25	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
26	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
27	C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
28	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
29	#107×C₁₀₉	C₂²	1	2	yes	—
30	#106×C₁₀₉	C₂²	1	1	yes	characteristic
31	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
32	C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
33	#103×C₁₀₉	C₂²	1	1	yes	characteristic
34	#103×C₁₀₉	C₂²	1	2	yes	—
35	#103×C₁₀₉	C₂²	1	2	yes	—
36	#103×C₁₀₉	C₂²	1	2	yes	—
37	#103×C₁₀₉	C₂²	1	2	yes	—
38	#102×C₁₀₉	C₂²	1	1	yes	characteristic
39	#97×C₁₀₉	C₂²	1	2	yes	—
40	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
41	C₂×Q₈×C₁₀₉	—	2	1	no	—
42	C₂×Q₈×C₁₀₉	D₄	1	1	yes	characteristic
43	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
44	C₂×Q₈×C₁₀₉	C₂³	1	2	yes	—
45	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₁₀₉	—	2	1	no	—
47	C₂²×C₄×C₁₀₉	—	2	2	no	—
48	SD₁₆×C₁₀₉	—	2	2	no	—
49	SD₁₆×C₁₀₉	—	2	4	no	—
50	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
51	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
52	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
53	C₄⋊C₄×C₁₀₉	—	2	2	no	—
54	C₄⋊C₄×C₁₀₉	—	2	1	no	—
55	C₄⋊C₄×C₁₀₉	—	2	2	no	—
56	C₄⋊C₄×C₁₀₉	—	2	1	no	—
57	C₄⋊C₄×C₁₀₉	—	2	1	no	—
58	C₄⋊C₄×C₁₀₉	—	2	2	no	—
59	C₄⋊C₄×C₁₀₉	—	2	2	no	—
60	C₄⋊C₄×C₁₀₉	—	2	1	no	—
61	C₄⋊C₄×C₁₀₉	—	2	2	no	—
62	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
63	C₄⋊C₄×C₁₀₉	D₄	1	2	yes	—
64	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
65	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
66	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
67	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
68	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
69	C₄²×C₁₀₉	—	2	2	no	—
70	C₄²×C₁₀₉	—	2	1	no	—
71	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
72	C₄²×C₁₀₉	C₂³	1	2	yes	—
73	C₂³×C₁₀₉	—	2	1	no	—
74	Q₈×C₁₀₉	—	2	2	no	—
75	Q₈×C₁₀₉	—	2	1	no	—
76	Q₈×C₁₀₉	—	2	2	no	—
77	Q₈×C₁₀₉	—	2	1	no	—
78	Q₈×C₁₀₉	—	2	2	no	—
79	Q₈×C₁₀₉	—	8	1	no	—
80	D₄×C₁₀₉	—	2	1	no	—
81	D₄×C₁₀₉	#41	1	2	yes	—
82	C₂×C₄×C₁₀₉	—	2	1	no	—
83	C₂×C₄×C₁₀₉	—	2	2	no	—
84	C₂×C₄×C₁₀₉	—	2	1	no	—
85	C₂×C₄×C₁₀₉	—	2	1	no	—
86	C₂×C₄×C₁₀₉	—	2	2	no	—
87	C₂×C₄×C₁₀₉	—	4	2	no	—
88	C₂×C₄×C₁₀₉	—	8	2	no	—
89	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
90	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
92	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
94	C₈×C₁₀₉	—	2	1	no	—
95	C₈×C₁₀₉	—	2	2	no	—
96	C₈×C₁₀₉	—	2	1	no	—
97	C₂²×C₁₀₉	—	2	2	no	—
98	C₂²×C₁₀₉	—	4	1	no	—
99	C₂²×C₁₀₉	—	4	1	no	—
100	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
101	C₄×C₁₀₉	—	2	2	no	—
102	C₄×C₁₀₉	—	2	2	no	—
103	C₄×C₁₀₉	—	2	1	no	—
104	C₄×C₁₀₉	—	4	2	no	—
105	C₄×C₁₀₉	—	4	1	no	—
106	C₄×C₁₀₉	—	4	1	no	—
107	C₄×C₁₀₉	—	4	1	no	—
108	C₄×C₁₀₉	—	4	2	no	—
109	C₄×C₁₀₉	#137	1	2	yes	—
110	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
111	C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
112	C₂×C₁₀₉	—	4	2	no	—
113	C₂×C₁₀₉	#548	1	1	yes	characteristic
114	C₂×C₁₀₉	C₂×#137	1	1	yes	characteristic
115	C₂×C₁₀₉	#577	1	1	yes	characteristic
116	#3372	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	C₁₀₉	#3372	1	1	yes	characteristic, direct product factor, Sylow
118	#557	C₂×C₁₀₉	1	1	yes	characteristic, complemented
119	#554	C₂×C₁₀₉	1	1	yes	characteristic
120	#491	C₂×C₁₀₉	1	2	yes	complemented
121	#487	C₂×C₁₀₉	1	1	yes	characteristic, complemented
122	#479	C₂×C₁₀₉	1	2	yes	complemented
123	#477	C₂×C₁₀₉	1	1	yes	characteristic, complemented
124	#461	C₂×C₁₀₉	1	2	yes	—
125	#461	C₂×C₁₀₉	1	1	yes	characteristic
126	#446	C₂×C₁₀₉	1	1	yes	characteristic, complemented
127	#440	C₂×C₁₀₉	1	2	yes	—
128	C₄×SD₁₆	C₂×C₁₀₉	1	1	yes	characteristic
129	C₂×SD₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
130	C₂×SD₁₆	C₂²×C₁₀₉	1	2	yes	—
131	C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
132	C₄⋊Q₈	C₂²×C₁₀₉	1	2	yes	—
133	C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
134	#125	—	2	1	no	—
135	#125	—	2	2	no	—
136	#125	C₂²×C₁₀₉	1	2	yes	—
137	C₂²⋊Q₈	—	2	1	no	—
138	C₂²⋊Q₈	—	2	2	no	—
139	C₄×Q₈	—	2	1	no	—
140	C₄×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
141	C₄×Q₈	C₂²×C₁₀₉	1	2	yes	—
142	C₄×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
143	C₄×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
144	C₄×D₄	C₂²×C₁₀₉	1	2	yes	—
145	C₂×C₄⋊C₄	—	2	1	no	—
146	#107	C₂²×C₁₀₉	1	2	yes	—
147	#106	C₂²×C₁₀₉	1	1	yes	characteristic
148	C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
149	C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
150	#103	C₂²×C₁₀₉	1	1	yes	characteristic
151	#103	C₂²×C₁₀₉	1	2	yes	—
152	#103	C₂²×C₁₀₉	1	2	yes	—
153	#103	C₂²×C₁₀₉	1	2	yes	—
154	#103	C₂²×C₁₀₉	1	2	yes	—
155	#102	C₂²×C₁₀₉	1	1	yes	characteristic
156	#97	C₂²×C₁₀₉	1	2	yes	—
157	C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
158	C₂×Q₈	—	2	1	no	—
159	C₂×Q₈	D₄×C₁₀₉	1	1	yes	characteristic
160	C₂×Q₈	C₂³×C₁₀₉	1	1	yes	characteristic
161	C₂×Q₈	C₂³×C₁₀₉	1	2	yes	—
162	C₂×D₄	C₂³×C₁₀₉	1	1	yes	characteristic
163	C₂²×C₄	—	2	1	no	—
164	C₂²×C₄	—	2	2	no	—
165	SD₁₆	—	2	2	no	—
166	SD₁₆	—	2	4	no	—
167	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
168	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
169	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
170	C₄⋊C₄	—	2	2	no	—
171	C₄⋊C₄	—	2	1	no	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	1	no	—
174	C₄⋊C₄	—	2	1	no	—
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	2	1	no	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
180	C₄⋊C₄	D₄×C₁₀₉	1	2	yes	—
181	C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
182	C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
183	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
184	C₂²⋊C₄	—	2	1	no	—
185	C₂²⋊C₄	—	2	2	no	—
186	C₄²	—	2	2	no	—
187	C₄²	—	2	1	no	—
188	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
189	C₄²	C₂³×C₁₀₉	1	2	yes	—
190	C₂³	—	2	1	no	—
191	Q₈	—	2	2	no	—
192	Q₈	—	2	1	no	—
193	Q₈	—	2	2	no	—
194	Q₈	—	2	1	no	—
195	Q₈	—	2	2	no	—
196	Q₈	—	8	1	no	—
197	D₄	—	2	1	no	—
198	D₄	#41×C₁₀₉	1	2	yes	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	—	8	2	no	—
206	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
207	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
208	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
209	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
210	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
211	C₈	—	2	1	no	—
212	C₈	—	2	2	no	—
213	C₈	—	2	1	no	—
214	C₂²	—	2	2	no	—
215	C₂²	—	4	1	no	—
216	C₂²	—	4	1	no	—
217	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
218	C₄	—	2	2	no	—
219	C₄	—	2	2	no	—
220	C₄	—	2	1	no	—
221	C₄	—	4	2	no	—
222	C₄	—	4	1	no	—
223	C₄	—	4	1	no	—
224	C₄	—	4	2	no	—
225	C₄	—	4	1	no	—
226	C₄	#137×C₁₀₉	1	2	yes	—
227	C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
228	C₄	2₋¹⁺⁴×C₁₀₉	1	1	yes	characteristic
229	C₂	—	4	2	no	—
230	C₂	#548×C₁₀₉	1	1	yes	characteristic
231	C₂	C₂×#137×C₁₀₉	1	1	yes	characteristic
232	C₂	#577×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₀₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₀₉
prime radical	C₂×D₄×C₁₀₉	C₂³
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. #557×C₁₀₉	C₂	semidirect
3. #491×C₁₀₉	C₂	semidirect
4. #487×C₁₀₉	C₂	semidirect
5. #479×C₁₀₉	C₂	semidirect
6. #477×C₁₀₉	C₂	semidirect
9. #446×C₁₀₉	C₂	semidirect
116. #3372	C₁₀₉	direct (N × H)
117. C₁₀₉	#3372	direct (N × H)
118. #557	C₂×C₁₀₉	semidirect
120. #491	C₂×C₁₀₉	semidirect
121. #487	C₂×C₁₀₉	semidirect
122. #479	C₂×C₁₀₉	semidirect
123. #477	C₂×C₁₀₉	semidirect
126. #446	C₂×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#557×C₁₀₉	2
#554×C₁₀₉	2
#491×C₁₀₉	2
#487×C₁₀₉	2
#479×C₁₀₉	2
#477×C₁₀₉	2
#461×C₁₀₉	2
#446×C₁₀₉	2
#440×C₁₀₉	2
C₄×SD₁₆×C₁₀₉	2
#3372	109

Subgroups — C109×#3372 #28360626

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#3372 #28360626