Subgroups — C₁₀₉×#17837 #42135831

Order 27904.

612 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₉×#2991	C₂	1	2	yes	maximal
2	C₈×#41×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#1574	C₂	1	6	yes	maximal, complemented
4	C₁₀₉×#1567	C₂	1	3	yes	maximal, complemented
5	C₁₀₉×#1564	C₂	1	3	yes	maximal, complemented
6	C₄×#41×C₁₀₉	C₂²	1	1	yes	characteristic
7	C₈×Q₈×C₁₀₉	—	2	2	no	—
8	C₈×Q₈×C₁₀₉	C₄	1	1	yes	characteristic
9	C₈×Q₈×C₁₀₉	C₄	1	1	yes	characteristic
10	#436×C₁₀₉	—	2	6	no	—
11	C₈×D₄×C₁₀₉	C₄	1	3	yes	—
12	C₈×D₄×C₁₀₉	C₄	1	3	yes	—
13	#431×C₁₀₉	C₂²	1	6	yes	—
14	#431×C₁₀₉	C₂²	1	3	yes	—
15	#423×C₁₀₉	C₂²	1	6	yes	—
16	C₂×C₄×C₈×C₁₀₉	C₂²	1	3	yes	—
17	#335×C₁₀₉	C₂²	1	1	yes	characteristic
18	#335×C₁₀₉	C₂²	1	3	yes	—
19	#335×C₁₀₉	C₂²	1	6	yes	—
20	#334×C₁₀₉	C₂²	1	6	yes	—
21	C₂×#41×C₁₀₉	Q₈	1	1	yes	characteristic
22	C₂×M₁₆×C₁₀₉	—	2	6	no	—
23	C₂²×C₈×C₁₀₉	C₂×C₄	1	3	yes	—
24	C₂²×C₈×C₁₀₉	C₂×C₄	1	3	yes	—
25	C₄×Q₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
26	C₄×Q₈×C₁₀₉	D₄	1	1	yes	characteristic
27	C₄×D₄×C₁₀₉	C₂×C₄	1	3	yes	—
28	C₄×D₄×C₁₀₉	D₄	1	3	yes	—
29	#117×C₁₀₉	Q₈	1	3	yes	—
30	C₂×C₄²×C₁₀₉	C₂³	1	3	yes	—
31	C₄⋊C₈×C₁₀₉	—	2	6	no	—
32	C₄⋊C₈×C₁₀₉	C₂×C₄	1	3	yes	—
33	C₄⋊C₈×C₁₀₉	C₂×C₄	1	3	yes	—
34	C₄⋊C₈×C₁₀₉	C₂³	1	2	yes	—
35	C₄⋊C₈×C₁₀₉	C₂³	1	6	yes	—
36	C₂²⋊C₈×C₁₀₉	—	2	6	no	—
37	C₂²⋊C₈×C₁₀₉	C₂×C₄	1	6	yes	—
38	C₄×C₈×C₁₀₉	—	2	6	no	—
39	C₄×C₈×C₁₀₉	C₂×C₄	1	3	yes	—
40	C₄×C₈×C₁₀₉	C₂×C₄	1	3	yes	—
41	C₄×C₈×C₁₀₉	C₂³	1	3	yes	—
42	C₄×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
43	#41×C₁₀₉	—	2	4	no	—
44	C₂×Q₈×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
45	C₂×D₄×C₁₀₉	C₄⋊C₄	1	3	yes	—
46	C₂²×C₄×C₁₀₉	C₂²×C₄	1	3	yes	—
47	C₂²×C₄×C₁₀₉	C₂×D₄	1	3	yes	—
48	C₂²×C₄×C₁₀₉	C₂×Q₈	1	3	yes	—
49	M₁₆×C₁₀₉	—	8	6	no	—
50	C₂×C₈×C₁₀₉	—	2	6	no	—
51	C₂×C₈×C₁₀₉	—	2	2	no	—
52	C₂×C₈×C₁₀₉	—	2	6	no	—
53	C₂×C₈×C₁₀₉	—	4	3	no	—
54	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₁₀₉	C₂²×C₄	1	3	yes	—
56	C₂×C₈×C₁₀₉	C₂²×C₄	1	3	yes	—
57	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
58	C₄⋊C₄×C₁₀₉	C₄⋊C₄	1	3	yes	—
59	C₄⋊C₄×C₁₀₉	C₄⋊C₄	1	3	yes	—
60	C₂²⋊C₄×C₁₀₉	C₄⋊C₄	1	6	yes	—
61	C₄²×C₁₀₉	C₂²×C₄	1	3	yes	—
62	C₄²×C₁₀₉	C₂×D₄	1	3	yes	—
63	C₄²×C₁₀₉	C₂×Q₈	1	3	yes	—
64	C₄²×C₁₀₉	C₂⁴	1	1	yes	characteristic
65	C₂³×C₁₀₉	C₂×C₄⋊C₄	1	3	yes	—
66	Q₈×C₁₀₉	#108	1	4	yes	—
67	D₄×C₁₀₉	—	2	6	no	—
68	C₂×C₄×C₁₀₉	—	2	3	no	—
69	C₂×C₄×C₁₀₉	—	2	3	no	—
70	C₂×C₄×C₁₀₉	—	2	3	no	—
71	C₂×C₄×C₁₀₉	—	2	3	no	—
72	C₂×C₄×C₁₀₉	—	2	6	no	—
73	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	3	yes	—
74	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	3	yes	—
75	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	3	yes	—
76	C₂×C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
77	C₂×C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
78	C₂×C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
79	C₈×C₁₀₉	—	2	3	no	—
80	C₈×C₁₀₉	—	2	1	no	—
81	C₈×C₁₀₉	—	2	3	no	—
82	C₈×C₁₀₉	—	4	6	no	—
83	C₈×C₁₀₉	—	4	2	no	—
84	C₈×C₁₀₉	#131	1	2	yes	—
85	C₂²×C₁₀₉	—	2	3	no	—
86	C₂²×C₁₀₉	—	2	3	no	—
87	C₂²×C₁₀₉	—	2	3	no	—
88	C₂²×C₁₀₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
89	C₄×C₁₀₉	—	2	3	no	—
90	C₄×C₁₀₉	—	2	3	no	—
91	C₄×C₁₀₉	—	2	1	no	—
92	C₄×C₁₀₉	C₂×#108	1	6	yes	—
93	C₄×C₁₀₉	C₂×#131	1	1	yes	characteristic
94	C₄×C₁₀₉	#568	1	1	yes	characteristic
95	C₄×C₁₀₉	#576	1	1	yes	characteristic
96	C₄×C₁₀₉	#578	1	1	yes	characteristic
97	#17837	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₂×C₁₀₉	—	4	3	no	—
99	C₂×C₁₀₉	#2906	1	1	yes	characteristic
100	C₂×C₁₀₉	#2913	1	1	yes	characteristic
101	C₂×C₁₀₉	C₂²×#108	1	1	yes	characteristic
102	#2991	C₂×C₁₀₉	1	2	yes	—
103	C₈×#41	C₂×C₁₀₉	1	1	yes	characteristic
104	#1574	C₂×C₁₀₉	1	6	yes	complemented
105	#1567	C₂×C₁₀₉	1	3	yes	complemented
106	#1564	C₂×C₁₀₉	1	3	yes	complemented
107	C₁₀₉	#17837	1	1	yes	characteristic, direct product factor, Sylow
108	C₄×#41	C₂²×C₁₀₉	1	1	yes	characteristic
109	C₈×Q₈	—	2	2	no	—
110	C₈×Q₈	C₄×C₁₀₉	1	1	yes	characteristic
111	C₈×Q₈	C₄×C₁₀₉	1	1	yes	characteristic
112	#436	—	2	6	no	—
113	C₈×D₄	C₄×C₁₀₉	1	3	yes	—
114	C₈×D₄	C₄×C₁₀₉	1	3	yes	—
115	#431	C₂²×C₁₀₉	1	6	yes	—
116	#431	C₂²×C₁₀₉	1	3	yes	—
117	#423	C₂²×C₁₀₉	1	6	yes	—
118	C₂×C₄×C₈	C₂²×C₁₀₉	1	3	yes	—
119	#335	C₂²×C₁₀₉	1	1	yes	characteristic
120	#335	C₂²×C₁₀₉	1	3	yes	—
121	#335	C₂²×C₁₀₉	1	6	yes	—
122	#334	C₂²×C₁₀₉	1	6	yes	—
123	C₂×#41	Q₈×C₁₀₉	1	1	yes	characteristic
124	C₂×M₁₆	—	2	6	no	—
125	C₂²×C₈	C₂×C₄×C₁₀₉	1	3	yes	—
126	C₂²×C₈	C₂×C₄×C₁₀₉	1	3	yes	—
127	C₄×Q₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
128	C₄×Q₈	D₄×C₁₀₉	1	1	yes	characteristic
129	C₄×D₄	C₂×C₄×C₁₀₉	1	3	yes	—
130	C₄×D₄	D₄×C₁₀₉	1	3	yes	—
131	#117	Q₈×C₁₀₉	1	3	yes	—
132	C₂×C₄²	C₂³×C₁₀₉	1	3	yes	—
133	C₄⋊C₈	—	2	6	no	—
134	C₄⋊C₈	C₂×C₄×C₁₀₉	1	3	yes	—
135	C₄⋊C₈	C₂×C₄×C₁₀₉	1	3	yes	—
136	C₄⋊C₈	C₂³×C₁₀₉	1	2	yes	—
137	C₄⋊C₈	C₂³×C₁₀₉	1	6	yes	—
138	C₂²⋊C₈	—	2	6	no	—
139	C₂²⋊C₈	C₂×C₄×C₁₀₉	1	6	yes	—
140	C₄×C₈	—	2	6	no	—
141	C₄×C₈	C₂×C₄×C₁₀₉	1	3	yes	—
142	C₄×C₈	C₂×C₄×C₁₀₉	1	3	yes	—
143	C₄×C₈	C₂³×C₁₀₉	1	3	yes	—
144	C₄×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
145	#41	—	2	4	no	—
146	C₂×Q₈	C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
147	C₂×D₄	C₄⋊C₄×C₁₀₉	1	3	yes	—
148	C₂²×C₄	C₂²×C₄×C₁₀₉	1	3	yes	—
149	C₂²×C₄	C₂×D₄×C₁₀₉	1	3	yes	—
150	C₂²×C₄	C₂×Q₈×C₁₀₉	1	3	yes	—
151	M₁₆	—	8	6	no	—
152	C₂×C₈	—	2	6	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	2	6	no	—
155	C₂×C₈	—	4	3	no	—
156	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
157	C₂×C₈	C₂²×C₄×C₁₀₉	1	3	yes	—
158	C₂×C₈	C₂²×C₄×C₁₀₉	1	3	yes	—
159	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
160	C₄⋊C₄	C₄⋊C₄×C₁₀₉	1	3	yes	—
161	C₄⋊C₄	C₄⋊C₄×C₁₀₉	1	3	yes	—
162	C₂²⋊C₄	C₄⋊C₄×C₁₀₉	1	6	yes	—
163	C₄²	C₂²×C₄×C₁₀₉	1	3	yes	—
164	C₄²	C₂×D₄×C₁₀₉	1	3	yes	—
165	C₄²	C₂×Q₈×C₁₀₉	1	3	yes	—
166	C₄²	C₂⁴×C₁₀₉	1	1	yes	characteristic
167	C₂³	C₂×C₄⋊C₄×C₁₀₉	1	3	yes	—
168	Q₈	#108×C₁₀₉	1	4	yes	—
169	D₄	—	2	6	no	—
170	C₂×C₄	—	2	3	no	—
171	C₂×C₄	—	2	3	no	—
172	C₂×C₄	—	2	3	no	—
173	C₂×C₄	—	2	3	no	—
174	C₂×C₄	—	2	6	no	—
175	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	3	yes	—
176	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	3	yes	—
177	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	3	yes	—
178	C₂×C₄	C₂³×C₄×C₁₀₉	1	1	yes	characteristic
179	C₂×C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
180	C₂×C₄	C₂²×Q₈×C₁₀₉	1	1	yes	characteristic
181	C₈	—	2	3	no	—
182	C₈	—	2	1	no	—
183	C₈	—	2	3	no	—
184	C₈	—	4	6	no	—
185	C₈	—	4	2	no	—
186	C₈	#131×C₁₀₉	1	2	yes	—
187	C₂²	—	2	3	no	—
188	C₂²	—	2	3	no	—
189	C₂²	—	2	3	no	—
190	C₂²	C₂²×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
191	C₄	—	2	3	no	—
192	C₄	—	2	3	no	—
193	C₄	—	2	1	no	—
194	C₄	C₂×#108×C₁₀₉	1	6	yes	—
195	C₄	C₂×#131×C₁₀₉	1	1	yes	characteristic
196	C₄	#568×C₁₀₉	1	1	yes	characteristic
197	C₄	#576×C₁₀₉	1	1	yes	characteristic
198	C₄	#578×C₁₀₉	1	1	yes	characteristic
199	C₂	—	4	3	no	—
200	C₂	C₁₀₉×#2906	1	1	yes	characteristic
201	C₂	C₁₀₉×#2913	1	1	yes	characteristic
202	C₂	C₂²×#108×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₀₉	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₀₉
Frattini, cosocle	C₄²	C₂⁴×C₁₀₉
socle	C₂²×C₁₀₉	C₂²×C₄⋊C₄
prime radical	C₂×#41×C₁₀₉	Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₉ ◁ C₄×#41×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₀₉×#1574	C₂	semidirect
4. C₁₀₉×#1567	C₂	semidirect
5. C₁₀₉×#1564	C₂	semidirect
97. #17837	C₁₀₉	direct (N × H)
104. #1574	C₂×C₁₀₉	semidirect
105. #1567	C₂×C₁₀₉	semidirect
106. #1564	C₂×C₁₀₉	semidirect
107. C₁₀₉	#17837	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₀₉×#2991	2
C₈×#41×C₁₀₉	2
C₁₀₉×#1574	2
C₁₀₉×#1567	2
C₁₀₉×#1564	2
#17837	109

Subgroups — C109×#17837 #42135831

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#17837 #42135831