Subgroups — C₁₀₉×#20570 #42138564

Order 27904.

878 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#446×C₁₀₉	C₂	1	2	yes	maximal
2	C₄²×Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#1771	C₂	1	4	yes	maximal
4	C₁₀₉×#1770	C₂	1	2	yes	maximal
5	C₁₀₉×#1767	C₂	1	2	yes	maximal
6	C₄×C₄⋊C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
7	C₁₀₉×#1745	C₂	1	2	yes	maximal
8	C₁₀₉×#1744	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Q₈×C₁₀₉	C₄	1	2	yes	—
10	C₂×C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
11	#446×C₁₀₉	C₄	1	16	yes	complemented
12	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
13	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	4	yes	—
14	C₂×#97×C₁₀₉	C₂²	1	4	yes	—
15	C₂×#97×C₁₀₉	C₂²	1	4	yes	—
16	C₂×C₄×C₈×C₁₀₉	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₁₀₉	C₂²	1	2	yes	—
18	C₄×C₄⋊C₄×C₁₀₉	C₄	1	4	yes	—
19	C₄×C₄⋊C₄×C₁₀₉	C₄	1	2	yes	—
20	C₄×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
21	C₄×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
22	C₄³×C₁₀₉	C₂²	1	1	yes	characteristic
23	C₄³×C₁₀₉	C₂²	1	2	yes	—
24	#336×C₁₀₉	C₂²	1	1	yes	characteristic
25	#336×C₁₀₉	C₂²	1	1	yes	characteristic
26	#336×C₁₀₉	C₂²	1	2	yes	—
27	#336×C₁₀₉	C₂²	1	2	yes	—
28	#336×C₁₀₉	C₂²	1	4	yes	—
29	C₂²×Q₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₀₉	C₂³	1	2	yes	—
31	C₂²×C₈×C₁₀₉	C₂³	1	4	yes	—
32	C₂²×C₈×C₁₀₉	C₂³	1	2	yes	—
33	C₄×Q₈×C₁₀₉	—	2	8	no	—
34	C₄×Q₈×C₁₀₉	C₂×C₄	1	8	yes	—
35	C₂×C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
36	C₂×C₄⋊C₄×C₁₀₉	C₂×C₄	1	4	yes	—
37	C₂×C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
38	C₂×C₄⋊C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₀₉	C₂×C₄	1	4	yes	—
40	C₂×C₄²×C₁₀₉	C₂×C₄	1	4	yes	—
41	C₂×C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₁₀₉	C₂³	1	2	yes	—
46	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₁₀₉	C₂³	1	2	yes	—
48	C₄⋊C₈×C₁₀₉	C₂×C₄	1	8	yes	—
49	C₄⋊C₈×C₁₀₉	C₂×C₄	1	16	yes	—
50	#97×C₁₀₉	—	4	4	no	—
51	#97×C₁₀₉	C₂×C₄	1	16	yes	—
52	C₄×C₈×C₁₀₉	—	4	2	no	—
53	C₄×C₈×C₁₀₉	C₂×C₄	1	8	yes	—
54	#95×C₁₀₉	C₂×C₄	1	2	yes	—
55	#95×C₁₀₉	C₂×C₄	1	4	yes	—
56	C₂×Q₈×C₁₀₉	—	2	4	no	—
57	C₂×Q₈×C₁₀₉	C₄²	1	4	yes	—
58	C₂²×C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
59	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
66	C₂×C₈×C₁₀₉	—	2	4	no	—
67	C₂×C₈×C₁₀₉	—	2	2	no	—
68	C₂×C₈×C₁₀₉	—	2	2	no	—
69	C₂×C₈×C₁₀₉	—	2	2	no	—
70	C₂×C₈×C₁₀₉	—	2	4	no	—
71	C₂×C₈×C₁₀₉	—	2	2	no	—
72	C₂×C₈×C₁₀₉	C₂²×C₄	1	4	yes	—
73	C₂×C₈×C₁₀₉	C₂²×C₄	1	4	yes	—
74	C₂×C₈×C₁₀₉	C₂²×C₄	1	8	yes	—
75	C₄⋊C₄×C₁₀₉	—	2	8	no	—
76	C₄⋊C₄×C₁₀₉	—	2	4	no	—
77	C₄⋊C₄×C₁₀₉	C₄²	1	8	yes	—
78	C₄⋊C₄×C₁₀₉	C₄²	1	4	yes	—
79	C₄²×C₁₀₉	—	2	8	no	—
80	C₄²×C₁₀₉	—	2	1	no	—
81	C₄²×C₁₀₉	—	2	2	no	—
82	C₄²×C₁₀₉	—	2	2	no	—
83	C₄²×C₁₀₉	—	2	1	no	—
84	C₄²×C₁₀₉	—	2	2	no	—
85	C₄²×C₁₀₉	—	4	2	no	—
86	C₄²×C₁₀₉	—	4	4	no	—
87	C₄²×C₁₀₉	M₁₆	1	8	yes	—
88	C₄²×C₁₀₉	C₂²×C₄	1	4	yes	—
89	C₄²×C₁₀₉	C₂²×C₄	1	8	yes	—
90	C₂³×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
91	Q₈×C₁₀₉	—	2	8	no	—
92	C₂×C₄×C₁₀₉	—	2	2	no	—
93	C₂×C₄×C₁₀₉	—	2	2	no	—
94	C₂×C₄×C₁₀₉	—	2	4	no	—
95	C₂×C₄×C₁₀₉	—	2	2	no	—
96	C₂×C₄×C₁₀₉	—	2	2	no	—
97	C₂×C₄×C₁₀₉	—	2	4	no	—
98	C₂×C₄×C₁₀₉	—	2	2	no	—
99	C₂×C₄×C₁₀₉	—	2	4	no	—
100	C₂×C₄×C₁₀₉	—	2	4	no	—
101	C₂×C₄×C₁₀₉	—	2	4	no	—
102	C₂×C₄×C₁₀₉	C₂×C₄²	1	2	yes	—
103	C₂×C₄×C₁₀₉	C₂×C₄²	1	4	yes	—
104	C₂×C₄×C₁₀₉	C₂×C₄²	1	4	yes	—
105	C₂×C₄×C₁₀₉	C₂×C₄²	1	2	yes	—
106	C₂×C₄×C₁₀₉	C₂×M₁₆	1	4	yes	—
107	C₂×C₄×C₁₀₉	C₂×M₁₆	1	4	yes	—
108	C₂×C₄×C₁₀₉	C₂×M₁₆	1	4	yes	—
109	C₂×C₄×C₁₀₉	C₂³×C₄	1	2	yes	—
110	C₂×C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
111	C₂×C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
112	C₂×C₄×C₁₀₉	2₋¹⁺⁴	1	2	yes	—
113	C₈×C₁₀₉	—	2	4	no	—
114	C₈×C₁₀₉	—	2	8	no	—
115	C₈×C₁₀₉	—	2	4	no	—
116	C₂²×C₁₀₉	C₂²×C₄²	1	1	yes	characteristic
117	C₂²×C₁₀₉	#519	1	1	yes	characteristic
118	C₂²×C₁₀₉	#520	1	1	yes	characteristic
119	C₂²×C₁₀₉	C₂²×M₁₆	1	2	yes	—
120	C₂²×C₁₀₉	#568	1	2	yes	—
121	C₄×C₁₀₉	—	2	4	no	—
122	C₄×C₁₀₉	—	2	2	no	—
123	C₄×C₁₀₉	—	2	4	no	—
124	C₄×C₁₀₉	—	2	4	no	—
125	C₄×C₁₀₉	—	2	4	no	—
126	C₄×C₁₀₉	—	4	4	no	—
127	C₄×C₁₀₉	C₄×M₁₆	1	4	yes	—
128	C₄×C₁₀₉	#519	1	2	yes	—
129	C₄×C₁₀₉	#520	1	2	yes	—
130	#20570	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
131	C₂×C₁₀₉	#2276	1	1	yes	characteristic
132	C₂×C₁₀₉	C₂×C₄×M₁₆	1	1	yes	characteristic
133	C₂×C₁₀₉	#2875	1	1	yes	characteristic
134	C₂×C₁₀₉	#2983	1	2	yes	—
135	C₂×C₁₀₉	#2989	1	2	yes	—
136	C₂×#446	C₂×C₁₀₉	1	2	yes	—
137	C₄²×Q₈	C₂×C₁₀₉	1	1	yes	characteristic
138	#1771	C₂×C₁₀₉	1	4	yes	—
139	#1770	C₂×C₁₀₉	1	2	yes	—
140	#1767	C₂×C₁₀₉	1	2	yes	—
141	C₄×C₄⋊C₈	C₂×C₁₀₉	1	1	yes	characteristic
142	#1745	C₂×C₁₀₉	1	2	yes	—
143	#1744	C₂×C₁₀₉	1	1	yes	characteristic
144	C₁₀₉	#20570	1	1	yes	characteristic, direct product factor, Sylow
145	C₂×C₄×Q₈	C₄×C₁₀₉	1	2	yes	—
146	C₂×C₄×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
147	#446	C₄×C₁₀₉	1	16	yes	complemented
148	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
149	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	4	yes	—
150	C₂×#97	C₂²×C₁₀₉	1	4	yes	—
151	C₂×#97	C₂²×C₁₀₉	1	4	yes	—
152	C₂×C₄×C₈	C₂²×C₁₀₉	1	2	yes	—
153	C₂×C₄×C₈	C₂²×C₁₀₉	1	2	yes	—
154	C₄×C₄⋊C₄	C₄×C₁₀₉	1	4	yes	—
155	C₄×C₄⋊C₄	C₄×C₁₀₉	1	2	yes	—
156	C₄×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
157	C₄×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
158	C₄³	C₂²×C₁₀₉	1	1	yes	characteristic
159	C₄³	C₂²×C₁₀₉	1	2	yes	—
160	#336	C₂²×C₁₀₉	1	1	yes	characteristic
161	#336	C₂²×C₁₀₉	1	1	yes	characteristic
162	#336	C₂²×C₁₀₉	1	2	yes	—
163	#336	C₂²×C₁₀₉	1	2	yes	—
164	#336	C₂²×C₁₀₉	1	4	yes	—
165	C₂²×Q₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
166	C₂²×C₈	C₂³×C₁₀₉	1	2	yes	—
167	C₂²×C₈	C₂³×C₁₀₉	1	4	yes	—
168	C₂²×C₈	C₂³×C₁₀₉	1	2	yes	—
169	C₄×Q₈	—	2	8	no	—
170	C₄×Q₈	C₂×C₄×C₁₀₉	1	8	yes	—
171	C₂×C₄⋊C₄	C₂×C₄×C₁₀₉	1	2	yes	—
172	C₂×C₄⋊C₄	C₂×C₄×C₁₀₉	1	4	yes	—
173	C₂×C₄⋊C₄	C₂×C₄×C₁₀₉	1	2	yes	—
174	C₂×C₄⋊C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
175	C₂×C₄²	C₂×C₄×C₁₀₉	1	4	yes	—
176	C₂×C₄²	C₂×C₄×C₁₀₉	1	4	yes	—
177	C₂×C₄²	C₂×C₄×C₁₀₉	1	2	yes	—
178	C₂×C₄²	C₂×C₄×C₁₀₉	1	2	yes	—
179	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
180	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
181	C₂×C₄²	C₂³×C₁₀₉	1	2	yes	—
182	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
183	C₂×C₄²	C₂³×C₁₀₉	1	2	yes	—
184	C₄⋊C₈	C₂×C₄×C₁₀₉	1	8	yes	—
185	C₄⋊C₈	C₂×C₄×C₁₀₉	1	16	yes	—
186	#97	—	4	4	no	—
187	#97	C₂×C₄×C₁₀₉	1	16	yes	—
188	C₄×C₈	—	4	2	no	—
189	C₄×C₈	C₂×C₄×C₁₀₉	1	8	yes	—
190	#95	C₂×C₄×C₁₀₉	1	2	yes	—
191	#95	C₂×C₄×C₁₀₉	1	4	yes	—
192	C₂×Q₈	—	2	4	no	—
193	C₂×Q₈	C₄²×C₁₀₉	1	4	yes	—
194	C₂²×C₄	C₂²×C₄×C₁₀₉	1	4	yes	—
195	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
196	C₂²×C₄	C₂²×C₄×C₁₀₉	1	2	yes	—
197	C₂²×C₄	C₂²×C₄×C₁₀₉	1	2	yes	—
198	C₂²×C₄	C₂²×C₄×C₁₀₉	1	2	yes	—
199	C₂²×C₄	C₂²×C₄×C₁₀₉	1	2	yes	—
200	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
201	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
202	C₂×C₈	—	2	4	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	C₂²×C₄×C₁₀₉	1	4	yes	—
209	C₂×C₈	C₂²×C₄×C₁₀₉	1	4	yes	—
210	C₂×C₈	C₂²×C₄×C₁₀₉	1	8	yes	—
211	C₄⋊C₄	—	2	8	no	—
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	C₄²×C₁₀₉	1	8	yes	—
214	C₄⋊C₄	C₄²×C₁₀₉	1	4	yes	—
215	C₄²	—	2	8	no	—
216	C₄²	—	2	1	no	—
217	C₄²	—	2	2	no	—
218	C₄²	—	2	2	no	—
219	C₄²	—	2	1	no	—
220	C₄²	—	2	2	no	—
221	C₄²	—	4	2	no	—
222	C₄²	—	4	4	no	—
223	C₄²	M₁₆×C₁₀₉	1	8	yes	—
224	C₄²	C₂²×C₄×C₁₀₉	1	4	yes	—
225	C₄²	C₂²×C₄×C₁₀₉	1	8	yes	—
226	C₂³	C₂³×C₄×C₁₀₉	1	1	yes	characteristic
227	Q₈	—	2	8	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	C₂×C₄²×C₁₀₉	1	2	yes	—
239	C₂×C₄	C₂×C₄²×C₁₀₉	1	4	yes	—
240	C₂×C₄	C₂×C₄²×C₁₀₉	1	4	yes	—
241	C₂×C₄	C₂×C₄²×C₁₀₉	1	2	yes	—
242	C₂×C₄	C₂×M₁₆×C₁₀₉	1	4	yes	—
243	C₂×C₄	C₂×M₁₆×C₁₀₉	1	4	yes	—
244	C₂×C₄	C₂×M₁₆×C₁₀₉	1	4	yes	—
245	C₂×C₄	C₂³×C₄×C₁₀₉	1	2	yes	—
246	C₂×C₄	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
247	C₂×C₄	2₋¹⁺⁴×C₁₀₉	1	1	yes	characteristic
248	C₂×C₄	2₋¹⁺⁴×C₁₀₉	1	2	yes	—
249	C₈	—	2	4	no	—
250	C₈	—	2	8	no	—
251	C₈	—	2	4	no	—
252	C₂²	C₂²×C₄²×C₁₀₉	1	1	yes	characteristic
253	C₂²	#519×C₁₀₉	1	1	yes	characteristic
254	C₂²	#520×C₁₀₉	1	1	yes	characteristic
255	C₂²	C₂²×M₁₆×C₁₀₉	1	2	yes	—
256	C₂²	#568×C₁₀₉	1	2	yes	—
257	C₄	—	2	4	no	—
258	C₄	—	2	4	no	—
259	C₄	—	2	4	no	—
260	C₄	—	2	2	no	—
261	C₄	—	2	4	no	—
262	C₄	—	4	4	no	—
263	C₄	C₄×M₁₆×C₁₀₉	1	4	yes	—
264	C₄	#519×C₁₀₉	1	2	yes	—
265	C₄	#520×C₁₀₉	1	2	yes	—
266	C₂	C₁₀₉×#2276	1	1	yes	characteristic
267	C₂	C₂×C₄×M₁₆×C₁₀₉	1	1	yes	characteristic
268	C₂	C₁₀₉×#2875	1	1	yes	characteristic
269	C₂	C₁₀₉×#2983	1	2	yes	—
270	C₂	C₁₀₉×#2989	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₀₉	C₂⁴
commutator	C₂²	C₂²×C₄²×C₁₀₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₀₉
socle, prime radical	C₂³×C₁₀₉	C₂³×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
11. #446×C₁₀₉	C₄	semidirect
130. #20570	C₁₀₉	direct (N × H)
144. C₁₀₉	#20570	direct (N × H)
147. #446	C₄×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#446×C₁₀₉	2
C₄²×Q₈×C₁₀₉	2
C₁₀₉×#1771	2
C₁₀₉×#1770	2
C₁₀₉×#1767	2
C₄×C₄⋊C₈×C₁₀₉	2
C₁₀₉×#1745	2
C₁₀₉×#1744	2
#20570	109

Subgroups — C109×#20570 #42138564

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#20570 #42138564