Subgroups — C₁₀₉×#17325 #42135319

Order 27904.

772 conjugacy classes of subgroups.

The subgroup lattice has 236 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	1	yes	maximal, characteristic
2	C₂×#431×C₁₀₉	C₂	1	2	yes	maximal
3	C₁₀₉×#1453	C₂	1	4	yes	maximal, complemented
4	C₁₀₉×#1452	C₂	1	4	yes	maximal, complemented
5	C₁₀₉×#1450	C₂	1	4	yes	maximal, complemented
6	C₂²×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
7	#432×C₁₀₉	C₄	1	4	yes	—
8	#431×C₁₀₉	C₄	1	4	yes	—
9	#431×C₁₀₉	C₄	1	4	yes	—
10	#431×C₁₀₉	C₂²	1	2	yes	—
11	#431×C₁₀₉	C₂²	1	4	yes	—
12	#431×C₁₀₉	C₂²	1	2	yes	—
13	#431×C₁₀₉	C₂²	1	4	yes	—
14	#423×C₁₀₉	C₄	1	2	yes	—
15	C₂×C₄⋊C₈×C₁₀₉	C₄	1	4	yes	—
16	#407×C₁₀₉	C₄	1	2	yes	—
17	C₂×C₂²⋊C₈×C₁₀₉	C₄	1	4	yes	—
18	C₄×M₁₆×C₁₀₉	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₁₀₉	C₂²	1	4	yes	—
20	#322×C₁₀₉	C₂²	1	8	yes	complemented
21	#322×C₁₀₉	C₂²	1	4	yes	complemented
22	C₈²×C₁₀₉	C₂²	1	4	yes	complemented
23	C₂³×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
24	C₂³×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
25	C₂×M₁₆×C₁₀₉	C₂×C₄	1	4	yes	—
26	C₂²×C₈×C₁₀₉	C₂×C₄	1	8	yes	—
27	C₂×C₄²×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	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₁₀₉	C₂³	1	2	yes	—
33	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₁₀₉	C₂×C₄	1	4	yes	—
35	C₄⋊C₈×C₁₀₉	C₂×C₄	1	4	yes	—
36	C₄⋊C₈×C₁₀₉	C₂×C₄	1	8	yes	—
37	C₄⋊C₈×C₁₀₉	C₂×C₄	1	8	yes	—
38	C₂²⋊C₈×C₁₀₉	C₂×C₄	1	4	yes	—
39	C₂²⋊C₈×C₁₀₉	C₂×C₄	1	4	yes	—
40	C₂²⋊C₈×C₁₀₉	C₂×C₄	1	8	yes	—
41	C₂²⋊C₈×C₁₀₉	C₂×C₄	1	8	yes	—
42	#97×C₁₀₉	C₂×C₄	1	4	yes	—
43	C₄×C₈×C₁₀₉	C₂×C₄	1	8	yes	—
44	C₄×C₈×C₁₀₉	C₂³	1	4	yes	—
45	C₄×C₈×C₁₀₉	C₂³	1	4	yes	—
46	C₄×C₈×C₁₀₉	C₂³	1	4	yes	—
47	C₂⁴×C₁₀₉	C₄²	1	1	yes	characteristic
48	C₂²×C₄×C₁₀₉	—	2	8	no	—
49	C₂²×C₄×C₁₀₉	—	4	2	no	—
50	C₂²×C₄×C₁₀₉	C₄²	1	2	yes	—
51	C₂²×C₄×C₁₀₉	C₄²	1	2	yes	—
52	C₂²×C₄×C₁₀₉	C₄²	1	2	yes	—
53	C₂²×C₄×C₁₀₉	C₄²	1	1	yes	characteristic
54	C₂²×C₄×C₁₀₉	C₄²	1	2	yes	—
55	C₂²×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
60	M₁₆×C₁₀₉	—	4	4	no	—
61	C₂×C₈×C₁₀₉	—	2	16	no	—
62	C₂×C₈×C₁₀₉	C₂²×C₄	1	8	yes	—
63	C₂×C₈×C₁₀₉	C₂²×C₄	1	8	yes	—
64	C₂×C₈×C₁₀₉	C₂²×C₄	1	8	yes	—
65	C₄²×C₁₀₉	C₄²	1	1	yes	characteristic
66	C₄²×C₁₀₉	C₄²	1	2	yes	—
67	C₄²×C₁₀₉	C₄²	1	2	yes	—
68	C₄²×C₁₀₉	C₄²	1	1	yes	characteristic
69	C₄²×C₁₀₉	C₂²×C₄	1	2	yes	—
70	C₄²×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
71	C₄²×C₁₀₉	C₂²×C₄	1	2	yes	—
72	C₄²×C₁₀₉	C₂²×C₄	1	2	yes	—
73	C₄²×C₁₀₉	C₂²×C₄	1	2	yes	—
74	C₄²×C₁₀₉	C₂⁴	1	1	yes	characteristic
75	C₂³×C₁₀₉	—	2	4	no	—
76	C₂³×C₁₀₉	—	4	1	no	—
77	C₂³×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
78	C₂³×C₁₀₉	C₂×C₄²	1	2	yes	—
79	C₂×C₄×C₁₀₉	—	2	4	no	—
80	C₂×C₄×C₁₀₉	—	2	4	no	—
81	C₂×C₄×C₁₀₉	—	2	4	no	—
82	C₂×C₄×C₁₀₉	—	2	2	no	—
83	C₂×C₄×C₁₀₉	—	2	4	no	—
84	C₂×C₄×C₁₀₉	—	2	4	no	—
85	C₂×C₄×C₁₀₉	—	2	4	no	—
86	C₂×C₄×C₁₀₉	—	4	2	no	—
87	C₂×C₄×C₁₀₉	—	4	1	no	—
88	C₂×C₄×C₁₀₉	C₂×C₄²	1	2	yes	—
89	C₂×C₄×C₁₀₉	C₂×C₄²	1	2	yes	—
90	C₂×C₄×C₁₀₉	C₂×C₄²	1	2	yes	—
91	C₂×C₄×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
92	C₂×C₄×C₁₀₉	C₂×C₄²	1	2	yes	—
93	C₂×C₄×C₁₀₉	#131	1	8	yes	—
94	C₂×C₄×C₁₀₉	C₂³×C₄	1	2	yes	—
95	C₂×C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
96	C₈×C₁₀₉	—	2	8	no	—
97	C₈×C₁₀₉	—	2	8	no	—
98	C₈×C₁₀₉	—	2	8	no	—
99	C₂²×C₁₀₉	—	2	2	no	—
100	C₂²×C₁₀₉	—	2	2	no	—
101	C₂²×C₁₀₉	—	2	2	no	—
102	C₂²×C₁₀₉	—	2	1	no	—
103	C₂²×C₁₀₉	—	4	4	no	—
104	C₂²×C₁₀₉	#405	1	4	yes	—
105	C₂²×C₁₀₉	C₂²×C₄²	1	1	yes	characteristic
106	C₄×C₁₀₉	—	2	2	no	—
107	C₄×C₁₀₉	—	2	2	no	—
108	C₄×C₁₀₉	—	2	1	no	—
109	C₄×C₁₀₉	—	2	2	no	—
110	C₄×C₁₀₉	#405	1	4	yes	—
111	C₄×C₁₀₉	C₂×#131	1	4	yes	—
112	C₄×C₁₀₉	#568	1	2	yes	—
113	#17325	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
114	C₂×C₁₀₉	—	2	4	no	—
115	C₂×C₁₀₉	—	4	1	no	—
116	C₂×C₁₀₉	C₂×#405	1	2	yes	—
117	C₂×C₁₀₉	#2873	1	1	yes	characteristic
118	#2920	C₂×C₁₀₉	1	1	yes	characteristic
119	C₂×#431	C₂×C₁₀₉	1	2	yes	—
120	#1453	C₂×C₁₀₉	1	4	yes	complemented
121	#1452	C₂×C₁₀₉	1	4	yes	complemented
122	#1450	C₂×C₁₀₉	1	4	yes	complemented
123	C₁₀₉	#17325	1	1	yes	characteristic, direct product factor, Sylow
124	C₂²×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
125	#432	C₄×C₁₀₉	1	4	yes	—
126	#431	C₄×C₁₀₉	1	4	yes	—
127	#431	C₄×C₁₀₉	1	4	yes	—
128	#431	C₂²×C₁₀₉	1	2	yes	—
129	#431	C₂²×C₁₀₉	1	2	yes	—
130	#431	C₂²×C₁₀₉	1	4	yes	—
131	#431	C₂²×C₁₀₉	1	4	yes	—
132	#423	C₄×C₁₀₉	1	2	yes	—
133	C₂×C₄⋊C₈	C₄×C₁₀₉	1	4	yes	—
134	#407	C₄×C₁₀₉	1	2	yes	—
135	C₂×C₂²⋊C₈	C₄×C₁₀₉	1	4	yes	—
136	C₄×M₁₆	C₂²×C₁₀₉	1	2	yes	—
137	C₂×C₄×C₈	C₂²×C₁₀₉	1	4	yes	—
138	#322	C₂²×C₁₀₉	1	8	yes	complemented
139	#322	C₂²×C₁₀₉	1	4	yes	complemented
140	C₈²	C₂²×C₁₀₉	1	4	yes	complemented
141	C₂³×C₄	C₂×C₄×C₁₀₉	1	2	yes	—
142	C₂³×C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
143	C₂×M₁₆	C₂×C₄×C₁₀₉	1	4	yes	—
144	C₂²×C₈	C₂×C₄×C₁₀₉	1	8	yes	—
145	C₂×C₄²	C₂×C₄×C₁₀₉	1	2	yes	—
146	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
147	C₂×C₄²	C₂×C₄×C₁₀₉	1	2	yes	—
148	C₂×C₄²	C₂×C₄×C₁₀₉	1	2	yes	—
149	C₂×C₄²	C₂×C₄×C₁₀₉	1	2	yes	—
150	C₂×C₄²	C₂³×C₁₀₉	1	2	yes	—
151	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
152	C₄⋊C₈	C₂×C₄×C₁₀₉	1	4	yes	—
153	C₄⋊C₈	C₂×C₄×C₁₀₉	1	4	yes	—
154	C₄⋊C₈	C₂×C₄×C₁₀₉	1	8	yes	—
155	C₄⋊C₈	C₂×C₄×C₁₀₉	1	8	yes	—
156	C₂²⋊C₈	C₂×C₄×C₁₀₉	1	4	yes	—
157	C₂²⋊C₈	C₂×C₄×C₁₀₉	1	4	yes	—
158	C₂²⋊C₈	C₂×C₄×C₁₀₉	1	8	yes	—
159	C₂²⋊C₈	C₂×C₄×C₁₀₉	1	8	yes	—
160	#97	C₂×C₄×C₁₀₉	1	4	yes	—
161	C₄×C₈	C₂×C₄×C₁₀₉	1	8	yes	—
162	C₄×C₈	C₂³×C₁₀₉	1	4	yes	—
163	C₄×C₈	C₂³×C₁₀₉	1	4	yes	—
164	C₄×C₈	C₂³×C₁₀₉	1	4	yes	—
165	C₂⁴	C₄²×C₁₀₉	1	1	yes	characteristic
166	C₂²×C₄	—	2	8	no	—
167	C₂²×C₄	—	4	2	no	—
168	C₂²×C₄	C₄²×C₁₀₉	1	2	yes	—
169	C₂²×C₄	C₄²×C₁₀₉	1	2	yes	—
170	C₂²×C₄	C₄²×C₁₀₉	1	2	yes	—
171	C₂²×C₄	C₄²×C₁₀₉	1	1	yes	characteristic
172	C₂²×C₄	C₄²×C₁₀₉	1	2	yes	—
173	C₂²×C₄	C₂²×C₄×C₁₀₉	1	2	yes	—
174	C₂²×C₄	C₂²×C₄×C₁₀₉	1	2	yes	—
175	C₂²×C₄	C₂²×C₄×C₁₀₉	1	2	yes	—
176	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
177	C₂²×C₄	C₂²×C₄×C₁₀₉	1	2	yes	—
178	M₁₆	—	4	4	no	—
179	C₂×C₈	—	2	16	no	—
180	C₂×C₈	C₂²×C₄×C₁₀₉	1	8	yes	—
181	C₂×C₈	C₂²×C₄×C₁₀₉	1	8	yes	—
182	C₂×C₈	C₂²×C₄×C₁₀₉	1	8	yes	—
183	C₄²	C₄²×C₁₀₉	1	1	yes	characteristic
184	C₄²	C₄²×C₁₀₉	1	2	yes	—
185	C₄²	C₄²×C₁₀₉	1	2	yes	—
186	C₄²	C₄²×C₁₀₉	1	1	yes	characteristic
187	C₄²	C₂²×C₄×C₁₀₉	1	2	yes	—
188	C₄²	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
189	C₄²	C₂²×C₄×C₁₀₉	1	2	yes	—
190	C₄²	C₂²×C₄×C₁₀₉	1	2	yes	—
191	C₄²	C₂²×C₄×C₁₀₉	1	2	yes	—
192	C₄²	C₂⁴×C₁₀₉	1	1	yes	characteristic
193	C₂³	—	2	4	no	—
194	C₂³	—	4	1	no	—
195	C₂³	C₂×C₄²×C₁₀₉	1	1	yes	characteristic
196	C₂³	C₂×C₄²×C₁₀₉	1	2	yes	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	—	4	1	no	—
206	C₂×C₄	C₂×C₄²×C₁₀₉	1	2	yes	—
207	C₂×C₄	C₂×C₄²×C₁₀₉	1	2	yes	—
208	C₂×C₄	C₂×C₄²×C₁₀₉	1	2	yes	—
209	C₂×C₄	C₂×C₄²×C₁₀₉	1	1	yes	characteristic
210	C₂×C₄	C₂×C₄²×C₁₀₉	1	2	yes	—
211	C₂×C₄	#131×C₁₀₉	1	8	yes	—
212	C₂×C₄	C₂³×C₄×C₁₀₉	1	2	yes	—
213	C₂×C₄	C₂³×C₄×C₁₀₉	1	1	yes	characteristic
214	C₈	—	2	8	no	—
215	C₈	—	2	8	no	—
216	C₈	—	2	8	no	—
217	C₂²	—	2	2	no	—
218	C₂²	—	2	2	no	—
219	C₂²	—	2	2	no	—
220	C₂²	—	2	1	no	—
221	C₂²	—	4	4	no	—
222	C₂²	#405×C₁₀₉	1	4	yes	—
223	C₂²	C₂²×C₄²×C₁₀₉	1	1	yes	characteristic
224	C₄	—	2	2	no	—
225	C₄	—	2	2	no	—
226	C₄	—	2	1	no	—
227	C₄	—	2	2	no	—
228	C₄	#405×C₁₀₉	1	4	yes	—
229	C₄	C₂×#131×C₁₀₉	1	4	yes	—
230	C₄	#568×C₁₀₉	1	2	yes	—
231	C₂	—	2	4	no	—
232	C₂	—	4	1	no	—
233	C₂	C₂×#405×C₁₀₉	1	2	yes	—
234	C₂	C₁₀₉×#2873	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₁₀₉×#1453	C₂	semidirect
4. C₁₀₉×#1452	C₂	semidirect
5. C₁₀₉×#1450	C₂	semidirect
20. #322×C₁₀₉	C₂²	semidirect
21. #322×C₁₀₉	C₂²	semidirect
22. C₈²×C₁₀₉	C₂²	semidirect
113. #17325	C₁₀₉	direct (N × H)
120. #1453	C₂×C₁₀₉	semidirect
121. #1452	C₂×C₁₀₉	semidirect
122. #1450	C₂×C₁₀₉	semidirect
123. C₁₀₉	#17325	direct (N × H)
138. #322	C₂²×C₁₀₉	semidirect
139. #322	C₂²×C₁₀₉	semidirect
140. C₈²	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×#1453	2
C₁₀₉×#1452	2
C₁₀₉×#1450	2
#17325	109

Subgroups — C109×#17325 #42135319

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#17325 #42135319