Subgroups — C₁₀₉×#8837 #42126831

Order 27904.

454 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₉×#2975	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#1954	C₂	1	1	yes	maximal, characteristic
3	C₄×#100×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₁₀₉×#1565	C₂	1	2	yes	maximal, complemented
5	C₁₀₉×#1278	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
7	#436×C₁₀₉	—	2	2	no	—
8	#436×C₁₀₉	—	2	2	no	—
9	#431×C₁₀₉	C₂²	1	2	yes	—
10	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₂×#100×C₁₀₉	C₄	1	2	yes	—
12	#407×C₁₀₉	—	2	2	no	—
13	C₂×C₂²⋊C₈×C₁₀₉	—	2	2	no	—
14	C₄×M₁₆×C₁₀₉	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
16	#343×C₁₀₉	C₄	1	1	yes	characteristic
17	#343×C₁₀₉	C₄	1	1	yes	characteristic
18	#335×C₁₀₉	—	2	2	no	—
19	#334×C₁₀₉	—	2	2	no	—
20	C₂²×D₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₁₀₉	D₄	1	2	yes	—
22	C₂×M₁₆×C₁₀₉	—	2	2	no	—
23	C₂×M₁₆×C₁₀₉	—	2	2	no	—
24	C₂×M₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
25	C₂×M₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₁₀₉	—	2	1	no	—
27	C₂²×C₈×C₁₀₉	—	2	1	no	—
28	C₄×D₄×C₁₀₉	—	2	2	no	—
29	C₄×D₄×C₁₀₉	—	2	2	no	—
30	C₂×C₄⋊C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₂²⋊C₄×C₁₀₉	Q₈	1	2	yes	—
32	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₁₀₉	—	2	2	no	—
34	C₄⋊C₈×C₁₀₉	D₄	1	2	yes	—
35	C₄⋊C₈×C₁₀₉	D₄	1	2	yes	—
36	#100×C₁₀₉	—	2	4	no	—
37	C₂²⋊C₈×C₁₀₉	—	2	4	no	—
38	C₂²⋊C₈×C₁₀₉	—	2	2	no	—
39	C₂²⋊C₈×C₁₀₉	—	4	4	no	—
40	#97×C₁₀₉	—	2	2	no	—
41	C₄×C₈×C₁₀₉	—	2	2	no	—
42	C₄×C₈×C₁₀₉	—	2	2	no	—
43	C₄×C₈×C₁₀₉	—	2	2	no	—
44	C₂⁴×C₁₀₉	C₄⋊C₄	1	2	yes	—
45	C₂×D₄×C₁₀₉	—	2	2	no	—
46	C₂×D₄×C₁₀₉	—	2	2	no	—
47	C₂×D₄×C₁₀₉	—	2	1	no	—
48	C₂×D₄×C₁₀₉	—	2	1	no	—
49	C₂²×C₄×C₁₀₉	—	2	2	no	—
50	C₂²×C₄×C₁₀₉	—	2	2	no	—
51	C₂²×C₄×C₁₀₉	—	4	2	no	—
52	C₂²×C₄×C₁₀₉	—	4	2	no	—
53	C₂²×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
57	M₁₆×C₁₀₉	—	2	2	no	—
58	M₁₆×C₁₀₉	—	2	2	no	—
59	M₁₆×C₁₀₉	—	2	2	no	—
60	M₁₆×C₁₀₉	—	8	2	no	—
61	M₁₆×C₁₀₉	—	8	2	no	—
62	M₁₆×C₁₀₉	M₁₆	1	4	yes	complemented
63	C₂×C₈×C₁₀₉	—	2	2	no	—
64	C₂×C₈×C₁₀₉	—	2	2	no	—
65	C₂×C₈×C₁₀₉	—	2	2	no	—
66	C₂×C₈×C₁₀₉	—	2	2	no	—
67	C₂×C₈×C₁₀₉	—	2	2	no	—
68	C₂×C₈×C₁₀₉	—	4	1	no	—
69	C₂×C₈×C₁₀₉	—	4	1	no	—
70	C₂×C₈×C₁₀₉	—	4	2	no	—
71	C₂×C₈×C₁₀₉	—	4	1	no	—
72	C₄⋊C₄×C₁₀₉	—	2	2	no	—
73	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
74	C₄²×C₁₀₉	C₂×D₄	1	2	yes	—
75	C₄²×C₁₀₉	#41	1	2	yes	—
76	C₂³×C₁₀₉	—	2	2	no	—
77	C₂³×C₁₀₉	—	2	2	no	—
78	C₂³×C₁₀₉	—	2	2	no	—
79	C₂³×C₁₀₉	—	4	2	no	—
80	C₂³×C₁₀₉	—	4	2	no	—
81	C₂³×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
82	D₄×C₁₀₉	—	4	2	no	—
83	D₄×C₁₀₉	—	4	2	no	—
84	C₂×C₄×C₁₀₉	—	2	1	no	—
85	C₂×C₄×C₁₀₉	—	2	1	no	—
86	C₂×C₄×C₁₀₉	—	2	2	no	—
87	C₂×C₄×C₁₀₉	—	4	1	no	—
88	C₂×C₄×C₁₀₉	—	4	2	no	—
89	C₂×C₄×C₁₀₉	—	4	2	no	—
90	C₂×C₄×C₁₀₉	—	4	2	no	—
91	C₂×C₄×C₁₀₉	—	8	2	no	—
92	C₂×C₄×C₁₀₉	C₄×D₄	1	2	yes	—
93	C₂×C₄×C₁₀₉	#120	1	1	yes	characteristic
94	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
95	C₂×C₄×C₁₀₉	#123	1	1	yes	characteristic
96	C₂×C₄×C₁₀₉	C₂×M₁₆	1	2	yes	—
97	C₂×C₄×C₁₀₉	#131	1	1	yes	characteristic
98	C₂×C₄×C₁₀₉	#131	1	1	yes	characteristic
99	C₈×C₁₀₉	—	2	4	no	—
100	C₈×C₁₀₉	—	4	2	no	—
101	C₈×C₁₀₉	—	4	2	no	—
102	C₈×C₁₀₉	—	4	2	no	—
103	C₈×C₁₀₉	—	4	2	no	—
104	C₈×C₁₀₉	—	8	1	no	—
105	C₂²×C₁₀₉	—	2	1	no	—
106	C₂²×C₁₀₉	—	2	1	no	—
107	C₂²×C₁₀₉	—	4	4	no	—
108	C₂²×C₁₀₉	—	4	2	no	—
109	C₂²×C₁₀₉	—	4	2	no	—
110	C₂²×C₁₀₉	—	4	2	no	—
111	C₂²×C₁₀₉	—	8	2	no	—
112	C₂²×C₁₀₉	#385	1	1	yes	characteristic
113	C₂²×C₁₀₉	#423	1	1	yes	characteristic
114	C₂²×C₁₀₉	#424	1	1	yes	characteristic
115	C₄×C₁₀₉	—	2	1	no	—
116	C₄×C₁₀₉	—	2	2	no	—
117	C₄×C₁₀₉	—	4	2	no	—
118	C₄×C₁₀₉	#435	1	2	yes	—
119	C₄×C₁₀₉	#436	1	2	yes	—
120	C₄×C₁₀₉	#453	1	1	yes	characteristic
121	C₄×C₁₀₉	#455	1	1	yes	characteristic
122	#8837	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
123	C₂×C₁₀₉	—	2	1	no	—
124	C₂×C₁₀₉	—	2	1	no	—
125	C₂×C₁₀₉	—	8	2	no	—
126	C₂×C₁₀₉	#1851	1	1	yes	characteristic
127	C₂×C₁₀₉	#1856	1	1	yes	characteristic
128	C₂×C₁₀₉	#1858	1	1	yes	characteristic
129	#2975	C₂×C₁₀₉	1	1	yes	characteristic
130	#1954	C₂×C₁₀₉	1	1	yes	characteristic
131	C₄×#100	C₂×C₁₀₉	1	1	yes	characteristic
132	#1565	C₂×C₁₀₉	1	2	yes	complemented
133	#1278	C₂×C₁₀₉	1	2	yes	complemented
134	C₁₀₉	#8837	1	1	yes	characteristic, direct product factor, Sylow
135	C₂×C₄×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
136	#436	—	2	2	no	—
137	#436	—	2	2	no	—
138	#431	C₂²×C₁₀₉	1	2	yes	—
139	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
140	C₂×#100	C₄×C₁₀₉	1	2	yes	—
141	#407	—	2	2	no	—
142	C₂×C₂²⋊C₈	—	2	2	no	—
143	C₄×M₁₆	C₂²×C₁₀₉	1	2	yes	—
144	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
145	#343	C₄×C₁₀₉	1	1	yes	characteristic
146	#343	C₄×C₁₀₉	1	1	yes	characteristic
147	#335	—	2	2	no	—
148	#334	—	2	2	no	—
149	C₂²×D₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
150	C₂³×C₄	D₄×C₁₀₉	1	2	yes	—
151	C₂×M₁₆	—	2	2	no	—
152	C₂×M₁₆	—	2	2	no	—
153	C₂×M₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
154	C₂×M₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
155	C₂²×C₈	—	2	1	no	—
156	C₂²×C₈	—	2	1	no	—
157	C₄×D₄	—	2	2	no	—
158	C₄×D₄	—	2	2	no	—
159	C₂×C₄⋊C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
160	C₂×C₂²⋊C₄	Q₈×C₁₀₉	1	2	yes	—
161	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
162	C₄⋊C₈	—	2	2	no	—
163	C₄⋊C₈	D₄×C₁₀₉	1	2	yes	—
164	C₄⋊C₈	D₄×C₁₀₉	1	2	yes	—
165	#100	—	2	4	no	—
166	C₂²⋊C₈	—	2	4	no	—
167	C₂²⋊C₈	—	2	2	no	—
168	C₂²⋊C₈	—	4	4	no	—
169	#97	—	2	2	no	—
170	C₄×C₈	—	2	2	no	—
171	C₄×C₈	—	2	2	no	—
172	C₄×C₈	—	2	2	no	—
173	C₂⁴	C₄⋊C₄×C₁₀₉	1	2	yes	—
174	C₂×D₄	—	2	2	no	—
175	C₂×D₄	—	2	2	no	—
176	C₂×D₄	—	2	1	no	—
177	C₂×D₄	—	2	1	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	—	2	2	no	—
180	C₂²×C₄	—	4	2	no	—
181	C₂²×C₄	—	4	2	no	—
182	C₂²×C₄	C₄⋊C₄×C₁₀₉	1	2	yes	—
183	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
184	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
185	C₂²×C₄	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
186	M₁₆	—	2	2	no	—
187	M₁₆	—	2	2	no	—
188	M₁₆	—	2	2	no	—
189	M₁₆	—	8	2	no	—
190	M₁₆	—	8	2	no	—
191	M₁₆	M₁₆×C₁₀₉	1	4	yes	complemented
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	4	1	no	—
198	C₂×C₈	—	4	1	no	—
199	C₂×C₈	—	4	2	no	—
200	C₂×C₈	—	4	1	no	—
201	C₄⋊C₄	—	2	2	no	—
202	C₂²⋊C₄	—	2	4	no	—
203	C₄²	C₂×D₄×C₁₀₉	1	2	yes	—
204	C₄²	#41×C₁₀₉	1	2	yes	—
205	C₂³	—	2	2	no	—
206	C₂³	—	2	2	no	—
207	C₂³	—	2	2	no	—
208	C₂³	—	4	2	no	—
209	C₂³	—	4	2	no	—
210	C₂³	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
211	D₄	—	4	2	no	—
212	D₄	—	4	2	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	4	1	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	8	2	no	—
221	C₂×C₄	C₄×D₄×C₁₀₉	1	2	yes	—
222	C₂×C₄	#120×C₁₀₉	1	1	yes	characteristic
223	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	2	yes	—
224	C₂×C₄	#123×C₁₀₉	1	1	yes	characteristic
225	C₂×C₄	C₂×M₁₆×C₁₀₉	1	2	yes	—
226	C₂×C₄	#131×C₁₀₉	1	1	yes	characteristic
227	C₂×C₄	#131×C₁₀₉	1	1	yes	characteristic
228	C₈	—	2	4	no	—
229	C₈	—	4	2	no	—
230	C₈	—	4	2	no	—
231	C₈	—	4	2	no	—
232	C₈	—	4	2	no	—
233	C₈	—	8	1	no	—
234	C₂²	—	2	1	no	—
235	C₂²	—	2	1	no	—
236	C₂²	—	4	4	no	—
237	C₂²	—	4	2	no	—
238	C₂²	—	4	2	no	—
239	C₂²	—	4	2	no	—
240	C₂²	—	8	2	no	—
241	C₂²	#385×C₁₀₉	1	1	yes	characteristic
242	C₂²	#423×C₁₀₉	1	1	yes	characteristic
243	C₂²	#424×C₁₀₉	1	1	yes	characteristic
244	C₄	—	2	1	no	—
245	C₄	—	2	2	no	—
246	C₄	—	4	2	no	—
247	C₄	#435×C₁₀₉	1	2	yes	—
248	C₄	#436×C₁₀₉	1	2	yes	—
249	C₄	#453×C₁₀₉	1	1	yes	characteristic
250	C₄	#455×C₁₀₉	1	1	yes	characteristic
251	C₂	—	2	1	no	—
252	C₂	—	2	1	no	—
253	C₂	—	8	2	no	—
254	C₂	C₁₀₉×#1851	1	1	yes	characteristic
255	C₂	C₁₀₉×#1856	1	1	yes	characteristic
256	C₂	C₁₀₉×#1858	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₀₉	#120
commutator	C₂²×C₄	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₉
socle	C₂²×C₁₀₉	#385
prime radical	C₂²×D₄×C₁₀₉	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₀₉×#1565	C₂	semidirect
5. C₁₀₉×#1278	C₂	semidirect
62. M₁₆×C₁₀₉	M₁₆	semidirect
122. #8837	C₁₀₉	direct (N × H)
132. #1565	C₂×C₁₀₉	semidirect
133. #1278	C₂×C₁₀₉	semidirect
134. C₁₀₉	#8837	direct (N × H)
191. M₁₆	M₁₆×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₁₀₉×#2975	2
C₁₀₉×#1954	2
C₄×#100×C₁₀₉	2
C₁₀₉×#1565	2
C₁₀₉×#1278	2
#8837	109

Subgroups — C109×#8837 #42126831

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#8837 #42126831