Subgroups — C₁₀₉×#11662 #42129656

Order 27904.

358 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#2162	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#1568	C₂	1	1	yes	maximal, characteristic
4	C₁₀₉×#1332	C₂	1	4	yes	maximal, complemented
5	C₂²×Q₁₆×C₁₀₉	C₄	1	1	yes	characteristic
6	C₂×C₄×Q₈×C₁₀₉	—	2	1	no	—
7	C₄×Q₁₆×C₁₀₉	—	2	1	no	—
8	C₄×Q₁₆×C₁₀₉	—	2	1	no	—
9	C₄×Q₁₆×C₁₀₉	C₄	1	2	yes	—
10	C₄×Q₁₆×C₁₀₉	C₂²	1	2	yes	—
11	#432×C₁₀₉	—	2	1	no	—
12	C₂×#107×C₁₀₉	C₄	1	1	yes	characteristic
13	C₂×#103×C₁₀₉	—	2	1	no	—
14	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₄⋊C₁₆×C₁₀₉	—	2	1	no	—
16	C₂²⋊C₁₆×C₁₀₉	—	2	1	no	—
17	C₄×C₁₆×C₁₀₉	C₂²	1	2	yes	—
18	#335×C₁₀₉	C₂²	1	2	yes	—
19	#334×C₁₀₉	—	2	1	no	—
20	C₂²×Q₈×C₁₀₉	—	2	1	no	—
21	C₂×Q₁₆×C₁₀₉	—	2	4	no	—
22	C₂×Q₁₆×C₁₀₉	—	2	1	no	—
23	C₂×Q₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₀₉	D₄	1	1	yes	characteristic
26	C₄×Q₈×C₁₀₉	—	2	2	no	—
27	C₄×Q₈×C₁₀₉	—	2	2	no	—
28	C₄×Q₈×C₁₀₉	—	4	1	no	—
29	C₄×Q₈×C₁₀₉	—	4	1	no	—
30	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
31	C₂×C₄⋊C₄×C₁₀₉	—	4	1	no	—
32	C₂×C₄²×C₁₀₉	—	4	1	no	—
33	C₂×C₄²×C₁₀₉	D₄	1	1	yes	characteristic
34	C₂×C₁₆×C₁₀₉	—	2	2	no	—
35	#107×C₁₀₉	—	2	1	no	—
36	#107×C₁₀₉	C₂×C₄	1	2	yes	—
37	C₄⋊C₈×C₁₀₉	—	2	2	no	—
38	#103×C₁₀₉	—	2	2	no	—
39	#103×C₁₀₉	—	2	2	no	—
40	C₂²⋊C₈×C₁₀₉	—	4	1	no	—
41	#97×C₁₀₉	—	4	1	no	—
42	C₄×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₁₀₉	D₄	1	1	yes	characteristic
44	C₄×C₈×C₁₀₉	D₄	1	1	yes	characteristic
45	C₄×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
46	C₂×Q₈×C₁₀₉	—	2	4	no	—
47	C₂×Q₈×C₁₀₉	—	2	2	no	—
48	C₂×Q₈×C₁₀₉	—	4	2	no	—
49	C₂×Q₈×C₁₀₉	—	4	1	no	—
50	C₂²×C₄×C₁₀₉	—	4	1	no	—
51	C₂²×C₄×C₁₀₉	—	4	1	no	—
52	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₀₉	D₈	1	1	yes	characteristic
54	C₂²×C₄×C₁₀₉	SD₁₆	1	1	yes	characteristic
55	Q₁₆×C₁₀₉	—	2	4	no	—
56	Q₁₆×C₁₀₉	—	2	2	no	—
57	Q₁₆×C₁₀₉	M₁₆	1	4	yes	—
58	C₂×C₈×C₁₀₉	—	2	2	no	—
59	C₂×C₈×C₁₀₉	—	4	1	no	—
60	C₂×C₈×C₁₀₉	—	4	2	no	—
61	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
65	C₄⋊C₄×C₁₀₉	—	2	2	no	—
66	C₄⋊C₄×C₁₀₉	—	4	1	no	—
67	C₄⋊C₄×C₁₀₉	—	4	2	no	—
68	C₄⋊C₄×C₁₀₉	—	4	2	no	—
69	C₄²×C₁₀₉	—	4	2	no	—
70	C₄²×C₁₀₉	—	4	2	no	—
71	C₄²×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
72	C₄²×C₁₀₉	D₈	1	1	yes	characteristic
73	C₄²×C₁₀₉	SD₁₆	1	1	yes	characteristic
74	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
75	C₁₆×C₁₀₉	—	4	2	no	—
76	C₂³×C₁₀₉	#102	1	1	yes	characteristic
77	Q₈×C₁₀₉	—	2	4	no	—
78	Q₈×C₁₀₉	—	4	4	no	—
79	Q₈×C₁₀₉	—	4	2	no	—
80	C₂×C₄×C₁₀₉	—	2	2	no	—
81	C₂×C₄×C₁₀₉	—	2	1	no	—
82	C₂×C₄×C₁₀₉	—	2	1	no	—
83	C₂×C₄×C₁₀₉	—	2	2	no	—
84	C₂×C₄×C₁₀₉	—	4	4	no	—
85	C₂×C₄×C₁₀₉	—	4	2	no	—
86	C₂×C₄×C₁₀₉	—	4	2	no	—
87	C₂×C₄×C₁₀₉	—	8	2	no	—
88	C₂×C₄×C₁₀₉	#102	1	1	yes	characteristic
89	C₂×C₄×C₁₀₉	#102	1	1	yes	characteristic
90	C₂×C₄×C₁₀₉	#102	1	1	yes	characteristic
91	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₁₀₉	C₂×D₈	1	1	yes	characteristic
93	C₂×C₄×C₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
94	C₈×C₁₀₉	—	2	1	no	—
95	C₈×C₁₀₉	—	2	1	no	—
96	C₈×C₁₀₉	—	2	1	no	—
97	C₈×C₁₀₉	—	8	2	no	—
98	C₈×C₁₀₉	C₂×M₁₆	1	2	yes	—
99	C₂²×C₁₀₉	—	2	1	no	—
100	C₂²×C₁₀₉	—	2	1	no	—
101	C₂²×C₁₀₉	#359	1	2	yes	—
102	C₂²×C₁₀₉	C₂×#102	1	1	yes	characteristic
103	C₄×C₁₀₉	—	2	1	no	—
104	C₄×C₁₀₉	—	2	1	no	—
105	C₄×C₁₀₉	—	2	1	no	—
106	C₄×C₁₀₉	—	2	1	no	—
107	C₄×C₁₀₉	—	4	4	no	—
108	C₄×C₁₀₉	—	8	2	no	—
109	C₄×C₁₀₉	#407	1	1	yes	characteristic
110	C₄×C₁₀₉	#421	1	1	yes	characteristic
111	C₄×C₁₀₉	#510	1	2	yes	—
112	#11662	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂×C₁₀₉	—	2	2	no	—
114	C₂×C₁₀₉	#1486	1	1	yes	characteristic
115	C₂×C₁₀₉	#2141	1	1	yes	characteristic
116	C₂×C₁₀₉	C₂×#359	1	1	yes	characteristic
117	C₂×C₄×Q₁₆	C₂×C₁₀₉	1	1	yes	characteristic
118	#2162	C₂×C₁₀₉	1	1	yes	characteristic
119	#1568	C₂×C₁₀₉	1	1	yes	characteristic
120	#1332	C₂×C₁₀₉	1	4	yes	complemented
121	C₁₀₉	#11662	1	1	yes	characteristic, direct product factor, Sylow
122	C₂²×Q₁₆	C₄×C₁₀₉	1	1	yes	characteristic
123	C₂×C₄×Q₈	—	2	1	no	—
124	C₄×Q₁₆	—	2	1	no	—
125	C₄×Q₁₆	—	2	1	no	—
126	C₄×Q₁₆	C₄×C₁₀₉	1	2	yes	—
127	C₄×Q₁₆	C₂²×C₁₀₉	1	2	yes	—
128	#432	—	2	1	no	—
129	C₂×#107	C₄×C₁₀₉	1	1	yes	characteristic
130	C₂×#103	—	2	1	no	—
131	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
132	C₄⋊C₁₆	—	2	1	no	—
133	C₂²⋊C₁₆	—	2	1	no	—
134	C₄×C₁₆	C₂²×C₁₀₉	1	2	yes	—
135	#335	C₂²×C₁₀₉	1	2	yes	—
136	#334	—	2	1	no	—
137	C₂²×Q₈	—	2	1	no	—
138	C₂×Q₁₆	—	2	4	no	—
139	C₂×Q₁₆	—	2	1	no	—
140	C₂×Q₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
141	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
142	C₂²×C₈	D₄×C₁₀₉	1	1	yes	characteristic
143	C₄×Q₈	—	2	2	no	—
144	C₄×Q₈	—	2	2	no	—
145	C₄×Q₈	—	4	1	no	—
146	C₄×Q₈	—	4	1	no	—
147	C₂×C₄⋊C₄	—	2	1	no	—
148	C₂×C₄⋊C₄	—	4	1	no	—
149	C₂×C₄²	—	4	1	no	—
150	C₂×C₄²	D₄×C₁₀₉	1	1	yes	characteristic
151	C₂×C₁₆	—	2	2	no	—
152	#107	—	2	1	no	—
153	#107	C₂×C₄×C₁₀₉	1	2	yes	—
154	C₄⋊C₈	—	2	2	no	—
155	#103	—	2	2	no	—
156	#103	—	2	2	no	—
157	C₂²⋊C₈	—	4	1	no	—
158	#97	—	4	1	no	—
159	C₄×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
160	C₄×C₈	D₄×C₁₀₉	1	1	yes	characteristic
161	C₄×C₈	D₄×C₁₀₉	1	1	yes	characteristic
162	C₄×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
163	C₂×Q₈	—	2	4	no	—
164	C₂×Q₈	—	2	2	no	—
165	C₂×Q₈	—	4	2	no	—
166	C₂×Q₈	—	4	1	no	—
167	C₂²×C₄	—	4	1	no	—
168	C₂²×C₄	—	4	1	no	—
169	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
170	C₂²×C₄	D₈×C₁₀₉	1	1	yes	characteristic
171	C₂²×C₄	SD₁₆×C₁₀₉	1	1	yes	characteristic
172	Q₁₆	—	2	4	no	—
173	Q₁₆	—	2	2	no	—
174	Q₁₆	M₁₆×C₁₀₉	1	4	yes	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	4	1	no	—
177	C₂×C₈	—	4	2	no	—
178	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
179	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
180	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
181	C₂×C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	—	4	1	no	—
184	C₄⋊C₄	—	4	2	no	—
185	C₄⋊C₄	—	4	2	no	—
186	C₄²	—	4	2	no	—
187	C₄²	—	4	2	no	—
188	C₄²	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
189	C₄²	D₈×C₁₀₉	1	1	yes	characteristic
190	C₄²	SD₁₆×C₁₀₉	1	1	yes	characteristic
191	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
192	C₁₆	—	4	2	no	—
193	C₂³	#102×C₁₀₉	1	1	yes	characteristic
194	Q₈	—	2	4	no	—
195	Q₈	—	4	4	no	—
196	Q₈	—	4	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	4	4	no	—
202	C₂×C₄	—	4	2	no	—
203	C₂×C₄	—	4	2	no	—
204	C₂×C₄	—	8	2	no	—
205	C₂×C₄	#102×C₁₀₉	1	1	yes	characteristic
206	C₂×C₄	#102×C₁₀₉	1	1	yes	characteristic
207	C₂×C₄	#102×C₁₀₉	1	1	yes	characteristic
208	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
209	C₂×C₄	C₂×D₈×C₁₀₉	1	1	yes	characteristic
210	C₂×C₄	C₂×SD₁₆×C₁₀₉	1	1	yes	characteristic
211	C₈	—	2	1	no	—
212	C₈	—	2	1	no	—
213	C₈	—	2	1	no	—
214	C₈	—	8	2	no	—
215	C₈	C₂×M₁₆×C₁₀₉	1	2	yes	—
216	C₂²	—	2	1	no	—
217	C₂²	—	2	1	no	—
218	C₂²	#359×C₁₀₉	1	2	yes	—
219	C₂²	C₂×#102×C₁₀₉	1	1	yes	characteristic
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	4	4	no	—
225	C₄	—	8	2	no	—
226	C₄	#407×C₁₀₉	1	1	yes	characteristic
227	C₄	#421×C₁₀₉	1	1	yes	characteristic
228	C₄	#510×C₁₀₉	1	2	yes	—
229	C₂	—	2	2	no	—
230	C₂	C₁₀₉×#1486	1	1	yes	characteristic
231	C₂	C₁₀₉×#2141	1	1	yes	characteristic
232	C₂	C₂×#359×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₀₉	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₄×C₈	C₂³×C₁₀₉
socle	C₂²×C₁₀₉	C₂×#102
prime radical	C₂³×C₁₀₉	#102
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ C₂×C₄×C₈×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₀₉×#1332	C₂	semidirect
112. #11662	C₁₀₉	direct (N × H)
120. #1332	C₂×C₁₀₉	semidirect
121. C₁₀₉	#11662	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×Q₁₆×C₁₀₉	2
C₁₀₉×#2162	2
C₁₀₉×#1568	2
C₁₀₉×#1332	2
#11662	109

Subgroups — C109×#11662 #42129656

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#11662 #42129656