Subgroups — C₁₀₉×#21959 #42139953

Order 27904.

906 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#2285	C₂	1	2	yes	maximal
3	C₁₀₉×#1948	C₂	1	8	yes	maximal, complemented
4	C₁₀₉×#1807	C₂	1	4	yes	maximal
5	C₂³×C₈×C₁₀₉	C₂²	1	2	yes	—
6	C₂×#117×C₁₀₉	—	2	2	no	—
7	C₂²×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
8	C₂×#106×C₁₀₉	C₂²	1	8	yes	—
9	C₂×C₄⋊C₈×C₁₀₉	C₄	1	8	yes	complemented
10	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	4	yes	—
11	#384×C₁₀₉	C₂²	1	4	yes	—
12	#382×C₁₀₉	—	2	2	no	—
13	#380×C₁₀₉	—	2	2	no	—
14	#380×C₁₀₉	C₂²	1	4	yes	—
15	C₄×C₄⋊C₄×C₁₀₉	—	2	2	no	—
16	C₄×C₄⋊C₄×C₁₀₉	C₂²	1	4	yes	—
17	#340×C₁₀₉	C₂²	1	8	yes	—
18	C₂³×C₄×C₁₀₉	D₄	1	1	yes	characteristic
19	C₂³×C₄×C₁₀₉	D₄	1	1	yes	characteristic
20	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
21	C₂²×C₈×C₁₀₉	—	4	4	no	—
22	C₂²×C₈×C₁₀₉	C₂×C₄	1	8	yes	—
23	C₂²×C₈×C₁₀₉	C₂³	1	4	yes	—
24	#117×C₁₀₉	—	4	8	no	—
25	C₂×C₄⋊C₄×C₁₀₉	—	2	4	no	—
26	C₂×C₄⋊C₄×C₁₀₉	—	2	4	no	—
27	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	8	yes	—
28	C₂×C₂²⋊C₄×C₁₀₉	—	2	4	no	—
29	C₂×C₄²×C₁₀₉	—	2	4	no	—
30	C₂×C₄²×C₁₀₉	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₁₀₉	D₄	1	2	yes	—
32	C₂×C₄²×C₁₀₉	Q₈	1	4	yes	—
33	C₂×C₄²×C₁₀₉	C₂³	1	2	yes	—
34	#106×C₁₀₉	—	4	8	no	—
35	C₄⋊C₈×C₁₀₉	C₂×C₄	1	16	yes	—
36	#95×C₁₀₉	—	2	4	no	—
37	#95×C₁₀₉	—	2	4	no	—
38	C₂⁴×C₁₀₉	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₀₉	—	2	2	no	—
40	C₂²×C₄×C₁₀₉	—	2	2	no	—
41	C₂²×C₄×C₁₀₉	—	2	8	no	—
42	C₂²×C₄×C₁₀₉	—	4	2	no	—
43	C₂²×C₄×C₁₀₉	C₄⋊C₄	1	4	yes	—
44	C₂²×C₄×C₁₀₉	C₄⋊C₄	1	4	yes	—
45	C₂²×C₄×C₁₀₉	SD₁₆	1	8	yes	—
46	C₂²×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
47	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
49	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
50	C₂²×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
51	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
52	C₂×C₈×C₁₀₉	—	2	8	no	—
53	C₂×C₈×C₁₀₉	—	4	8	no	—
54	C₂×C₈×C₁₀₉	C₂²×C₄	1	8	yes	—
55	C₄⋊C₄×C₁₀₉	—	2	8	no	—
56	C₄⋊C₄×C₁₀₉	—	2	8	no	—
57	C₄⋊C₄×C₁₀₉	—	4	8	no	—
58	C₄⋊C₄×C₁₀₉	—	4	8	no	—
59	C₂²⋊C₄×C₁₀₉	—	4	8	no	—
60	C₄²×C₁₀₉	—	4	8	no	—
61	C₄²×C₁₀₉	C₄⋊C₄	1	8	yes	—
62	C₄²×C₁₀₉	C₂²×C₄	1	4	yes	—
63	C₄²×C₁₀₉	C₂×Q₈	1	4	yes	—
64	C₂³×C₁₀₉	—	2	2	no	—
65	C₂³×C₁₀₉	—	2	2	no	—
66	C₂³×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
67	C₂³×C₁₀₉	C₂×SD₁₆	1	4	yes	—
68	C₂³×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
69	C₂×C₄×C₁₀₉	—	2	2	no	—
70	C₂×C₄×C₁₀₉	—	2	4	no	—
71	C₂×C₄×C₁₀₉	—	2	2	no	—
72	C₂×C₄×C₁₀₉	—	2	2	no	—
73	C₂×C₄×C₁₀₉	—	2	4	no	—
74	C₂×C₄×C₁₀₉	—	4	4	no	—
75	C₂×C₄×C₁₀₉	—	4	8	no	—
76	C₂×C₄×C₁₀₉	—	4	8	no	—
77	C₂×C₄×C₁₀₉	—	4	8	no	—
78	C₂×C₄×C₁₀₉	#106	1	16	yes	complemented
79	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
80	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	4	yes	—
81	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	4	yes	—
82	C₂×C₄×C₁₀₉	C₂×SD₁₆	1	4	yes	—
83	C₂×C₄×C₁₀₉	C₂×SD₁₆	1	4	yes	—
84	C₂×C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
86	C₂×C₄×C₁₀₉	H(5,𝔽₂)	1	2	yes	—
87	C₂×C₄×C₁₀₉	2₋¹⁺⁴	1	2	yes	—
88	C₈×C₁₀₉	—	2	8	no	—
89	C₂²×C₁₀₉	—	2	2	no	—
90	C₂²×C₁₀₉	—	2	2	no	—
91	C₂²×C₁₀₉	—	2	8	no	—
92	C₂²×C₁₀₉	C₂×#106	1	4	yes	—
93	C₂²×C₁₀₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
94	C₂²×C₁₀₉	#537	1	2	yes	—
95	C₂²×C₁₀₉	C₂²×SD₁₆	1	2	yes	—
96	C₂²×C₁₀₉	#575	1	2	yes	—
97	C₄×C₁₀₉	—	2	2	no	—
98	C₄×C₁₀₉	—	2	4	no	—
99	C₄×C₁₀₉	—	8	8	no	—
100	C₄×C₁₀₉	C₂×#106	1	8	yes	—
101	C₄×C₁₀₉	#518	1	1	yes	characteristic
102	C₄×C₁₀₉	#519	1	1	yes	characteristic
103	C₄×C₁₀₉	#544	1	2	yes	—
104	#21959	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
105	C₂×C₁₀₉	—	2	4	no	—
106	C₂×C₁₀₉	#2316	1	1	yes	characteristic
107	C₂×C₁₀₉	C₂²×#106	1	1	yes	characteristic
108	C₂×C₁₀₉	#2911	1	1	yes	characteristic
109	C₂×C₁₀₉	#3246	1	4	yes	—
110	C₂²×C₄⋊C₈	C₂×C₁₀₉	1	1	yes	characteristic
111	#2285	C₂×C₁₀₉	1	2	yes	—
112	#1948	C₂×C₁₀₉	1	8	yes	complemented
113	#1807	C₂×C₁₀₉	1	4	yes	—
114	C₁₀₉	#21959	1	1	yes	characteristic, direct product factor, Sylow
115	C₂³×C₈	C₂²×C₁₀₉	1	2	yes	—
116	C₂×#117	—	2	2	no	—
117	C₂²×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
118	C₂×#106	C₂²×C₁₀₉	1	8	yes	—
119	C₂×C₄⋊C₈	C₄×C₁₀₉	1	8	yes	complemented
120	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	4	yes	—
121	#384	C₂²×C₁₀₉	1	4	yes	—
122	#382	—	2	2	no	—
123	#380	—	2	2	no	—
124	#380	C₂²×C₁₀₉	1	4	yes	—
125	C₄×C₄⋊C₄	—	2	2	no	—
126	C₄×C₄⋊C₄	C₂²×C₁₀₉	1	4	yes	—
127	#340	C₂²×C₁₀₉	1	8	yes	—
128	C₂³×C₄	D₄×C₁₀₉	1	1	yes	characteristic
129	C₂³×C₄	D₄×C₁₀₉	1	1	yes	characteristic
130	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
131	C₂²×C₈	—	4	4	no	—
132	C₂²×C₈	C₂×C₄×C₁₀₉	1	8	yes	—
133	C₂²×C₈	C₂³×C₁₀₉	1	4	yes	—
134	#117	—	4	8	no	—
135	C₂×C₄⋊C₄	—	2	4	no	—
136	C₂×C₄⋊C₄	—	2	4	no	—
137	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	8	yes	—
138	C₂×C₂²⋊C₄	—	2	4	no	—
139	C₂×C₄²	—	2	4	no	—
140	C₂×C₄²	C₂×C₄×C₁₀₉	1	4	yes	—
141	C₂×C₄²	D₄×C₁₀₉	1	2	yes	—
142	C₂×C₄²	Q₈×C₁₀₉	1	4	yes	—
143	C₂×C₄²	C₂³×C₁₀₉	1	2	yes	—
144	#106	—	4	8	no	—
145	C₄⋊C₈	C₂×C₄×C₁₀₉	1	16	yes	—
146	#95	—	2	4	no	—
147	#95	—	2	4	no	—
148	C₂⁴	C₂×D₄×C₁₀₉	1	1	yes	characteristic
149	C₂²×C₄	—	2	2	no	—
150	C₂²×C₄	—	2	2	no	—
151	C₂²×C₄	—	2	8	no	—
152	C₂²×C₄	—	4	2	no	—
153	C₂²×C₄	C₄⋊C₄×C₁₀₉	1	4	yes	—
154	C₂²×C₄	C₄⋊C₄×C₁₀₉	1	4	yes	—
155	C₂²×C₄	SD₁₆×C₁₀₉	1	8	yes	—
156	C₂²×C₄	C₂²×C₄×C₁₀₉	1	2	yes	—
157	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
158	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
159	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
160	C₂²×C₄	C₂×Q₈×C₁₀₉	1	2	yes	—
161	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
162	C₂×C₈	—	2	8	no	—
163	C₂×C₈	—	4	8	no	—
164	C₂×C₈	C₂²×C₄×C₁₀₉	1	8	yes	—
165	C₄⋊C₄	—	2	8	no	—
166	C₄⋊C₄	—	2	8	no	—
167	C₄⋊C₄	—	4	8	no	—
168	C₄⋊C₄	—	4	8	no	—
169	C₂²⋊C₄	—	4	8	no	—
170	C₄²	—	4	8	no	—
171	C₄²	C₄⋊C₄×C₁₀₉	1	8	yes	—
172	C₄²	C₂²×C₄×C₁₀₉	1	4	yes	—
173	C₄²	C₂×Q₈×C₁₀₉	1	4	yes	—
174	C₂³	—	2	2	no	—
175	C₂³	—	2	2	no	—
176	C₂³	C₂×C₄⋊C₄×C₁₀₉	1	2	yes	—
177	C₂³	C₂×SD₁₆×C₁₀₉	1	4	yes	—
178	C₂³	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	4	4	no	—
185	C₂×C₄	—	4	8	no	—
186	C₂×C₄	—	4	8	no	—
187	C₂×C₄	—	4	8	no	—
188	C₂×C₄	#106×C₁₀₉	1	16	yes	complemented
189	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	2	yes	—
190	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	4	yes	—
191	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	4	yes	—
192	C₂×C₄	C₂×SD₁₆×C₁₀₉	1	4	yes	—
193	C₂×C₄	C₂×SD₁₆×C₁₀₉	1	4	yes	—
194	C₂×C₄	C₂³×C₄×C₁₀₉	1	1	yes	characteristic
195	C₂×C₄	C₂²×Q₈×C₁₀₉	1	1	yes	characteristic
196	C₂×C₄	H(5,𝔽₂)×C₁₀₉	1	2	yes	—
197	C₂×C₄	2₋¹⁺⁴×C₁₀₉	1	2	yes	—
198	C₈	—	2	8	no	—
199	C₂²	—	2	8	no	—
200	C₂²	—	2	2	no	—
201	C₂²	—	2	2	no	—
202	C₂²	C₂×#106×C₁₀₉	1	4	yes	—
203	C₂²	C₂²×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
204	C₂²	#537×C₁₀₉	1	2	yes	—
205	C₂²	C₂²×SD₁₆×C₁₀₉	1	2	yes	—
206	C₂²	#575×C₁₀₉	1	2	yes	—
207	C₄	—	2	2	no	—
208	C₄	—	2	4	no	—
209	C₄	—	8	8	no	—
210	C₄	C₂×#106×C₁₀₉	1	8	yes	—
211	C₄	#518×C₁₀₉	1	1	yes	characteristic
212	C₄	#519×C₁₀₉	1	1	yes	characteristic
213	C₄	#544×C₁₀₉	1	2	yes	—
214	C₂	—	2	4	no	—
215	C₂	C₁₀₉×#2316	1	1	yes	characteristic
216	C₂	C₂²×#106×C₁₀₉	1	1	yes	characteristic
217	C₂	C₁₀₉×#2911	1	1	yes	characteristic
218	C₂	C₁₀₉×#3246	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₁₀₉×#1948	C₂	semidirect
9. C₂×C₄⋊C₈×C₁₀₉	C₄	semidirect
78. C₂×C₄×C₁₀₉	#106	semidirect
104. #21959	C₁₀₉	direct (N × H)
112. #1948	C₂×C₁₀₉	semidirect
114. C₁₀₉	#21959	direct (N × H)
119. C₂×C₄⋊C₈	C₄×C₁₀₉	semidirect
188. C₂×C₄	#106×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊C₈×C₁₀₉	2
C₁₀₉×#2285	2
C₁₀₉×#1948	2
C₁₀₉×#1807	2
#21959	109

Subgroups — C109×#21959 #42139953

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#21959 #42139953