Subgroups — C₁₀₉×#12850 #42130844

Order 27904.

430 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#405×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#349×C₁₀₉	C₂	1	2	yes	maximal
3	C₁₀₉×#1376	C₂	1	2	yes	maximal, complemented
4	C₁₀₉×#1375	C₂	1	2	yes	maximal, complemented
5	C₂²×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
6	C₂³×C₈×C₁₀₉	C₄	1	1	yes	characteristic
7	C₂×#117×C₁₀₉	C₄	1	1	yes	characteristic
8	C₂×M₃₂×C₁₀₉	C₂²	1	4	yes	—
9	#405×C₁₀₉	—	2	4	no	—
10	C₂×#97×C₁₀₉	C₄	1	1	yes	characteristic
11	C₂×#97×C₁₀₉	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₀₉	C₄	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
14	#349×C₁₀₉	—	2	4	no	—
15	C₂³×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
16	C₂×M₁₆×C₁₀₉	—	2	2	no	—
17	C₂×M₁₆×C₁₀₉	—	2	2	no	—
18	C₂×M₁₆×C₁₀₉	D₄	1	4	yes	—
19	C₂²×C₈×C₁₀₉	—	2	2	no	—
20	C₂²×C₈×C₁₀₉	—	2	2	no	—
21	C₂²×C₈×C₁₀₉	—	2	2	no	—
22	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
26	#117×C₁₀₉	—	2	2	no	—
27	#117×C₁₀₉	C₈	1	4	yes	—
28	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
29	C₂×C₂²⋊C₄×C₁₀₉	—	2	1	no	—
30	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
32	M₃₂×C₁₀₉	—	2	4	no	—
33	M₃₂×C₁₀₉	—	2	4	no	—
34	C₂×C₁₆×C₁₀₉	—	2	4	no	—
35	#97×C₁₀₉	—	2	2	no	—
36	#97×C₁₀₉	—	2	2	no	—
37	C₄×C₈×C₁₀₉	—	2	2	no	—
38	C₄×C₈×C₁₀₉	—	2	2	no	—
39	C₂⁴×C₁₀₉	M₁₆	1	1	yes	characteristic
40	C₂²×C₄×C₁₀₉	—	2	1	no	—
41	C₂²×C₄×C₁₀₉	—	2	1	no	—
42	C₂²×C₄×C₁₀₉	—	2	1	no	—
43	C₂²×C₄×C₁₀₉	—	2	1	no	—
44	C₂²×C₄×C₁₀₉	—	2	1	no	—
45	C₂²×C₄×C₁₀₉	—	2	1	no	—
46	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₁₀₉	C₂×C₈	1	2	yes	—
48	C₂²×C₄×C₁₀₉	M₁₆	1	1	yes	characteristic
49	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
50	M₁₆×C₁₀₉	—	2	8	no	—
51	C₂×C₈×C₁₀₉	—	2	8	no	—
52	C₂×C₈×C₁₀₉	—	2	1	no	—
53	C₂×C₈×C₁₀₉	—	2	1	no	—
54	C₂×C₈×C₁₀₉	—	2	1	no	—
55	C₂×C₈×C₁₀₉	—	2	1	no	—
56	C₂×C₈×C₁₀₉	—	2	1	no	—
57	C₂×C₈×C₁₀₉	—	2	1	no	—
58	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	2	yes	—
59	C₂×C₈×C₁₀₉	M₁₆	1	2	yes	—
60	C₂×C₈×C₁₀₉	M₁₆	1	2	yes	—
61	C₂×C₈×C₁₀₉	C₂×D₄	1	2	yes	—
62	C₂×C₈×C₁₀₉	#41	1	2	yes	—
63	C₂×C₈×C₁₀₉	#41	1	2	yes	—
64	C₄⋊C₄×C₁₀₉	—	2	4	no	—
65	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
66	C₄²×C₁₀₉	—	2	1	no	—
67	C₄²×C₁₀₉	—	2	1	no	—
68	C₄²×C₁₀₉	C₂×C₈	1	2	yes	—
69	C₄²×C₁₀₉	C₂×C₈	1	2	yes	—
70	C₁₆×C₁₀₉	—	4	4	no	—
71	C₂³×C₁₀₉	—	2	1	no	—
72	C₂³×C₁₀₉	—	2	1	no	—
73	C₂³×C₁₀₉	—	4	2	no	—
74	C₂³×C₁₀₉	C₂²⋊C₈	1	2	yes	—
75	C₂³×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
76	C₂×C₄×C₁₀₉	—	2	1	no	—
77	C₂×C₄×C₁₀₉	—	2	1	no	—
78	C₂×C₄×C₁₀₉	—	2	2	no	—
79	C₂×C₄×C₁₀₉	—	2	2	no	—
80	C₂×C₄×C₁₀₉	—	2	2	no	—
81	C₂×C₄×C₁₀₉	—	2	2	no	—
82	C₂×C₄×C₁₀₉	—	2	2	no	—
83	C₂×C₄×C₁₀₉	—	2	2	no	—
84	C₂×C₄×C₁₀₉	—	4	2	no	—
85	C₂×C₄×C₁₀₉	—	4	2	no	—
86	C₂×C₄×C₁₀₉	C₂²⋊C₈	1	2	yes	—
87	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₁₀₉	#117	1	1	yes	characteristic
89	C₂×C₄×C₁₀₉	#117	1	1	yes	characteristic
90	C₂×C₄×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
91	C₂×C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
92	C₂×C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
93	C₈×C₁₀₉	—	2	2	no	—
94	C₈×C₁₀₉	—	2	2	no	—
95	C₈×C₁₀₉	—	2	2	no	—
96	C₈×C₁₀₉	#123	1	4	yes	—
97	C₂²×C₁₀₉	—	2	2	no	—
98	C₂²×C₁₀₉	—	2	2	no	—
99	C₂²×C₁₀₉	—	4	2	no	—
100	C₂²×C₁₀₉	—	4	2	no	—
101	C₂²×C₁₀₉	—	4	2	no	—
102	C₂²×C₁₀₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
103	C₂²×C₁₀₉	#432	1	1	yes	characteristic
104	C₂²×C₁₀₉	#432	1	1	yes	characteristic
105	C₄×C₁₀₉	—	2	2	no	—
106	C₄×C₁₀₉	—	4	2	no	—
107	C₄×C₁₀₉	—	4	2	no	—
108	C₄×C₁₀₉	#381	1	1	yes	characteristic
109	C₄×C₁₀₉	#407	1	1	yes	characteristic
110	C₄×C₁₀₉	#431	1	1	yes	characteristic
111	C₄×C₁₀₉	#431	1	1	yes	characteristic
112	#12850	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂×C₁₀₉	—	2	1	no	—
114	C₂×C₁₀₉	—	2	1	no	—
115	C₂×C₁₀₉	—	4	2	no	—
116	C₂×C₁₀₉	#1817	1	1	yes	characteristic
117	C₂×C₁₀₉	#2166	1	2	yes	—
118	C₂×#405	C₂×C₁₀₉	1	1	yes	characteristic
119	C₂×#349	C₂×C₁₀₉	1	2	yes	—
120	#1376	C₂×C₁₀₉	1	2	yes	complemented
121	#1375	C₂×C₁₀₉	1	2	yes	complemented
122	C₁₀₉	#12850	1	1	yes	characteristic, direct product factor, Sylow
123	C₂²×M₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
124	C₂³×C₈	C₄×C₁₀₉	1	1	yes	characteristic
125	C₂×#117	C₄×C₁₀₉	1	1	yes	characteristic
126	C₂×M₃₂	C₂²×C₁₀₉	1	4	yes	—
127	#405	—	2	4	no	—
128	C₂×#97	C₄×C₁₀₉	1	1	yes	characteristic
129	C₂×#97	C₂²×C₁₀₉	1	1	yes	characteristic
130	C₂×C₄×C₈	C₄×C₁₀₉	1	1	yes	characteristic
131	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
132	#349	—	2	4	no	—
133	C₂³×C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
134	C₂×M₁₆	—	2	2	no	—
135	C₂×M₁₆	—	2	2	no	—
136	C₂×M₁₆	D₄×C₁₀₉	1	4	yes	—
137	C₂²×C₈	—	2	2	no	—
138	C₂²×C₈	—	2	2	no	—
139	C₂²×C₈	—	2	2	no	—
140	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
141	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
142	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
143	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
144	#117	—	2	2	no	—
145	#117	C₈×C₁₀₉	1	4	yes	—
146	C₂×C₄⋊C₄	—	2	1	no	—
147	C₂×C₂²⋊C₄	—	2	1	no	—
148	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
149	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
150	M₃₂	—	2	4	no	—
151	M₃₂	—	2	4	no	—
152	C₂×C₁₆	—	2	4	no	—
153	#97	—	2	2	no	—
154	#97	—	2	2	no	—
155	C₄×C₈	—	2	2	no	—
156	C₄×C₈	—	2	2	no	—
157	C₂⁴	M₁₆×C₁₀₉	1	1	yes	characteristic
158	C₂²×C₄	—	2	1	no	—
159	C₂²×C₄	—	2	1	no	—
160	C₂²×C₄	—	2	1	no	—
161	C₂²×C₄	—	2	1	no	—
162	C₂²×C₄	—	2	1	no	—
163	C₂²×C₄	—	2	1	no	—
164	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	2	yes	—
165	C₂²×C₄	C₂×C₈×C₁₀₉	1	2	yes	—
166	C₂²×C₄	M₁₆×C₁₀₉	1	1	yes	characteristic
167	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
168	M₁₆	—	2	8	no	—
169	C₂×C₈	—	2	8	no	—
170	C₂×C₈	—	2	1	no	—
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	2	yes	—
177	C₂×C₈	M₁₆×C₁₀₉	1	2	yes	—
178	C₂×C₈	M₁₆×C₁₀₉	1	2	yes	—
179	C₂×C₈	C₂×D₄×C₁₀₉	1	2	yes	—
180	C₂×C₈	#41×C₁₀₉	1	2	yes	—
181	C₂×C₈	#41×C₁₀₉	1	2	yes	—
182	C₄⋊C₄	—	2	4	no	—
183	C₂²⋊C₄	—	2	4	no	—
184	C₄²	—	2	1	no	—
185	C₄²	—	2	1	no	—
186	C₄²	C₂×C₈×C₁₀₉	1	2	yes	—
187	C₄²	C₂×C₈×C₁₀₉	1	2	yes	—
188	C₁₆	—	4	4	no	—
189	C₂³	—	2	1	no	—
190	C₂³	—	2	1	no	—
191	C₂³	—	4	2	no	—
192	C₂³	C₂²⋊C₈×C₁₀₉	1	2	yes	—
193	C₂³	C₂×M₁₆×C₁₀₉	1	1	yes	characteristic
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	4	2	no	—
203	C₂×C₄	—	4	2	no	—
204	C₂×C₄	C₂²⋊C₈×C₁₀₉	1	2	yes	—
205	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
206	C₂×C₄	#117×C₁₀₉	1	1	yes	characteristic
207	C₂×C₄	#117×C₁₀₉	1	1	yes	characteristic
208	C₂×C₄	C₂²×C₈×C₁₀₉	1	1	yes	characteristic
209	C₂×C₄	C₂×M₁₆×C₁₀₉	1	1	yes	characteristic
210	C₂×C₄	C₂×M₁₆×C₁₀₉	1	1	yes	characteristic
211	C₈	—	2	2	no	—
212	C₈	—	2	2	no	—
213	C₈	—	2	2	no	—
214	C₈	#123×C₁₀₉	1	4	yes	—
215	C₂²	—	2	2	no	—
216	C₂²	—	2	2	no	—
217	C₂²	—	4	2	no	—
218	C₂²	—	4	2	no	—
219	C₂²	—	4	2	no	—
220	C₂²	C₂×C₂²⋊C₈×C₁₀₉	1	1	yes	characteristic
221	C₂²	#432×C₁₀₉	1	1	yes	characteristic
222	C₂²	#432×C₁₀₉	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	4	2	no	—
225	C₄	—	4	2	no	—
226	C₄	#381×C₁₀₉	1	1	yes	characteristic
227	C₄	#407×C₁₀₉	1	1	yes	characteristic
228	C₄	#431×C₁₀₉	1	1	yes	characteristic
229	C₄	#431×C₁₀₉	1	1	yes	characteristic
230	C₂	—	2	1	no	—
231	C₂	—	2	1	no	—
232	C₂	—	4	2	no	—
233	C₂	C₁₀₉×#1817	1	1	yes	characteristic
234	C₂	C₁₀₉×#2166	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₀₉	#117
commutator	C₂×C₄	C₂²×C₈×C₁₀₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₀₉
socle	C₂²×C₁₀₉	#432
prime radical	C₂⁴×C₁₀₉	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₉ ◁ C₂×#97×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₀₉×#1376	C₂	semidirect
4. C₁₀₉×#1375	C₂	semidirect
112. #12850	C₁₀₉	direct (N × H)
120. #1376	C₂×C₁₀₉	semidirect
121. #1375	C₂×C₁₀₉	semidirect
122. C₁₀₉	#12850	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#405×C₁₀₉	2
C₂×#349×C₁₀₉	2
C₁₀₉×#1376	2
C₁₀₉×#1375	2
#12850	109

Subgroups — C109×#12850 #42130844

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#12850 #42130844