Subgroups — C₁₀₉×#2498 #28359752

Order 13952.

546 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#123×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#117×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#400×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#399×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#398×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#397×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
7	#388×C₁₀₉	C₂	1	2	yes	maximal, complemented
8	#387×C₁₀₉	C₂	1	2	yes	maximal, complemented
9	#386×C₁₀₉	C₂	1	2	yes	maximal
10	#380×C₁₀₉	C₂	1	2	yes	maximal, complemented
11	C₂×#95×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₁₀₉	C₂²	1	2	yes	—
14	C₂³×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
15	#123×C₁₀₉	—	2	4	no	—
16	#117×C₁₀₉	—	2	4	no	—
17	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
22	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
23	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	2	yes	—
24	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
27	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	2	yes	—
28	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	2	yes	—
29	C₂×C₄²×C₁₀₉	C₂²	1	2	yes	—
30	#95×C₁₀₉	C₂²	1	2	yes	—
31	#95×C₁₀₉	C₂²	1	1	yes	characteristic
32	#95×C₁₀₉	C₂²	1	1	yes	characteristic
33	#95×C₁₀₉	C₂²	1	2	yes	—
34	#95×C₁₀₉	C₂²	1	2	yes	—
35	#95×C₁₀₉	C₂²	1	2	yes	—
36	#95×C₁₀₉	C₂²	1	2	yes	—
37	C₂⁴×C₁₀₉	C₂³	1	1	yes	characteristic
38	C₂⁴×C₁₀₉	C₂³	1	1	yes	characteristic
39	C₂×D₄×C₁₀₉	—	2	2	no	—
40	C₂×D₄×C₁₀₉	—	2	4	no	—
41	C₂²×C₄×C₁₀₉	—	2	4	no	—
42	C₂²×C₄×C₁₀₉	—	2	4	no	—
43	C₂²×C₄×C₁₀₉	—	2	4	no	—
44	C₂²×C₄×C₁₀₉	—	4	2	no	—
45	C₂²×C₄×C₁₀₉	D₄	1	4	yes	—
46	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
47	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
49	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
50	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
55	C₄⋊C₄×C₁₀₉	—	2	2	no	—
56	C₄⋊C₄×C₁₀₉	—	2	2	no	—
57	C₄⋊C₄×C₁₀₉	—	2	2	no	—
58	C₄⋊C₄×C₁₀₉	—	2	2	no	—
59	C₄⋊C₄×C₁₀₉	—	2	4	no	—
60	C₄⋊C₄×C₁₀₉	—	4	2	no	—
61	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
62	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
63	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
64	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
65	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
66	C₂²⋊C₄×C₁₀₉	—	4	2	no	—
67	C₄²×C₁₀₉	—	2	4	no	—
68	C₂³×C₁₀₉	—	2	2	no	—
69	C₂³×C₁₀₉	—	2	1	no	—
70	C₂³×C₁₀₉	—	2	1	no	—
71	C₂³×C₁₀₉	—	2	1	no	—
72	C₂³×C₁₀₉	—	2	2	no	—
73	C₂³×C₁₀₉	—	2	2	no	—
74	C₂³×C₁₀₉	—	2	1	no	—
75	C₂³×C₁₀₉	—	2	1	no	—
76	C₂³×C₁₀₉	C₂×D₄	1	2	yes	—
77	C₂³×C₁₀₉	#41	1	4	yes	—
78	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
79	D₄×C₁₀₉	—	4	4	no	—
80	C₂×C₄×C₁₀₉	—	2	8	no	—
81	C₂×C₄×C₁₀₉	—	2	2	no	—
82	C₂×C₄×C₁₀₉	—	2	2	no	—
83	C₂×C₄×C₁₀₉	—	2	2	no	—
84	C₂×C₄×C₁₀₉	—	2	2	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	1	no	—
89	C₂×C₄×C₁₀₉	—	2	1	no	—
90	C₂×C₄×C₁₀₉	—	2	2	no	—
91	C₂×C₄×C₁₀₉	—	2	2	no	—
92	C₂×C₄×C₁₀₉	—	2	2	no	—
93	C₂×C₄×C₁₀₉	—	2	2	no	—
94	C₂×C₄×C₁₀₉	—	2	2	no	—
95	C₂×C₄×C₁₀₉	—	2	2	no	—
96	C₂×C₄×C₁₀₉	—	2	2	no	—
97	C₂×C₄×C₁₀₉	—	2	2	no	—
98	C₂×C₄×C₁₀₉	—	2	1	no	—
99	C₂×C₄×C₁₀₉	—	2	2	no	—
100	C₂×C₄×C₁₀₉	—	2	1	no	—
101	C₂×C₄×C₁₀₉	—	4	8	no	—
102	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
103	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
104	C₂×C₄×C₁₀₉	#41	1	4	yes	—
105	C₂²×C₁₀₉	—	2	2	no	—
106	C₂²×C₁₀₉	—	2	2	no	—
107	C₂²×C₁₀₉	—	2	4	no	—
108	C₂²×C₁₀₉	—	2	4	no	—
109	C₂²×C₁₀₉	—	4	1	no	—
110	C₂²×C₁₀₉	—	4	1	no	—
111	C₂²×C₁₀₉	—	4	1	no	—
112	C₂²×C₁₀₉	—	4	2	no	—
113	C₂²×C₁₀₉	—	4	2	no	—
114	C₂²×C₁₀₉	#123	1	4	yes	—
115	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
116	C₂²×C₁₀₉	C₂×#41	1	2	yes	—
117	C₂²×C₁₀₉	C₂×#41	1	2	yes	—
118	C₂²×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
119	C₂²×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
120	C₄×C₁₀₉	—	2	2	no	—
121	C₄×C₁₀₉	—	2	2	no	—
122	C₄×C₁₀₉	—	2	4	no	—
123	C₄×C₁₀₉	—	4	1	no	—
124	C₄×C₁₀₉	—	4	2	no	—
125	C₄×C₁₀₉	—	4	1	no	—
126	C₄×C₁₀₉	—	4	2	no	—
127	C₄×C₁₀₉	—	4	1	no	—
128	C₄×C₁₀₉	—	4	2	no	—
129	C₂×C₁₀₉	—	2	4	no	—
130	C₂×C₁₀₉	—	8	1	no	—
131	C₂×C₁₀₉	C₂×#123	1	1	yes	characteristic
132	C₂×C₁₀₉	#532	1	1	yes	characteristic
133	C₂×C₁₀₉	#537	1	1	yes	characteristic
134	C₂×C₁₀₉	#551	1	2	yes	—
135	C₂×C₁₀₉	#552	1	2	yes	—
136	#2498	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
137	C₁₀₉	#2498	1	1	yes	characteristic, direct product factor, Sylow
138	C₂×#123	C₂×C₁₀₉	1	1	yes	characteristic
139	C₂×#117	C₂×C₁₀₉	1	1	yes	characteristic, complemented
140	#400	C₂×C₁₀₉	1	1	yes	characteristic, complemented
141	#399	C₂×C₁₀₉	1	1	yes	characteristic, complemented
142	#398	C₂×C₁₀₉	1	1	yes	characteristic, complemented
143	#397	C₂×C₁₀₉	1	1	yes	characteristic, complemented
144	#388	C₂×C₁₀₉	1	2	yes	complemented
145	#387	C₂×C₁₀₉	1	2	yes	complemented
146	#386	C₂×C₁₀₉	1	2	yes	—
147	#380	C₂×C₁₀₉	1	2	yes	complemented
148	C₂×#95	C₂×C₁₀₉	1	1	yes	characteristic, complemented
149	C₂²×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
150	C₂³×C₄	C₂²×C₁₀₉	1	2	yes	—
151	C₂³×C₄	C₂²×C₁₀₉	1	1	yes	characteristic
152	#123	—	2	4	no	—
153	#117	—	2	4	no	—
154	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
155	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
156	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
157	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
158	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
159	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
160	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	2	yes	—
161	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
162	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
163	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
164	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	2	yes	—
165	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	2	yes	—
166	C₂×C₄²	C₂²×C₁₀₉	1	2	yes	—
167	#95	C₂²×C₁₀₉	1	2	yes	—
168	#95	C₂²×C₁₀₉	1	1	yes	characteristic
169	#95	C₂²×C₁₀₉	1	1	yes	characteristic
170	#95	C₂²×C₁₀₉	1	2	yes	—
171	#95	C₂²×C₁₀₉	1	2	yes	—
172	#95	C₂²×C₁₀₉	1	2	yes	—
173	#95	C₂²×C₁₀₉	1	2	yes	—
174	C₂⁴	C₂³×C₁₀₉	1	1	yes	characteristic
175	C₂⁴	C₂³×C₁₀₉	1	1	yes	characteristic
176	C₂×D₄	—	2	2	no	—
177	C₂×D₄	—	2	4	no	—
178	C₂²×C₄	—	2	4	no	—
179	C₂²×C₄	—	2	4	no	—
180	C₂²×C₄	—	2	4	no	—
181	C₂²×C₄	—	4	2	no	—
182	C₂²×C₄	D₄×C₁₀₉	1	4	yes	—
183	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
184	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
185	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
186	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
187	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
188	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
189	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
190	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
191	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
192	C₄⋊C₄	—	2	2	no	—
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	—	2	2	no	—
196	C₄⋊C₄	—	2	4	no	—
197	C₄⋊C₄	—	4	2	no	—
198	C₂²⋊C₄	—	2	4	no	—
199	C₂²⋊C₄	—	2	2	no	—
200	C₂²⋊C₄	—	2	2	no	—
201	C₂²⋊C₄	—	2	2	no	—
202	C₂²⋊C₄	—	2	4	no	—
203	C₂²⋊C₄	—	4	2	no	—
204	C₄²	—	2	4	no	—
205	C₂³	—	2	2	no	—
206	C₂³	—	2	1	no	—
207	C₂³	—	2	1	no	—
208	C₂³	—	2	1	no	—
209	C₂³	—	2	2	no	—
210	C₂³	—	2	2	no	—
211	C₂³	—	2	1	no	—
212	C₂³	—	2	1	no	—
213	C₂³	C₂×D₄×C₁₀₉	1	2	yes	—
214	C₂³	#41×C₁₀₉	1	4	yes	—
215	C₂³	C₂⁴×C₁₀₉	1	1	yes	characteristic
216	D₄	—	4	4	no	—
217	C₂×C₄	—	2	8	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	4	8	no	—
239	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
240	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
241	C₂×C₄	#41×C₁₀₉	1	4	yes	—
242	C₂²	—	2	2	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	4	no	—
245	C₂²	—	2	4	no	—
246	C₂²	—	4	1	no	—
247	C₂²	—	4	1	no	—
248	C₂²	—	4	1	no	—
249	C₂²	—	4	2	no	—
250	C₂²	—	4	2	no	—
251	C₂²	#123×C₁₀₉	1	4	yes	—
252	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
253	C₂²	C₂×#41×C₁₀₉	1	2	yes	—
254	C₂²	C₂×#41×C₁₀₉	1	2	yes	—
255	C₂²	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
256	C₂²	2₋¹⁺⁴×C₁₀₉	1	1	yes	characteristic
257	C₄	—	2	2	no	—
258	C₄	—	2	2	no	—
259	C₄	—	2	4	no	—
260	C₄	—	4	1	no	—
261	C₄	—	4	2	no	—
262	C₄	—	4	1	no	—
263	C₄	—	4	2	no	—
264	C₄	—	4	1	no	—
265	C₄	—	4	2	no	—
266	C₂	—	2	4	no	—
267	C₂	—	8	1	no	—
268	C₂	C₂×#123×C₁₀₉	1	1	yes	characteristic
269	C₂	#532×C₁₀₉	1	1	yes	characteristic
270	C₂	#537×C₁₀₉	1	1	yes	characteristic
271	C₂	#551×C₁₀₉	1	2	yes	—
272	C₂	#552×C₁₀₉	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₀₉	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₁₀₉
prime radical	C₂²×D₄×C₁₀₉	C₂²
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. #400×C₁₀₉	C₂	semidirect
4. #399×C₁₀₉	C₂	semidirect
5. #398×C₁₀₉	C₂	semidirect
6. #397×C₁₀₉	C₂	semidirect
7. #388×C₁₀₉	C₂	semidirect
8. #387×C₁₀₉	C₂	semidirect
10. #380×C₁₀₉	C₂	semidirect
11. C₂×#95×C₁₀₉	C₂	semidirect
136. #2498	C₁₀₉	direct (N × H)
137. C₁₀₉	#2498	direct (N × H)
139. C₂×#117	C₂×C₁₀₉	semidirect
140. #400	C₂×C₁₀₉	semidirect
141. #399	C₂×C₁₀₉	semidirect
142. #398	C₂×C₁₀₉	semidirect
143. #397	C₂×C₁₀₉	semidirect
144. #388	C₂×C₁₀₉	semidirect
145. #387	C₂×C₁₀₉	semidirect
147. #380	C₂×C₁₀₉	semidirect
148. C₂×#95	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×#123×C₁₀₉	2
C₂×#117×C₁₀₉	2
#400×C₁₀₉	2
#399×C₁₀₉	2
#398×C₁₀₉	2
#397×C₁₀₉	2
#388×C₁₀₉	2
#387×C₁₀₉	2
#386×C₁₀₉	2
#380×C₁₀₉	2
C₂×#95×C₁₀₉	2
#2498	109

Subgroups — C109×#2498 #28359752

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#2498 #28359752