Subgroups — C₂×C₁₀₉×#1779 #42138571

Order 27904.

6110 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×C₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₂²⋊C₈×C₁₀₉	C₂	1	6	yes	maximal, complemented
3	C₁₀₉×#1779	C₂	1	8	yes	maximal, direct product factor
4	C₂⁶×C₁₀₉	C₄	1	1	yes	characteristic
5	C₂⁴×C₄×C₁₀₉	—	2	24	no	—
6	C₂⁴×C₄×C₁₀₉	C₄	1	3	yes	—
7	C₂⁴×C₄×C₁₀₉	C₄	1	4	yes	—
8	C₂⁴×C₄×C₁₀₉	C₂²	1	4	yes	—
9	C₂⁴×C₄×C₁₀₉	C₂²	1	3	yes	—
10	C₂³×C₈×C₁₀₉	C₂²	1	4	yes	complemented
11	C₂×C₂²⋊C₈×C₁₀₉	—	2	24	no	—
12	C₂×C₂²⋊C₈×C₁₀₉	C₂²	1	24	yes	complemented
13	C₂⁵×C₁₀₉	—	2	12	no	—
14	C₂⁵×C₁₀₉	—	2	12	no	—
15	C₂⁵×C₁₀₉	C₈	1	8	yes	complemented
16	C₂⁵×C₁₀₉	C₂×C₄	1	4	yes	—
17	C₂⁵×C₁₀₉	C₂×C₄	1	3	yes	—
18	C₂³×C₄×C₁₀₉	—	2	128	no	—
19	C₂³×C₄×C₁₀₉	—	2	12	no	—
20	C₂³×C₄×C₁₀₉	—	2	24	no	—
21	C₂³×C₄×C₁₀₉	—	2	48	no	—
22	C₂³×C₄×C₁₀₉	—	2	48	no	—
23	C₂³×C₄×C₁₀₉	—	2	24	no	—
24	C₂³×C₄×C₁₀₉	C₂×C₄	1	12	yes	—
25	C₂³×C₄×C₁₀₉	C₂×C₄	1	3	yes	—
26	C₂³×C₄×C₁₀₉	C₂×C₄	1	6	yes	—
27	C₂³×C₄×C₁₀₉	D₄	1	24	yes	—
28	C₂³×C₄×C₁₀₉	C₂³	1	6	yes	—
29	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
30	C₂²×C₈×C₁₀₉	—	2	24	no	—
31	C₂²×C₈×C₁₀₉	C₂³	1	8	yes	complemented
32	C₂²⋊C₈×C₁₀₉	—	2	48	no	—
33	C₂⁴×C₁₀₉	—	2	32	no	—
34	C₂⁴×C₁₀₉	—	2	96	no	—
35	C₂⁴×C₁₀₉	—	2	96	no	—
36	C₂⁴×C₁₀₉	—	2	24	no	—
37	C₂⁴×C₁₀₉	—	2	12	no	—
38	C₂⁴×C₁₀₉	—	2	24	no	—
39	C₂⁴×C₁₀₉	—	2	12	no	—
40	C₂⁴×C₁₀₉	C₂²⋊C₄	1	12	yes	—
41	C₂⁴×C₁₀₉	C₂×C₈	1	12	yes	—
42	C₂⁴×C₁₀₉	C₂×C₈	1	16	yes	complemented
43	C₂⁴×C₁₀₉	M₁₆	1	12	yes	—
44	C₂⁴×C₁₀₉	C₂²×C₄	1	6	yes	—
45	C₂⁴×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₀₉	—	2	96	no	—
47	C₂²×C₄×C₁₀₉	—	2	128	no	—
48	C₂²×C₄×C₁₀₉	—	2	128	no	—
49	C₂²×C₄×C₁₀₉	—	2	24	no	—
50	C₂²×C₄×C₁₀₉	—	2	12	no	—
51	C₂²×C₄×C₁₀₉	—	2	48	no	—
52	C₂²×C₄×C₁₀₉	—	2	24	no	—
53	C₂²×C₄×C₁₀₉	—	2	96	no	—
54	C₂²×C₄×C₁₀₉	—	2	24	no	—
55	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	12	yes	—
56	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	24	yes	—
57	C₂²×C₄×C₁₀₉	C₂²×C₄	1	6	yes	—
58	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₀₉	C₂×D₄	1	24	yes	—
60	C₂²×C₄×C₁₀₉	C₂×D₄	1	12	yes	—
61	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
62	C₂×C₈×C₁₀₉	—	2	24	no	—
63	C₂×C₈×C₁₀₉	—	4	16	no	—
64	C₂³×C₁₀₉	—	2	32	no	—
65	C₂³×C₁₀₉	—	2	96	no	—
66	C₂³×C₁₀₉	—	2	256	no	—
67	C₂³×C₁₀₉	—	2	12	no	—
68	C₂³×C₁₀₉	—	2	12	no	—
69	C₂³×C₁₀₉	—	2	24	no	—
70	C₂³×C₁₀₉	—	2	48	no	—
71	C₂³×C₁₀₉	—	2	96	no	—
72	C₂³×C₁₀₉	—	2	24	no	—
73	C₂³×C₁₀₉	—	2	48	no	—
74	C₂³×C₁₀₉	C₂²⋊C₈	1	48	yes	complemented
75	C₂³×C₁₀₉	C₂×C₂²⋊C₄	1	12	yes	—
76	C₂³×C₁₀₉	C₂×C₂²⋊C₄	1	6	yes	—
77	C₂³×C₁₀₉	C₂²×C₈	1	12	yes	—
78	C₂³×C₁₀₉	C₂²×C₈	1	2	yes	—
79	C₂³×C₁₀₉	C₂×M₁₆	1	12	yes	—
80	C₂³×C₁₀₉	C₂×M₁₆	1	6	yes	—
81	C₂³×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
82	C₂×C₄×C₁₀₉	—	2	96	no	—
83	C₂×C₄×C₁₀₉	—	2	12	no	—
84	C₂×C₄×C₁₀₉	—	2	24	no	—
85	C₂×C₄×C₁₀₉	—	2	48	no	—
86	C₂×C₄×C₁₀₉	—	2	48	no	—
87	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	12	yes	—
88	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	6	yes	—
89	C₂×C₄×C₁₀₉	#120	1	16	yes	—
90	C₂×C₄×C₁₀₉	C₂²×D₄	1	6	yes	—
91	C₈×C₁₀₉	—	4	8	no	—
92	C₂²×C₁₀₉	—	2	128	no	—
93	C₂²×C₁₀₉	—	2	24	no	—
94	C₂²×C₁₀₉	—	2	12	no	—
95	C₂²×C₁₀₉	—	2	24	no	—
96	C₂²×C₁₀₉	—	2	12	no	—
97	C₂²×C₁₀₉	—	2	96	no	—
98	C₂²×C₁₀₉	#379	1	4	yes	—
99	C₂²×C₁₀₉	C₂×C₂²⋊C₈	1	12	yes	—
100	C₂²×C₁₀₉	C₂×C₂²⋊C₈	1	24	yes	complemented
101	C₂²×C₁₀₉	#407	1	12	yes	—
102	C₂²×C₁₀₉	C₂²×C₂²⋊C₄	1	3	yes	—
103	C₂²×C₁₀₉	C₂³×C₈	1	1	yes	characteristic
104	C₂²×C₁₀₉	C₂²×M₁₆	1	3	yes	—
105	C₄×C₁₀₉	—	2	12	no	—
106	C₄×C₁₀₉	#379	1	4	yes	—
107	C₄×C₁₀₉	C₂×#120	1	4	yes	—
108	C₂×#1779	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
109	C₂×C₁₀₉	—	2	24	no	—
110	C₂×C₁₀₉	#1779	1	8	yes	direct product factor
111	C₂×C₁₀₉	C₂×#379	1	1	yes	characteristic
112	C₂×C₁₀₉	C₂²×C₂²⋊C₈	1	3	yes	—
113	C₂×C₁₀₉	C₂×#407	1	3	yes	—
114	C₂⁵×C₄	C₂×C₁₀₉	1	1	yes	characteristic
115	C₂²×C₂²⋊C₈	C₂×C₁₀₉	1	6	yes	complemented
116	#1779	C₂×C₁₀₉	1	8	yes	direct product factor
117	C₁₀₉	C₂×#1779	1	1	yes	characteristic, direct product factor, Sylow
118	C₂⁶	C₄×C₁₀₉	1	1	yes	characteristic
119	C₂⁴×C₄	—	2	24	no	—
120	C₂⁴×C₄	C₄×C₁₀₉	1	3	yes	—
121	C₂⁴×C₄	C₄×C₁₀₉	1	4	yes	—
122	C₂⁴×C₄	C₂²×C₁₀₉	1	4	yes	—
123	C₂⁴×C₄	C₂²×C₁₀₉	1	3	yes	—
124	C₂³×C₈	C₂²×C₁₀₉	1	4	yes	complemented
125	C₂×C₂²⋊C₈	—	2	24	no	—
126	C₂×C₂²⋊C₈	C₂²×C₁₀₉	1	24	yes	complemented
127	C₂⁵	—	2	12	no	—
128	C₂⁵	—	2	12	no	—
129	C₂⁵	C₈×C₁₀₉	1	8	yes	complemented
130	C₂⁵	C₂×C₄×C₁₀₉	1	4	yes	—
131	C₂⁵	C₂×C₄×C₁₀₉	1	3	yes	—
132	C₂³×C₄	—	2	128	no	—
133	C₂³×C₄	—	2	12	no	—
134	C₂³×C₄	—	2	24	no	—
135	C₂³×C₄	—	2	48	no	—
136	C₂³×C₄	—	2	48	no	—
137	C₂³×C₄	—	2	24	no	—
138	C₂³×C₄	C₂×C₄×C₁₀₉	1	12	yes	—
139	C₂³×C₄	C₂×C₄×C₁₀₉	1	3	yes	—
140	C₂³×C₄	C₂×C₄×C₁₀₉	1	6	yes	—
141	C₂³×C₄	D₄×C₁₀₉	1	24	yes	—
142	C₂³×C₄	C₂³×C₁₀₉	1	6	yes	—
143	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
144	C₂²×C₈	—	2	24	no	—
145	C₂²×C₈	C₂³×C₁₀₉	1	8	yes	complemented
146	C₂²⋊C₈	—	2	48	no	—
147	C₂⁴	—	2	32	no	—
148	C₂⁴	—	2	96	no	—
149	C₂⁴	—	2	96	no	—
150	C₂⁴	—	2	24	no	—
151	C₂⁴	—	2	12	no	—
152	C₂⁴	—	2	24	no	—
153	C₂⁴	—	2	12	no	—
154	C₂⁴	C₂²⋊C₄×C₁₀₉	1	12	yes	—
155	C₂⁴	C₂×C₈×C₁₀₉	1	12	yes	—
156	C₂⁴	C₂×C₈×C₁₀₉	1	16	yes	complemented
157	C₂⁴	M₁₆×C₁₀₉	1	12	yes	—
158	C₂⁴	C₂²×C₄×C₁₀₉	1	6	yes	—
159	C₂⁴	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
160	C₂²×C₄	—	2	96	no	—
161	C₂²×C₄	—	2	128	no	—
162	C₂²×C₄	—	2	128	no	—
163	C₂²×C₄	—	2	24	no	—
164	C₂²×C₄	—	2	12	no	—
165	C₂²×C₄	—	2	48	no	—
166	C₂²×C₄	—	2	24	no	—
167	C₂²×C₄	—	2	96	no	—
168	C₂²×C₄	—	2	24	no	—
169	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	12	yes	—
170	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	24	yes	—
171	C₂²×C₄	C₂²×C₄×C₁₀₉	1	6	yes	—
172	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
173	C₂²×C₄	C₂×D₄×C₁₀₉	1	24	yes	—
174	C₂²×C₄	C₂×D₄×C₁₀₉	1	12	yes	—
175	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
176	C₂×C₈	—	2	24	no	—
177	C₂×C₈	—	4	16	no	—
178	C₂³	—	2	32	no	—
179	C₂³	—	2	256	no	—
180	C₂³	—	2	96	no	—
181	C₂³	—	2	12	no	—
182	C₂³	—	2	12	no	—
183	C₂³	—	2	24	no	—
184	C₂³	—	2	48	no	—
185	C₂³	—	2	96	no	—
186	C₂³	—	2	24	no	—
187	C₂³	—	2	48	no	—
188	C₂³	C₂²⋊C₈×C₁₀₉	1	48	yes	complemented
189	C₂³	C₂×C₂²⋊C₄×C₁₀₉	1	12	yes	—
190	C₂³	C₂×C₂²⋊C₄×C₁₀₉	1	6	yes	—
191	C₂³	C₂²×C₈×C₁₀₉	1	12	yes	—
192	C₂³	C₂²×C₈×C₁₀₉	1	2	yes	—
193	C₂³	C₂×M₁₆×C₁₀₉	1	12	yes	—
194	C₂³	C₂×M₁₆×C₁₀₉	1	6	yes	—
195	C₂³	C₂³×C₄×C₁₀₉	1	1	yes	characteristic
196	C₂×C₄	—	2	96	no	—
197	C₂×C₄	—	2	12	no	—
198	C₂×C₄	—	2	24	no	—
199	C₂×C₄	—	2	48	no	—
200	C₂×C₄	—	2	48	no	—
201	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	12	yes	—
202	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	6	yes	—
203	C₂×C₄	#120×C₁₀₉	1	16	yes	—
204	C₂×C₄	C₂²×D₄×C₁₀₉	1	6	yes	—
205	C₈	—	4	8	no	—
206	C₂²	—	2	128	no	—
207	C₂²	—	2	12	no	—
208	C₂²	—	2	96	no	—
209	C₂²	—	2	24	no	—
210	C₂²	—	2	12	no	—
211	C₂²	—	2	24	no	—
212	C₂²	#379×C₁₀₉	1	4	yes	—
213	C₂²	C₂×C₂²⋊C₈×C₁₀₉	1	12	yes	—
214	C₂²	C₂×C₂²⋊C₈×C₁₀₉	1	24	yes	complemented
215	C₂²	#407×C₁₀₉	1	12	yes	—
216	C₂²	C₂²×C₂²⋊C₄×C₁₀₉	1	3	yes	—
217	C₂²	C₂³×C₈×C₁₀₉	1	1	yes	characteristic
218	C₂²	C₂²×M₁₆×C₁₀₉	1	3	yes	—
219	C₄	—	2	12	no	—
220	C₄	#379×C₁₀₉	1	4	yes	—
221	C₄	C₂×#120×C₁₀₉	1	4	yes	—
222	C₂	—	2	24	no	—
223	C₂	C₁₀₉×#1779	1	8	yes	direct product factor
224	C₂	C₂×#379×C₁₀₉	1	1	yes	characteristic
225	C₂	C₂²×C₂²⋊C₈×C₁₀₉	1	3	yes	—
226	C₂	C₂×#407×C₁₀₉	1	3	yes	—

	subgroup	G/H
center	C₂³×C₄×C₁₀₉	C₂³
commutator	C₂²	C₂³×C₈×C₁₀₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₀₉
socle	C₂⁴×C₁₀₉	C₂²×C₄
prime radical	C₂⁶×C₁₀₉	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₂²⋊C₈×C₁₀₉	C₂	semidirect
3. C₁₀₉×#1779	C₂	direct (N × H)
10. C₂³×C₈×C₁₀₉	C₂²	semidirect
12. C₂×C₂²⋊C₈×C₁₀₉	C₂²	semidirect
15. C₂⁵×C₁₀₉	C₈	semidirect
31. C₂²×C₈×C₁₀₉	C₂³	semidirect
42. C₂⁴×C₁₀₉	C₂×C₈	semidirect
74. C₂³×C₁₀₉	C₂²⋊C₈	semidirect
100. C₂²×C₁₀₉	C₂×C₂²⋊C₈	semidirect
108. C₂×#1779	C₁₀₉	direct (N × H)
110. C₂×C₁₀₉	#1779	direct (N × H)
115. C₂²×C₂²⋊C₈	C₂×C₁₀₉	semidirect
116. #1779	C₂×C₁₀₉	direct (N × H)
117. C₁₀₉	C₂×#1779	direct (N × H)
124. C₂³×C₈	C₂²×C₁₀₉	semidirect
126. C₂×C₂²⋊C₈	C₂²×C₁₀₉	semidirect
129. C₂⁵	C₈×C₁₀₉	semidirect
145. C₂²×C₈	C₂³×C₁₀₉	semidirect
156. C₂⁴	C₂×C₈×C₁₀₉	semidirect
188. C₂³	C₂²⋊C₈×C₁₀₉	semidirect
214. C₂²	C₂×C₂²⋊C₈×C₁₀₉	semidirect
223. C₂	C₁₀₉×#1779	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁵×C₄×C₁₀₉	2
C₂²×C₂²⋊C₈×C₁₀₉	2
C₁₀₉×#1779	2
C₂×#1779	109

Subgroups — C2×C109×#1779 #42138571

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₀₉×#1779 #42138571