Subgroups — C₁₀₉×#8836 #42126830

Order 27904.

410 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₉×#2982	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#1984	C₂	1	1	yes	maximal, characteristic
3	C₄×#101×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₁₀₉×#1565	C₂	1	2	yes	maximal
5	C₁₀₉×#1278	C₂	1	2	yes	maximal
6	C₂×C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
7	C₈×Q₈×C₁₀₉	—	2	2	no	—
8	C₈×Q₈×C₁₀₉	—	2	2	no	—
9	#431×C₁₀₉	—	2	2	no	—
10	#431×C₁₀₉	C₂²	1	2	yes	—
11	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
12	C₂×#101×C₁₀₉	C₄	1	1	yes	characteristic
13	C₂×#101×C₁₀₉	C₄	1	1	yes	characteristic
14	C₄×M₁₆×C₁₀₉	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
16	#343×C₁₀₉	C₄	1	2	yes	—
17	#336×C₁₀₉	—	2	2	no	—
18	#335×C₁₀₉	—	2	2	no	—
19	#334×C₁₀₉	—	2	2	no	—
20	C₂²×Q₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
21	C₂×M₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
22	C₂×M₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₁₀₉	—	2	1	no	—
24	C₂²×C₈×C₁₀₉	—	2	1	no	—
25	C₄×Q₈×C₁₀₉	—	2	2	no	—
26	C₄×Q₈×C₁₀₉	—	2	2	no	—
27	C₂×C₄⋊C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₁₀₉	Q₈	1	2	yes	—
29	C₂×C₄²×C₁₀₉	D₄	1	2	yes	—
30	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
31	C₄⋊C₈×C₁₀₉	—	2	2	no	—
32	C₄⋊C₈×C₁₀₉	—	2	2	no	—
33	C₄⋊C₈×C₁₀₉	—	2	2	no	—
34	C₄⋊C₈×C₁₀₉	D₄	1	2	yes	—
35	C₄⋊C₈×C₁₀₉	D₄	1	2	yes	—
36	#101×C₁₀₉	—	2	2	no	—
37	#101×C₁₀₉	C₈	1	4	yes	complemented
38	C₂²⋊C₈×C₁₀₉	—	2	2	no	—
39	#97×C₁₀₉	—	2	2	no	—
40	C₄×C₈×C₁₀₉	—	2	4	no	—
41	C₄×C₈×C₁₀₉	—	2	2	no	—
42	C₄×C₈×C₁₀₉	—	2	2	no	—
43	C₄×C₈×C₁₀₉	—	2	2	no	—
44	C₂×Q₈×C₁₀₉	—	2	2	no	—
45	C₂×Q₈×C₁₀₉	—	2	2	no	—
46	C₂×Q₈×C₁₀₉	C₂×C₈	1	2	yes	—
47	C₂×Q₈×C₁₀₉	M₁₆	1	2	yes	—
48	C₂²×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
50	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
53	M₁₆×C₁₀₉	—	2	2	no	—
54	M₁₆×C₁₀₉	—	2	2	no	—
55	M₁₆×C₁₀₉	—	2	2	no	—
56	M₁₆×C₁₀₉	C₂×C₈	1	4	yes	—
57	C₂×C₈×C₁₀₉	—	2	2	no	—
58	C₂×C₈×C₁₀₉	—	2	2	no	—
59	C₂×C₈×C₁₀₉	—	2	2	no	—
60	C₂×C₈×C₁₀₉	—	2	2	no	—
61	C₂×C₈×C₁₀₉	—	2	2	no	—
62	C₂×C₈×C₁₀₉	—	4	1	no	—
63	C₂×C₈×C₁₀₉	—	4	1	no	—
64	C₂×C₈×C₁₀₉	—	4	2	no	—
65	C₂×C₈×C₁₀₉	—	4	1	no	—
66	C₄⋊C₄×C₁₀₉	—	2	2	no	—
67	C₄⋊C₄×C₁₀₉	—	2	4	no	—
68	C₄²×C₁₀₉	—	2	2	no	—
69	C₄²×C₁₀₉	—	2	2	no	—
70	C₄²×C₁₀₉	C₂×D₄	1	2	yes	—
71	C₄²×C₁₀₉	#41	1	2	yes	—
72	C₂³×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
73	Q₈×C₁₀₉	—	2	4	no	—
74	Q₈×C₁₀₉	—	2	4	no	—
75	C₂×C₄×C₁₀₉	—	2	2	no	—
76	C₂×C₄×C₁₀₉	—	2	2	no	—
77	C₂×C₄×C₁₀₉	—	2	1	no	—
78	C₂×C₄×C₁₀₉	—	2	1	no	—
79	C₂×C₄×C₁₀₉	—	2	2	no	—
80	C₂×C₄×C₁₀₉	—	4	1	no	—
81	C₂×C₄×C₁₀₉	—	4	2	no	—
82	C₂×C₄×C₁₀₉	C₄⋊C₈	1	4	yes	—
83	C₂×C₄×C₁₀₉	C₄×D₄	1	2	yes	—
84	C₂×C₄×C₁₀₉	#120	1	1	yes	characteristic
85	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
86	C₂×C₄×C₁₀₉	#123	1	1	yes	characteristic
87	C₂×C₄×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
88	C₂×C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
89	C₂×C₄×C₁₀₉	#131	1	2	yes	—
90	C₈×C₁₀₉	—	2	4	no	—
91	C₈×C₁₀₉	—	4	2	no	—
92	C₈×C₁₀₉	—	4	2	no	—
93	C₈×C₁₀₉	—	4	2	no	—
94	C₈×C₁₀₉	—	4	2	no	—
95	C₈×C₁₀₉	—	8	1	no	—
96	C₂²×C₁₀₉	—	2	1	no	—
97	C₂²×C₁₀₉	—	2	1	no	—
98	C₂²×C₁₀₉	#385	1	1	yes	characteristic
99	C₂²×C₁₀₉	C₂×C₄⋊C₈	1	1	yes	characteristic
100	C₂²×C₁₀₉	#424	1	1	yes	characteristic
101	C₄×C₁₀₉	—	2	4	no	—
102	C₄×C₁₀₉	—	2	1	no	—
103	C₄×C₁₀₉	—	2	2	no	—
104	C₄×C₁₀₉	—	4	2	no	—
105	C₄×C₁₀₉	C₈×D₄	1	2	yes	—
106	C₄×C₁₀₉	#435	1	2	yes	—
107	C₄×C₁₀₉	#454	1	1	yes	characteristic
108	C₄×C₁₀₉	#456	1	1	yes	characteristic
109	#8836	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₂×C₁₀₉	—	2	1	no	—
111	C₂×C₁₀₉	—	2	1	no	—
112	C₂×C₁₀₉	#1850	1	1	yes	characteristic
113	C₂×C₁₀₉	#1857	1	1	yes	characteristic
114	C₂×C₁₀₉	#1858	1	1	yes	characteristic
115	#2982	C₂×C₁₀₉	1	1	yes	characteristic
116	#1984	C₂×C₁₀₉	1	1	yes	characteristic
117	C₄×#101	C₂×C₁₀₉	1	1	yes	characteristic
118	#1565	C₂×C₁₀₉	1	2	yes	—
119	#1278	C₂×C₁₀₉	1	2	yes	—
120	C₁₀₉	#8836	1	1	yes	characteristic, direct product factor, Sylow
121	C₂×C₄×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
122	C₈×Q₈	—	2	2	no	—
123	C₈×Q₈	—	2	2	no	—
124	#431	—	2	2	no	—
125	#431	C₂²×C₁₀₉	1	2	yes	—
126	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
127	C₂×#101	C₄×C₁₀₉	1	1	yes	characteristic
128	C₂×#101	C₄×C₁₀₉	1	1	yes	characteristic
129	C₄×M₁₆	C₂²×C₁₀₉	1	2	yes	—
130	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
131	#343	C₄×C₁₀₉	1	2	yes	—
132	#336	—	2	2	no	—
133	#335	—	2	2	no	—
134	#334	—	2	2	no	—
135	C₂²×Q₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
136	C₂×M₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
137	C₂×M₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
138	C₂²×C₈	—	2	1	no	—
139	C₂²×C₈	—	2	1	no	—
140	C₄×Q₈	—	2	2	no	—
141	C₄×Q₈	—	2	2	no	—
142	C₂×C₄⋊C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
143	C₂×C₄⋊C₄	Q₈×C₁₀₉	1	2	yes	—
144	C₂×C₄²	D₄×C₁₀₉	1	2	yes	—
145	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
146	C₄⋊C₈	—	2	2	no	—
147	C₄⋊C₈	—	2	2	no	—
148	C₄⋊C₈	—	2	2	no	—
149	C₄⋊C₈	D₄×C₁₀₉	1	2	yes	—
150	C₄⋊C₈	D₄×C₁₀₉	1	2	yes	—
151	#101	—	2	2	no	—
152	#101	C₈×C₁₀₉	1	4	yes	complemented
153	C₂²⋊C₈	—	2	2	no	—
154	#97	—	2	2	no	—
155	C₄×C₈	—	2	4	no	—
156	C₄×C₈	—	2	2	no	—
157	C₄×C₈	—	2	2	no	—
158	C₄×C₈	—	2	2	no	—
159	C₂×Q₈	—	2	2	no	—
160	C₂×Q₈	—	2	2	no	—
161	C₂×Q₈	C₂×C₈×C₁₀₉	1	2	yes	—
162	C₂×Q₈	M₁₆×C₁₀₉	1	2	yes	—
163	C₂²×C₄	C₄⋊C₄×C₁₀₉	1	2	yes	—
164	C₂²×C₄	C₄⋊C₄×C₁₀₉	1	2	yes	—
165	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
166	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
167	C₂²×C₄	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
168	M₁₆	—	2	2	no	—
169	M₁₆	—	2	2	no	—
170	M₁₆	—	2	2	no	—
171	M₁₆	C₂×C₈×C₁₀₉	1	4	yes	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	—	4	1	no	—
178	C₂×C₈	—	4	1	no	—
179	C₂×C₈	—	4	2	no	—
180	C₂×C₈	—	4	1	no	—
181	C₄⋊C₄	—	2	2	no	—
182	C₄⋊C₄	—	2	4	no	—
183	C₄²	—	2	2	no	—
184	C₄²	—	2	2	no	—
185	C₄²	C₂×D₄×C₁₀₉	1	2	yes	—
186	C₄²	#41×C₁₀₉	1	2	yes	—
187	C₂³	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
188	Q₈	—	2	4	no	—
189	Q₈	—	2	4	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	C₄⋊C₈×C₁₀₉	1	4	yes	—
198	C₂×C₄	C₄×D₄×C₁₀₉	1	2	yes	—
199	C₂×C₄	#120×C₁₀₉	1	1	yes	characteristic
200	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	2	yes	—
201	C₂×C₄	#123×C₁₀₉	1	1	yes	characteristic
202	C₂×C₄	C₂²×C₈×C₁₀₉	1	1	yes	characteristic
203	C₂×C₄	C₂×M₁₆×C₁₀₉	1	1	yes	characteristic
204	C₂×C₄	#131×C₁₀₉	1	2	yes	—
205	C₈	—	2	4	no	—
206	C₈	—	4	2	no	—
207	C₈	—	4	2	no	—
208	C₈	—	4	2	no	—
209	C₈	—	4	2	no	—
210	C₈	—	8	1	no	—
211	C₂²	—	2	1	no	—
212	C₂²	—	2	1	no	—
213	C₂²	#385×C₁₀₉	1	1	yes	characteristic
214	C₂²	C₂×C₄⋊C₈×C₁₀₉	1	1	yes	characteristic
215	C₂²	#424×C₁₀₉	1	1	yes	characteristic
216	C₄	—	2	4	no	—
217	C₄	—	2	1	no	—
218	C₄	—	2	2	no	—
219	C₄	—	4	2	no	—
220	C₄	C₈×D₄×C₁₀₉	1	2	yes	—
221	C₄	#435×C₁₀₉	1	2	yes	—
222	C₄	#454×C₁₀₉	1	1	yes	characteristic
223	C₄	#456×C₁₀₉	1	1	yes	characteristic
224	C₂	—	2	1	no	—
225	C₂	—	2	1	no	—
226	C₂	C₁₀₉×#1850	1	1	yes	characteristic
227	C₂	C₁₀₉×#1857	1	1	yes	characteristic
228	C₂	C₁₀₉×#1858	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₀₉	#120
commutator	C₂×C₄	C₂²×C₈×C₁₀₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₉
socle	C₂²×C₁₀₉	#385
prime radical	C₂³×C₁₀₉	C₂×C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
37. #101×C₁₀₉	C₈	semidirect
109. #8836	C₁₀₉	direct (N × H)
120. C₁₀₉	#8836	direct (N × H)
152. #101	C₈×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₁₀₉×#2982	2
C₁₀₉×#1984	2
C₄×#101×C₁₀₉	2
C₁₀₉×#1565	2
C₁₀₉×#1278	2
#8836	109

Subgroups — C109×#8836 #42126830

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#8836 #42126830