Subgroups — C₁₀₉×#8282 #42126276

Order 27904.

414 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#107×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#1770	C₂	1	2	yes	maximal
3	C₂×#335×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈²×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₁₀₉×#1279	C₂	1	2	yes	maximal
6	C₂×#107×C₁₀₉	C₄	1	2	yes	—
7	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
8	C₂×#97×C₁₀₉	—	2	2	no	—
9	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
12	C₄×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
13	#340×C₁₀₉	C₄	1	2	yes	—
14	#336×C₁₀₉	—	2	2	no	—
15	#335×C₁₀₉	C₄	1	4	yes	complemented
16	C₈²×C₁₀₉	C₄	1	4	yes	complemented
17	C₂²×C₈×C₁₀₉	—	2	1	no	—
18	C₂²×C₈×C₁₀₉	—	2	2	no	—
19	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₀₉	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₀₉	D₄	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₁₀₉	C₂×C₄	1	4	yes	—
24	C₂×C₄²×C₁₀₉	—	2	2	no	—
25	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
26	#107×C₁₀₉	—	2	4	no	—
27	C₄⋊C₈×C₁₀₉	—	2	2	no	—
28	C₄⋊C₈×C₁₀₉	C₂×C₄	1	4	yes	—
29	#97×C₁₀₉	—	4	4	no	—
30	C₄×C₈×C₁₀₉	—	2	1	no	—
31	C₄×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₁₀₉	Q₈	1	2	yes	—
36	#95×C₁₀₉	—	2	2	no	—
37	C₂²×C₄×C₁₀₉	—	2	4	no	—
38	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₀₉	#41	1	1	yes	characteristic
41	C₂×C₈×C₁₀₉	—	2	4	no	—
42	C₂×C₈×C₁₀₉	—	2	2	no	—
43	C₂×C₈×C₁₀₉	—	2	2	no	—
44	C₂×C₈×C₁₀₉	—	2	1	no	—
45	C₂×C₈×C₁₀₉	—	2	1	no	—
46	C₂×C₈×C₁₀₉	—	2	4	no	—
47	C₂×C₈×C₁₀₉	—	4	2	no	—
48	C₂×C₈×C₁₀₉	—	4	2	no	—
49	C₂×C₈×C₁₀₉	C₄²	1	2	yes	—
50	C₂×C₈×C₁₀₉	C₄²	1	2	yes	—
51	C₂×C₈×C₁₀₉	C₄⋊C₄	1	2	yes	—
52	C₂×C₈×C₁₀₉	C₄⋊C₄	1	2	yes	—
53	C₂×C₈×C₁₀₉	M₁₆	1	4	yes	—
54	C₂×C₈×C₁₀₉	D₈	1	2	yes	—
55	C₂×C₈×C₁₀₉	Q₁₆	1	2	yes	—
56	C₄⋊C₄×C₁₀₉	—	2	4	no	—
57	C₄⋊C₄×C₁₀₉	—	2	4	no	—
58	C₄²×C₁₀₉	—	4	2	no	—
59	C₄²×C₁₀₉	—	4	2	no	—
60	C₄²×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
61	C₄²×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
62	C₄²×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
63	C₄²×C₁₀₉	#41	1	1	yes	characteristic
64	C₂³×C₁₀₉	C₄×D₄	1	1	yes	characteristic
65	C₂×C₄×C₁₀₉	—	2	2	no	—
66	C₂×C₄×C₁₀₉	—	4	4	no	—
67	C₂×C₄×C₁₀₉	—	4	4	no	—
68	C₂×C₄×C₁₀₉	—	4	4	no	—
69	C₂×C₄×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
70	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₁₀₉	#117	1	1	yes	characteristic
72	C₂×C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
73	C₂×C₄×C₁₀₉	C₄×Q₈	1	1	yes	characteristic
74	C₂×C₄×C₁₀₉	C₄×Q₈	1	1	yes	characteristic
75	C₂×C₄×C₁₀₉	C₂×M₁₆	1	2	yes	—
76	C₂×C₄×C₁₀₉	#131	1	2	yes	—
77	C₂×C₄×C₁₀₉	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
79	C₂×C₄×C₁₀₉	#135	1	1	yes	characteristic
80	C₂×C₄×C₁₀₉	#135	1	1	yes	characteristic
81	C₈×C₁₀₉	—	2	4	no	—
82	C₈×C₁₀₉	—	2	2	no	—
83	C₈×C₁₀₉	—	2	2	no	—
84	C₈×C₁₀₉	—	4	4	no	—
85	C₈×C₁₀₉	#97	1	4	yes	complemented
86	C₈×C₁₀₉	#107	1	4	yes	complemented
87	C₂²×C₁₀₉	C₄×C₄⋊C₄	1	1	yes	characteristic
88	C₂²×C₁₀₉	#436	1	2	yes	—
89	C₂²×C₁₀₉	C₄×D₈	1	1	yes	characteristic
90	C₂²×C₁₀₉	C₄×Q₁₆	1	1	yes	characteristic
91	C₂²×C₁₀₉	#443	1	2	yes	—
92	C₄×C₁₀₉	—	2	1	no	—
93	C₄×C₁₀₉	—	2	1	no	—
94	C₄×C₁₀₉	—	8	4	no	—
95	C₄×C₁₀₉	C₂×#97	1	1	yes	characteristic
96	C₄×C₁₀₉	#405	1	1	yes	characteristic
97	C₄×C₁₀₉	C₂×#107	1	1	yes	characteristic
98	C₄×C₁₀₉	#427	1	1	yes	characteristic
99	C₄×C₁₀₉	C₄×D₈	1	1	yes	characteristic
100	C₄×C₁₀₉	C₄×Q₁₆	1	1	yes	characteristic
101	C₄×C₁₀₉	#446	1	2	yes	—
102	#8282	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₂×C₁₀₉	#1589	1	2	yes	—
104	C₂×C₁₀₉	#1591	1	2	yes	—
105	C₂×C₁₀₉	#1770	1	1	yes	characteristic
106	C₂×C₁₀₉	#1772	1	1	yes	characteristic
107	C₂×C₁₀₉	C₄×#107	1	1	yes	characteristic
108	C₄×#107	C₂×C₁₀₉	1	1	yes	characteristic
109	#1770	C₂×C₁₀₉	1	2	yes	—
110	C₂×#335	C₂×C₁₀₉	1	1	yes	characteristic
111	C₂×C₈²	C₂×C₁₀₉	1	1	yes	characteristic
112	#1279	C₂×C₁₀₉	1	2	yes	—
113	C₁₀₉	#8282	1	1	yes	characteristic, direct product factor, Sylow
114	C₂×#107	C₄×C₁₀₉	1	2	yes	—
115	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
116	C₂×#97	—	2	2	no	—
117	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
118	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
119	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
120	C₄×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
121	#340	C₄×C₁₀₉	1	2	yes	—
122	#336	—	2	2	no	—
123	#335	C₄×C₁₀₉	1	4	yes	complemented
124	C₈²	C₄×C₁₀₉	1	4	yes	complemented
125	C₂²×C₈	—	2	1	no	—
126	C₂²×C₈	—	2	2	no	—
127	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
128	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
129	C₂²×C₈	D₄×C₁₀₉	1	1	yes	characteristic
130	C₂²×C₈	D₄×C₁₀₉	1	1	yes	characteristic
131	C₂×C₄⋊C₄	C₂×C₄×C₁₀₉	1	4	yes	—
132	C₂×C₄²	—	2	2	no	—
133	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
134	#107	—	2	4	no	—
135	C₄⋊C₈	—	2	2	no	—
136	C₄⋊C₈	C₂×C₄×C₁₀₉	1	4	yes	—
137	#97	—	4	4	no	—
138	C₄×C₈	—	2	1	no	—
139	C₄×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
140	C₄×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
141	C₄×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
142	C₄×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
143	C₄×C₈	Q₈×C₁₀₉	1	2	yes	—
144	#95	—	2	2	no	—
145	C₂²×C₄	—	2	4	no	—
146	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
147	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
148	C₂²×C₄	#41×C₁₀₉	1	1	yes	characteristic
149	C₂×C₈	—	2	4	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	2	1	no	—
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	—	2	4	no	—
155	C₂×C₈	—	4	2	no	—
156	C₂×C₈	—	4	2	no	—
157	C₂×C₈	C₄²×C₁₀₉	1	2	yes	—
158	C₂×C₈	C₄²×C₁₀₉	1	2	yes	—
159	C₂×C₈	C₄⋊C₄×C₁₀₉	1	2	yes	—
160	C₂×C₈	C₄⋊C₄×C₁₀₉	1	2	yes	—
161	C₂×C₈	M₁₆×C₁₀₉	1	4	yes	—
162	C₂×C₈	D₈×C₁₀₉	1	2	yes	—
163	C₂×C₈	Q₁₆×C₁₀₉	1	2	yes	—
164	C₄⋊C₄	—	2	4	no	—
165	C₄⋊C₄	—	2	4	no	—
166	C₄²	—	4	2	no	—
167	C₄²	—	4	2	no	—
168	C₄²	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
169	C₄²	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
170	C₄²	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
171	C₄²	#41×C₁₀₉	1	1	yes	characteristic
172	C₂³	C₄×D₄×C₁₀₉	1	1	yes	characteristic
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	4	4	no	—
175	C₂×C₄	—	4	4	no	—
176	C₂×C₄	—	4	4	no	—
177	C₂×C₄	C₂×C₄²×C₁₀₉	1	1	yes	characteristic
178	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
179	C₂×C₄	#117×C₁₀₉	1	1	yes	characteristic
180	C₂×C₄	C₄×D₄×C₁₀₉	1	1	yes	characteristic
181	C₂×C₄	C₄×Q₈×C₁₀₉	1	1	yes	characteristic
182	C₂×C₄	C₄×Q₈×C₁₀₉	1	1	yes	characteristic
183	C₂×C₄	C₂×M₁₆×C₁₀₉	1	2	yes	—
184	C₂×C₄	#131×C₁₀₉	1	2	yes	—
185	C₂×C₄	C₂×D₈×C₁₀₉	1	1	yes	characteristic
186	C₂×C₄	C₂×Q₁₆×C₁₀₉	1	1	yes	characteristic
187	C₂×C₄	#135×C₁₀₉	1	1	yes	characteristic
188	C₂×C₄	#135×C₁₀₉	1	1	yes	characteristic
189	C₈	—	2	4	no	—
190	C₈	—	2	2	no	—
191	C₈	—	2	2	no	—
192	C₈	—	4	4	no	—
193	C₈	#97×C₁₀₉	1	4	yes	complemented
194	C₈	#107×C₁₀₉	1	4	yes	complemented
195	C₂²	C₄×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
196	C₂²	#436×C₁₀₉	1	2	yes	—
197	C₂²	C₄×D₈×C₁₀₉	1	1	yes	characteristic
198	C₂²	C₄×Q₁₆×C₁₀₉	1	1	yes	characteristic
199	C₂²	#443×C₁₀₉	1	2	yes	—
200	C₄	—	2	1	no	—
201	C₄	—	2	1	no	—
202	C₄	—	8	4	no	—
203	C₄	C₂×#97×C₁₀₉	1	1	yes	characteristic
204	C₄	#405×C₁₀₉	1	1	yes	characteristic
205	C₄	C₂×#107×C₁₀₉	1	1	yes	characteristic
206	C₄	#427×C₁₀₉	1	1	yes	characteristic
207	C₄	C₄×D₈×C₁₀₉	1	1	yes	characteristic
208	C₄	C₄×Q₁₆×C₁₀₉	1	1	yes	characteristic
209	C₄	#446×C₁₀₉	1	2	yes	—
210	C₂	C₁₀₉×#1589	1	2	yes	—
211	C₂	C₁₀₉×#1591	1	2	yes	—
212	C₂	C₁₀₉×#1770	1	1	yes	characteristic
213	C₂	C₁₀₉×#1772	1	1	yes	characteristic
214	C₂	C₄×#107×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₀₉	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₁₀₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₉
socle, prime radical	C₂³×C₁₀₉	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₀₉ ◁ C₂×C₄×C₈×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
15. #335×C₁₀₉	C₄	semidirect
16. C₈²×C₁₀₉	C₄	semidirect
85. C₈×C₁₀₉	#97	semidirect
86. C₈×C₁₀₉	#107	semidirect
102. #8282	C₁₀₉	direct (N × H)
113. C₁₀₉	#8282	direct (N × H)
123. #335	C₄×C₁₀₉	semidirect
124. C₈²	C₄×C₁₀₉	semidirect
193. C₈	#97×C₁₀₉	semidirect
194. C₈	#107×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₄×#107×C₁₀₉	2
C₁₀₉×#1770	2
C₂×#335×C₁₀₉	2
C₂×C₈²×C₁₀₉	2
C₁₀₉×#1279	2
#8282	109

Subgroups — C109×#8282 #42126276

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#8282 #42126276