Subgroups — C₁₀₉×#17692 #42135686

Order 27904.

872 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₉×#3504	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#2971	C₂	1	2	yes	maximal
3	C₁₀₉×#1537	C₂	1	3	yes	maximal, complemented
4	C₁₀₉×#1528	C₂	1	6	yes	maximal, complemented
5	C₁₀₉×#1526	C₂	1	3	yes	maximal, complemented
6	#557×C₁₀₉	—	2	2	no	—
7	#554×C₁₀₉	—	2	6	no	—
8	#544×C₁₀₉	C₄	1	6	yes	—
9	#533×C₁₀₉	C₂²	1	3	yes	—
10	C₂×#125×C₁₀₉	C₂²	1	3	yes	—
11	#520×C₁₀₉	C₄	1	2	yes	—
12	C₄×#41×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₈×Q₈×C₁₀₉	—	2	2	no	—
14	#436×C₁₀₉	—	2	6	no	—
15	#432×C₁₀₉	C₂²	1	6	yes	—
16	C₄×M₁₆×C₁₀₉	C₂²	1	6	yes	—
17	#330×C₁₀₉	C₂²	1	1	yes	characteristic
18	#330×C₁₀₉	C₂²	1	6	yes	—
19	#330×C₁₀₉	C₂²	1	3	yes	—
20	#330×C₁₀₉	C₂²	1	6	yes	—
21	C₂×#41×C₁₀₉	C₂×C₄	1	1	yes	characteristic
22	C₂×M₁₆×C₁₀₉	—	2	6	no	—
23	C₄⋊Q₈×C₁₀₉	—	2	6	no	—
24	C₄⋊Q₈×C₁₀₉	C₂×C₄	1	6	yes	—
25	#125×C₁₀₉	—	2	6	no	—
26	#125×C₁₀₉	—	2	6	no	—
27	#125×C₁₀₉	C₂×C₄	1	6	yes	—
28	#125×C₁₀₉	C₂³	1	3	yes	—
29	#125×C₁₀₉	C₂³	1	1	yes	characteristic
30	C₂²⋊Q₈×C₁₀₉	—	2	6	no	—
31	C₂²⋊Q₈×C₁₀₉	—	2	12	no	—
32	C₄×Q₈×C₁₀₉	—	2	2	no	—
33	C₄×Q₈×C₁₀₉	—	2	6	no	—
34	C₄×Q₈×C₁₀₉	D₄	1	2	yes	—
35	C₄×D₄×C₁₀₉	—	2	6	no	—
36	C₄×D₄×C₁₀₉	D₄	1	6	yes	—
37	#117×C₁₀₉	C₂×C₄	1	6	yes	—
38	#117×C₁₀₉	C₂×C₄	1	3	yes	—
39	C₂×C₄⋊C₄×C₁₀₉	—	2	6	no	—
40	C₂×C₄⋊C₄×C₁₀₉	C₂×C₄	1	6	yes	—
41	C₂×C₄²×C₁₀₉	C₂³	1	3	yes	—
42	C₄⋊C₈×C₁₀₉	—	2	6	no	—
43	C₂²⋊C₈×C₁₀₉	—	2	6	no	—
44	#97×C₁₀₉	C₂³	1	2	yes	—
45	#97×C₁₀₉	C₂³	1	6	yes	—
46	C₄×C₈×C₁₀₉	—	2	6	no	—
47	#41×C₁₀₉	—	2	4	no	—
48	C₂×Q₈×C₁₀₉	—	2	6	no	—
49	C₂×Q₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×D₄×C₁₀₉	C₂²⋊C₄	1	3	yes	—
51	C₂²×C₄×C₁₀₉	—	2	6	no	—
52	C₂²×C₄×C₁₀₉	C₂²×C₄	1	3	yes	—
53	C₂²×C₄×C₁₀₉	C₂×D₄	1	6	yes	—
54	M₁₆×C₁₀₉	—	4	12	no	—
55	C₂×C₈×C₁₀₉	—	2	2	no	—
56	C₂×C₈×C₁₀₉	—	2	6	no	—
57	C₄⋊C₄×C₁₀₉	—	2	12	no	—
58	C₄⋊C₄×C₁₀₉	—	2	6	no	—
59	C₄⋊C₄×C₁₀₉	—	2	6	no	—
60	C₄⋊C₄×C₁₀₉	—	2	6	no	—
61	C₄⋊C₄×C₁₀₉	—	2	2	no	—
62	C₄⋊C₄×C₁₀₉	C₂²⋊C₄	1	6	yes	—
63	C₄⋊C₄×C₁₀₉	C₂²×C₄	1	6	yes	—
64	C₄⋊C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
65	C₂²⋊C₄×C₁₀₉	—	2	6	no	—
66	C₂²⋊C₄×C₁₀₉	C₂²⋊C₄	1	6	yes	—
67	C₄²×C₁₀₉	—	2	6	no	—
68	C₄²×C₁₀₉	C₂²×C₄	1	3	yes	—
69	C₄²×C₁₀₉	C₂×D₄	1	6	yes	—
70	C₄²×C₁₀₉	C₂⁴	1	1	yes	characteristic
71	C₂³×C₁₀₉	C₂×C₂²⋊C₄	1	3	yes	—
72	Q₈×C₁₀₉	—	8	6	no	—
73	Q₈×C₁₀₉	#101	1	4	yes	—
74	D₄×C₁₀₉	—	2	6	no	—
75	C₂×C₄×C₁₀₉	—	2	6	no	—
76	C₂×C₄×C₁₀₉	—	2	2	no	—
77	C₂×C₄×C₁₀₉	—	2	6	no	—
78	C₂×C₄×C₁₀₉	—	4	6	no	—
79	C₂×C₄×C₁₀₉	—	8	6	no	—
80	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	6	yes	—
81	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	3	yes	—
82	C₂×C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
83	C₂×C₄×C₁₀₉	C₂²×D₄	1	2	yes	—
84	C₈×C₁₀₉	—	2	4	no	—
85	C₈×C₁₀₉	—	4	6	no	—
86	C₂²×C₁₀₉	—	2	3	no	—
87	C₂²×C₁₀₉	—	2	6	no	—
88	C₂²×C₁₀₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
89	C₄×C₁₀₉	—	2	1	no	—
90	C₄×C₁₀₉	—	2	2	no	—
91	C₄×C₁₀₉	—	2	6	no	—
92	C₄×C₁₀₉	—	4	6	no	—
93	C₄×C₁₀₉	—	4	2	no	—
94	C₄×C₁₀₉	C₂×#101	1	6	yes	—
95	#17692	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂×C₁₀₉	—	4	3	no	—
97	C₂×C₁₀₉	C₂²×#101	1	1	yes	characteristic
98	C₂×C₁₀₉	#2901	1	2	yes	—
99	#3504	C₂×C₁₀₉	1	1	yes	characteristic
100	#2971	C₂×C₁₀₉	1	2	yes	—
101	#1537	C₂×C₁₀₉	1	3	yes	complemented
102	#1528	C₂×C₁₀₉	1	6	yes	complemented
103	#1526	C₂×C₁₀₉	1	3	yes	complemented
104	C₁₀₉	#17692	1	1	yes	characteristic, direct product factor, Sylow
105	#557	—	2	2	no	—
106	#554	—	2	6	no	—
107	#544	C₄×C₁₀₉	1	6	yes	—
108	#533	C₂²×C₁₀₉	1	3	yes	—
109	C₂×#125	C₂²×C₁₀₉	1	3	yes	—
110	#520	C₄×C₁₀₉	1	2	yes	—
111	C₄×#41	C₂²×C₁₀₉	1	1	yes	characteristic
112	C₈×Q₈	—	2	2	no	—
113	#436	—	2	6	no	—
114	#432	C₂²×C₁₀₉	1	6	yes	—
115	C₄×M₁₆	C₂²×C₁₀₉	1	6	yes	—
116	#330	C₂²×C₁₀₉	1	3	yes	—
117	#330	C₂²×C₁₀₉	1	6	yes	—
118	#330	C₂²×C₁₀₉	1	1	yes	characteristic
119	#330	C₂²×C₁₀₉	1	6	yes	—
120	C₂×#41	C₂×C₄×C₁₀₉	1	1	yes	characteristic
121	C₂×M₁₆	—	2	6	no	—
122	C₄⋊Q₈	—	2	6	no	—
123	C₄⋊Q₈	C₂×C₄×C₁₀₉	1	6	yes	—
124	#125	—	2	6	no	—
125	#125	—	2	6	no	—
126	#125	C₂×C₄×C₁₀₉	1	6	yes	—
127	#125	C₂³×C₁₀₉	1	3	yes	—
128	#125	C₂³×C₁₀₉	1	1	yes	characteristic
129	C₂²⋊Q₈	—	2	6	no	—
130	C₂²⋊Q₈	—	2	12	no	—
131	C₄×Q₈	—	2	2	no	—
132	C₄×Q₈	—	2	6	no	—
133	C₄×Q₈	D₄×C₁₀₉	1	2	yes	—
134	C₄×D₄	—	2	6	no	—
135	C₄×D₄	D₄×C₁₀₉	1	6	yes	—
136	#117	C₂×C₄×C₁₀₉	1	6	yes	—
137	#117	C₂×C₄×C₁₀₉	1	3	yes	—
138	C₂×C₄⋊C₄	—	2	6	no	—
139	C₂×C₄⋊C₄	C₂×C₄×C₁₀₉	1	6	yes	—
140	C₂×C₄²	C₂³×C₁₀₉	1	3	yes	—
141	C₄⋊C₈	—	2	6	no	—
142	C₂²⋊C₈	—	2	6	no	—
143	#97	C₂³×C₁₀₉	1	2	yes	—
144	#97	C₂³×C₁₀₉	1	6	yes	—
145	C₄×C₈	—	2	6	no	—
146	#41	—	2	4	no	—
147	C₂×Q₈	—	2	6	no	—
148	C₂×Q₈	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
149	C₂×D₄	C₂²⋊C₄×C₁₀₉	1	3	yes	—
150	C₂²×C₄	—	2	6	no	—
151	C₂²×C₄	C₂²×C₄×C₁₀₉	1	3	yes	—
152	C₂²×C₄	C₂×D₄×C₁₀₉	1	6	yes	—
153	M₁₆	—	4	12	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	2	6	no	—
156	C₄⋊C₄	—	2	12	no	—
157	C₄⋊C₄	—	2	6	no	—
158	C₄⋊C₄	—	2	6	no	—
159	C₄⋊C₄	—	2	6	no	—
160	C₄⋊C₄	—	2	2	no	—
161	C₄⋊C₄	C₂²⋊C₄×C₁₀₉	1	6	yes	—
162	C₄⋊C₄	C₂²×C₄×C₁₀₉	1	6	yes	—
163	C₄⋊C₄	C₂²×C₄×C₁₀₉	1	2	yes	—
164	C₂²⋊C₄	—	2	6	no	—
165	C₂²⋊C₄	C₂²⋊C₄×C₁₀₉	1	6	yes	—
166	C₄²	—	2	6	no	—
167	C₄²	C₂²×C₄×C₁₀₉	1	3	yes	—
168	C₄²	C₂×D₄×C₁₀₉	1	6	yes	—
169	C₄²	C₂⁴×C₁₀₉	1	1	yes	characteristic
170	C₂³	C₂×C₂²⋊C₄×C₁₀₉	1	3	yes	—
171	Q₈	—	8	6	no	—
172	Q₈	#101×C₁₀₉	1	4	yes	—
173	D₄	—	2	6	no	—
174	C₂×C₄	—	2	6	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	6	no	—
177	C₂×C₄	—	4	6	no	—
178	C₂×C₄	—	8	6	no	—
179	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	6	yes	—
180	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	3	yes	—
181	C₂×C₄	C₂³×C₄×C₁₀₉	1	1	yes	characteristic
182	C₂×C₄	C₂²×D₄×C₁₀₉	1	2	yes	—
183	C₈	—	2	4	no	—
184	C₈	—	4	6	no	—
185	C₂²	—	2	3	no	—
186	C₂²	—	2	6	no	—
187	C₂²	C₂²×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
188	C₄	—	2	1	no	—
189	C₄	—	2	2	no	—
190	C₄	—	2	6	no	—
191	C₄	—	4	6	no	—
192	C₄	—	4	2	no	—
193	C₄	C₂×#101×C₁₀₉	1	6	yes	—
194	C₂	—	4	3	no	—
195	C₂	C₂²×#101×C₁₀₉	1	1	yes	characteristic
196	C₂	C₁₀₉×#2901	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₁₀₉×#1537	C₂	semidirect
4. C₁₀₉×#1528	C₂	semidirect
5. C₁₀₉×#1526	C₂	semidirect
95. #17692	C₁₀₉	direct (N × H)
101. #1537	C₂×C₁₀₉	semidirect
102. #1528	C₂×C₁₀₉	semidirect
103. #1526	C₂×C₁₀₉	semidirect
104. C₁₀₉	#17692	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×#3504	2
C₁₀₉×#2971	2
C₁₀₉×#1537	2
C₁₀₉×#1528	2
C₁₀₉×#1526	2
#17692	109

Subgroups — C109×#17692 #42135686

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#17692 #42135686