Subgroups — C₁₀₉×#17815 #42135809

Order 27904.

638 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₉×#2920	C₂	1	2	yes	maximal
2	C₂×#431×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#1572	C₂	1	8	yes	maximal, complemented
4	C₁₀₉×#1567	C₂	1	2	yes	maximal, complemented
5	C₁₀₉×#1566	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
7	#432×C₁₀₉	—	2	2	no	—
8	#432×C₁₀₉	C₂²	1	4	yes	—
9	#431×C₁₀₉	—	2	2	no	—
10	#431×C₁₀₉	C₄	1	4	yes	—
11	#431×C₁₀₉	C₂²	1	4	yes	—
12	#431×C₁₀₉	C₂²	1	4	yes	—
13	#423×C₁₀₉	C₂²	1	4	yes	—
14	C₂×C₄⋊C₈×C₁₀₉	C₄	1	2	yes	—
15	#407×C₁₀₉	—	2	2	no	—
16	C₂×C₂²⋊C₈×C₁₀₉	C₄	1	2	yes	—
17	C₄×M₁₆×C₁₀₉	—	2	2	no	—
18	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
20	#335×C₁₀₉	C₂²	1	8	yes	complemented
21	#334×C₁₀₉	C₂²	1	8	yes	complemented
22	C₂³×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₀₉	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₁₀₉	Q₈	1	1	yes	characteristic
25	C₂×M₁₆×C₁₀₉	—	2	4	no	—
26	C₂²×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₁₀₉	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₀₉	D₄	1	2	yes	—
32	C₂×C₄²×C₁₀₉	Q₈	1	1	yes	characteristic
33	C₂×C₄²×C₁₀₉	Q₈	1	2	yes	—
34	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₁₀₉	C₂³	1	2	yes	—
36	C₄⋊C₈×C₁₀₉	—	2	4	no	—
37	C₄⋊C₈×C₁₀₉	C₂×C₄	1	8	yes	—
38	C₄⋊C₈×C₁₀₉	C₂³	1	8	yes	—
39	C₂²⋊C₈×C₁₀₉	—	2	4	no	—
40	C₂²⋊C₈×C₁₀₉	—	2	4	no	—
41	C₂²⋊C₈×C₁₀₉	C₂×C₄	1	8	yes	—
42	#97×C₁₀₉	—	2	4	no	—
43	C₄×C₈×C₁₀₉	—	2	4	no	—
44	C₄×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
45	C₄×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₁₀₉	C₂³	1	2	yes	—
47	C₄×C₈×C₁₀₉	C₂³	1	2	yes	—
48	C₂⁴×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₀₉	—	2	4	no	—
50	C₂²×C₄×C₁₀₉	—	4	2	no	—
51	C₂²×C₄×C₁₀₉	—	4	2	no	—
52	C₂²×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
57	C₂²×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
63	C₂²×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
64	M₁₆×C₁₀₉	—	4	8	no	—
65	C₂×C₈×C₁₀₉	—	2	8	no	—
66	C₂×C₈×C₁₀₉	—	4	2	no	—
67	C₂×C₈×C₁₀₉	—	4	2	no	—
68	C₂×C₈×C₁₀₉	C₂²×C₄	1	4	yes	—
69	C₂×C₈×C₁₀₉	C₂²×C₄	1	4	yes	—
70	C₄²×C₁₀₉	C₄⋊C₄	1	2	yes	—
71	C₄²×C₁₀₉	C₄⋊C₄	1	2	yes	—
72	C₄²×C₁₀₉	C₄⋊C₄	1	2	yes	—
73	C₄²×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
74	C₄²×C₁₀₉	C₂²×C₄	1	2	yes	—
75	C₄²×C₁₀₉	C₂×D₄	1	2	yes	—
76	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₁₀₉	C₂×Q₈	1	2	yes	—
78	C₄²×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
79	C₄²×C₁₀₉	C₂⁴	1	1	yes	characteristic
80	C₂³×C₁₀₉	—	2	2	no	—
81	C₂³×C₁₀₉	—	4	1	no	—
82	C₂³×C₁₀₉	—	4	1	no	—
83	C₂³×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
84	C₂³×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₀₉	—	2	2	no	—
86	C₂×C₄×C₁₀₉	—	2	2	no	—
87	C₂×C₄×C₁₀₉	—	2	4	no	—
88	C₂×C₄×C₁₀₉	—	2	4	no	—
89	C₂×C₄×C₁₀₉	—	2	4	no	—
90	C₂×C₄×C₁₀₉	—	2	4	no	—
91	C₂×C₄×C₁₀₉	—	2	2	no	—
92	C₂×C₄×C₁₀₉	—	4	1	no	—
93	C₂×C₄×C₁₀₉	—	4	2	no	—
94	C₂×C₄×C₁₀₉	—	4	1	no	—
95	C₂×C₄×C₁₀₉	—	4	2	no	—
96	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
97	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
98	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
99	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₁₀₉	#135	1	4	yes	—
103	C₂×C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
104	C₂×C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
106	C₈×C₁₀₉	—	2	4	no	—
107	C₈×C₁₀₉	—	2	4	no	—
108	C₈×C₁₀₉	—	4	8	no	—
109	C₂²×C₁₀₉	—	2	1	no	—
110	C₂²×C₁₀₉	—	2	1	no	—
111	C₂²×C₁₀₉	—	2	2	no	—
112	C₂²×C₁₀₉	—	2	2	no	—
113	C₂²×C₁₀₉	—	2	2	no	—
114	C₂²×C₁₀₉	—	4	4	no	—
115	C₂²×C₁₀₉	#427	1	2	yes	—
116	C₂²×C₁₀₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
117	C₄×C₁₀₉	—	2	1	no	—
118	C₄×C₁₀₉	—	2	2	no	—
119	C₄×C₁₀₉	—	2	2	no	—
120	C₄×C₁₀₉	—	2	2	no	—
121	C₄×C₁₀₉	—	2	1	no	—
122	C₄×C₁₀₉	#427	1	2	yes	—
123	C₄×C₁₀₉	#430	1	2	yes	—
124	C₄×C₁₀₉	#568	1	2	yes	—
125	C₄×C₁₀₉	C₂×#135	1	2	yes	—
126	#17815	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
127	C₂×C₁₀₉	—	2	2	no	—
128	C₂×C₁₀₉	—	4	2	no	—
129	C₂×C₁₀₉	#2905	1	1	yes	characteristic
130	C₂×C₁₀₉	C₂×#427	1	1	yes	characteristic
131	C₂×C₁₀₉	C₂×#430	1	1	yes	characteristic
132	#2920	C₂×C₁₀₉	1	2	yes	—
133	C₂×#431	C₂×C₁₀₉	1	1	yes	characteristic
134	#1572	C₂×C₁₀₉	1	8	yes	complemented
135	#1567	C₂×C₁₀₉	1	2	yes	complemented
136	#1566	C₂×C₁₀₉	1	2	yes	complemented
137	C₁₀₉	#17815	1	1	yes	characteristic, direct product factor, Sylow
138	C₂²×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
139	#432	—	2	2	no	—
140	#432	C₂²×C₁₀₉	1	4	yes	—
141	#431	—	2	2	no	—
142	#431	C₄×C₁₀₉	1	4	yes	—
143	#431	C₂²×C₁₀₉	1	4	yes	—
144	#431	C₂²×C₁₀₉	1	4	yes	—
145	#423	C₂²×C₁₀₉	1	4	yes	—
146	C₂×C₄⋊C₈	C₄×C₁₀₉	1	2	yes	—
147	#407	—	2	2	no	—
148	C₂×C₂²⋊C₈	C₄×C₁₀₉	1	2	yes	—
149	C₄×M₁₆	—	2	2	no	—
150	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
151	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
152	#335	C₂²×C₁₀₉	1	8	yes	complemented
153	#334	C₂²×C₁₀₉	1	8	yes	complemented
154	C₂³×C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
155	C₂³×C₄	D₄×C₁₀₉	1	1	yes	characteristic
156	C₂³×C₄	Q₈×C₁₀₉	1	1	yes	characteristic
157	C₂×M₁₆	—	2	4	no	—
158	C₂²×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
159	C₂²×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
160	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
161	C₂×C₄²	C₂×C₄×C₁₀₉	1	2	yes	—
162	C₂×C₄²	D₄×C₁₀₉	1	1	yes	characteristic
163	C₂×C₄²	D₄×C₁₀₉	1	2	yes	—
164	C₂×C₄²	Q₈×C₁₀₉	1	1	yes	characteristic
165	C₂×C₄²	Q₈×C₁₀₉	1	2	yes	—
166	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
167	C₂×C₄²	C₂³×C₁₀₉	1	2	yes	—
168	C₄⋊C₈	—	2	4	no	—
169	C₄⋊C₈	C₂×C₄×C₁₀₉	1	8	yes	—
170	C₄⋊C₈	C₂³×C₁₀₉	1	8	yes	—
171	C₂²⋊C₈	—	2	4	no	—
172	C₂²⋊C₈	—	2	4	no	—
173	C₂²⋊C₈	C₂×C₄×C₁₀₉	1	8	yes	—
174	#97	—	2	4	no	—
175	C₄×C₈	—	2	4	no	—
176	C₄×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
177	C₄×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
178	C₄×C₈	C₂³×C₁₀₉	1	2	yes	—
179	C₄×C₈	C₂³×C₁₀₉	1	2	yes	—
180	C₂⁴	C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
181	C₂²×C₄	—	2	4	no	—
182	C₂²×C₄	—	4	2	no	—
183	C₂²×C₄	—	4	2	no	—
184	C₂²×C₄	C₄⋊C₄×C₁₀₉	1	2	yes	—
185	C₂²×C₄	C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
186	C₂²×C₄	C₄⋊C₄×C₁₀₉	1	2	yes	—
187	C₂²×C₄	C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
188	C₂²×C₄	C₄⋊C₄×C₁₀₉	1	2	yes	—
189	C₂²×C₄	C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
190	C₂²×C₄	C₂²×C₄×C₁₀₉	1	2	yes	—
191	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
192	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
193	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
194	C₂²×C₄	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
195	C₂²×C₄	C₂×Q₈×C₁₀₉	1	2	yes	—
196	M₁₆	—	4	8	no	—
197	C₂×C₈	—	2	8	no	—
198	C₂×C₈	—	4	2	no	—
199	C₂×C₈	—	4	2	no	—
200	C₂×C₈	C₂²×C₄×C₁₀₉	1	4	yes	—
201	C₂×C₈	C₂²×C₄×C₁₀₉	1	4	yes	—
202	C₄²	C₄⋊C₄×C₁₀₉	1	2	yes	—
203	C₄²	C₄⋊C₄×C₁₀₉	1	2	yes	—
204	C₄²	C₄⋊C₄×C₁₀₉	1	2	yes	—
205	C₄²	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
206	C₄²	C₂²×C₄×C₁₀₉	1	2	yes	—
207	C₄²	C₂×D₄×C₁₀₉	1	2	yes	—
208	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
209	C₄²	C₂×Q₈×C₁₀₉	1	2	yes	—
210	C₄²	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
211	C₄²	C₂⁴×C₁₀₉	1	1	yes	characteristic
212	C₂³	—	2	2	no	—
213	C₂³	—	4	1	no	—
214	C₂³	—	4	1	no	—
215	C₂³	C₂×C₄⋊C₄×C₁₀₉	1	2	yes	—
216	C₂³	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	4	1	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	2	yes	—
229	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	2	yes	—
230	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	2	yes	—
231	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
232	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
233	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
234	C₂×C₄	#135×C₁₀₉	1	4	yes	—
235	C₂×C₄	C₂³×C₄×C₁₀₉	1	1	yes	characteristic
236	C₂×C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
237	C₂×C₄	C₂²×Q₈×C₁₀₉	1	1	yes	characteristic
238	C₈	—	2	4	no	—
239	C₈	—	2	4	no	—
240	C₈	—	4	8	no	—
241	C₂²	—	2	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	4	4	no	—
247	C₂²	#427×C₁₀₉	1	2	yes	—
248	C₂²	C₂²×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
249	C₄	—	2	1	no	—
250	C₄	—	2	2	no	—
251	C₄	—	2	2	no	—
252	C₄	—	2	2	no	—
253	C₄	—	2	1	no	—
254	C₄	#427×C₁₀₉	1	2	yes	—
255	C₄	#430×C₁₀₉	1	2	yes	—
256	C₄	#568×C₁₀₉	1	2	yes	—
257	C₄	C₂×#135×C₁₀₉	1	2	yes	—
258	C₂	—	2	2	no	—
259	C₂	—	4	2	no	—
260	C₂	C₁₀₉×#2905	1	1	yes	characteristic
261	C₂	C₂×#427×C₁₀₉	1	1	yes	characteristic
262	C₂	C₂×#430×C₁₀₉	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₁₀₉×#1572	C₂	semidirect
4. C₁₀₉×#1567	C₂	semidirect
5. C₁₀₉×#1566	C₂	semidirect
20. #335×C₁₀₉	C₂²	semidirect
21. #334×C₁₀₉	C₂²	semidirect
126. #17815	C₁₀₉	direct (N × H)
134. #1572	C₂×C₁₀₉	semidirect
135. #1567	C₂×C₁₀₉	semidirect
136. #1566	C₂×C₁₀₉	semidirect
137. C₁₀₉	#17815	direct (N × H)
152. #335	C₂²×C₁₀₉	semidirect
153. #334	C₂²×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₁₀₉×#2920	2
C₂×#431×C₁₀₉	2
C₁₀₉×#1572	2
C₁₀₉×#1567	2
C₁₀₉×#1566	2
#17815	109

Subgroups — C109×#17815 #42135809

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#17815 #42135809