Subgroups — C₁₀₉×#8294 #42126288

Order 27904.

358 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#108×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#1564	C₂	1	1	yes	maximal, characteristic
3	C₈×M₁₆×C₁₀₉	C₂	1	2	yes	maximal
4	C₂×#322×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₁₀₉×#1278	C₂	1	2	yes	maximal
6	#431×C₁₀₉	C₂²	1	2	yes	—
7	C₂×#108×C₁₀₉	C₄	1	2	yes	—
8	C₄×M₁₆×C₁₀₉	C₂²	1	2	yes	—
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	#341×C₁₀₉	C₄	1	2	yes	—
13	#335×C₁₀₉	C₄	1	2	yes	—
14	#334×C₁₀₉	C₄	1	2	yes	—
15	#322×C₁₀₉	—	2	2	no	—
16	#322×C₁₀₉	C₄	1	2	yes	—
17	#322×C₁₀₉	C₄	1	2	yes	—
18	C₈²×C₁₀₉	—	2	2	no	—
19	C₂×M₁₆×C₁₀₉	C₂×C₄	1	4	yes	—
20	C₂²×C₈×C₁₀₉	—	2	1	no	—
21	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₀₉	Q₈	1	2	yes	—
24	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
25	#108×C₁₀₉	C₈	1	8	yes	complemented
26	C₄⋊C₈×C₁₀₉	C₂×C₄	1	4	yes	—
27	C₂²⋊C₈×C₁₀₉	—	2	2	no	—
28	#97×C₁₀₉	—	2	2	no	—
29	C₄×C₈×C₁₀₉	—	2	2	no	—
30	C₄×C₈×C₁₀₉	—	2	2	no	—
31	C₄×C₈×C₁₀₉	—	2	1	no	—
32	C₄×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₁₀₉	D₄	1	2	yes	—
39	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
41	C₂²×C₄×C₁₀₉	#41	1	1	yes	characteristic
42	M₁₆×C₁₀₉	—	2	4	no	—
43	M₁₆×C₁₀₉	C₂×C₈	1	8	yes	—
44	C₂×C₈×C₁₀₉	—	2	4	no	—
45	C₂×C₈×C₁₀₉	—	2	2	no	—
46	C₂×C₈×C₁₀₉	—	2	2	no	—
47	C₂×C₈×C₁₀₉	—	2	4	no	—
48	C₂×C₈×C₁₀₉	—	4	2	no	—
49	C₂×C₈×C₁₀₉	—	4	2	no	—
50	C₂×C₈×C₁₀₉	C₄²	1	1	yes	characteristic
51	C₂×C₈×C₁₀₉	C₄²	1	1	yes	characteristic
52	C₂×C₈×C₁₀₉	C₄²	1	1	yes	characteristic
53	C₂×C₈×C₁₀₉	C₄²	1	1	yes	characteristic
54	C₂×C₈×C₁₀₉	C₄⋊C₄	1	4	yes	—
55	C₂×C₈×C₁₀₉	C₂×C₈	1	4	yes	—
56	C₄²×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
58	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
59	C₄²×C₁₀₉	#41	1	1	yes	characteristic
60	C₂³×C₁₀₉	C₄×Q₈	1	1	yes	characteristic
61	C₂×C₄×C₁₀₉	—	2	2	no	—
62	C₂×C₄×C₁₀₉	—	2	2	no	—
63	C₂×C₄×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
64	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₁₀₉	#117	1	1	yes	characteristic
66	C₂×C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
67	C₂×C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
68	C₂×C₄×C₁₀₉	C₄×Q₈	1	1	yes	characteristic
69	C₂×C₄×C₁₀₉	C₂²×C₈	1	2	yes	—
70	C₂×C₄×C₁₀₉	#131	1	2	yes	—
71	C₈×C₁₀₉	—	2	1	no	—
72	C₈×C₁₀₉	—	2	1	no	—
73	C₈×C₁₀₉	—	2	4	no	—
74	C₈×C₁₀₉	—	2	8	no	—
75	C₈×C₁₀₉	—	4	2	no	—
76	C₈×C₁₀₉	—	4	4	no	—
77	C₈×C₁₀₉	C₄×C₈	1	2	yes	—
78	C₈×C₁₀₉	C₄×C₈	1	2	yes	—
79	C₂²×C₁₀₉	—	2	1	no	—
80	C₂²×C₁₀₉	—	2	1	no	—
81	C₂²×C₁₀₉	C₄×C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₁₀₉	C₈×Q₈	1	2	yes	—
83	C₄×C₁₀₉	—	2	1	no	—
84	C₄×C₁₀₉	—	2	1	no	—
85	C₄×C₁₀₉	C₂×C₄×C₈	1	1	yes	characteristic
86	C₄×C₁₀₉	#405	1	1	yes	characteristic
87	C₄×C₁₀₉	#430	1	2	yes	—
88	C₄×C₁₀₉	C₈×D₄	1	2	yes	—
89	C₄×C₁₀₉	#444	1	2	yes	—
90	#8294	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂×C₁₀₉	—	2	2	no	—
92	C₂×C₁₀₉	C₈×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₁₀₉	#1773	1	1	yes	characteristic
94	C₂×C₁₀₉	#1778	1	1	yes	characteristic
95	C₄×#108	C₂×C₁₀₉	1	1	yes	characteristic
96	#1564	C₂×C₁₀₉	1	1	yes	characteristic
97	C₈×M₁₆	C₂×C₁₀₉	1	2	yes	—
98	C₂×#322	C₂×C₁₀₉	1	1	yes	characteristic
99	#1278	C₂×C₁₀₉	1	2	yes	—
100	C₁₀₉	#8294	1	1	yes	characteristic, direct product factor, Sylow
101	#431	C₂²×C₁₀₉	1	2	yes	—
102	C₂×#108	C₄×C₁₀₉	1	2	yes	—
103	C₄×M₁₆	C₂²×C₁₀₉	1	2	yes	—
104	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
105	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
106	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
107	#341	C₄×C₁₀₉	1	2	yes	—
108	#335	C₄×C₁₀₉	1	2	yes	—
109	#334	C₄×C₁₀₉	1	2	yes	—
110	#322	—	2	2	no	—
111	#322	C₄×C₁₀₉	1	2	yes	—
112	#322	C₄×C₁₀₉	1	2	yes	—
113	C₈²	—	2	2	no	—
114	C₂×M₁₆	C₂×C₄×C₁₀₉	1	4	yes	—
115	C₂²×C₈	—	2	1	no	—
116	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
117	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
118	C₂²×C₈	Q₈×C₁₀₉	1	2	yes	—
119	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
120	#108	C₈×C₁₀₉	1	8	yes	complemented
121	C₄⋊C₈	C₂×C₄×C₁₀₉	1	4	yes	—
122	C₂²⋊C₈	—	2	2	no	—
123	#97	—	2	2	no	—
124	C₄×C₈	—	2	2	no	—
125	C₄×C₈	—	2	2	no	—
126	C₄×C₈	—	2	1	no	—
127	C₄×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
128	C₄×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
129	C₄×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
130	C₄×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
131	C₄×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
132	C₄×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
133	C₄×C₈	D₄×C₁₀₉	1	2	yes	—
134	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
135	C₂²×C₄	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
136	C₂²×C₄	#41×C₁₀₉	1	1	yes	characteristic
137	M₁₆	—	2	4	no	—
138	M₁₆	C₂×C₈×C₁₀₉	1	8	yes	—
139	C₂×C₈	—	2	4	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	4	no	—
143	C₂×C₈	—	4	2	no	—
144	C₂×C₈	—	4	2	no	—
145	C₂×C₈	C₄²×C₁₀₉	1	1	yes	characteristic
146	C₂×C₈	C₄²×C₁₀₉	1	1	yes	characteristic
147	C₂×C₈	C₄²×C₁₀₉	1	1	yes	characteristic
148	C₂×C₈	C₄²×C₁₀₉	1	1	yes	characteristic
149	C₂×C₈	C₄⋊C₄×C₁₀₉	1	4	yes	—
150	C₂×C₈	C₂×C₈×C₁₀₉	1	4	yes	—
151	C₄²	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
152	C₄²	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
153	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
154	C₄²	#41×C₁₀₉	1	1	yes	characteristic
155	C₂³	C₄×Q₈×C₁₀₉	1	1	yes	characteristic
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	C₂×C₄²×C₁₀₉	1	1	yes	characteristic
159	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
160	C₂×C₄	#117×C₁₀₉	1	1	yes	characteristic
161	C₂×C₄	C₄×D₄×C₁₀₉	1	1	yes	characteristic
162	C₂×C₄	C₄×D₄×C₁₀₉	1	1	yes	characteristic
163	C₂×C₄	C₄×Q₈×C₁₀₉	1	1	yes	characteristic
164	C₂×C₄	C₂²×C₈×C₁₀₉	1	2	yes	—
165	C₂×C₄	#131×C₁₀₉	1	2	yes	—
166	C₈	—	2	1	no	—
167	C₈	—	2	1	no	—
168	C₈	—	2	4	no	—
169	C₈	—	2	8	no	—
170	C₈	—	4	2	no	—
171	C₈	—	4	4	no	—
172	C₈	C₄×C₈×C₁₀₉	1	2	yes	—
173	C₈	C₄×C₈×C₁₀₉	1	2	yes	—
174	C₂²	—	2	1	no	—
175	C₂²	—	2	1	no	—
176	C₂²	C₄×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
177	C₂²	C₈×Q₈×C₁₀₉	1	2	yes	—
178	C₄	—	2	1	no	—
179	C₄	—	2	1	no	—
180	C₄	C₂×C₄×C₈×C₁₀₉	1	1	yes	characteristic
181	C₄	#405×C₁₀₉	1	1	yes	characteristic
182	C₄	#430×C₁₀₉	1	2	yes	—
183	C₄	C₈×D₄×C₁₀₉	1	2	yes	—
184	C₄	#444×C₁₀₉	1	2	yes	—
185	C₂	—	2	2	no	—
186	C₂	C₈×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
187	C₂	C₁₀₉×#1773	1	1	yes	characteristic
188	C₂	C₁₀₉×#1778	1	1	yes	characteristic

	subgroup	G/H
center	C₄²×C₁₀₉	C₂×D₄
commutator	C₄	C₂×C₄×C₈×C₁₀₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₉
socle	C₂²×C₁₀₉	C₄×C₄⋊C₄
prime radical	C₂³×C₁₀₉	C₄×Q₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₁₀₉ ◁ C₂×C₄×C₈×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
25. #108×C₁₀₉	C₈	semidirect
90. #8294	C₁₀₉	direct (N × H)
100. C₁₀₉	#8294	direct (N × H)
120. #108	C₈×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₄×#108×C₁₀₉	2
C₁₀₉×#1564	2
C₈×M₁₆×C₁₀₉	2
C₂×#322×C₁₀₉	2
C₁₀₉×#1278	2
#8294	109

Subgroups — C109×#8294 #42126288

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#8294 #42126288