Subgroups — C₁₀₉×#20348 #42138342

Order 27904.

2158 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#131×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#520×C₁₀₉	C₂	1	2	yes	maximal
3	C₁₀₉×#1736	C₂	1	6	yes	maximal, complemented
4	C₁₀₉×#1735	C₂	1	3	yes	maximal, complemented
5	C₂×#340×C₁₀₉	C₂	1	3	yes	maximal, complemented
6	C₂²×#41×C₁₀₉	C₂²	1	1	yes	characteristic
7	C₂×#131×C₁₀₉	—	2	8	no	—
8	C₂×#131×C₁₀₉	C₄	1	8	yes	complemented
9	C₂²×M₁₆×C₁₀₉	C₂²	1	3	yes	—
10	C₂³×C₈×C₁₀₉	C₂²	1	3	yes	—
11	#520×C₁₀₉	—	2	8	no	—
12	#520×C₁₀₉	C₄	1	16	yes	complemented
13	C₂×C₄×Q₈×C₁₀₉	—	2	2	no	—
14	C₂×C₄×D₄×C₁₀₉	—	2	6	no	—
15	C₂×#117×C₁₀₉	C₂²	1	6	yes	—
16	C₂²×C₄⋊C₄×C₁₀₉	C₂²	1	6	yes	—
17	#340×C₁₀₉	C₂²	1	3	yes	—
18	#340×C₁₀₉	C₂²	1	6	yes	—
19	#340×C₁₀₉	C₂²	1	1	yes	characteristic
20	#340×C₁₀₉	C₂²	1	6	yes	—
21	C₂×#41×C₁₀₉	C₂×C₄	1	8	yes	—
22	C₂×#41×C₁₀₉	C₂×C₄	1	4	yes	—
23	C₂×#41×C₁₀₉	D₄	1	8	yes	—
24	C₂×#41×C₁₀₉	Q₈	1	4	yes	—
25	C₂²×Q₈×C₁₀₉	D₄	1	1	yes	characteristic
26	C₂²×D₄×C₁₀₉	D₄	1	3	yes	—
27	C₂³×C₄×C₁₀₉	—	2	6	no	—
28	C₂³×C₄×C₁₀₉	C₂³	1	3	yes	—
29	#131×C₁₀₉	—	2	32	no	—
30	C₂×M₁₆×C₁₀₉	—	2	12	no	—
31	C₂×M₁₆×C₁₀₉	C₂×C₄	1	12	yes	complemented
32	C₂²×C₈×C₁₀₉	—	2	12	no	—
33	C₂²×C₈×C₁₀₉	C₂×C₄	1	12	yes	complemented
34	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₁₀₉	C₂³	1	3	yes	—
36	C₄×Q₈×C₁₀₉	—	2	16	no	—
37	C₄×D₄×C₁₀₉	—	2	48	no	—
38	#117×C₁₀₉	—	2	12	no	—
39	#117×C₁₀₉	C₂×C₄	1	24	yes	complemented
40	C₂×C₄⋊C₄×C₁₀₉	—	2	6	no	—
41	C₂×C₄⋊C₄×C₁₀₉	—	2	12	no	—
42	C₂×C₄⋊C₄×C₁₀₉	C₂×C₄	1	24	yes	complemented
43	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	6	yes	—
44	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
45	C₂×C₂²⋊C₄×C₁₀₉	—	2	6	no	—
46	C₂×C₄²×C₁₀₉	—	2	6	no	—
47	C₂⁴×C₁₀₉	C₂×D₄	1	3	yes	—
48	#41×C₁₀₉	C₄²	1	16	yes	—
49	#41×C₁₀₉	C₂²⋊C₄	1	16	yes	—
50	#41×C₁₀₉	C₄⋊C₄	1	32	yes	—
51	C₂×Q₈×C₁₀₉	C₂²⋊C₄	1	8	yes	—
52	C₂×Q₈×C₁₀₉	C₄⋊C₄	1	4	yes	—
53	C₂×D₄×C₁₀₉	C₂²⋊C₄	1	24	yes	—
54	C₂×D₄×C₁₀₉	C₄⋊C₄	1	12	yes	—
55	C₂²×C₄×C₁₀₉	—	2	2	no	—
56	C₂²×C₄×C₁₀₉	—	2	6	no	—
57	C₂²×C₄×C₁₀₉	—	2	24	no	—
58	C₂²×C₄×C₁₀₉	—	8	6	no	—
59	C₂²×C₄×C₁₀₉	—	8	6	no	—
60	C₂²×C₄×C₁₀₉	C₂²×C₄	1	6	yes	—
61	C₂²×C₄×C₁₀₉	C₂²×C₄	1	12	yes	—
62	C₂²×C₄×C₁₀₉	C₂×D₄	1	3	yes	—
63	C₂²×C₄×C₁₀₉	C₂×D₄	1	12	yes	—
64	C₂²×C₄×C₁₀₉	C₂×Q₈	1	6	yes	—
65	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
66	M₁₆×C₁₀₉	—	2	24	no	—
67	C₂×C₈×C₁₀₉	—	2	24	no	—
68	C₂×C₈×C₁₀₉	—	2	2	no	—
69	C₂×C₈×C₁₀₉	—	2	6	no	—
70	C₂×C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₁₀₉	C₂²×C₄	1	6	yes	—
72	C₄⋊C₄×C₁₀₉	—	2	24	no	—
73	C₄⋊C₄×C₁₀₉	—	2	6	no	—
74	C₄⋊C₄×C₁₀₉	—	2	2	no	—
75	C₄⋊C₄×C₁₀₉	C₂²×C₄	1	12	yes	—
76	C₄⋊C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
77	C₂²⋊C₄×C₁₀₉	—	2	24	no	—
78	C₄²×C₁₀₉	—	2	24	no	—
79	C₂³×C₁₀₉	—	2	3	no	—
80	C₂³×C₁₀₉	—	2	6	no	—
81	C₂³×C₁₀₉	—	2	3	no	—
82	C₂³×C₁₀₉	C₂×C₂²⋊C₄	1	12	yes	—
83	C₂³×C₁₀₉	C₂×C₄⋊C₄	1	6	yes	—
84	C₂³×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
85	Q₈×C₁₀₉	#95	1	16	yes	—
86	D₄×C₁₀₉	#95	1	48	yes	—
87	C₂×C₄×C₁₀₉	—	2	4	no	—
88	C₂×C₄×C₁₀₉	—	2	12	no	—
89	C₂×C₄×C₁₀₉	—	4	6	no	—
90	C₂×C₄×C₁₀₉	—	4	2	no	—
91	C₂×C₄×C₁₀₉	—	4	6	no	—
92	C₂×C₄×C₁₀₉	—	4	2	no	—
93	C₂×C₄×C₁₀₉	—	8	24	no	—
94	C₂×C₄×C₁₀₉	C₂×C₄²	1	12	yes	—
95	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	12	yes	—
96	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	12	yes	—
97	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	24	yes	—
98	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	6	yes	—
99	C₂×C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
100	C₂×C₄×C₁₀₉	C₂³×C₄	1	2	yes	—
101	C₂×C₄×C₁₀₉	C₂²×D₄	1	2	yes	—
102	C₂×C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
103	C₈×C₁₀₉	—	2	2	no	—
104	C₈×C₁₀₉	—	2	6	no	—
105	C₂²×C₁₀₉	—	2	12	no	—
106	C₂²×C₁₀₉	—	2	12	no	—
107	C₂²×C₁₀₉	—	2	6	no	—
108	C₂²×C₁₀₉	—	2	6	no	—
109	C₂²×C₁₀₉	C₂×#95	1	12	yes	—
110	C₂²×C₁₀₉	C₂²×C₂²⋊C₄	1	2	yes	—
111	C₂²×C₁₀₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
112	C₂²×C₁₀₉	#576	1	1	yes	characteristic
113	C₂²×C₁₀₉	#577	1	2	yes	—
114	C₂²×C₁₀₉	#578	1	1	yes	characteristic
115	C₄×C₁₀₉	—	4	12	no	—
116	C₄×C₁₀₉	—	4	4	no	—
117	C₄×C₁₀₉	C₂×#95	1	12	yes	—
118	C₄×C₁₀₉	C₂²×C₄²	1	1	yes	characteristic
119	C₄×C₁₀₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
120	C₄×C₁₀₉	C₂²×C₄⋊C₄	1	2	yes	—
121	#20348	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
122	C₂×C₁₀₉	—	2	12	no	—
123	C₂×C₁₀₉	C₂²×#95	1	1	yes	characteristic
124	C₂×C₁₀₉	#2897	1	2	yes	—
125	C₂×C₁₀₉	#2898	1	2	yes	—
126	C₂×C₁₀₉	#2912	1	1	yes	characteristic
127	C₂×C₁₀₉	#2913	1	1	yes	characteristic
128	C₂²×#131	C₂×C₁₀₉	1	1	yes	characteristic
129	C₂×#520	C₂×C₁₀₉	1	2	yes	—
130	#1736	C₂×C₁₀₉	1	6	yes	complemented
131	#1735	C₂×C₁₀₉	1	3	yes	complemented
132	C₂×#340	C₂×C₁₀₉	1	3	yes	complemented
133	C₁₀₉	#20348	1	1	yes	characteristic, direct product factor, Sylow
134	C₂²×#41	C₂²×C₁₀₉	1	1	yes	characteristic
135	C₂×#131	—	2	8	no	—
136	C₂×#131	C₄×C₁₀₉	1	8	yes	complemented
137	C₂²×M₁₆	C₂²×C₁₀₉	1	3	yes	—
138	C₂³×C₈	C₂²×C₁₀₉	1	3	yes	—
139	#520	—	2	8	no	—
140	#520	C₄×C₁₀₉	1	16	yes	complemented
141	C₂×C₄×Q₈	—	2	2	no	—
142	C₂×C₄×D₄	—	2	6	no	—
143	C₂×#117	C₂²×C₁₀₉	1	6	yes	—
144	C₂²×C₄⋊C₄	C₂²×C₁₀₉	1	6	yes	—
145	#340	C₂²×C₁₀₉	1	3	yes	—
146	#340	C₂²×C₁₀₉	1	6	yes	—
147	#340	C₂²×C₁₀₉	1	1	yes	characteristic
148	#340	C₂²×C₁₀₉	1	6	yes	—
149	C₂×#41	C₂×C₄×C₁₀₉	1	8	yes	—
150	C₂×#41	C₂×C₄×C₁₀₉	1	4	yes	—
151	C₂×#41	D₄×C₁₀₉	1	8	yes	—
152	C₂×#41	Q₈×C₁₀₉	1	4	yes	—
153	C₂²×Q₈	D₄×C₁₀₉	1	1	yes	characteristic
154	C₂²×D₄	D₄×C₁₀₉	1	3	yes	—
155	C₂³×C₄	—	2	6	no	—
156	C₂³×C₄	C₂³×C₁₀₉	1	3	yes	—
157	#131	—	2	32	no	—
158	C₂×M₁₆	—	2	12	no	—
159	C₂×M₁₆	C₂×C₄×C₁₀₉	1	12	yes	complemented
160	C₂²×C₈	—	2	12	no	—
161	C₂²×C₈	C₂×C₄×C₁₀₉	1	12	yes	complemented
162	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
163	C₂²×C₈	C₂³×C₁₀₉	1	3	yes	—
164	C₄×Q₈	—	2	16	no	—
165	C₄×D₄	—	2	48	no	—
166	#117	—	2	12	no	—
167	#117	C₂×C₄×C₁₀₉	1	24	yes	complemented
168	C₂×C₄⋊C₄	—	2	6	no	—
169	C₂×C₄⋊C₄	—	2	12	no	—
170	C₂×C₄⋊C₄	C₂×C₄×C₁₀₉	1	24	yes	complemented
171	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	6	yes	—
172	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
173	C₂×C₂²⋊C₄	—	2	6	no	—
174	C₂×C₄²	—	2	6	no	—
175	C₂⁴	C₂×D₄×C₁₀₉	1	3	yes	—
176	#41	C₄²×C₁₀₉	1	16	yes	—
177	#41	C₂²⋊C₄×C₁₀₉	1	16	yes	—
178	#41	C₄⋊C₄×C₁₀₉	1	32	yes	—
179	C₂×Q₈	C₂²⋊C₄×C₁₀₉	1	8	yes	—
180	C₂×Q₈	C₄⋊C₄×C₁₀₉	1	4	yes	—
181	C₂×D₄	C₂²⋊C₄×C₁₀₉	1	24	yes	—
182	C₂×D₄	C₄⋊C₄×C₁₀₉	1	12	yes	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	—	2	6	no	—
185	C₂²×C₄	—	2	24	no	—
186	C₂²×C₄	—	8	6	no	—
187	C₂²×C₄	—	8	6	no	—
188	C₂²×C₄	C₂²×C₄×C₁₀₉	1	6	yes	—
189	C₂²×C₄	C₂²×C₄×C₁₀₉	1	12	yes	—
190	C₂²×C₄	C₂×D₄×C₁₀₉	1	3	yes	—
191	C₂²×C₄	C₂×D₄×C₁₀₉	1	12	yes	—
192	C₂²×C₄	C₂×Q₈×C₁₀₉	1	6	yes	—
193	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
194	M₁₆	—	2	24	no	—
195	C₂×C₈	—	2	24	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	2	6	no	—
198	C₂×C₈	C₂²×C₄×C₁₀₉	1	2	yes	—
199	C₂×C₈	C₂²×C₄×C₁₀₉	1	6	yes	—
200	C₄⋊C₄	—	2	24	no	—
201	C₄⋊C₄	—	2	6	no	—
202	C₄⋊C₄	—	2	2	no	—
203	C₄⋊C₄	C₂²×C₄×C₁₀₉	1	12	yes	—
204	C₄⋊C₄	C₂²×C₄×C₁₀₉	1	4	yes	—
205	C₂²⋊C₄	—	2	24	no	—
206	C₄²	—	2	24	no	—
207	C₂³	—	2	3	no	—
208	C₂³	—	2	6	no	—
209	C₂³	—	2	3	no	—
210	C₂³	C₂×C₂²⋊C₄×C₁₀₉	1	12	yes	—
211	C₂³	C₂×C₄⋊C₄×C₁₀₉	1	6	yes	—
212	C₂³	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
213	Q₈	#95×C₁₀₉	1	16	yes	—
214	D₄	#95×C₁₀₉	1	48	yes	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	12	no	—
217	C₂×C₄	—	4	6	no	—
218	C₂×C₄	—	4	6	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	—	8	24	no	—
222	C₂×C₄	C₂×C₄²×C₁₀₉	1	12	yes	—
223	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	12	yes	—
224	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	12	yes	—
225	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	24	yes	—
226	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	6	yes	—
227	C₂×C₄	C₂³×C₄×C₁₀₉	1	1	yes	characteristic
228	C₂×C₄	C₂³×C₄×C₁₀₉	1	2	yes	—
229	C₂×C₄	C₂²×D₄×C₁₀₉	1	2	yes	—
230	C₂×C₄	C₂²×Q₈×C₁₀₉	1	1	yes	characteristic
231	C₈	—	2	2	no	—
232	C₈	—	2	6	no	—
233	C₂²	—	2	12	no	—
234	C₂²	—	2	12	no	—
235	C₂²	—	2	6	no	—
236	C₂²	—	2	6	no	—
237	C₂²	C₂×#95×C₁₀₉	1	12	yes	—
238	C₂²	C₂²×C₂²⋊C₄×C₁₀₉	1	2	yes	—
239	C₂²	C₂²×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
240	C₂²	#576×C₁₀₉	1	1	yes	characteristic
241	C₂²	#577×C₁₀₉	1	2	yes	—
242	C₂²	#578×C₁₀₉	1	1	yes	characteristic
243	C₄	—	4	12	no	—
244	C₄	—	4	4	no	—
245	C₄	C₂×#95×C₁₀₉	1	12	yes	—
246	C₄	C₂²×C₄²×C₁₀₉	1	1	yes	characteristic
247	C₄	C₂²×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
248	C₄	C₂²×C₄⋊C₄×C₁₀₉	1	2	yes	—
249	C₂	—	2	12	no	—
250	C₂	C₂²×#95×C₁₀₉	1	1	yes	characteristic
251	C₂	C₁₀₉×#2897	1	2	yes	—
252	C₂	C₁₀₉×#2898	1	2	yes	—
253	C₂	C₁₀₉×#2912	1	1	yes	characteristic
254	C₂	C₁₀₉×#2913	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₀₉	C₂²×D₄
commutator	C₄	C₂²×C₄²×C₁₀₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₀₉
prime radical	C₂²×#41×C₁₀₉	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₉ ◁ C₂²×#41×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₀₉×#1736	C₂	semidirect
4. C₁₀₉×#1735	C₂	semidirect
5. C₂×#340×C₁₀₉	C₂	semidirect
8. C₂×#131×C₁₀₉	C₄	semidirect
12. #520×C₁₀₉	C₄	semidirect
31. C₂×M₁₆×C₁₀₉	C₂×C₄	semidirect
33. C₂²×C₈×C₁₀₉	C₂×C₄	semidirect
39. #117×C₁₀₉	C₂×C₄	semidirect
42. C₂×C₄⋊C₄×C₁₀₉	C₂×C₄	semidirect
121. #20348	C₁₀₉	direct (N × H)
130. #1736	C₂×C₁₀₉	semidirect
131. #1735	C₂×C₁₀₉	semidirect
132. C₂×#340	C₂×C₁₀₉	semidirect
133. C₁₀₉	#20348	direct (N × H)
136. C₂×#131	C₄×C₁₀₉	semidirect
140. #520	C₄×C₁₀₉	semidirect
159. C₂×M₁₆	C₂×C₄×C₁₀₉	semidirect
161. C₂²×C₈	C₂×C₄×C₁₀₉	semidirect
167. #117	C₂×C₄×C₁₀₉	semidirect
170. C₂×C₄⋊C₄	C₂×C₄×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#131×C₁₀₉	2
C₂×#520×C₁₀₉	2
C₁₀₉×#1736	2
C₁₀₉×#1735	2
C₂×#340×C₁₀₉	2
#20348	109

Subgroups — C109×#20348 #42138342

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#20348 #42138342