Subgroups — C₁₀₉×#10650 #42128644

Order 27904.

350 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₉×#1947	C₂	1	2	yes	maximal
2	C₁₀₉×#1939	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#1845	C₂	1	1	yes	maximal, characteristic
4	C₁₀₉×#1296	C₂	1	2	yes	maximal
5	C₁₀₉×#1277	C₂	1	1	yes	maximal, characteristic
6	C₂×#107×C₁₀₉	—	2	1	no	—
7	C₂×#107×C₁₀₉	—	2	1	no	—
8	C₂×#107×C₁₀₉	—	2	2	no	—
9	C₂×C₄⋊C₈×C₁₀₉	—	2	1	no	—
10	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
13	#384×C₁₀₉	C₂²	1	2	yes	—
14	C₄×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
15	#340×C₁₀₉	—	2	2	no	—
16	#336×C₁₀₉	—	2	1	no	—
17	C₂²×C₈×C₁₀₉	—	2	2	no	—
18	C₂²×C₈×C₁₀₉	—	2	1	no	—
19	C₂²×C₈×C₁₀₉	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₀₉	D₄	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
22	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
23	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
24	C₂×C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
25	C₂×C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₁₀₉	—	2	1	no	—
27	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
28	#107×C₁₀₉	—	4	4	no	—
29	#107×C₁₀₉	—	4	2	no	—
30	#107×C₁₀₉	—	4	2	no	—
31	C₄⋊C₈×C₁₀₉	—	2	2	no	—
32	C₄⋊C₈×C₁₀₉	—	2	2	no	—
33	C₄⋊C₈×C₁₀₉	—	4	2	no	—
34	C₄⋊C₈×C₁₀₉	Q₈	1	4	yes	—
35	C₄×C₈×C₁₀₉	—	2	2	no	—
36	#95×C₁₀₉	—	2	1	no	—
37	C₂²×C₄×C₁₀₉	—	2	1	no	—
38	C₂²×C₄×C₁₀₉	—	2	1	no	—
39	C₂²×C₄×C₁₀₉	—	4	2	no	—
40	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₀₉	#41	1	1	yes	characteristic
43	C₂×C₈×C₁₀₉	—	2	1	no	—
44	C₂×C₈×C₁₀₉	—	2	1	no	—
45	C₂×C₈×C₁₀₉	—	2	1	no	—
46	C₂×C₈×C₁₀₉	—	2	1	no	—
47	C₂×C₈×C₁₀₉	—	2	4	no	—
48	C₂×C₈×C₁₀₉	—	4	2	no	—
49	C₂×C₈×C₁₀₉	—	4	2	no	—
50	C₂×C₈×C₁₀₉	—	4	1	no	—
51	C₂×C₈×C₁₀₉	—	4	1	no	—
52	C₂×C₈×C₁₀₉	—	4	1	no	—
53	C₂×C₈×C₁₀₉	D₈	1	2	yes	—
54	C₂×C₈×C₁₀₉	Q₁₆	1	2	yes	—
55	C₄⋊C₄×C₁₀₉	—	2	1	no	—
56	C₄⋊C₄×C₁₀₉	—	2	2	no	—
57	C₄⋊C₄×C₁₀₉	—	4	2	no	—
58	C₄⋊C₄×C₁₀₉	—	4	2	no	—
59	C₄⋊C₄×C₁₀₉	—	4	2	no	—
60	C₄⋊C₄×C₁₀₉	—	4	2	no	—
61	C₄⋊C₄×C₁₀₉	—	8	2	no	—
62	C₄⋊C₄×C₁₀₉	Q₁₆	1	2	yes	—
63	C₄²×C₁₀₉	—	4	1	no	—
64	C₄²×C₁₀₉	—	4	1	no	—
65	C₄²×C₁₀₉	C₂×Q₈	1	2	yes	—
66	C₄²×C₁₀₉	#41	1	2	yes	—
67	C₂³×C₁₀₉	#121	1	1	yes	characteristic
68	C₂×C₄×C₁₀₉	—	2	1	no	—
69	C₂×C₄×C₁₀₉	—	2	1	no	—
70	C₂×C₄×C₁₀₉	—	2	2	no	—
71	C₂×C₄×C₁₀₉	—	4	1	no	—
72	C₂×C₄×C₁₀₉	—	4	1	no	—
73	C₂×C₄×C₁₀₉	—	4	1	no	—
74	C₂×C₄×C₁₀₉	—	4	1	no	—
75	C₂×C₄×C₁₀₉	—	4	1	no	—
76	C₂×C₄×C₁₀₉	—	8	2	no	—
77	C₂×C₄×C₁₀₉	—	8	2	no	—
78	C₂×C₄×C₁₀₉	—	8	2	no	—
79	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
80	C₂×C₄×C₁₀₉	#123	1	1	yes	characteristic
81	C₂×C₄×C₁₀₉	#125	1	2	yes	—
82	C₂×C₄×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
83	C₂×C₄×C₁₀₉	C₂×D₈	1	1	yes	characteristic
84	C₂×C₄×C₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
85	C₂×C₄×C₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
86	C₂×C₄×C₁₀₉	#135	1	1	yes	characteristic
87	C₂×C₄×C₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
88	C₂×C₄×C₁₀₉	AGL(1,ℤ/8)	1	1	yes	characteristic
89	C₂×C₄×C₁₀₉	#137	1	1	yes	characteristic
90	C₈×C₁₀₉	—	2	2	no	—
91	C₈×C₁₀₉	—	2	2	no	—
92	C₈×C₁₀₉	—	4	4	no	—
93	C₈×C₁₀₉	—	8	1	no	—
94	C₂²×C₁₀₉	#398	1	1	yes	characteristic
95	C₂²×C₁₀₉	#459	1	1	yes	characteristic
96	C₂²×C₁₀₉	#462	1	1	yes	characteristic
97	C₂²×C₁₀₉	#467	1	1	yes	characteristic
98	C₂²×C₁₀₉	#469	1	1	yes	characteristic
99	C₂²×C₁₀₉	#472	1	1	yes	characteristic
100	C₂²×C₁₀₉	#473	1	1	yes	characteristic
101	C₄×C₁₀₉	—	2	2	no	—
102	C₄×C₁₀₉	—	4	2	no	—
103	C₄×C₁₀₉	—	8	1	no	—
104	C₄×C₁₀₉	—	16	2	no	—
105	C₄×C₁₀₉	#477	1	2	yes	—
106	C₄×C₁₀₉	#478	1	2	yes	—
107	C₄×C₁₀₉	#480	1	1	yes	characteristic
108	C₄×C₁₀₉	#484	1	1	yes	characteristic
109	C₄×C₁₀₉	#500	1	1	yes	characteristic
110	C₄×C₁₀₉	#501	1	1	yes	characteristic
111	#10650	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
112	C₂×C₁₀₉	#1669	1	1	yes	characteristic
113	C₂×C₁₀₉	#1676	1	1	yes	characteristic
114	C₂×C₁₀₉	#1687	1	1	yes	characteristic
115	C₂×C₁₀₉	#1694	1	1	yes	characteristic
116	C₂×C₁₀₉	#2083	1	1	yes	characteristic
117	C₂×C₁₀₉	#2087	1	1	yes	characteristic
118	C₂×C₁₀₉	#2089	1	1	yes	characteristic
119	#1947	C₂×C₁₀₉	1	2	yes	—
120	#1939	C₂×C₁₀₉	1	1	yes	characteristic
121	#1845	C₂×C₁₀₉	1	1	yes	characteristic
122	#1296	C₂×C₁₀₉	1	2	yes	—
123	#1277	C₂×C₁₀₉	1	1	yes	characteristic
124	C₁₀₉	#10650	1	1	yes	characteristic, direct product factor, Sylow
125	C₂×#107	—	2	1	no	—
126	C₂×#107	—	2	1	no	—
127	C₂×#107	—	2	2	no	—
128	C₂×C₄⋊C₈	—	2	1	no	—
129	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
130	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
131	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
132	#384	C₂²×C₁₀₉	1	2	yes	—
133	C₄×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
134	#340	—	2	2	no	—
135	#336	—	2	1	no	—
136	C₂²×C₈	—	2	2	no	—
137	C₂²×C₈	—	2	1	no	—
138	C₂²×C₈	D₄×C₁₀₉	1	1	yes	characteristic
139	C₂²×C₈	D₄×C₁₀₉	1	1	yes	characteristic
140	C₂×C₄⋊C₄	—	2	2	no	—
141	C₂×C₄⋊C₄	—	2	2	no	—
142	C₂×C₄⋊C₄	—	2	2	no	—
143	C₂×C₄⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
144	C₂×C₄⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
145	C₂×C₄²	—	2	1	no	—
146	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
147	#107	—	4	4	no	—
148	#107	—	4	2	no	—
149	#107	—	4	2	no	—
150	C₄⋊C₈	—	2	2	no	—
151	C₄⋊C₈	—	2	2	no	—
152	C₄⋊C₈	—	4	2	no	—
153	C₄⋊C₈	Q₈×C₁₀₉	1	4	yes	—
154	C₄×C₈	—	2	2	no	—
155	#95	—	2	1	no	—
156	C₂²×C₄	—	2	1	no	—
157	C₂²×C₄	—	2	1	no	—
158	C₂²×C₄	—	4	2	no	—
159	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
160	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
161	C₂²×C₄	#41×C₁₀₉	1	1	yes	characteristic
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	4	no	—
167	C₂×C₈	—	4	2	no	—
168	C₂×C₈	—	4	2	no	—
169	C₂×C₈	—	4	1	no	—
170	C₂×C₈	—	4	1	no	—
171	C₂×C₈	—	4	1	no	—
172	C₂×C₈	D₈×C₁₀₉	1	2	yes	—
173	C₂×C₈	Q₁₆×C₁₀₉	1	2	yes	—
174	C₄⋊C₄	—	2	1	no	—
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	—	4	2	no	—
177	C₄⋊C₄	—	4	2	no	—
178	C₄⋊C₄	—	4	2	no	—
179	C₄⋊C₄	—	4	2	no	—
180	C₄⋊C₄	—	8	2	no	—
181	C₄⋊C₄	Q₁₆×C₁₀₉	1	2	yes	—
182	C₄²	—	4	1	no	—
183	C₄²	—	4	1	no	—
184	C₄²	C₂×Q₈×C₁₀₉	1	2	yes	—
185	C₄²	#41×C₁₀₉	1	2	yes	—
186	C₂³	#121×C₁₀₉	1	1	yes	characteristic
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	4	1	no	—
191	C₂×C₄	—	4	1	no	—
192	C₂×C₄	—	4	1	no	—
193	C₂×C₄	—	4	1	no	—
194	C₂×C₄	—	4	1	no	—
195	C₂×C₄	—	8	2	no	—
196	C₂×C₄	—	8	2	no	—
197	C₂×C₄	—	8	2	no	—
198	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	2	yes	—
199	C₂×C₄	#123×C₁₀₉	1	1	yes	characteristic
200	C₂×C₄	#125×C₁₀₉	1	2	yes	—
201	C₂×C₄	C₄⋊Q₈×C₁₀₉	1	1	yes	characteristic
202	C₂×C₄	C₂×D₈×C₁₀₉	1	1	yes	characteristic
203	C₂×C₄	C₂×Q₁₆×C₁₀₉	1	1	yes	characteristic
204	C₂×C₄	C₂×Q₁₆×C₁₀₉	1	1	yes	characteristic
205	C₂×C₄	#135×C₁₀₉	1	1	yes	characteristic
206	C₂×C₄	AGL(1,ℤ/8)×C₁₀₉	1	1	yes	characteristic
207	C₂×C₄	AGL(1,ℤ/8)×C₁₀₉	1	2	yes	—
208	C₂×C₄	#137×C₁₀₉	1	1	yes	characteristic
209	C₈	—	2	2	no	—
210	C₈	—	2	2	no	—
211	C₈	—	4	4	no	—
212	C₈	—	8	1	no	—
213	C₂²	#398×C₁₀₉	1	1	yes	characteristic
214	C₂²	#459×C₁₀₉	1	1	yes	characteristic
215	C₂²	#462×C₁₀₉	1	1	yes	characteristic
216	C₂²	#467×C₁₀₉	1	1	yes	characteristic
217	C₂²	#469×C₁₀₉	1	1	yes	characteristic
218	C₂²	#472×C₁₀₉	1	1	yes	characteristic
219	C₂²	#473×C₁₀₉	1	1	yes	characteristic
220	C₄	—	2	2	no	—
221	C₄	—	4	2	no	—
222	C₄	—	8	1	no	—
223	C₄	—	16	2	no	—
224	C₄	#477×C₁₀₉	1	2	yes	—
225	C₄	#478×C₁₀₉	1	2	yes	—
226	C₄	#480×C₁₀₉	1	1	yes	characteristic
227	C₄	#484×C₁₀₉	1	1	yes	characteristic
228	C₄	#500×C₁₀₉	1	1	yes	characteristic
229	C₄	#501×C₁₀₉	1	1	yes	characteristic
230	C₂	C₁₀₉×#1669	1	1	yes	characteristic
231	C₂	C₁₀₉×#1676	1	1	yes	characteristic
232	C₂	C₁₀₉×#1687	1	1	yes	characteristic
233	C₂	C₁₀₉×#1694	1	1	yes	characteristic
234	C₂	C₁₀₉×#2083	1	1	yes	characteristic
235	C₂	C₁₀₉×#2087	1	1	yes	characteristic
236	C₂	C₁₀₉×#2089	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₀₉	#121
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₉
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
111. #10650	C₁₀₉	direct (N × H)
124. C₁₀₉	#10650	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₀₉×#1947	2
C₁₀₉×#1939	2
C₁₀₉×#1845	2
C₁₀₉×#1296	2
C₁₀₉×#1277	2
#10650	109

Subgroups — C109×#10650 #42128644

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#10650 #42128644