Subgroups — C₁₀₉×#12547 #42130541

Order 27904.

586 conjugacy classes of subgroups.

The subgroup lattice has 230 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	C₂×#405×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#1375	C₂	1	4	yes	maximal, complemented
4	C₂²×M₁₆×C₁₀₉	C₄	1	1	yes	characteristic
5	C₂³×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
6	C₂×#117×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	4	yes	complemented
10	C₂×M₃₂×C₁₀₉	C₂²	1	4	yes	—
11	C₂²×C₁₆×C₁₀₉	C₄	1	4	yes	complemented
12	#405×C₁₀₉	—	2	4	no	—
13	C₂×#97×C₁₀₉	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₁₀₉	C₄	1	2	yes	—
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	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	2	yes	—
22	C₂²×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₁₀₉	D₄	1	2	yes	—
24	C₂²×C₈×C₁₀₉	D₄	1	4	yes	—
25	C₂²×C₈×C₁₀₉	Q₈	1	2	yes	—
26	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
27	#117×C₁₀₉	—	2	2	no	—
28	#117×C₁₀₉	C₈	1	4	yes	—
29	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
30	C₂×C₂²⋊C₄×C₁₀₉	—	2	1	no	—
31	C₂×C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
32	M₃₂×C₁₀₉	—	2	8	no	—
33	M₃₂×C₁₀₉	—	2	4	no	—
34	M₃₂×C₁₀₉	—	2	4	no	—
35	C₂×C₁₆×C₁₀₉	—	2	8	no	—
36	C₂×C₁₆×C₁₀₉	C₂×C₄	1	8	yes	—
37	#97×C₁₀₉	—	2	4	no	—
38	C₄×C₈×C₁₀₉	—	2	4	no	—
39	C₂⁴×C₁₀₉	M₁₆	1	1	yes	characteristic
40	C₂²×C₄×C₁₀₉	—	2	2	no	—
41	C₂²×C₄×C₁₀₉	C₄²	1	2	yes	—
42	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
43	C₂²×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
44	C₂²×C₄×C₁₀₉	C₂×C₈	1	2	yes	—
45	C₂²×C₄×C₁₀₉	C₂×C₈	1	2	yes	—
46	C₂²×C₄×C₁₀₉	M₁₆	1	1	yes	characteristic
47	C₂²×C₄×C₁₀₉	M₁₆	1	2	yes	—
48	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
49	M₁₆×C₁₀₉	—	2	8	no	—
50	C₂×C₈×C₁₀₉	—	2	2	no	—
51	C₂×C₈×C₁₀₉	—	2	2	no	—
52	C₂×C₈×C₁₀₉	C₄²	1	1	yes	characteristic
53	C₂×C₈×C₁₀₉	C₄²	1	1	yes	characteristic
54	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	2	yes	—
55	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	4	yes	—
56	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	4	yes	—
57	C₂×C₈×C₁₀₉	C₄⋊C₄	1	2	yes	—
58	C₂×C₈×C₁₀₉	C₄⋊C₄	1	4	yes	—
59	C₂×C₈×C₁₀₉	C₄⋊C₄	1	4	yes	—
60	C₂×C₈×C₁₀₉	C₂×C₈	1	4	yes	—
61	C₂×C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
62	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₁₀₉	C₂×D₄	1	2	yes	—
64	C₂×C₈×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
65	C₄⋊C₄×C₁₀₉	—	2	4	no	—
66	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
67	C₄²×C₁₀₉	—	2	2	no	—
68	C₄²×C₁₀₉	C₂×C₈	1	4	yes	—
69	C₁₆×C₁₀₉	—	2	8	no	—
70	C₂³×C₁₀₉	—	4	2	no	—
71	C₂³×C₁₀₉	#97	1	2	yes	—
72	C₂³×C₁₀₉	C₂²⋊C₈	1	2	yes	—
73	C₂³×C₁₀₉	C₄⋊C₈	1	2	yes	—
74	C₂³×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
75	C₂×C₄×C₁₀₉	—	2	4	no	—
76	C₂×C₄×C₁₀₉	—	4	4	no	—
77	C₂×C₄×C₁₀₉	#95	1	4	yes	—
78	C₂×C₄×C₁₀₉	C₄×C₈	1	4	yes	—
79	C₂×C₄×C₁₀₉	#97	1	2	yes	—
80	C₂×C₄×C₁₀₉	C₂²⋊C₈	1	2	yes	—
81	C₂×C₄×C₁₀₉	C₂²⋊C₈	1	4	yes	—
82	C₂×C₄×C₁₀₉	C₄⋊C₈	1	2	yes	—
83	C₂×C₄×C₁₀₉	C₄⋊C₈	1	4	yes	—
84	C₂×C₄×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
85	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
88	C₂×C₄×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
89	C₂×C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
90	C₈×C₁₀₉	—	2	4	no	—
91	C₈×C₁₀₉	#95	1	2	yes	—
92	C₈×C₁₀₉	#95	1	2	yes	—
93	C₈×C₁₀₉	C₂×C₂²⋊C₄	1	2	yes	—
94	C₈×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
95	C₂²×C₁₀₉	—	2	2	no	—
96	C₂²×C₁₀₉	—	2	4	no	—
97	C₂²×C₁₀₉	—	4	2	no	—
98	C₂²×C₁₀₉	—	4	2	no	—
99	C₂²×C₁₀₉	#336	1	4	yes	—
100	C₂²×C₁₀₉	C₂×#97	1	1	yes	characteristic
101	C₂²×C₁₀₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
102	C₂²×C₁₀₉	C₂×C₄⋊C₈	1	1	yes	characteristic
103	C₄×C₁₀₉	—	4	4	no	—
104	C₄×C₁₀₉	#336	1	4	yes	—
105	C₄×C₁₀₉	C₂×#95	1	1	yes	characteristic
106	C₄×C₁₀₉	C₂×C₄×C₈	1	1	yes	characteristic
107	C₄×C₁₀₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
108	C₄×C₁₀₉	C₂×C₄⋊C₈	1	1	yes	characteristic
109	#12547	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₂×C₁₀₉	—	2	1	no	—
111	C₂×C₁₀₉	—	2	4	no	—
112	C₂×C₁₀₉	—	2	1	no	—
113	C₂×C₁₀₉	C₂×#336	1	1	yes	characteristic
114	C₂×C₁₀₉	#2110	1	2	yes	—
115	C₂²×M₃₂	C₂×C₁₀₉	1	2	yes	—
116	C₂×#405	C₂×C₁₀₉	1	1	yes	characteristic
117	#1375	C₂×C₁₀₉	1	4	yes	complemented
118	C₁₀₉	#12547	1	1	yes	characteristic, direct product factor, Sylow
119	C₂²×M₁₆	C₄×C₁₀₉	1	1	yes	characteristic
120	C₂³×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
121	C₂×#117	C₄×C₁₀₉	1	1	yes	characteristic
122	C₂×M₃₂	—	2	4	no	—
123	C₂×M₃₂	—	2	4	no	—
124	C₂×M₃₂	C₄×C₁₀₉	1	4	yes	complemented
125	C₂×M₃₂	C₂²×C₁₀₉	1	4	yes	—
126	C₂²×C₁₆	C₄×C₁₀₉	1	4	yes	complemented
127	#405	—	2	4	no	—
128	C₂×#97	C₂²×C₁₀₉	1	2	yes	—
129	C₂×C₄×C₈	C₄×C₁₀₉	1	2	yes	—
130	C₂³×C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
131	C₂×M₁₆	—	2	2	no	—
132	C₂×M₁₆	—	2	2	no	—
133	C₂×M₁₆	C₈×C₁₀₉	1	4	yes	—
134	C₂²×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
135	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
136	C₂²×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
137	C₂²×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
138	C₂²×C₈	D₄×C₁₀₉	1	2	yes	—
139	C₂²×C₈	D₄×C₁₀₉	1	4	yes	—
140	C₂²×C₈	Q₈×C₁₀₉	1	2	yes	—
141	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
142	#117	—	2	2	no	—
143	#117	C₈×C₁₀₉	1	4	yes	—
144	C₂×C₄⋊C₄	—	2	1	no	—
145	C₂×C₂²⋊C₄	—	2	1	no	—
146	C₂×C₄²	C₂×C₄×C₁₀₉	1	2	yes	—
147	M₃₂	—	2	8	no	—
148	M₃₂	—	2	4	no	—
149	M₃₂	—	2	4	no	—
150	C₂×C₁₆	—	2	8	no	—
151	C₂×C₁₆	C₂×C₄×C₁₀₉	1	8	yes	—
152	#97	—	2	4	no	—
153	C₄×C₈	—	2	4	no	—
154	C₂⁴	M₁₆×C₁₀₉	1	1	yes	characteristic
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	C₄²×C₁₀₉	1	2	yes	—
157	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	2	yes	—
158	C₂²×C₄	C₄⋊C₄×C₁₀₉	1	2	yes	—
159	C₂²×C₄	C₂×C₈×C₁₀₉	1	2	yes	—
160	C₂²×C₄	C₂×C₈×C₁₀₉	1	2	yes	—
161	C₂²×C₄	M₁₆×C₁₀₉	1	1	yes	characteristic
162	C₂²×C₄	M₁₆×C₁₀₉	1	2	yes	—
163	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
164	M₁₆	—	2	8	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	C₄²×C₁₀₉	1	1	yes	characteristic
168	C₂×C₈	C₄²×C₁₀₉	1	1	yes	characteristic
169	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	2	yes	—
170	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	4	yes	—
171	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	4	yes	—
172	C₂×C₈	C₄⋊C₄×C₁₀₉	1	2	yes	—
173	C₂×C₈	C₄⋊C₄×C₁₀₉	1	4	yes	—
174	C₂×C₈	C₄⋊C₄×C₁₀₉	1	4	yes	—
175	C₂×C₈	C₂×C₈×C₁₀₉	1	4	yes	—
176	C₂×C₈	C₂²×C₄×C₁₀₉	1	2	yes	—
177	C₂×C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
178	C₂×C₈	C₂×D₄×C₁₀₉	1	2	yes	—
179	C₂×C₈	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
180	C₄⋊C₄	—	2	4	no	—
181	C₂²⋊C₄	—	2	4	no	—
182	C₄²	—	2	2	no	—
183	C₄²	C₂×C₈×C₁₀₉	1	4	yes	—
184	C₁₆	—	2	8	no	—
185	C₂³	—	4	2	no	—
186	C₂³	#97×C₁₀₉	1	2	yes	—
187	C₂³	C₂²⋊C₈×C₁₀₉	1	2	yes	—
188	C₂³	C₄⋊C₈×C₁₀₉	1	2	yes	—
189	C₂³	C₂×M₁₆×C₁₀₉	1	1	yes	characteristic
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	4	4	no	—
192	C₂×C₄	#95×C₁₀₉	1	4	yes	—
193	C₂×C₄	C₄×C₈×C₁₀₉	1	4	yes	—
194	C₂×C₄	#97×C₁₀₉	1	2	yes	—
195	C₂×C₄	C₂²⋊C₈×C₁₀₉	1	2	yes	—
196	C₂×C₄	C₂²⋊C₈×C₁₀₉	1	4	yes	—
197	C₂×C₄	C₄⋊C₈×C₁₀₉	1	2	yes	—
198	C₂×C₄	C₄⋊C₈×C₁₀₉	1	4	yes	—
199	C₂×C₄	C₂×C₄²×C₁₀₉	1	1	yes	characteristic
200	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
201	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
202	C₂×C₄	C₂²×C₈×C₁₀₉	1	1	yes	characteristic
203	C₂×C₄	C₂²×C₈×C₁₀₉	1	1	yes	characteristic
204	C₂×C₄	C₂×M₁₆×C₁₀₉	1	1	yes	characteristic
205	C₈	—	2	4	no	—
206	C₈	#95×C₁₀₉	1	2	yes	—
207	C₈	#95×C₁₀₉	1	2	yes	—
208	C₈	C₂×C₂²⋊C₄×C₁₀₉	1	2	yes	—
209	C₈	C₂×C₄⋊C₄×C₁₀₉	1	2	yes	—
210	C₂²	—	2	2	no	—
211	C₂²	—	2	4	no	—
212	C₂²	—	4	2	no	—
213	C₂²	—	4	2	no	—
214	C₂²	#336×C₁₀₉	1	4	yes	—
215	C₂²	C₂×#97×C₁₀₉	1	1	yes	characteristic
216	C₂²	C₂×C₂²⋊C₈×C₁₀₉	1	1	yes	characteristic
217	C₂²	C₂×C₄⋊C₈×C₁₀₉	1	1	yes	characteristic
218	C₄	—	4	4	no	—
219	C₄	#336×C₁₀₉	1	4	yes	—
220	C₄	C₂×#95×C₁₀₉	1	1	yes	characteristic
221	C₄	C₂×C₄×C₈×C₁₀₉	1	1	yes	characteristic
222	C₄	C₂×C₂²⋊C₈×C₁₀₉	1	1	yes	characteristic
223	C₄	C₂×C₄⋊C₈×C₁₀₉	1	1	yes	characteristic
224	C₂	—	2	1	no	—
225	C₂	—	2	4	no	—
226	C₂	—	2	1	no	—
227	C₂	C₂×#336×C₁₀₉	1	1	yes	characteristic
228	C₂	C₁₀₉×#2110	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₁₀₉	C₄²
commutator	C₄	C₂×C₄×C₈×C₁₀₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₀₉
socle	C₂²×C₁₀₉	C₂×#97
prime radical	C₂⁴×C₁₀₉	M₁₆
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₀₉×#1375	C₂	semidirect
9. C₂×M₃₂×C₁₀₉	C₄	semidirect
11. C₂²×C₁₆×C₁₀₉	C₄	semidirect
109. #12547	C₁₀₉	direct (N × H)
117. #1375	C₂×C₁₀₉	semidirect
118. C₁₀₉	#12547	direct (N × H)
124. C₂×M₃₂	C₄×C₁₀₉	semidirect
126. C₂²×C₁₆	C₄×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
C₂×#405×C₁₀₉	2
C₁₀₉×#1375	2
#12547	109

Subgroups — C109×#12547 #42130541

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#12547 #42130541