Subgroups — C₁₀₉×#9691 #42127685

Order 27904.

570 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₉×#2868	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#1842	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#333×C₁₀₉	C₂	1	4	yes	maximal
5	C₂×C₄⋊Q₈×C₁₀₉	C₄	1	4	yes	—
6	C₂×C₄⋊Q₈×C₁₀₉	C₂²	1	2	yes	—
7	C₂×C₄⋊C₈×C₁₀₉	—	2	2	no	—
8	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
9	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₁₀₉	—	2	1	no	—
11	#384×C₁₀₉	—	2	1	no	—
12	#384×C₁₀₉	—	2	1	no	—
13	#384×C₁₀₉	—	2	2	no	—
14	C₄³×C₁₀₉	C₂²	1	1	yes	characteristic
15	#336×C₁₀₉	—	2	1	no	—
16	#333×C₁₀₉	—	2	8	no	—
17	C₂²×Q₈×C₁₀₉	C₂×C₄	1	4	yes	—
18	C₂²×C₈×C₁₀₉	—	2	2	no	—
19	C₂²×C₈×C₁₀₉	—	2	2	no	—
20	C₄⋊Q₈×C₁₀₉	—	2	4	no	—
21	C₄⋊Q₈×C₁₀₉	—	2	4	no	—
22	C₄⋊Q₈×C₁₀₉	—	4	8	no	—
23	C₂×C₄⋊C₄×C₁₀₉	—	2	4	no	—
24	C₂×C₄⋊C₄×C₁₀₉	—	2	4	no	—
25	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
26	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
27	C₂×C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₁₀₉	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₀₉	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₀₉	D₄	1	2	yes	—
31	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
32	C₄⋊C₈×C₁₀₉	—	2	4	no	—
33	C₄⋊C₈×C₁₀₉	—	2	2	no	—
34	C₄⋊C₈×C₁₀₉	—	4	4	no	—
35	C₄⋊C₈×C₁₀₉	D₄	1	4	yes	—
36	C₄×C₈×C₁₀₉	—	4	2	no	—
37	C₂×Q₈×C₁₀₉	—	2	8	no	—
38	C₂×Q₈×C₁₀₉	—	4	8	no	—
39	C₂²×C₄×C₁₀₉	—	2	4	no	—
40	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
41	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
42	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈×C₁₀₉	—	2	4	no	—
46	C₂×C₈×C₁₀₉	—	4	2	no	—
47	C₂×C₈×C₁₀₉	—	4	2	no	—
48	C₂×C₈×C₁₀₉	—	4	2	no	—
49	C₂×C₈×C₁₀₉	—	4	4	no	—
50	C₄⋊C₄×C₁₀₉	—	4	8	no	—
51	C₄⋊C₄×C₁₀₉	—	4	8	no	—
52	C₄⋊C₄×C₁₀₉	—	4	4	no	—
53	C₄⋊C₄×C₁₀₉	—	4	2	no	—
54	C₄⋊C₄×C₁₀₉	—	4	2	no	—
55	C₄²×C₁₀₉	—	2	4	no	—
56	C₄²×C₁₀₉	—	2	2	no	—
57	C₄²×C₁₀₉	SD₁₆	1	2	yes	—
58	C₄²×C₁₀₉	SD₁₆	1	4	yes	—
59	C₄²×C₁₀₉	Q₁₆	1	4	yes	—
60	C₄²×C₁₀₉	Q₁₆	1	2	yes	—
61	C₄²×C₁₀₉	C₂×D₄	1	2	yes	—
62	C₄²×C₁₀₉	#41	1	2	yes	—
63	C₂³×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	Q₈×C₁₀₉	—	8	8	no	—
65	C₂×C₄×C₁₀₉	—	2	2	no	—
66	C₂×C₄×C₁₀₉	—	2	2	no	—
67	C₂×C₄×C₁₀₉	—	2	2	no	—
68	C₂×C₄×C₁₀₉	—	4	4	no	—
69	C₂×C₄×C₁₀₉	—	4	8	no	—
70	C₂×C₄×C₁₀₉	#100	1	4	yes	—
71	C₂×C₄×C₁₀₉	#103	1	4	yes	—
72	C₂×C₄×C₁₀₉	#103	1	4	yes	—
73	C₂×C₄×C₁₀₉	#103	1	4	yes	—
74	C₂×C₄×C₁₀₉	C₄×D₄	1	2	yes	—
75	C₂×C₄×C₁₀₉	#121	1	2	yes	—
76	C₂×C₄×C₁₀₉	#124	1	1	yes	characteristic
77	C₂×C₄×C₁₀₉	#127	1	1	yes	characteristic
78	C₂×C₄×C₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×C₄×C₁₀₉	C₂×SD₁₆	1	2	yes	—
80	C₂×C₄×C₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
81	C₂×C₄×C₁₀₉	C₂×Q₁₆	1	2	yes	—
82	C₂×C₄×C₁₀₉	#137	1	2	yes	—
83	C₈×C₁₀₉	—	4	4	no	—
84	C₈×C₁₀₉	—	8	2	no	—
85	C₂²×C₁₀₉	#390	1	1	yes	characteristic
86	C₂²×C₁₀₉	C₂×#100	1	2	yes	—
87	C₂²×C₁₀₉	C₂×#103	1	1	yes	characteristic
88	C₂²×C₁₀₉	C₂×#103	1	2	yes	—
89	C₂²×C₁₀₉	#419	1	1	yes	characteristic
90	C₄×C₁₀₉	—	2	4	no	—
91	C₄×C₁₀₉	—	2	2	no	—
92	C₄×C₁₀₉	—	8	4	no	—
93	C₄×C₁₀₉	#333	1	8	yes	complemented
94	C₄×C₁₀₉	#461	1	2	yes	—
95	C₄×C₁₀₉	#462	1	2	yes	—
96	C₄×C₁₀₉	#487	1	2	yes	—
97	C₄×C₁₀₉	#492	1	1	yes	characteristic
98	C₄×C₁₀₉	#494	1	1	yes	characteristic
99	#9691	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂×C₁₀₉	C₂×#333	1	2	yes	—
101	C₂×C₁₀₉	#1548	1	2	yes	—
102	C₂×C₁₀₉	#1956	1	1	yes	characteristic
103	C₂×C₁₀₉	#1965	1	1	yes	characteristic
104	C₂×C₁₀₉	#1981	1	1	yes	characteristic
105	#2868	C₂×C₁₀₉	1	1	yes	characteristic
106	#1842	C₂×C₁₀₉	1	1	yes	characteristic
107	C₄×C₄⋊C₈	C₂×C₁₀₉	1	1	yes	characteristic
108	C₂×#333	C₂×C₁₀₉	1	4	yes	—
109	C₁₀₉	#9691	1	1	yes	characteristic, direct product factor, Sylow
110	C₂×C₄⋊Q₈	C₄×C₁₀₉	1	4	yes	—
111	C₂×C₄⋊Q₈	C₂²×C₁₀₉	1	2	yes	—
112	C₂×C₄⋊C₈	—	2	2	no	—
113	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
114	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
115	C₂×C₄×C₈	—	2	1	no	—
116	#384	—	2	1	no	—
117	#384	—	2	1	no	—
118	#384	—	2	2	no	—
119	C₄³	C₂²×C₁₀₉	1	1	yes	characteristic
120	#336	—	2	1	no	—
121	#333	—	2	8	no	—
122	C₂²×Q₈	C₂×C₄×C₁₀₉	1	4	yes	—
123	C₂²×C₈	—	2	2	no	—
124	C₂²×C₈	—	2	2	no	—
125	C₄⋊Q₈	—	2	4	no	—
126	C₄⋊Q₈	—	2	4	no	—
127	C₄⋊Q₈	—	4	8	no	—
128	C₂×C₄⋊C₄	—	2	4	no	—
129	C₂×C₄⋊C₄	—	2	4	no	—
130	C₂×C₄⋊C₄	—	2	2	no	—
131	C₂×C₄⋊C₄	—	2	2	no	—
132	C₂×C₄²	C₂×C₄×C₁₀₉	1	2	yes	—
133	C₂×C₄²	D₄×C₁₀₉	1	1	yes	characteristic
134	C₂×C₄²	D₄×C₁₀₉	1	1	yes	characteristic
135	C₂×C₄²	D₄×C₁₀₉	1	2	yes	—
136	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
137	C₄⋊C₈	—	2	4	no	—
138	C₄⋊C₈	—	2	2	no	—
139	C₄⋊C₈	—	4	4	no	—
140	C₄⋊C₈	D₄×C₁₀₉	1	4	yes	—
141	C₄×C₈	—	4	2	no	—
142	C₂×Q₈	—	2	8	no	—
143	C₂×Q₈	—	4	8	no	—
144	C₂²×C₄	—	2	4	no	—
145	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	2	yes	—
146	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	2	yes	—
147	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
148	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
149	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
150	C₂×C₈	—	2	4	no	—
151	C₂×C₈	—	4	2	no	—
152	C₂×C₈	—	4	2	no	—
153	C₂×C₈	—	4	2	no	—
154	C₂×C₈	—	4	4	no	—
155	C₄⋊C₄	—	4	8	no	—
156	C₄⋊C₄	—	4	8	no	—
157	C₄⋊C₄	—	4	4	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄⋊C₄	—	4	2	no	—
160	C₄²	—	2	4	no	—
161	C₄²	—	2	2	no	—
162	C₄²	SD₁₆×C₁₀₉	1	2	yes	—
163	C₄²	SD₁₆×C₁₀₉	1	4	yes	—
164	C₄²	Q₁₆×C₁₀₉	1	4	yes	—
165	C₄²	Q₁₆×C₁₀₉	1	2	yes	—
166	C₄²	C₂×D₄×C₁₀₉	1	2	yes	—
167	C₄²	#41×C₁₀₉	1	2	yes	—
168	C₂³	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
169	Q₈	—	8	8	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	4	4	no	—
174	C₂×C₄	—	4	8	no	—
175	C₂×C₄	#100×C₁₀₉	1	4	yes	—
176	C₂×C₄	#103×C₁₀₉	1	4	yes	—
177	C₂×C₄	#103×C₁₀₉	1	4	yes	—
178	C₂×C₄	#103×C₁₀₉	1	4	yes	—
179	C₂×C₄	C₄×D₄×C₁₀₉	1	2	yes	—
180	C₂×C₄	#121×C₁₀₉	1	2	yes	—
181	C₂×C₄	#124×C₁₀₉	1	1	yes	characteristic
182	C₂×C₄	#127×C₁₀₉	1	1	yes	characteristic
183	C₂×C₄	C₂×SD₁₆×C₁₀₉	1	1	yes	characteristic
184	C₂×C₄	C₂×SD₁₆×C₁₀₉	1	2	yes	—
185	C₂×C₄	C₂×Q₁₆×C₁₀₉	1	1	yes	characteristic
186	C₂×C₄	C₂×Q₁₆×C₁₀₉	1	2	yes	—
187	C₂×C₄	#137×C₁₀₉	1	2	yes	—
188	C₈	—	4	4	no	—
189	C₈	—	8	2	no	—
190	C₂²	#390×C₁₀₉	1	1	yes	characteristic
191	C₂²	C₂×#100×C₁₀₉	1	2	yes	—
192	C₂²	C₂×#103×C₁₀₉	1	2	yes	—
193	C₂²	C₂×#103×C₁₀₉	1	1	yes	characteristic
194	C₂²	#419×C₁₀₉	1	1	yes	characteristic
195	C₄	—	2	2	no	—
196	C₄	—	2	4	no	—
197	C₄	—	8	4	no	—
198	C₄	#333×C₁₀₉	1	8	yes	complemented
199	C₄	#461×C₁₀₉	1	2	yes	—
200	C₄	#462×C₁₀₉	1	2	yes	—
201	C₄	#487×C₁₀₉	1	2	yes	—
202	C₄	#492×C₁₀₉	1	1	yes	characteristic
203	C₄	#494×C₁₀₉	1	1	yes	characteristic
204	C₂	C₂×#333×C₁₀₉	1	2	yes	—
205	C₂	C₁₀₉×#1548	1	2	yes	—
206	C₂	C₁₀₉×#1956	1	1	yes	characteristic
207	C₂	C₁₀₉×#1965	1	1	yes	characteristic
208	C₂	C₁₀₉×#1981	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₀₉	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₉
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₉ ◁ C₄³×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
93. C₄×C₁₀₉	#333	semidirect
99. #9691	C₁₀₉	direct (N × H)
109. C₁₀₉	#9691	direct (N × H)
198. C₄	#333×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₁₀₉×#2868	2
C₁₀₉×#1842	2
C₄×C₄⋊C₈×C₁₀₉	2
C₂×#333×C₁₀₉	2
#9691	109

Subgroups — C109×#9691 #42127685

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#9691 #42127685