Subgroups — C₁₀₉×#17819 #42135813

Order 27904.

638 conjugacy classes of subgroups.

The subgroup lattice has 248 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×#1574	C₂	1	8	yes	maximal, complemented
4	C₁₀₉×#1568	C₂	1	4	yes	maximal, complemented
5	C₂²×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
6	#432×C₁₀₉	—	2	2	no	—
7	#432×C₁₀₉	C₂²	1	4	yes	—
8	#431×C₁₀₉	—	2	2	no	—
9	#431×C₁₀₉	C₄	1	4	yes	—
10	#431×C₁₀₉	C₂²	1	4	yes	—
11	#431×C₁₀₉	C₂²	1	4	yes	—
12	#423×C₁₀₉	C₂²	1	4	yes	—
13	C₂×C₄⋊C₈×C₁₀₉	C₄	1	2	yes	—
14	#407×C₁₀₉	—	2	2	no	—
15	C₂×C₂²⋊C₈×C₁₀₉	C₄	1	2	yes	—
16	C₄×M₁₆×C₁₀₉	—	2	2	no	—
17	C₂×C₄×C₈×C₁₀₉	C₂²	1	2	yes	—
18	#335×C₁₀₉	C₂²	1	8	yes	complemented
19	#334×C₁₀₉	C₂²	1	8	yes	complemented
20	C₂³×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₁₀₉	D₄	1	1	yes	characteristic
22	C₂³×C₄×C₁₀₉	Q₈	1	1	yes	characteristic
23	C₂×M₁₆×C₁₀₉	—	2	4	no	—
24	C₂²×C₈×C₁₀₉	C₂×C₄	1	4	yes	—
25	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₁₀₉	D₄	1	2	yes	—
28	C₂×C₄²×C₁₀₉	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₀₉	Q₈	1	1	yes	characteristic
30	C₂×C₄²×C₁₀₉	Q₈	1	2	yes	—
31	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₁₀₉	C₂³	1	2	yes	—
33	C₄⋊C₈×C₁₀₉	—	2	4	no	—
34	C₄⋊C₈×C₁₀₉	C₂×C₄	1	8	yes	—
35	C₄⋊C₈×C₁₀₉	C₂³	1	8	yes	—
36	C₂²⋊C₈×C₁₀₉	—	2	4	no	—
37	C₂²⋊C₈×C₁₀₉	—	2	4	no	—
38	C₂²⋊C₈×C₁₀₉	C₂×C₄	1	8	yes	—
39	#97×C₁₀₉	—	2	4	no	—
40	C₄×C₈×C₁₀₉	—	2	4	no	—
41	C₄×C₈×C₁₀₉	C₂×C₄	1	4	yes	—
42	C₄×C₈×C₁₀₉	C₂³	1	4	yes	—
43	C₂⁴×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₀₉	—	2	4	no	—
45	C₂²×C₄×C₁₀₉	—	4	2	no	—
46	C₂²×C₄×C₁₀₉	—	4	2	no	—
47	C₂²×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
51	C₂²×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₀₉	C₂²×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×D₄	1	2	yes	—
57	C₂²×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
58	C₂²×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
59	M₁₆×C₁₀₉	—	4	8	no	—
60	C₂×C₈×C₁₀₉	—	2	8	no	—
61	C₂×C₈×C₁₀₉	—	4	4	no	—
62	C₂×C₈×C₁₀₉	C₂²×C₄	1	8	yes	—
63	C₄²×C₁₀₉	C₄⋊C₄	1	2	yes	—
64	C₄²×C₁₀₉	C₄⋊C₄	1	2	yes	—
65	C₄²×C₁₀₉	C₄⋊C₄	1	2	yes	—
66	C₄²×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
67	C₄²×C₁₀₉	C₂²×C₄	1	2	yes	—
68	C₄²×C₁₀₉	C₂×D₄	1	2	yes	—
69	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
71	C₄²×C₁₀₉	C₂×Q₈	1	2	yes	—
72	C₄²×C₁₀₉	C₂⁴	1	1	yes	characteristic
73	C₂³×C₁₀₉	—	2	2	no	—
74	C₂³×C₁₀₉	—	4	1	no	—
75	C₂³×C₁₀₉	—	4	1	no	—
76	C₂³×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
77	C₂³×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₁₀₉	—	2	2	no	—
79	C₂×C₄×C₁₀₉	—	2	2	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	—	4	1	no	—
86	C₂×C₄×C₁₀₉	—	4	2	no	—
87	C₂×C₄×C₁₀₉	—	4	1	no	—
88	C₂×C₄×C₁₀₉	—	4	2	no	—
89	C₂×C₄×C₁₀₉	#108	1	4	yes	—
90	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
93	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
94	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
96	C₂×C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
97	C₂×C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
98	C₂×C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
99	C₈×C₁₀₉	—	2	8	no	—
100	C₈×C₁₀₉	—	4	8	no	—
101	C₂²×C₁₀₉	—	2	1	no	—
102	C₂²×C₁₀₉	—	2	1	no	—
103	C₂²×C₁₀₉	—	2	2	no	—
104	C₂²×C₁₀₉	—	2	2	no	—
105	C₂²×C₁₀₉	—	2	2	no	—
106	C₂²×C₁₀₉	—	4	4	no	—
107	C₂²×C₁₀₉	C₂×#108	1	2	yes	—
108	C₂²×C₁₀₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
109	C₄×C₁₀₉	—	2	1	no	—
110	C₄×C₁₀₉	—	2	2	no	—
111	C₄×C₁₀₉	—	2	2	no	—
112	C₄×C₁₀₉	—	2	1	no	—
113	C₄×C₁₀₉	—	2	1	no	—
114	C₄×C₁₀₉	—	2	1	no	—
115	C₄×C₁₀₉	C₂×#108	1	4	yes	—
116	C₄×C₁₀₉	#568	1	2	yes	—
117	C₄×C₁₀₉	#575	1	2	yes	—
118	#17819	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
119	C₂×C₁₀₉	—	2	2	no	—
120	C₂×C₁₀₉	—	4	2	no	—
121	C₂×C₁₀₉	#2905	1	1	yes	characteristic
122	C₂×C₁₀₉	#2911	1	1	yes	characteristic
123	C₂×C₁₀₉	C₂²×#108	1	1	yes	characteristic
124	#2920	C₂×C₁₀₉	1	2	yes	—
125	C₂×#431	C₂×C₁₀₉	1	1	yes	characteristic
126	#1574	C₂×C₁₀₉	1	8	yes	complemented
127	#1568	C₂×C₁₀₉	1	4	yes	complemented
128	C₁₀₉	#17819	1	1	yes	characteristic, direct product factor, Sylow
129	C₂²×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
130	#432	—	2	2	no	—
131	#432	C₂²×C₁₀₉	1	4	yes	—
132	#431	—	2	2	no	—
133	#431	C₄×C₁₀₉	1	4	yes	—
134	#431	C₂²×C₁₀₉	1	4	yes	—
135	#431	C₂²×C₁₀₉	1	4	yes	—
136	#423	C₂²×C₁₀₉	1	4	yes	—
137	C₂×C₄⋊C₈	C₄×C₁₀₉	1	2	yes	—
138	#407	—	2	2	no	—
139	C₂×C₂²⋊C₈	C₄×C₁₀₉	1	2	yes	—
140	C₄×M₁₆	—	2	2	no	—
141	C₂×C₄×C₈	C₂²×C₁₀₉	1	2	yes	—
142	#335	C₂²×C₁₀₉	1	8	yes	complemented
143	#334	C₂²×C₁₀₉	1	8	yes	complemented
144	C₂³×C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
145	C₂³×C₄	D₄×C₁₀₉	1	1	yes	characteristic
146	C₂³×C₄	Q₈×C₁₀₉	1	1	yes	characteristic
147	C₂×M₁₆	—	2	4	no	—
148	C₂²×C₈	C₂×C₄×C₁₀₉	1	4	yes	—
149	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
150	C₂×C₄²	C₂×C₄×C₁₀₉	1	2	yes	—
151	C₂×C₄²	D₄×C₁₀₉	1	2	yes	—
152	C₂×C₄²	D₄×C₁₀₉	1	1	yes	characteristic
153	C₂×C₄²	Q₈×C₁₀₉	1	1	yes	characteristic
154	C₂×C₄²	Q₈×C₁₀₉	1	2	yes	—
155	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
156	C₂×C₄²	C₂³×C₁₀₉	1	2	yes	—
157	C₄⋊C₈	—	2	4	no	—
158	C₄⋊C₈	C₂×C₄×C₁₀₉	1	8	yes	—
159	C₄⋊C₈	C₂³×C₁₀₉	1	8	yes	—
160	C₂²⋊C₈	—	2	4	no	—
161	C₂²⋊C₈	—	2	4	no	—
162	C₂²⋊C₈	C₂×C₄×C₁₀₉	1	8	yes	—
163	#97	—	2	4	no	—
164	C₄×C₈	—	2	4	no	—
165	C₄×C₈	C₂×C₄×C₁₀₉	1	4	yes	—
166	C₄×C₈	C₂³×C₁₀₉	1	4	yes	—
167	C₂⁴	C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
168	C₂²×C₄	—	2	4	no	—
169	C₂²×C₄	—	4	2	no	—
170	C₂²×C₄	—	4	2	no	—
171	C₂²×C₄	C₄⋊C₄×C₁₀₉	1	2	yes	—
172	C₂²×C₄	C₄⋊C₄×C₁₀₉	1	2	yes	—
173	C₂²×C₄	C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
174	C₂²×C₄	C₄⋊C₄×C₁₀₉	1	2	yes	—
175	C₂²×C₄	C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
176	C₂²×C₄	C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
177	C₂²×C₄	C₂²×C₄×C₁₀₉	1	2	yes	—
178	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
179	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
180	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
181	C₂²×C₄	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
182	C₂²×C₄	C₂×Q₈×C₁₀₉	1	2	yes	—
183	M₁₆	—	4	8	no	—
184	C₂×C₈	—	2	8	no	—
185	C₂×C₈	—	4	4	no	—
186	C₂×C₈	C₂²×C₄×C₁₀₉	1	8	yes	—
187	C₄²	C₄⋊C₄×C₁₀₉	1	2	yes	—
188	C₄²	C₄⋊C₄×C₁₀₉	1	2	yes	—
189	C₄²	C₄⋊C₄×C₁₀₉	1	2	yes	—
190	C₄²	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
191	C₄²	C₂²×C₄×C₁₀₉	1	2	yes	—
192	C₄²	C₂×D₄×C₁₀₉	1	2	yes	—
193	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
194	C₄²	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
195	C₄²	C₂×Q₈×C₁₀₉	1	2	yes	—
196	C₄²	C₂⁴×C₁₀₉	1	1	yes	characteristic
197	C₂³	—	2	2	no	—
198	C₂³	—	4	1	no	—
199	C₂³	—	4	1	no	—
200	C₂³	C₂×C₄⋊C₄×C₁₀₉	1	2	yes	—
201	C₂³	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	4	1	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	4	1	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	#108×C₁₀₉	1	4	yes	—
214	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
215	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
216	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	2	yes	—
217	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	2	yes	—
218	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
219	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	2	yes	—
220	C₂×C₄	C₂³×C₄×C₁₀₉	1	1	yes	characteristic
221	C₂×C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
222	C₂×C₄	C₂²×Q₈×C₁₀₉	1	1	yes	characteristic
223	C₈	—	2	8	no	—
224	C₈	—	4	8	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	2	1	no	—
227	C₂²	—	2	2	no	—
228	C₂²	—	2	2	no	—
229	C₂²	—	2	2	no	—
230	C₂²	—	4	4	no	—
231	C₂²	C₂×#108×C₁₀₉	1	2	yes	—
232	C₂²	C₂²×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
233	C₄	—	2	1	no	—
234	C₄	—	2	2	no	—
235	C₄	—	2	2	no	—
236	C₄	—	2	1	no	—
237	C₄	—	2	1	no	—
238	C₄	—	2	1	no	—
239	C₄	C₂×#108×C₁₀₉	1	4	yes	—
240	C₄	#568×C₁₀₉	1	2	yes	—
241	C₄	#575×C₁₀₉	1	2	yes	—
242	C₂	—	2	2	no	—
243	C₂	—	4	2	no	—
244	C₂	C₁₀₉×#2905	1	1	yes	characteristic
245	C₂	C₁₀₉×#2911	1	1	yes	characteristic
246	C₂	C₂²×#108×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×#1574	C₂	semidirect
4. C₁₀₉×#1568	C₂	semidirect
18. #335×C₁₀₉	C₂²	semidirect
19. #334×C₁₀₉	C₂²	semidirect
118. #17819	C₁₀₉	direct (N × H)
126. #1574	C₂×C₁₀₉	semidirect
127. #1568	C₂×C₁₀₉	semidirect
128. C₁₀₉	#17819	direct (N × H)
142. #335	C₂²×C₁₀₉	semidirect
143. #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×#1574	2
C₁₀₉×#1568	2
#17819	109

Subgroups — C109×#17819 #42135813

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#17819 #42135813