Subgroups — C₁₀₉×#12544 #42130538

Order 27904.

678 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₃₂×C₁₀₉	C₂	1	2	yes	maximal
2	C₂²×C₄×C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#1374	C₂	1	4	yes	maximal, complemented
4	C₂³×C₈×C₁₀₉	C₄	1	1	yes	characteristic
5	C₂³×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
6	C₂²×C₄²×C₁₀₉	C₄	1	1	yes	characteristic
7	C₂×M₃₂×C₁₀₉	—	2	4	no	—
8	C₂×M₃₂×C₁₀₉	—	2	4	no	—
9	C₂×M₃₂×C₁₀₉	C₄	1	8	yes	complemented
10	C₂²×C₁₆×C₁₀₉	C₂²	1	4	yes	—
11	C₂×C₄×C₈×C₁₀₉	—	2	4	no	—
12	C₂×C₄×C₈×C₁₀₉	C₄	1	2	yes	—
13	C₂×C₄×C₈×C₁₀₉	C₂²	1	2	yes	—
14	C₂³×C₄×C₁₀₉	C₈	1	2	yes	—
15	C₂³×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₀₉	—	2	2	no	—
17	C₂²×C₈×C₁₀₉	—	2	2	no	—
18	C₂²×C₈×C₁₀₉	C₈	1	4	yes	—
19	C₂²×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₀₉	C₂×C₄	1	4	yes	—
22	C₂²×C₈×C₁₀₉	D₄	1	2	yes	—
23	C₂²×C₈×C₁₀₉	D₄	1	4	yes	—
24	C₂²×C₈×C₁₀₉	Q₈	1	2	yes	—
25	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₁₀₉	—	2	2	no	—
27	C₂×C₄²×C₁₀₉	—	2	2	no	—
28	C₂×C₄²×C₁₀₉	C₈	1	2	yes	—
29	C₂×C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
30	M₃₂×C₁₀₉	—	2	16	no	—
31	C₂×C₁₆×C₁₀₉	—	2	4	no	—
32	C₂×C₁₆×C₁₀₉	—	2	4	no	—
33	C₂×C₁₆×C₁₀₉	C₂×C₄	1	8	yes	complemented
34	C₄×C₈×C₁₀₉	—	2	4	no	—
35	C₄×C₈×C₁₀₉	—	2	4	no	—
36	C₂⁴×C₁₀₉	C₂×C₈	1	1	yes	characteristic
37	C₂²×C₄×C₁₀₉	—	2	4	no	—
38	C₂²×C₄×C₁₀₉	—	2	8	no	—
39	C₂²×C₄×C₁₀₉	C₄²	1	2	yes	—
40	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
41	C₂²×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
42	C₂²×C₄×C₁₀₉	C₂×C₈	1	4	yes	—
43	C₂²×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
44	C₂²×C₄×C₁₀₉	C₂×C₈	1	2	yes	—
45	C₂²×C₄×C₁₀₉	M₁₆	1	2	yes	—
46	C₂²×C₄×C₁₀₉	M₁₆	1	2	yes	—
47	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₁₀₉	—	2	8	no	—
49	C₂×C₈×C₁₀₉	—	2	2	no	—
50	C₂×C₈×C₁₀₉	—	2	2	no	—
51	C₂×C₈×C₁₀₉	C₄²	1	2	yes	—
52	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	2	yes	—
53	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	8	yes	—
54	C₂×C₈×C₁₀₉	C₄⋊C₄	1	2	yes	—
55	C₂×C₈×C₁₀₉	C₄⋊C₄	1	8	yes	—
56	C₂×C₈×C₁₀₉	C₂×C₈	1	4	yes	—
57	C₂×C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
58	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₁₀₉	C₂×D₄	1	2	yes	—
60	C₂×C₈×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
61	C₄²×C₁₀₉	—	2	4	no	—
62	C₄²×C₁₀₉	—	2	2	no	—
63	C₄²×C₁₀₉	—	2	2	no	—
64	C₁₆×C₁₀₉	—	2	8	no	—
65	C₂³×C₁₀₉	—	2	4	no	—
66	C₂³×C₁₀₉	C₄×C₈	1	2	yes	—
67	C₂³×C₁₀₉	C₂²⋊C₈	1	2	yes	—
68	C₂³×C₁₀₉	C₄⋊C₈	1	2	yes	—
69	C₂³×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
70	C₂×C₄×C₁₀₉	—	2	16	no	—
71	C₂×C₄×C₁₀₉	—	2	4	no	—
72	C₂×C₄×C₁₀₉	—	2	4	no	—
73	C₂×C₄×C₁₀₉	#95	1	4	yes	—
74	C₂×C₄×C₁₀₉	C₄×C₈	1	2	yes	—
75	C₂×C₄×C₁₀₉	#97	1	4	yes	—
76	C₂×C₄×C₁₀₉	C₂²⋊C₈	1	2	yes	—
77	C₂×C₄×C₁₀₉	C₂²⋊C₈	1	4	yes	—
78	C₂×C₄×C₁₀₉	C₄⋊C₈	1	2	yes	—
79	C₂×C₄×C₁₀₉	C₄⋊C₈	1	4	yes	—
80	C₂×C₄×C₁₀₉	M₃₂	1	8	yes	complemented
81	C₂×C₄×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
82	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
86	C₂×C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
87	C₈×C₁₀₉	—	2	4	no	—
88	C₈×C₁₀₉	#95	1	4	yes	—
89	C₈×C₁₀₉	C₂×C₂²⋊C₄	1	2	yes	—
90	C₈×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
91	C₂²×C₁₀₉	—	2	4	no	—
92	C₂²×C₁₀₉	—	2	4	no	—
93	C₂²×C₁₀₉	—	2	4	no	—
94	C₂²×C₁₀₉	#336	1	4	yes	—
95	C₂²×C₁₀₉	C₂×C₄×C₈	1	1	yes	characteristic
96	C₂²×C₁₀₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
97	C₂²×C₁₀₉	C₂×C₄⋊C₈	1	1	yes	characteristic
98	C₂²×C₁₀₉	C₂×M₃₂	1	4	yes	—
99	C₄×C₁₀₉	—	2	8	no	—
100	C₄×C₁₀₉	#336	1	4	yes	—
101	C₄×C₁₀₉	C₂×#95	1	1	yes	characteristic
102	C₄×C₁₀₉	C₂×#97	1	1	yes	characteristic
103	C₄×C₁₀₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
104	C₄×C₁₀₉	C₂×C₄⋊C₈	1	1	yes	characteristic
105	#12544	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
106	C₂×C₁₀₉	—	2	4	no	—
107	C₂×C₁₀₉	C₂×#336	1	1	yes	characteristic
108	C₂×C₁₀₉	C₄×M₃₂	1	2	yes	—
109	C₂×C₁₀₉	#2112	1	2	yes	—
110	C₂×C₁₀₉	#2150	1	2	yes	—
111	C₂²×M₃₂	C₂×C₁₀₉	1	2	yes	—
112	C₂²×C₄×C₈	C₂×C₁₀₉	1	1	yes	characteristic
113	#1374	C₂×C₁₀₉	1	4	yes	complemented
114	C₁₀₉	#12544	1	1	yes	characteristic, direct product factor, Sylow
115	C₂³×C₈	C₄×C₁₀₉	1	1	yes	characteristic
116	C₂³×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
117	C₂²×C₄²	C₄×C₁₀₉	1	1	yes	characteristic
118	C₂×M₃₂	—	2	4	no	—
119	C₂×M₃₂	—	2	4	no	—
120	C₂×M₃₂	C₄×C₁₀₉	1	8	yes	complemented
121	C₂²×C₁₆	C₂²×C₁₀₉	1	4	yes	—
122	C₂×C₄×C₈	—	2	4	no	—
123	C₂×C₄×C₈	C₄×C₁₀₉	1	2	yes	—
124	C₂×C₄×C₈	C₂²×C₁₀₉	1	2	yes	—
125	C₂³×C₄	C₈×C₁₀₉	1	2	yes	—
126	C₂³×C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
127	C₂²×C₈	—	2	2	no	—
128	C₂²×C₈	—	2	2	no	—
129	C₂²×C₈	C₈×C₁₀₉	1	4	yes	—
130	C₂²×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
131	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
132	C₂²×C₈	C₂×C₄×C₁₀₉	1	4	yes	—
133	C₂²×C₈	D₄×C₁₀₉	1	2	yes	—
134	C₂²×C₈	D₄×C₁₀₉	1	4	yes	—
135	C₂²×C₈	Q₈×C₁₀₉	1	2	yes	—
136	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
137	C₂×C₄²	—	2	2	no	—
138	C₂×C₄²	—	2	2	no	—
139	C₂×C₄²	C₈×C₁₀₉	1	2	yes	—
140	C₂×C₄²	C₂×C₄×C₁₀₉	1	2	yes	—
141	M₃₂	—	2	16	no	—
142	C₂×C₁₆	—	2	4	no	—
143	C₂×C₁₆	—	2	4	no	—
144	C₂×C₁₆	C₂×C₄×C₁₀₉	1	8	yes	complemented
145	C₄×C₈	—	2	4	no	—
146	C₄×C₈	—	2	4	no	—
147	C₂⁴	C₂×C₈×C₁₀₉	1	1	yes	characteristic
148	C₂²×C₄	—	2	4	no	—
149	C₂²×C₄	—	2	8	no	—
150	C₂²×C₄	C₄²×C₁₀₉	1	2	yes	—
151	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	2	yes	—
152	C₂²×C₄	C₄⋊C₄×C₁₀₉	1	2	yes	—
153	C₂²×C₄	C₂×C₈×C₁₀₉	1	4	yes	—
154	C₂²×C₄	C₂×C₈×C₁₀₉	1	1	yes	characteristic
155	C₂²×C₄	C₂×C₈×C₁₀₉	1	2	yes	—
156	C₂²×C₄	M₁₆×C₁₀₉	1	2	yes	—
157	C₂²×C₄	M₁₆×C₁₀₉	1	2	yes	—
158	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
159	C₂×C₈	—	2	8	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	C₄²×C₁₀₉	1	2	yes	—
163	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	2	yes	—
164	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	8	yes	—
165	C₂×C₈	C₄⋊C₄×C₁₀₉	1	2	yes	—
166	C₂×C₈	C₄⋊C₄×C₁₀₉	1	8	yes	—
167	C₂×C₈	C₂×C₈×C₁₀₉	1	4	yes	—
168	C₂×C₈	C₂²×C₄×C₁₀₉	1	2	yes	—
169	C₂×C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
170	C₂×C₈	C₂×D₄×C₁₀₉	1	2	yes	—
171	C₂×C₈	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
172	C₄²	—	2	4	no	—
173	C₄²	—	2	2	no	—
174	C₄²	—	2	2	no	—
175	C₁₆	—	2	8	no	—
176	C₂³	—	2	4	no	—
177	C₂³	C₄×C₈×C₁₀₉	1	2	yes	—
178	C₂³	C₂²⋊C₈×C₁₀₉	1	2	yes	—
179	C₂³	C₄⋊C₈×C₁₀₉	1	2	yes	—
180	C₂³	C₂²×C₈×C₁₀₉	1	1	yes	characteristic
181	C₂×C₄	—	2	16	no	—
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	#95×C₁₀₉	1	4	yes	—
185	C₂×C₄	C₄×C₈×C₁₀₉	1	2	yes	—
186	C₂×C₄	#97×C₁₀₉	1	4	yes	—
187	C₂×C₄	C₂²⋊C₈×C₁₀₉	1	2	yes	—
188	C₂×C₄	C₂²⋊C₈×C₁₀₉	1	4	yes	—
189	C₂×C₄	C₄⋊C₈×C₁₀₉	1	2	yes	—
190	C₂×C₄	C₄⋊C₈×C₁₀₉	1	4	yes	—
191	C₂×C₄	M₃₂×C₁₀₉	1	8	yes	complemented
192	C₂×C₄	C₂×C₄²×C₁₀₉	1	1	yes	characteristic
193	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
194	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
195	C₂×C₄	C₂²×C₈×C₁₀₉	1	1	yes	characteristic
196	C₂×C₄	C₂×M₁₆×C₁₀₉	1	1	yes	characteristic
197	C₂×C₄	C₂×M₁₆×C₁₀₉	1	1	yes	characteristic
198	C₈	—	2	4	no	—
199	C₈	#95×C₁₀₉	1	4	yes	—
200	C₈	C₂×C₂²⋊C₄×C₁₀₉	1	2	yes	—
201	C₈	C₂×C₄⋊C₄×C₁₀₉	1	2	yes	—
202	C₂²	—	2	4	no	—
203	C₂²	—	2	4	no	—
204	C₂²	—	2	4	no	—
205	C₂²	#336×C₁₀₉	1	4	yes	—
206	C₂²	C₂×C₄×C₈×C₁₀₉	1	1	yes	characteristic
207	C₂²	C₂×C₂²⋊C₈×C₁₀₉	1	1	yes	characteristic
208	C₂²	C₂×C₄⋊C₈×C₁₀₉	1	1	yes	characteristic
209	C₂²	C₂×M₃₂×C₁₀₉	1	4	yes	—
210	C₄	—	2	8	no	—
211	C₄	#336×C₁₀₉	1	4	yes	—
212	C₄	C₂×#95×C₁₀₉	1	1	yes	characteristic
213	C₄	C₂×#97×C₁₀₉	1	1	yes	characteristic
214	C₄	C₂×C₂²⋊C₈×C₁₀₉	1	1	yes	characteristic
215	C₄	C₂×C₄⋊C₈×C₁₀₉	1	1	yes	characteristic
216	C₂	—	2	4	no	—
217	C₂	C₂×#336×C₁₀₉	1	1	yes	characteristic
218	C₂	C₄×M₃₂×C₁₀₉	1	2	yes	—
219	C₂	C₁₀₉×#2112	1	2	yes	—
220	C₂	C₁₀₉×#2150	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₁₀₉×#1374	C₂	semidirect
9. C₂×M₃₂×C₁₀₉	C₄	semidirect
33. C₂×C₁₆×C₁₀₉	C₂×C₄	semidirect
80. C₂×C₄×C₁₀₉	M₃₂	semidirect
105. #12544	C₁₀₉	direct (N × H)
113. #1374	C₂×C₁₀₉	semidirect
114. C₁₀₉	#12544	direct (N × H)
120. C₂×M₃₂	C₄×C₁₀₉	semidirect
144. C₂×C₁₆	C₂×C₄×C₁₀₉	semidirect
191. C₂×C₄	M₃₂×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×M₃₂×C₁₀₉	2
C₂²×C₄×C₈×C₁₀₉	2
C₁₀₉×#1374	2
#12544	109

Subgroups — C109×#12544 #42130538

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#12544 #42130538