Subgroups — C₁₀₉×#12788 #42130782

Order 27904.

538 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#364×C₁₀₉	C₂	1	2	yes	maximal, complemented
3	C₁₀₉×#2112	C₂	1	2	yes	maximal
4	C₁₀₉×#1376	C₂	1	2	yes	maximal, complemented
5	C₂³×C₈×C₁₀₉	C₄	1	1	yes	characteristic
6	C₂³×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
7	C₂²×C₄²×C₁₀₉	C₄	1	1	yes	characteristic
8	C₂×M₃₂×C₁₀₉	C₂²	1	4	yes	—
9	C₂×C₄×C₈×C₁₀₉	—	2	4	no	—
10	C₂×C₄×C₈×C₁₀₉	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₀₉	C₄	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
14	#364×C₁₀₉	—	2	4	no	—
15	C₂²⋊C₁₆×C₁₀₉	—	2	4	no	—
16	C₂³×C₄×C₁₀₉	—	2	1	no	—
17	C₂³×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₀₉	—	2	2	no	—
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	C₈	1	4	yes	—
23	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₀₉	D₄	1	2	yes	—
27	C₂²×C₈×C₁₀₉	Q₈	1	2	yes	—
28	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
29	C₂×C₄²×C₁₀₉	—	2	1	no	—
30	C₂×C₄²×C₁₀₉	—	2	2	no	—
31	C₂×C₄²×C₁₀₉	—	2	2	no	—
32	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
34	M₃₂×C₁₀₉	—	2	4	no	—
35	M₃₂×C₁₀₉	—	2	4	no	—
36	C₂×C₁₆×C₁₀₉	—	2	4	no	—
37	C₄×C₈×C₁₀₉	—	2	2	no	—
38	C₄×C₈×C₁₀₉	—	2	2	no	—
39	C₄×C₈×C₁₀₉	—	2	2	no	—
40	C₄×C₈×C₁₀₉	D₄	1	4	yes	—
41	C₂⁴×C₁₀₉	M₁₆	1	1	yes	characteristic
42	C₂²×C₄×C₁₀₉	—	2	4	no	—
43	C₂²×C₄×C₁₀₉	—	2	8	no	—
44	C₂²×C₄×C₁₀₉	—	2	1	no	—
45	C₂²×C₄×C₁₀₉	—	2	1	no	—
46	C₂²×C₄×C₁₀₉	—	2	1	no	—
47	C₂²×C₄×C₁₀₉	—	2	1	no	—
48	C₂²×C₄×C₁₀₉	—	2	1	no	—
49	C₂²×C₄×C₁₀₉	—	2	1	no	—
50	C₂²×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
51	C₂²×C₄×C₁₀₉	C₂×C₈	1	2	yes	—
52	C₂²×C₄×C₁₀₉	M₁₆	1	1	yes	characteristic
53	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
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	—	2	8	no	—
59	C₂×C₈×C₁₀₉	—	2	4	no	—
60	C₂×C₈×C₁₀₉	—	2	4	no	—
61	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	2	yes	—
62	C₂×C₈×C₁₀₉	C₄⋊C₄	1	2	yes	—
63	C₂×C₈×C₁₀₉	C₂×C₈	1	2	yes	—
64	C₂×C₈×C₁₀₉	C₂×C₈	1	2	yes	—
65	C₂×C₈×C₁₀₉	M₁₆	1	2	yes	—
66	C₂×C₈×C₁₀₉	C₂×D₄	1	2	yes	—
67	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
69	C₂×C₈×C₁₀₉	#41	1	2	yes	—
70	C₄²×C₁₀₉	—	2	4	no	—
71	C₄²×C₁₀₉	—	2	1	no	—
72	C₄²×C₁₀₉	—	2	1	no	—
73	C₄²×C₁₀₉	C₂²⋊C₄	1	2	yes	—
74	C₄²×C₁₀₉	M₁₆	1	2	yes	—
75	C₁₆×C₁₀₉	—	4	4	no	—
76	C₂³×C₁₀₉	—	2	4	no	—
77	C₂³×C₁₀₉	—	2	1	no	—
78	C₂³×C₁₀₉	—	2	1	no	—
79	C₂³×C₁₀₉	C₄⋊C₈	1	2	yes	—
80	C₂³×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
81	C₂×C₄×C₁₀₉	—	2	16	no	—
82	C₂×C₄×C₁₀₉	—	2	4	no	—
83	C₂×C₄×C₁₀₉	—	2	4	no	—
84	C₂×C₄×C₁₀₉	—	2	1	no	—
85	C₂×C₄×C₁₀₉	—	2	2	no	—
86	C₂×C₄×C₁₀₉	—	2	2	no	—
87	C₂×C₄×C₁₀₉	—	2	1	no	—
88	C₂×C₄×C₁₀₉	—	2	2	no	—
89	C₂×C₄×C₁₀₉	—	2	2	no	—
90	C₂×C₄×C₁₀₉	—	2	2	no	—
91	C₂×C₄×C₁₀₉	—	2	2	no	—
92	C₂×C₄×C₁₀₉	C₄⋊C₈	1	2	yes	—
93	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₀₉	#117	1	1	yes	characteristic
96	C₂×C₄×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
97	C₂×C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
98	C₂×C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
99	C₈×C₁₀₉	—	2	2	no	—
100	C₈×C₁₀₉	—	2	2	no	—
101	C₈×C₁₀₉	C₂²⋊C₈	1	4	yes	—
102	C₈×C₁₀₉	#121	1	2	yes	—
103	C₈×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
104	C₂²×C₁₀₉	—	2	4	no	—
105	C₂²×C₁₀₉	—	2	4	no	—
106	C₂²×C₁₀₉	—	2	2	no	—
107	C₂²×C₁₀₉	—	2	2	no	—
108	C₂²×C₁₀₉	—	2	2	no	—
109	C₂²×C₁₀₉	#364	1	4	yes	complemented
110	C₂²×C₁₀₉	#407	1	1	yes	characteristic
111	C₂²×C₁₀₉	C₂×C₄⋊C₈	1	1	yes	characteristic
112	C₂²×C₁₀₉	#432	1	1	yes	characteristic
113	C₄×C₁₀₉	—	2	4	no	—
114	C₄×C₁₀₉	—	2	4	no	—
115	C₄×C₁₀₉	—	2	2	no	—
116	C₄×C₁₀₉	#380	1	1	yes	characteristic
117	C₄×C₁₀₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
118	C₄×C₁₀₉	#423	1	1	yes	characteristic
119	C₄×C₁₀₉	#431	1	1	yes	characteristic
120	#12788	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₂×C₁₀₉	—	2	1	no	—
122	C₂×C₁₀₉	—	2	4	no	—
123	C₂×C₁₀₉	—	2	1	no	—
124	C₂×C₁₀₉	#1797	1	1	yes	characteristic
125	C₂×C₁₀₉	C₂×#364	1	2	yes	—
126	C₂²×C₄×C₈	C₂×C₁₀₉	1	1	yes	characteristic
127	C₂×#364	C₂×C₁₀₉	1	2	yes	complemented
128	#2112	C₂×C₁₀₉	1	2	yes	—
129	#1376	C₂×C₁₀₉	1	2	yes	complemented
130	C₁₀₉	#12788	1	1	yes	characteristic, direct product factor, Sylow
131	C₂³×C₈	C₄×C₁₀₉	1	1	yes	characteristic
132	C₂³×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
133	C₂²×C₄²	C₄×C₁₀₉	1	1	yes	characteristic
134	C₂×M₃₂	C₂²×C₁₀₉	1	4	yes	—
135	C₂×C₄×C₈	—	2	4	no	—
136	C₂×C₄×C₈	C₄×C₁₀₉	1	1	yes	characteristic
137	C₂×C₄×C₈	C₄×C₁₀₉	1	1	yes	characteristic
138	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
139	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
140	#364	—	2	4	no	—
141	C₂²⋊C₁₆	—	2	4	no	—
142	C₂³×C₄	—	2	1	no	—
143	C₂³×C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
144	C₂²×C₈	—	2	2	no	—
145	C₂²×C₈	—	2	2	no	—
146	C₂²×C₈	—	2	2	no	—
147	C₂²×C₈	—	2	2	no	—
148	C₂²×C₈	C₈×C₁₀₉	1	4	yes	—
149	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
150	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
151	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
152	C₂²×C₈	D₄×C₁₀₉	1	2	yes	—
153	C₂²×C₈	Q₈×C₁₀₉	1	2	yes	—
154	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
155	C₂×C₄²	—	2	1	no	—
156	C₂×C₄²	—	2	2	no	—
157	C₂×C₄²	—	2	2	no	—
158	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
159	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
160	M₃₂	—	2	4	no	—
161	M₃₂	—	2	4	no	—
162	C₂×C₁₆	—	2	4	no	—
163	C₄×C₈	—	2	2	no	—
164	C₄×C₈	—	2	2	no	—
165	C₄×C₈	—	2	2	no	—
166	C₄×C₈	D₄×C₁₀₉	1	4	yes	—
167	C₂⁴	M₁₆×C₁₀₉	1	1	yes	characteristic
168	C₂²×C₄	—	2	4	no	—
169	C₂²×C₄	—	2	8	no	—
170	C₂²×C₄	—	2	1	no	—
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	—	2	1	no	—
174	C₂²×C₄	—	2	1	no	—
175	C₂²×C₄	—	2	1	no	—
176	C₂²×C₄	C₄⋊C₄×C₁₀₉	1	2	yes	—
177	C₂²×C₄	C₂×C₈×C₁₀₉	1	2	yes	—
178	C₂²×C₄	M₁₆×C₁₀₉	1	1	yes	characteristic
179	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	—	2	1	no	—
182	C₂×C₈	—	2	1	no	—
183	C₂×C₈	—	2	1	no	—
184	C₂×C₈	—	2	8	no	—
185	C₂×C₈	—	2	4	no	—
186	C₂×C₈	—	2	4	no	—
187	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	2	yes	—
188	C₂×C₈	C₄⋊C₄×C₁₀₉	1	2	yes	—
189	C₂×C₈	C₂×C₈×C₁₀₉	1	2	yes	—
190	C₂×C₈	C₂×C₈×C₁₀₉	1	2	yes	—
191	C₂×C₈	M₁₆×C₁₀₉	1	2	yes	—
192	C₂×C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
193	C₂×C₈	C₂×D₄×C₁₀₉	1	2	yes	—
194	C₂×C₈	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
195	C₂×C₈	#41×C₁₀₉	1	2	yes	—
196	C₄²	—	2	4	no	—
197	C₄²	—	2	1	no	—
198	C₄²	—	2	1	no	—
199	C₄²	C₂²⋊C₄×C₁₀₉	1	2	yes	—
200	C₄²	M₁₆×C₁₀₉	1	2	yes	—
201	C₁₆	—	4	4	no	—
202	C₂³	—	2	4	no	—
203	C₂³	—	2	1	no	—
204	C₂³	—	2	1	no	—
205	C₂³	C₄⋊C₈×C₁₀₉	1	2	yes	—
206	C₂³	C₂×M₁₆×C₁₀₉	1	1	yes	characteristic
207	C₂×C₄	—	2	16	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	C₄⋊C₈×C₁₀₉	1	2	yes	—
219	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
220	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
221	C₂×C₄	#117×C₁₀₉	1	1	yes	characteristic
222	C₂×C₄	C₂²×C₈×C₁₀₉	1	1	yes	characteristic
223	C₂×C₄	C₂×M₁₆×C₁₀₉	1	1	yes	characteristic
224	C₂×C₄	C₂×M₁₆×C₁₀₉	1	1	yes	characteristic
225	C₈	—	2	2	no	—
226	C₈	—	2	2	no	—
227	C₈	C₂²⋊C₈×C₁₀₉	1	4	yes	—
228	C₈	#121×C₁₀₉	1	2	yes	—
229	C₈	C₂²⋊Q₈×C₁₀₉	1	2	yes	—
230	C₂²	—	2	4	no	—
231	C₂²	—	2	4	no	—
232	C₂²	—	2	2	no	—
233	C₂²	—	2	2	no	—
234	C₂²	—	2	2	no	—
235	C₂²	#364×C₁₀₉	1	4	yes	complemented
236	C₂²	#407×C₁₀₉	1	1	yes	characteristic
237	C₂²	C₂×C₄⋊C₈×C₁₀₉	1	1	yes	characteristic
238	C₂²	#432×C₁₀₉	1	1	yes	characteristic
239	C₄	—	2	4	no	—
240	C₄	—	2	4	no	—
241	C₄	—	2	2	no	—
242	C₄	#380×C₁₀₉	1	1	yes	characteristic
243	C₄	C₂×C₂²⋊C₈×C₁₀₉	1	1	yes	characteristic
244	C₄	#423×C₁₀₉	1	1	yes	characteristic
245	C₄	#431×C₁₀₉	1	1	yes	characteristic
246	C₂	—	2	1	no	—
247	C₂	—	2	4	no	—
248	C₂	—	2	1	no	—
249	C₂	C₁₀₉×#1797	1	1	yes	characteristic
250	C₂	C₂×#364×C₁₀₉	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₁₀₉	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₁₀₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₀₉
socle	C₂²×C₁₀₉	C₂×C₄⋊C₈
prime radical	C₂⁴×C₁₀₉	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₀₉ ◁ C₂³×C₈×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#364×C₁₀₉	C₂	semidirect
4. C₁₀₉×#1376	C₂	semidirect
109. C₂²×C₁₀₉	#364	semidirect
120. #12788	C₁₀₉	direct (N × H)
127. C₂×#364	C₂×C₁₀₉	semidirect
129. #1376	C₂×C₁₀₉	semidirect
130. C₁₀₉	#12788	direct (N × H)
235. C₂²	#364×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₈×C₁₀₉	2
C₂×#364×C₁₀₉	2
C₁₀₉×#2112	2
C₁₀₉×#1376	2
#12788	109

Subgroups — C109×#12788 #42130782

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#12788 #42130782