Subgroups — C₁₀₉×#2500 #28359754

Order 13952.

568 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#121×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#117×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#394×C₁₀₉	C₂	1	4	yes	maximal, complemented
4	#390×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#390×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#388×C₁₀₉	C₂	1	4	yes	maximal, complemented
7	#384×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#382×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×C₂²⋊C₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₂²×D₄×C₁₀₉	C₂²	1	2	yes	—
12	C₂³×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
13	#121×C₁₀₉	—	2	4	no	—
14	#117×C₁₀₉	—	2	4	no	—
15	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	4	yes	—
19	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	2	yes	—
20	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	4	yes	—
23	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	4	yes	—
24	C₂×C₄²×C₁₀₉	C₂²	1	2	yes	—
25	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
26	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
27	#95×C₁₀₉	C₂²	1	2	yes	—
28	#95×C₁₀₉	C₂²	1	4	yes	—
29	C₂⁴×C₁₀₉	C₂³	1	2	yes	—
30	C₂⁴×C₁₀₉	C₂³	1	1	yes	characteristic
31	C₂×D₄×C₁₀₉	—	2	4	no	—
32	C₂×D₄×C₁₀₉	—	2	2	no	—
33	C₂×D₄×C₁₀₉	—	2	2	no	—
34	C₂×D₄×C₁₀₉	—	2	2	no	—
35	C₂×D₄×C₁₀₉	—	4	4	no	—
36	C₂²×C₄×C₁₀₉	—	2	2	no	—
37	C₂²×C₄×C₁₀₉	—	2	2	no	—
38	C₂²×C₄×C₁₀₉	D₄	1	4	yes	—
39	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
40	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
41	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
42	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
43	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
45	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
46	C₄⋊C₄×C₁₀₉	—	2	2	no	—
47	C₄⋊C₄×C₁₀₉	—	2	2	no	—
48	C₄⋊C₄×C₁₀₉	—	2	2	no	—
49	C₄⋊C₄×C₁₀₉	—	4	4	no	—
50	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
51	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
52	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
53	C₂²⋊C₄×C₁₀₉	—	2	8	no	—
54	C₂²⋊C₄×C₁₀₉	—	4	4	no	—
55	C₄²×C₁₀₉	—	2	4	no	—
56	C₄²×C₁₀₉	—	2	2	no	—
57	C₄²×C₁₀₉	—	2	2	no	—
58	C₂³×C₁₀₉	—	2	2	no	—
59	C₂³×C₁₀₉	—	2	1	no	—
60	C₂³×C₁₀₉	—	2	1	no	—
61	C₂³×C₁₀₉	—	2	2	no	—
62	C₂³×C₁₀₉	—	2	2	no	—
63	C₂³×C₁₀₉	—	2	4	no	—
64	C₂³×C₁₀₉	—	2	2	no	—
65	C₂³×C₁₀₉	—	2	2	no	—
66	C₂³×C₁₀₉	—	2	4	no	—
67	C₂³×C₁₀₉	C₂×D₄	1	2	yes	—
68	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
69	D₄×C₁₀₉	—	4	8	no	—
70	D₄×C₁₀₉	—	4	4	no	—
71	C₂×C₄×C₁₀₉	—	2	2	no	—
72	C₂×C₄×C₁₀₉	—	2	1	no	—
73	C₂×C₄×C₁₀₉	—	2	1	no	—
74	C₂×C₄×C₁₀₉	—	2	2	no	—
75	C₂×C₄×C₁₀₉	—	2	8	no	—
76	C₂×C₄×C₁₀₉	—	2	2	no	—
77	C₂×C₄×C₁₀₉	—	2	2	no	—
78	C₂×C₄×C₁₀₉	—	2	2	no	—
79	C₂×C₄×C₁₀₉	—	2	2	no	—
80	C₂×C₄×C₁₀₉	—	2	4	no	—
81	C₂×C₄×C₁₀₉	—	2	2	no	—
82	C₂×C₄×C₁₀₉	—	2	2	no	—
83	C₂×C₄×C₁₀₉	—	2	4	no	—
84	C₂×C₄×C₁₀₉	—	2	4	no	—
85	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
86	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
87	C₂×C₄×C₁₀₉	#41	1	4	yes	—
88	C₂×C₄×C₁₀₉	#41	1	2	yes	—
89	C₂×C₄×C₁₀₉	#41	1	2	yes	—
90	C₂²×C₁₀₉	—	2	2	no	—
91	C₂²×C₁₀₉	—	2	2	no	—
92	C₂²×C₁₀₉	—	2	2	no	—
93	C₂²×C₁₀₉	—	4	2	no	—
94	C₂²×C₁₀₉	—	4	2	no	—
95	C₂²×C₁₀₉	—	4	2	no	—
96	C₂²×C₁₀₉	—	4	2	no	—
97	C₂²×C₁₀₉	—	4	2	no	—
98	C₂²×C₁₀₉	—	4	4	no	—
99	C₂²×C₁₀₉	—	4	4	no	—
100	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
101	C₂²×C₁₀₉	C₂×#41	1	1	yes	characteristic
102	C₂²×C₁₀₉	C₂×#41	1	1	yes	characteristic
103	C₂²×C₁₀₉	C₂×#41	1	2	yes	—
104	C₂²×C₁₀₉	H(5,𝔽₂)	1	2	yes	—
105	C₄×C₁₀₉	—	2	2	no	—
106	C₄×C₁₀₉	—	2	4	no	—
107	C₄×C₁₀₉	—	4	2	no	—
108	C₄×C₁₀₉	—	4	2	no	—
109	C₄×C₁₀₉	—	4	2	no	—
110	C₄×C₁₀₉	—	4	2	no	—
111	C₄×C₁₀₉	#123	1	4	yes	—
112	C₂×C₁₀₉	—	4	2	no	—
113	C₂×C₁₀₉	—	8	2	no	—
114	C₂×C₁₀₉	C₂×#123	1	1	yes	characteristic
115	C₂×C₁₀₉	#532	1	1	yes	characteristic
116	C₂×C₁₀₉	#535	1	1	yes	characteristic
117	C₂×C₁₀₉	#553	1	2	yes	—
118	C₂×C₁₀₉	#555	1	2	yes	—
119	#2500	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₁₀₉	#2500	1	1	yes	characteristic, direct product factor, Sylow
121	C₂×#121	C₂×C₁₀₉	1	1	yes	characteristic
122	C₂×#117	C₂×C₁₀₉	1	1	yes	characteristic, complemented
123	#394	C₂×C₁₀₉	1	4	yes	complemented
124	#390	C₂×C₁₀₉	1	1	yes	characteristic, complemented
125	#390	C₂×C₁₀₉	1	1	yes	characteristic, complemented
126	#388	C₂×C₁₀₉	1	4	yes	complemented
127	#384	C₂×C₁₀₉	1	1	yes	characteristic, complemented
128	#382	C₂×C₁₀₉	1	1	yes	characteristic, complemented
129	C₄×C₂²⋊C₄	C₂×C₁₀₉	1	1	yes	characteristic, complemented
130	C₂²×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
131	C₂²×D₄	C₂²×C₁₀₉	1	2	yes	—
132	C₂³×C₄	C₂²×C₁₀₉	1	1	yes	characteristic
133	#121	—	2	4	no	—
134	#117	—	2	4	no	—
135	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
136	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
137	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
138	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	4	yes	—
139	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	2	yes	—
140	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	2	yes	—
141	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	2	yes	—
142	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	4	yes	—
143	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	4	yes	—
144	C₂×C₄²	C₂²×C₁₀₉	1	2	yes	—
145	C₂×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
146	C₂×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
147	#95	C₂²×C₁₀₉	1	2	yes	—
148	#95	C₂²×C₁₀₉	1	4	yes	—
149	C₂⁴	C₂³×C₁₀₉	1	2	yes	—
150	C₂⁴	C₂³×C₁₀₉	1	1	yes	characteristic
151	C₂×D₄	—	2	4	no	—
152	C₂×D₄	—	2	2	no	—
153	C₂×D₄	—	2	2	no	—
154	C₂×D₄	—	2	2	no	—
155	C₂×D₄	—	4	4	no	—
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	D₄×C₁₀₉	1	4	yes	—
159	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
160	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
161	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
162	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
163	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
164	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
165	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
166	C₄⋊C₄	—	2	2	no	—
167	C₄⋊C₄	—	2	2	no	—
168	C₄⋊C₄	—	2	2	no	—
169	C₄⋊C₄	—	4	4	no	—
170	C₂²⋊C₄	—	2	4	no	—
171	C₂²⋊C₄	—	2	4	no	—
172	C₂²⋊C₄	—	2	4	no	—
173	C₂²⋊C₄	—	2	8	no	—
174	C₂²⋊C₄	—	4	4	no	—
175	C₄²	—	2	4	no	—
176	C₄²	—	2	2	no	—
177	C₄²	—	2	2	no	—
178	C₂³	—	2	2	no	—
179	C₂³	—	2	1	no	—
180	C₂³	—	2	1	no	—
181	C₂³	—	2	2	no	—
182	C₂³	—	2	2	no	—
183	C₂³	—	2	4	no	—
184	C₂³	—	2	2	no	—
185	C₂³	—	2	2	no	—
186	C₂³	—	2	4	no	—
187	C₂³	C₂×D₄×C₁₀₉	1	2	yes	—
188	C₂³	C₂⁴×C₁₀₉	1	1	yes	characteristic
189	D₄	—	4	8	no	—
190	D₄	—	4	4	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	8	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	4	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
206	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
207	C₂×C₄	#41×C₁₀₉	1	4	yes	—
208	C₂×C₄	#41×C₁₀₉	1	2	yes	—
209	C₂×C₄	#41×C₁₀₉	1	2	yes	—
210	C₂²	—	2	2	no	—
211	C₂²	—	2	2	no	—
212	C₂²	—	2	2	no	—
213	C₂²	—	4	2	no	—
214	C₂²	—	4	2	no	—
215	C₂²	—	4	2	no	—
216	C₂²	—	4	2	no	—
217	C₂²	—	4	2	no	—
218	C₂²	—	4	4	no	—
219	C₂²	—	4	4	no	—
220	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
221	C₂²	C₂×#41×C₁₀₉	1	1	yes	characteristic
222	C₂²	C₂×#41×C₁₀₉	1	1	yes	characteristic
223	C₂²	C₂×#41×C₁₀₉	1	2	yes	—
224	C₂²	H(5,𝔽₂)×C₁₀₉	1	2	yes	—
225	C₄	—	2	2	no	—
226	C₄	—	2	4	no	—
227	C₄	—	4	2	no	—
228	C₄	—	4	2	no	—
229	C₄	—	4	2	no	—
230	C₄	—	4	2	no	—
231	C₄	#123×C₁₀₉	1	4	yes	—
232	C₂	—	4	2	no	—
233	C₂	—	8	2	no	—
234	C₂	C₂×#123×C₁₀₉	1	1	yes	characteristic
235	C₂	#532×C₁₀₉	1	1	yes	characteristic
236	C₂	#535×C₁₀₉	1	1	yes	characteristic
237	C₂	#553×C₁₀₉	1	2	yes	—
238	C₂	#555×C₁₀₉	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₀₉	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₁₀₉
prime radical	C₂×#121×C₁₀₉	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#117×C₁₀₉	C₂	semidirect
3. #394×C₁₀₉	C₂	semidirect
4. #390×C₁₀₉	C₂	semidirect
5. #390×C₁₀₉	C₂	semidirect
6. #388×C₁₀₉	C₂	semidirect
7. #384×C₁₀₉	C₂	semidirect
8. #382×C₁₀₉	C₂	semidirect
9. C₄×C₂²⋊C₄×C₁₀₉	C₂	semidirect
119. #2500	C₁₀₉	direct (N × H)
120. C₁₀₉	#2500	direct (N × H)
122. C₂×#117	C₂×C₁₀₉	semidirect
123. #394	C₂×C₁₀₉	semidirect
124. #390	C₂×C₁₀₉	semidirect
125. #390	C₂×C₁₀₉	semidirect
126. #388	C₂×C₁₀₉	semidirect
127. #384	C₂×C₁₀₉	semidirect
128. #382	C₂×C₁₀₉	semidirect
129. C₄×C₂²⋊C₄	C₂×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#121×C₁₀₉	2
C₂×#117×C₁₀₉	2
#394×C₁₀₉	2
#390×C₁₀₉	2
#388×C₁₀₉	2
#384×C₁₀₉	2
#382×C₁₀₉	2
C₄×C₂²⋊C₄×C₁₀₉	2
#2500	109

Subgroups — C109×#2500 #28359754

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#2500 #28359754