Subgroups — C₁₀₉×#2413 #28359667

Order 13952.

610 conjugacy classes of subgroups.

The subgroup lattice has 296 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×#123×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#117×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₄⋊C₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#398×C₁₀₉	C₂	1	2	yes	maximal, complemented
6	#396×C₁₀₉	C₂	1	2	yes	maximal, complemented
7	#389×C₁₀₉	C₂	1	2	yes	maximal, complemented
8	#388×C₁₀₉	C₂	1	2	yes	maximal, complemented
9	#386×C₁₀₉	C₂	1	1	yes	maximal, characteristic
10	#385×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
11	#381×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂³×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
16	#123×C₁₀₉	—	2	4	no	—
17	#123×C₁₀₉	—	2	4	no	—
18	#117×C₁₀₉	—	2	4	no	—
19	C₂×C₄⋊C₄×C₁₀₉	—	2	4	no	—
20	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
21	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
22	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
23	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
25	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
27	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
28	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
29	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
30	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	2	yes	—
31	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
32	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	2	yes	—
33	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
34	C₂×C₄²×C₁₀₉	C₂²	1	2	yes	—
35	#95×C₁₀₉	C₂²	1	2	yes	—
36	#95×C₁₀₉	C₂²	1	2	yes	—
37	#95×C₁₀₉	C₂²	1	2	yes	—
38	#95×C₁₀₉	C₂²	1	2	yes	—
39	C₂⁴×C₁₀₉	C₂³	1	1	yes	characteristic
40	C₂⁴×C₁₀₉	C₂³	1	1	yes	characteristic
41	C₂×D₄×C₁₀₉	—	2	2	no	—
42	C₂×D₄×C₁₀₉	—	2	2	no	—
43	C₂×D₄×C₁₀₉	—	2	2	no	—
44	C₂²×C₄×C₁₀₉	—	2	2	no	—
45	C₂²×C₄×C₁₀₉	—	2	2	no	—
46	C₂²×C₄×C₁₀₉	—	2	2	no	—
47	C₂²×C₄×C₁₀₉	—	2	4	no	—
48	C₂²×C₄×C₁₀₉	—	2	2	no	—
49	C₂²×C₄×C₁₀₉	—	2	2	no	—
50	C₂²×C₄×C₁₀₉	—	2	2	no	—
51	C₂²×C₄×C₁₀₉	D₄	1	4	yes	—
52	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
54	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
56	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
58	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
61	C₄⋊C₄×C₁₀₉	—	2	4	no	—
62	C₄⋊C₄×C₁₀₉	—	2	4	no	—
63	C₄⋊C₄×C₁₀₉	—	2	4	no	—
64	C₄⋊C₄×C₁₀₉	—	2	2	no	—
65	C₄⋊C₄×C₁₀₉	—	2	2	no	—
66	C₄⋊C₄×C₁₀₉	—	2	4	no	—
67	C₄⋊C₄×C₁₀₉	—	2	4	no	—
68	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
69	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
70	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
71	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
72	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
73	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
74	C₂²⋊C₄×C₁₀₉	D₄	1	4	yes	—
75	C₄²×C₁₀₉	—	2	4	no	—
76	C₂³×C₁₀₉	—	2	2	no	—
77	C₂³×C₁₀₉	—	2	2	no	—
78	C₂³×C₁₀₉	—	2	1	no	—
79	C₂³×C₁₀₉	—	2	1	no	—
80	C₂³×C₁₀₉	—	2	2	no	—
81	C₂³×C₁₀₉	—	2	1	no	—
82	C₂³×C₁₀₉	—	2	2	no	—
83	C₂³×C₁₀₉	C₂×D₄	1	2	yes	—
84	C₂³×C₁₀₉	C₂×D₄	1	2	yes	—
85	C₂³×C₁₀₉	#41	1	2	yes	—
86	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
87	D₄×C₁₀₉	—	4	4	no	—
88	C₂×C₄×C₁₀₉	—	2	8	no	—
89	C₂×C₄×C₁₀₉	—	2	8	no	—
90	C₂×C₄×C₁₀₉	—	2	1	no	—
91	C₂×C₄×C₁₀₉	—	2	1	no	—
92	C₂×C₄×C₁₀₉	—	2	2	no	—
93	C₂×C₄×C₁₀₉	—	2	2	no	—
94	C₂×C₄×C₁₀₉	—	2	1	no	—
95	C₂×C₄×C₁₀₉	—	2	1	no	—
96	C₂×C₄×C₁₀₉	—	2	2	no	—
97	C₂×C₄×C₁₀₉	—	2	2	no	—
98	C₂×C₄×C₁₀₉	—	2	2	no	—
99	C₂×C₄×C₁₀₉	—	2	1	no	—
100	C₂×C₄×C₁₀₉	—	2	1	no	—
101	C₂×C₄×C₁₀₉	—	2	2	no	—
102	C₂×C₄×C₁₀₉	—	2	2	no	—
103	C₂×C₄×C₁₀₉	—	2	1	no	—
104	C₂×C₄×C₁₀₉	—	2	1	no	—
105	C₂×C₄×C₁₀₉	—	2	2	no	—
106	C₂×C₄×C₁₀₉	—	2	2	no	—
107	C₂×C₄×C₁₀₉	—	2	2	no	—
108	C₂×C₄×C₁₀₉	—	2	1	no	—
109	C₂×C₄×C₁₀₉	—	2	2	no	—
110	C₂×C₄×C₁₀₉	—	2	2	no	—
111	C₂×C₄×C₁₀₉	—	4	4	no	—
112	C₂×C₄×C₁₀₉	C₂×D₄	1	4	yes	—
113	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
114	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
115	C₂×C₄×C₁₀₉	#41	1	4	yes	—
116	C₂²×C₁₀₉	—	2	2	no	—
117	C₂²×C₁₀₉	—	2	2	no	—
118	C₂²×C₁₀₉	—	2	4	no	—
119	C₂²×C₁₀₉	—	2	4	no	—
120	C₂²×C₁₀₉	—	4	1	no	—
121	C₂²×C₁₀₉	—	4	1	no	—
122	C₂²×C₁₀₉	—	4	1	no	—
123	C₂²×C₁₀₉	—	4	2	no	—
124	C₂²×C₁₀₉	—	4	2	no	—
125	C₂²×C₁₀₉	#121	1	4	yes	—
126	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
127	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
128	C₂²×C₁₀₉	C₂×#41	1	1	yes	characteristic
129	C₂²×C₁₀₉	C₂×#41	1	2	yes	—
130	C₂²×C₁₀₉	2₋¹⁺⁴	1	2	yes	—
131	C₄×C₁₀₉	—	2	2	no	—
132	C₄×C₁₀₉	—	2	2	no	—
133	C₄×C₁₀₉	—	2	4	no	—
134	C₄×C₁₀₉	—	2	4	no	—
135	C₄×C₁₀₉	—	4	2	no	—
136	C₄×C₁₀₉	—	4	1	no	—
137	C₄×C₁₀₉	—	4	1	no	—
138	C₄×C₁₀₉	—	4	1	no	—
139	C₄×C₁₀₉	—	4	2	no	—
140	C₂×C₁₀₉	—	2	4	no	—
141	C₂×C₁₀₉	—	8	1	no	—
142	C₂×C₁₀₉	C₂×#121	1	1	yes	characteristic
143	C₂×C₁₀₉	#525	1	1	yes	characteristic
144	C₂×C₁₀₉	#536	1	1	yes	characteristic
145	C₂×C₁₀₉	#547	1	2	yes	—
146	C₂×C₁₀₉	#552	1	2	yes	—
147	#2413	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
148	C₁₀₉	#2413	1	1	yes	characteristic, direct product factor, Sylow
149	C₂×#123	C₂×C₁₀₉	1	1	yes	characteristic
150	C₂×#123	C₂×C₁₀₉	1	1	yes	characteristic
151	C₂×#117	C₂×C₁₀₉	1	1	yes	characteristic, complemented
152	C₂²×C₄⋊C₄	C₂×C₁₀₉	1	1	yes	characteristic, complemented
153	#398	C₂×C₁₀₉	1	2	yes	complemented
154	#396	C₂×C₁₀₉	1	2	yes	complemented
155	#389	C₂×C₁₀₉	1	2	yes	complemented
156	#388	C₂×C₁₀₉	1	2	yes	complemented
157	#386	C₂×C₁₀₉	1	1	yes	characteristic
158	#385	C₂×C₁₀₉	1	1	yes	characteristic, complemented
159	#381	C₂×C₁₀₉	1	1	yes	characteristic, complemented
160	C₂²×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
161	C₂³×C₄	C₂²×C₁₀₉	1	1	yes	characteristic
162	C₂³×C₄	C₂²×C₁₀₉	1	1	yes	characteristic
163	C₂³×C₄	C₂²×C₁₀₉	1	1	yes	characteristic
164	#123	—	2	4	no	—
165	#123	—	2	4	no	—
166	#117	—	2	4	no	—
167	C₂×C₄⋊C₄	—	2	4	no	—
168	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
169	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
170	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
171	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
172	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
173	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
174	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
175	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
176	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
177	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
178	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	2	yes	—
179	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
180	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	2	yes	—
181	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
182	C₂×C₄²	C₂²×C₁₀₉	1	2	yes	—
183	#95	C₂²×C₁₀₉	1	2	yes	—
184	#95	C₂²×C₁₀₉	1	2	yes	—
185	#95	C₂²×C₁₀₉	1	2	yes	—
186	#95	C₂²×C₁₀₉	1	2	yes	—
187	C₂⁴	C₂³×C₁₀₉	1	1	yes	characteristic
188	C₂⁴	C₂³×C₁₀₉	1	1	yes	characteristic
189	C₂×D₄	—	2	2	no	—
190	C₂×D₄	—	2	2	no	—
191	C₂×D₄	—	2	2	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	D₄×C₁₀₉	1	4	yes	—
200	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
201	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
202	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
203	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
204	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
205	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
206	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
207	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
208	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
209	C₄⋊C₄	—	2	4	no	—
210	C₄⋊C₄	—	2	4	no	—
211	C₄⋊C₄	—	2	4	no	—
212	C₄⋊C₄	—	2	2	no	—
213	C₄⋊C₄	—	2	2	no	—
214	C₄⋊C₄	—	2	4	no	—
215	C₄⋊C₄	—	2	4	no	—
216	C₂²⋊C₄	—	2	2	no	—
217	C₂²⋊C₄	—	2	2	no	—
218	C₂²⋊C₄	—	2	2	no	—
219	C₂²⋊C₄	—	2	2	no	—
220	C₂²⋊C₄	—	2	4	no	—
221	C₂²⋊C₄	—	2	4	no	—
222	C₂²⋊C₄	D₄×C₁₀₉	1	4	yes	—
223	C₄²	—	2	4	no	—
224	C₂³	—	2	2	no	—
225	C₂³	—	2	2	no	—
226	C₂³	—	2	1	no	—
227	C₂³	—	2	1	no	—
228	C₂³	—	2	2	no	—
229	C₂³	—	2	1	no	—
230	C₂³	—	2	2	no	—
231	C₂³	C₂×D₄×C₁₀₉	1	2	yes	—
232	C₂³	C₂×D₄×C₁₀₉	1	2	yes	—
233	C₂³	#41×C₁₀₉	1	2	yes	—
234	C₂³	C₂⁴×C₁₀₉	1	1	yes	characteristic
235	D₄	—	4	4	no	—
236	C₂×C₄	—	2	8	no	—
237	C₂×C₄	—	2	8	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	C₂×D₄×C₁₀₉	1	4	yes	—
261	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
262	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
263	C₂×C₄	#41×C₁₀₉	1	4	yes	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	4	no	—
267	C₂²	—	2	4	no	—
268	C₂²	—	4	1	no	—
269	C₂²	—	4	1	no	—
270	C₂²	—	4	1	no	—
271	C₂²	—	4	2	no	—
272	C₂²	—	4	2	no	—
273	C₂²	#121×C₁₀₉	1	4	yes	—
274	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
275	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
276	C₂²	C₂×#41×C₁₀₉	1	1	yes	characteristic
277	C₂²	C₂×#41×C₁₀₉	1	2	yes	—
278	C₂²	2₋¹⁺⁴×C₁₀₉	1	2	yes	—
279	C₄	—	2	2	no	—
280	C₄	—	2	2	no	—
281	C₄	—	2	4	no	—
282	C₄	—	2	4	no	—
283	C₄	—	4	2	no	—
284	C₄	—	4	1	no	—
285	C₄	—	4	1	no	—
286	C₄	—	4	1	no	—
287	C₄	—	4	2	no	—
288	C₂	—	2	4	no	—
289	C₂	—	8	1	no	—
290	C₂	C₂×#121×C₁₀₉	1	1	yes	characteristic
291	C₂	#525×C₁₀₉	1	1	yes	characteristic
292	C₂	#536×C₁₀₉	1	1	yes	characteristic
293	C₂	#547×C₁₀₉	1	2	yes	—
294	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
3. C₂×#117×C₁₀₉	C₂	semidirect
4. C₂²×C₄⋊C₄×C₁₀₉	C₂	semidirect
5. #398×C₁₀₉	C₂	semidirect
6. #396×C₁₀₉	C₂	semidirect
7. #389×C₁₀₉	C₂	semidirect
8. #388×C₁₀₉	C₂	semidirect
10. #385×C₁₀₉	C₂	semidirect
11. #381×C₁₀₉	C₂	semidirect
147. #2413	C₁₀₉	direct (N × H)
148. C₁₀₉	#2413	direct (N × H)
151. C₂×#117	C₂×C₁₀₉	semidirect
152. C₂²×C₄⋊C₄	C₂×C₁₀₉	semidirect
153. #398	C₂×C₁₀₉	semidirect
154. #396	C₂×C₁₀₉	semidirect
155. #389	C₂×C₁₀₉	semidirect
156. #388	C₂×C₁₀₉	semidirect
158. #385	C₂×C₁₀₉	semidirect
159. #381	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
C₂²×C₄⋊C₄×C₁₀₉	2
#398×C₁₀₉	2
#396×C₁₀₉	2
#389×C₁₀₉	2
#388×C₁₀₉	2
#386×C₁₀₉	2
#385×C₁₀₉	2
#381×C₁₀₉	2
#2413	109

Subgroups — C109×#2413 #28359667

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#2413 #28359667