Subgroups — C₁₀₉×#12674 #42130668

Order 27904.

358 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#367×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#366×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#346×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₄×#107×C₁₀₉	C₂	1	2	yes	maximal
5	C₁₀₉×#1380	C₂	1	2	yes	maximal
6	C₂²×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
7	C₂²×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
8	C₂×#107×C₁₀₉	C₂²	1	2	yes	—
9	C₂×#107×C₁₀₉	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₄×C₄⋊C₄×C₁₀₉	—	2	2	no	—
12	#367×C₁₀₉	C₄	1	4	yes	complemented
13	#366×C₁₀₉	C₄	1	4	yes	complemented
14	#346×C₁₀₉	C₄	1	4	yes	complemented
15	#340×C₁₀₉	—	2	2	no	—
16	C₂²×C₈×C₁₀₉	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
19	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
20	C₂×C₄²×C₁₀₉	—	4	2	no	—
21	C₂×C₄²×C₁₀₉	D₄	1	1	yes	characteristic
22	C₂×C₁₆×C₁₀₉	—	2	1	no	—
23	C₂×C₁₆×C₁₀₉	—	2	1	no	—
24	C₂×C₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
25	C₂×C₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
26	C₂×C₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
27	C₂×C₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
28	#107×C₁₀₉	—	2	2	no	—
29	#107×C₁₀₉	—	2	2	no	—
30	#107×C₁₀₉	C₂×C₄	1	4	yes	—
31	#107×C₁₀₉	C₂×C₄	1	4	yes	—
32	C₄×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₁₀₉	Q₈	1	2	yes	—
34	#95×C₁₀₉	—	4	2	no	—
35	C₂²×C₄×C₁₀₉	—	4	2	no	—
36	C₂²×C₄×C₁₀₉	—	4	2	no	—
37	C₂²×C₄×C₁₀₉	D₈	1	2	yes	—
38	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₁₀₉	—	2	1	no	—
40	C₂×C₈×C₁₀₉	—	2	1	no	—
41	C₂×C₈×C₁₀₉	C₄⋊C₄	1	2	yes	—
42	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₈×C₁₀₉	#41	1	1	yes	characteristic
47	C₂×C₈×C₁₀₉	#41	1	1	yes	characteristic
48	C₄⋊C₄×C₁₀₉	—	2	4	no	—
49	C₄⋊C₄×C₁₀₉	—	2	4	no	—
50	C₄⋊C₄×C₁₀₉	—	4	2	no	—
51	C₄⋊C₄×C₁₀₉	—	4	2	no	—
52	C₄²×C₁₀₉	—	8	4	no	—
53	C₄²×C₁₀₉	C₄⋊C₄	1	2	yes	—
54	C₄²×C₁₀₉	Q₁₆	1	2	yes	—
55	C₁₆×C₁₀₉	—	2	2	no	—
56	C₁₆×C₁₀₉	C₄²	1	4	yes	complemented
57	C₂³×C₁₀₉	C₂×D₈	1	1	yes	characteristic
58	C₂×C₄×C₁₀₉	—	2	2	no	—
59	C₂×C₄×C₁₀₉	—	2	2	no	—
60	C₂×C₄×C₁₀₉	—	4	4	no	—
61	C₂×C₄×C₁₀₉	—	4	4	no	—
62	C₂×C₄×C₁₀₉	—	8	2	no	—
63	C₂×C₄×C₁₀₉	—	8	2	no	—
64	C₂×C₄×C₁₀₉	—	8	2	no	—
65	C₂×C₄×C₁₀₉	—	8	2	no	—
66	C₂×C₄×C₁₀₉	#107	1	4	yes	—
67	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
71	C₂×C₄×C₁₀₉	#135	1	1	yes	characteristic
72	C₂×C₄×C₁₀₉	#135	1	1	yes	characteristic
73	C₈×C₁₀₉	—	2	2	no	—
74	C₈×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
75	C₈×C₁₀₉	#117	1	1	yes	characteristic
76	C₈×C₁₀₉	C₄×Q₈	1	1	yes	characteristic
77	C₈×C₁₀₉	C₄×Q₈	1	1	yes	characteristic
78	C₂²×C₁₀₉	C₂×#107	1	1	yes	characteristic
79	C₂²×C₁₀₉	C₄×D₈	1	1	yes	characteristic
80	C₂²×C₁₀₉	C₄×D₈	1	1	yes	characteristic
81	C₂²×C₁₀₉	#509	1	2	yes	—
82	C₂²×C₁₀₉	#510	1	2	yes	—
83	C₄×C₁₀₉	—	2	4	no	—
84	C₄×C₁₀₉	—	8	4	no	—
85	C₄×C₁₀₉	—	8	4	no	—
86	C₄×C₁₀₉	C₄×C₄⋊C₄	1	1	yes	characteristic
87	C₄×C₁₀₉	#427	1	1	yes	characteristic
88	C₄×C₁₀₉	C₄×Q₁₆	1	1	yes	characteristic
89	C₄×C₁₀₉	C₄×Q₁₆	1	1	yes	characteristic
90	#12674	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂×C₁₀₉	C₄×#107	1	1	yes	characteristic
92	C₂×C₁₀₉	#2159	1	2	yes	—
93	C₂×C₁₀₉	#2175	1	2	yes	—
94	C₂×C₁₀₉	#2176	1	1	yes	characteristic
95	C₂×C₁₀₉	#2177	1	1	yes	characteristic
96	C₂×#367	C₂×C₁₀₉	1	1	yes	characteristic
97	C₂×#366	C₂×C₁₀₉	1	1	yes	characteristic
98	C₂×#346	C₂×C₁₀₉	1	1	yes	characteristic
99	C₄×#107	C₂×C₁₀₉	1	2	yes	—
100	#1380	C₂×C₁₀₉	1	2	yes	—
101	C₁₀₉	#12674	1	1	yes	characteristic, direct product factor, Sylow
102	C₂²×C₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
103	C₂²×C₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
104	C₂×#107	C₂²×C₁₀₉	1	2	yes	—
105	C₂×#107	C₂²×C₁₀₉	1	2	yes	—
106	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
107	C₄×C₄⋊C₄	—	2	2	no	—
108	#367	C₄×C₁₀₉	1	4	yes	complemented
109	#366	C₄×C₁₀₉	1	4	yes	complemented
110	#346	C₄×C₁₀₉	1	4	yes	complemented
111	#340	—	2	2	no	—
112	C₂²×C₈	D₄×C₁₀₉	1	1	yes	characteristic
113	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄⋊C₄	—	2	2	no	—
116	C₂×C₄²	—	4	2	no	—
117	C₂×C₄²	D₄×C₁₀₉	1	1	yes	characteristic
118	C₂×C₁₆	—	2	1	no	—
119	C₂×C₁₆	—	2	1	no	—
120	C₂×C₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
121	C₂×C₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
122	C₂×C₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
123	C₂×C₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
124	#107	—	2	2	no	—
125	#107	—	2	2	no	—
126	#107	C₂×C₄×C₁₀₉	1	4	yes	—
127	#107	C₂×C₄×C₁₀₉	1	4	yes	—
128	C₄×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
129	C₄×C₈	Q₈×C₁₀₉	1	2	yes	—
130	#95	—	4	2	no	—
131	C₂²×C₄	—	4	2	no	—
132	C₂²×C₄	—	4	2	no	—
133	C₂²×C₄	D₈×C₁₀₉	1	2	yes	—
134	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	C₄⋊C₄×C₁₀₉	1	2	yes	—
138	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
139	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
140	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
141	C₂×C₈	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
142	C₂×C₈	#41×C₁₀₉	1	1	yes	characteristic
143	C₂×C₈	#41×C₁₀₉	1	1	yes	characteristic
144	C₄⋊C₄	—	2	4	no	—
145	C₄⋊C₄	—	2	4	no	—
146	C₄⋊C₄	—	4	2	no	—
147	C₄⋊C₄	—	4	2	no	—
148	C₄²	—	8	4	no	—
149	C₄²	C₄⋊C₄×C₁₀₉	1	2	yes	—
150	C₄²	Q₁₆×C₁₀₉	1	2	yes	—
151	C₁₆	—	2	2	no	—
152	C₁₆	C₄²×C₁₀₉	1	4	yes	complemented
153	C₂³	C₂×D₈×C₁₀₉	1	1	yes	characteristic
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	4	4	no	—
157	C₂×C₄	—	4	4	no	—
158	C₂×C₄	—	8	2	no	—
159	C₂×C₄	—	8	2	no	—
160	C₂×C₄	—	8	2	no	—
161	C₂×C₄	—	8	2	no	—
162	C₂×C₄	#107×C₁₀₉	1	4	yes	—
163	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
164	C₂×C₄	C₄×D₄×C₁₀₉	1	1	yes	characteristic
165	C₂×C₄	C₄×D₄×C₁₀₉	1	1	yes	characteristic
166	C₂×C₄	C₂×Q₁₆×C₁₀₉	1	1	yes	characteristic
167	C₂×C₄	#135×C₁₀₉	1	1	yes	characteristic
168	C₂×C₄	#135×C₁₀₉	1	1	yes	characteristic
169	C₈	—	2	2	no	—
170	C₈	C₂×C₄²×C₁₀₉	1	1	yes	characteristic
171	C₈	#117×C₁₀₉	1	1	yes	characteristic
172	C₈	C₄×Q₈×C₁₀₉	1	1	yes	characteristic
173	C₈	C₄×Q₈×C₁₀₉	1	1	yes	characteristic
174	C₂²	C₂×#107×C₁₀₉	1	1	yes	characteristic
175	C₂²	C₄×D₈×C₁₀₉	1	1	yes	characteristic
176	C₂²	C₄×D₈×C₁₀₉	1	1	yes	characteristic
177	C₂²	#509×C₁₀₉	1	2	yes	—
178	C₂²	#510×C₁₀₉	1	2	yes	—
179	C₄	—	2	4	no	—
180	C₄	—	8	4	no	—
181	C₄	—	8	4	no	—
182	C₄	C₄×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
183	C₄	#427×C₁₀₉	1	1	yes	characteristic
184	C₄	C₄×Q₁₆×C₁₀₉	1	1	yes	characteristic
185	C₄	C₄×Q₁₆×C₁₀₉	1	1	yes	characteristic
186	C₂	C₄×#107×C₁₀₉	1	1	yes	characteristic
187	C₂	C₁₀₉×#2159	1	2	yes	—
188	C₂	C₁₀₉×#2175	1	2	yes	—
189	C₂	C₁₀₉×#2176	1	1	yes	characteristic
190	C₂	C₁₀₉×#2177	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₀₉	C₂×D₈
commutator	C₈	C₂×C₄²×C₁₀₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₀₉
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×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
12. #367×C₁₀₉	C₄	semidirect
13. #366×C₁₀₉	C₄	semidirect
14. #346×C₁₀₉	C₄	semidirect
56. C₁₆×C₁₀₉	C₄²	semidirect
90. #12674	C₁₀₉	direct (N × H)
101. C₁₀₉	#12674	direct (N × H)
108. #367	C₄×C₁₀₉	semidirect
109. #366	C₄×C₁₀₉	semidirect
110. #346	C₄×C₁₀₉	semidirect
152. C₁₆	C₄²×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#367×C₁₀₉	2
C₂×#366×C₁₀₉	2
C₂×#346×C₁₀₉	2
C₄×#107×C₁₀₉	2
C₁₀₉×#1380	2
#12674	109

Subgroups — C109×#12674 #42130668

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#12674 #42130668