Subgroups — C₁₀₉×#2447 #28359701

Order 13952.

540 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#124×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	#397×C₁₀₉	C₂	1	2	yes	maximal, complemented
4	#397×C₁₀₉	C₂	1	2	yes	maximal, complemented
5	#391×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#391×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
7	#390×C₁₀₉	C₂	1	1	yes	maximal, characteristic
8	#389×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
9	#385×C₁₀₉	C₂	1	2	yes	maximal, complemented
10	#381×C₁₀₉	C₂	1	2	yes	maximal, complemented
11	C₄×C₄⋊C₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₂²×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₁₀₉	C₂²	1	2	yes	—
15	#124×C₁₀₉	—	2	4	no	—
16	C₄×D₄×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×C₂²⋊C₄×C₁₀₉	C₂²	1	2	yes	—
23	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	2	yes	—
24	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	2	yes	—
25	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	2	yes	—
26	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	2	yes	—
27	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
28	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
29	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
30	#95×C₁₀₉	C₂²	1	1	yes	characteristic
31	#95×C₁₀₉	C₂²	1	1	yes	characteristic
32	#95×C₁₀₉	C₂²	1	2	yes	—
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	2	yes	—
38	C₂×Q₈×C₁₀₉	—	2	2	no	—
39	C₂×Q₈×C₁₀₉	—	2	2	no	—
40	C₂×Q₈×C₁₀₉	—	2	2	no	—
41	C₂×D₄×C₁₀₉	—	2	2	no	—
42	C₂×D₄×C₁₀₉	—	2	2	no	—
43	C₂×D₄×C₁₀₉	D₄	1	4	yes	—
44	C₂²×C₄×C₁₀₉	—	2	4	no	—
45	C₂²×C₄×C₁₀₉	—	2	4	no	—
46	C₂²×C₄×C₁₀₉	—	4	2	no	—
47	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
49	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
52	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
53	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
57	C₄⋊C₄×C₁₀₉	—	2	2	no	—
58	C₄⋊C₄×C₁₀₉	—	2	2	no	—
59	C₄⋊C₄×C₁₀₉	—	2	2	no	—
60	C₄⋊C₄×C₁₀₉	—	2	2	no	—
61	C₄⋊C₄×C₁₀₉	—	4	2	no	—
62	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
63	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
64	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
65	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
66	C₂²⋊C₄×C₁₀₉	—	4	2	no	—
67	C₄²×C₁₀₉	—	2	2	no	—
68	C₄²×C₁₀₉	—	2	2	no	—
69	C₄²×C₁₀₉	—	2	2	no	—
70	C₂³×C₁₀₉	—	2	2	no	—
71	C₂³×C₁₀₉	—	2	2	no	—
72	C₂³×C₁₀₉	—	2	2	no	—
73	C₂³×C₁₀₉	—	2	2	no	—
74	C₂³×C₁₀₉	—	2	2	no	—
75	C₂³×C₁₀₉	—	2	2	no	—
76	C₂³×C₁₀₉	C₂×D₄	1	4	yes	—
77	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
78	Q₈×C₁₀₉	—	4	4	no	—
79	D₄×C₁₀₉	—	2	8	no	—
80	C₂×C₄×C₁₀₉	—	2	1	no	—
81	C₂×C₄×C₁₀₉	—	2	1	no	—
82	C₂×C₄×C₁₀₉	—	2	1	no	—
83	C₂×C₄×C₁₀₉	—	2	1	no	—
84	C₂×C₄×C₁₀₉	—	2	1	no	—
85	C₂×C₄×C₁₀₉	—	2	1	no	—
86	C₂×C₄×C₁₀₉	—	2	1	no	—
87	C₂×C₄×C₁₀₉	—	2	2	no	—
88	C₂×C₄×C₁₀₉	—	2	2	no	—
89	C₂×C₄×C₁₀₉	—	2	1	no	—
90	C₂×C₄×C₁₀₉	—	2	2	no	—
91	C₂×C₄×C₁₀₉	—	2	1	no	—
92	C₂×C₄×C₁₀₉	—	2	1	no	—
93	C₂×C₄×C₁₀₉	—	2	1	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	—	2	2	no	—
102	C₂×C₄×C₁₀₉	—	2	1	no	—
103	C₂×C₄×C₁₀₉	—	2	1	no	—
104	C₂×C₄×C₁₀₉	—	2	1	no	—
105	C₂×C₄×C₁₀₉	—	4	8	no	—
106	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
107	C₂×C₄×C₁₀₉	#41	1	2	yes	—
108	C₂×C₄×C₁₀₉	#41	1	2	yes	—
109	C₂×C₄×C₁₀₉	#41	1	2	yes	—
110	C₂×C₄×C₁₀₉	#41	1	2	yes	—
111	C₂²×C₁₀₉	—	2	4	no	—
112	C₂²×C₁₀₉	—	2	4	no	—
113	C₂²×C₁₀₉	—	2	4	no	—
114	C₂²×C₁₀₉	—	4	2	no	—
115	C₂²×C₁₀₉	—	4	2	no	—
116	C₂²×C₁₀₉	—	4	2	no	—
117	C₂²×C₁₀₉	—	4	2	no	—
118	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
119	C₂²×C₁₀₉	C₂×#41	1	1	yes	characteristic
120	C₂²×C₁₀₉	C₂×#41	1	1	yes	characteristic
121	C₂²×C₁₀₉	C₂×#41	1	1	yes	characteristic
122	C₂²×C₁₀₉	C₂×#41	1	1	yes	characteristic
123	C₂²×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
124	C₂²×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
125	C₄×C₁₀₉	—	2	2	no	—
126	C₄×C₁₀₉	—	2	2	no	—
127	C₄×C₁₀₉	—	4	2	no	—
128	C₄×C₁₀₉	—	4	1	no	—
129	C₄×C₁₀₉	—	4	1	no	—
130	C₄×C₁₀₉	—	4	2	no	—
131	C₄×C₁₀₉	—	4	1	no	—
132	C₄×C₁₀₉	—	4	1	no	—
133	C₄×C₁₀₉	—	4	2	no	—
134	C₄×C₁₀₉	#123	1	4	yes	—
135	C₂×C₁₀₉	—	4	4	no	—
136	C₂×C₁₀₉	C₂×#123	1	1	yes	characteristic
137	C₂×C₁₀₉	#529	1	1	yes	characteristic
138	C₂×C₁₀₉	#542	1	1	yes	characteristic
139	C₂×C₁₀₉	#546	1	1	yes	characteristic
140	C₂×C₁₀₉	#547	1	1	yes	characteristic
141	C₂×C₁₀₉	#556	1	1	yes	characteristic
142	C₂×C₁₀₉	#559	1	1	yes	characteristic
143	#2447	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₁₀₉	#2447	1	1	yes	characteristic, direct product factor, Sylow
145	C₂×#124	C₂×C₁₀₉	1	1	yes	characteristic
146	C₂×C₄×D₄	C₂×C₁₀₉	1	1	yes	characteristic
147	#397	C₂×C₁₀₉	1	2	yes	complemented
148	#397	C₂×C₁₀₉	1	2	yes	complemented
149	#391	C₂×C₁₀₉	1	1	yes	characteristic, complemented
150	#391	C₂×C₁₀₉	1	1	yes	characteristic, complemented
151	#390	C₂×C₁₀₉	1	1	yes	characteristic
152	#389	C₂×C₁₀₉	1	1	yes	characteristic, complemented
153	#385	C₂×C₁₀₉	1	2	yes	complemented
154	#381	C₂×C₁₀₉	1	2	yes	complemented
155	C₄×C₄⋊C₄	C₂×C₁₀₉	1	1	yes	characteristic, complemented
156	C₂²×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
157	C₂²×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
158	C₂³×C₄	C₂²×C₁₀₉	1	2	yes	—
159	#124	—	2	4	no	—
160	C₄×D₄	—	2	4	no	—
161	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
162	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
163	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
164	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
165	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
166	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	2	yes	—
167	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	2	yes	—
168	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	2	yes	—
169	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	2	yes	—
170	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	2	yes	—
171	C₂×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
172	C₂×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
173	C₂×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
174	#95	C₂²×C₁₀₉	1	1	yes	characteristic
175	#95	C₂²×C₁₀₉	1	1	yes	characteristic
176	#95	C₂²×C₁₀₉	1	2	yes	—
177	#95	C₂²×C₁₀₉	1	2	yes	—
178	#95	C₂²×C₁₀₉	1	2	yes	—
179	#95	C₂²×C₁₀₉	1	2	yes	—
180	#95	C₂²×C₁₀₉	1	2	yes	—
181	C₂⁴	C₂³×C₁₀₉	1	2	yes	—
182	C₂×Q₈	—	2	2	no	—
183	C₂×Q₈	—	2	2	no	—
184	C₂×Q₈	—	2	2	no	—
185	C₂×D₄	—	2	2	no	—
186	C₂×D₄	—	2	2	no	—
187	C₂×D₄	D₄×C₁₀₉	1	4	yes	—
188	C₂²×C₄	—	2	4	no	—
189	C₂²×C₄	—	2	4	no	—
190	C₂²×C₄	—	4	2	no	—
191	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
192	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
193	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
194	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
195	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
196	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
197	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
198	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
199	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
200	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
201	C₄⋊C₄	—	2	2	no	—
202	C₄⋊C₄	—	2	2	no	—
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	—	4	2	no	—
206	C₂²⋊C₄	—	2	4	no	—
207	C₂²⋊C₄	—	2	4	no	—
208	C₂²⋊C₄	—	2	4	no	—
209	C₂²⋊C₄	—	2	4	no	—
210	C₂²⋊C₄	—	4	2	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	2	2	no	—
213	C₄²	—	2	2	no	—
214	C₂³	—	2	2	no	—
215	C₂³	—	2	2	no	—
216	C₂³	—	2	2	no	—
217	C₂³	—	2	2	no	—
218	C₂³	—	2	2	no	—
219	C₂³	—	2	2	no	—
220	C₂³	C₂×D₄×C₁₀₉	1	4	yes	—
221	C₂³	C₂⁴×C₁₀₉	1	1	yes	characteristic
222	Q₈	—	4	4	no	—
223	D₄	—	2	8	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	4	8	no	—
250	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
251	C₂×C₄	#41×C₁₀₉	1	2	yes	—
252	C₂×C₄	#41×C₁₀₉	1	2	yes	—
253	C₂×C₄	#41×C₁₀₉	1	2	yes	—
254	C₂×C₄	#41×C₁₀₉	1	2	yes	—
255	C₂²	—	2	4	no	—
256	C₂²	—	2	4	no	—
257	C₂²	—	2	4	no	—
258	C₂²	—	4	2	no	—
259	C₂²	—	4	2	no	—
260	C₂²	—	4	2	no	—
261	C₂²	—	4	2	no	—
262	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
263	C₂²	C₂×#41×C₁₀₉	1	1	yes	characteristic
264	C₂²	C₂×#41×C₁₀₉	1	1	yes	characteristic
265	C₂²	C₂×#41×C₁₀₉	1	1	yes	characteristic
266	C₂²	C₂×#41×C₁₀₉	1	1	yes	characteristic
267	C₂²	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
268	C₂²	2₋¹⁺⁴×C₁₀₉	1	1	yes	characteristic
269	C₄	—	2	2	no	—
270	C₄	—	2	2	no	—
271	C₄	—	4	2	no	—
272	C₄	—	4	1	no	—
273	C₄	—	4	2	no	—
274	C₄	—	4	1	no	—
275	C₄	—	4	1	no	—
276	C₄	—	4	1	no	—
277	C₄	—	4	2	no	—
278	C₄	#123×C₁₀₉	1	4	yes	—
279	C₂	—	4	4	no	—
280	C₂	C₂×#123×C₁₀₉	1	1	yes	characteristic
281	C₂	#529×C₁₀₉	1	1	yes	characteristic
282	C₂	#542×C₁₀₉	1	1	yes	characteristic
283	C₂	#546×C₁₀₉	1	1	yes	characteristic
284	C₂	#547×C₁₀₉	1	1	yes	characteristic
285	C₂	#556×C₁₀₉	1	1	yes	characteristic
286	C₂	#559×C₁₀₉	1	1	yes	characteristic

	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
3. #397×C₁₀₉	C₂	semidirect
4. #397×C₁₀₉	C₂	semidirect
5. #391×C₁₀₉	C₂	semidirect
6. #391×C₁₀₉	C₂	semidirect
8. #389×C₁₀₉	C₂	semidirect
9. #385×C₁₀₉	C₂	semidirect
10. #381×C₁₀₉	C₂	semidirect
11. C₄×C₄⋊C₄×C₁₀₉	C₂	semidirect
143. #2447	C₁₀₉	direct (N × H)
144. C₁₀₉	#2447	direct (N × H)
147. #397	C₂×C₁₀₉	semidirect
148. #397	C₂×C₁₀₉	semidirect
149. #391	C₂×C₁₀₉	semidirect
150. #391	C₂×C₁₀₉	semidirect
152. #389	C₂×C₁₀₉	semidirect
153. #385	C₂×C₁₀₉	semidirect
154. #381	C₂×C₁₀₉	semidirect
155. C₄×C₄⋊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×#124×C₁₀₉	2
C₂×C₄×D₄×C₁₀₉	2
#397×C₁₀₉	2
#391×C₁₀₉	2
#390×C₁₀₉	2
#389×C₁₀₉	2
#385×C₁₀₉	2
#381×C₁₀₉	2
C₄×C₄⋊C₄×C₁₀₉	2
#2447	109

Subgroups — C109×#2447 #28359701

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#2447 #28359701