Subgroups — C₂×C₁₀₉×#1818 #42138675

Order 27904.

1526 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#117×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₂²⋊C₈×C₁₀₉	C₂	1	2	yes	maximal
3	C₁₀₉×#1818	C₂	1	8	yes	maximal, direct product factor
4	C₂×#336×C₁₀₉	C₂	1	4	yes	maximal, complemented
5	C₂⁴×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
6	C₂³×C₈×C₁₀₉	C₂²	1	4	yes	—
7	C₂×#117×C₁₀₉	—	2	8	no	—
8	C₂×#117×C₁₀₉	C₄	1	4	yes	—
9	C₂×#117×C₁₀₉	C₂²	1	4	yes	—
10	C₂²×C₄⋊C₄×C₁₀₉	C₄	1	2	yes	—
11	C₂²×C₂²⋊C₄×C₁₀₉	C₄	1	2	yes	—
12	C₂²×C₄²×C₁₀₉	C₂²	1	2	yes	—
13	C₂×C₂²⋊C₈×C₁₀₉	—	2	8	no	—
14	C₂×C₂²⋊C₈×C₁₀₉	C₂²	1	8	yes	—
15	#336×C₁₀₉	C₂²	1	16	yes	complemented
16	C₂⁵×C₁₀₉	C₂×C₄	1	1	yes	characteristic
17	C₂³×C₄×C₁₀₉	—	2	8	no	—
18	C₂³×C₄×C₁₀₉	C₂×C₄	1	4	yes	—
19	C₂³×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
20	C₂³×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
21	C₂³×C₄×C₁₀₉	D₄	1	8	yes	—
22	C₂³×C₄×C₁₀₉	C₂³	1	2	yes	—
23	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
24	C₂²×C₈×C₁₀₉	—	2	8	no	—
25	C₂²×C₈×C₁₀₉	—	2	16	no	—
26	C₂²×C₈×C₁₀₉	C₂³	1	8	yes	—
27	#117×C₁₀₉	—	2	16	no	—
28	#117×C₁₀₉	—	2	16	no	—
29	C₂×C₄⋊C₄×C₁₀₉	—	2	8	no	—
30	C₂×C₄⋊C₄×C₁₀₉	C₂×C₄	1	8	yes	—
31	C₂×C₂²⋊C₄×C₁₀₉	—	2	8	no	—
32	C₂×C₂²⋊C₄×C₁₀₉	C₂×C₄	1	8	yes	—
33	C₂×C₄²×C₁₀₉	—	2	8	no	—
34	C₂×C₄²×C₁₀₉	C₂×C₄	1	4	yes	—
35	C₂×C₄²×C₁₀₉	C₂³	1	4	yes	—
36	C₂²⋊C₈×C₁₀₉	—	2	16	no	—
37	C₂⁴×C₁₀₉	—	2	4	no	—
38	C₂⁴×C₁₀₉	—	2	8	no	—
39	C₂⁴×C₁₀₉	C₂²⋊C₄	1	4	yes	—
40	C₂⁴×C₁₀₉	C₂²×C₄	1	2	yes	—
41	C₂⁴×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₀₉	—	2	4	no	—
43	C₂²×C₄×C₁₀₉	—	2	8	no	—
44	C₂²×C₄×C₁₀₉	—	2	32	no	—
45	C₂²×C₄×C₁₀₉	—	2	8	no	—
46	C₂²×C₄×C₁₀₉	—	2	8	no	—
47	C₂²×C₄×C₁₀₉	—	2	16	no	—
48	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	4	yes	—
49	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	8	yes	—
50	C₂²×C₄×C₁₀₉	C₂²×C₄	1	8	yes	—
51	C₂²×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
52	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₀₉	C₂×D₄	1	8	yes	—
54	C₂²×C₄×C₁₀₉	C₂×D₄	1	4	yes	—
55	C₂²×C₄×C₁₀₉	#41	1	8	yes	—
56	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
57	C₂×C₈×C₁₀₉	—	2	8	no	—
58	C₂×C₈×C₁₀₉	—	2	16	no	—
59	C₂×C₈×C₁₀₉	—	4	16	no	—
60	C₄⋊C₄×C₁₀₉	—	2	16	no	—
61	C₂²⋊C₄×C₁₀₉	—	2	16	no	—
62	C₄²×C₁₀₉	—	2	8	no	—
63	C₄²×C₁₀₉	—	2	8	no	—
64	C₂³×C₁₀₉	—	2	4	no	—
65	C₂³×C₁₀₉	—	2	8	no	—
66	C₂³×C₁₀₉	—	2	8	no	—
67	C₂³×C₁₀₉	—	2	16	no	—
68	C₂³×C₁₀₉	—	4	16	no	—
69	C₂³×C₁₀₉	C₂×C₂²⋊C₄	1	4	yes	—
70	C₂³×C₁₀₉	C₂×C₂²⋊C₄	1	2	yes	—
71	C₂³×C₁₀₉	#117	1	4	yes	—
72	C₂³×C₁₀₉	#131	1	8	yes	—
73	C₂³×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
74	C₂×C₄×C₁₀₉	—	2	4	no	—
75	C₂×C₄×C₁₀₉	—	2	8	no	—
76	C₂×C₄×C₁₀₉	—	2	16	no	—
77	C₂×C₄×C₁₀₉	—	2	16	no	—
78	C₂×C₄×C₁₀₉	—	2	8	no	—
79	C₂×C₄×C₁₀₉	—	2	16	no	—
80	C₂×C₄×C₁₀₉	—	4	16	no	—
81	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	4	yes	—
82	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	2	yes	—
83	C₂×C₄×C₁₀₉	#117	1	4	yes	—
84	C₂×C₄×C₁₀₉	#123	1	16	yes	—
85	C₂×C₄×C₁₀₉	C₂²×D₄	1	2	yes	—
86	C₂×C₄×C₁₀₉	C₂×#41	1	4	yes	—
87	C₈×C₁₀₉	—	4	8	no	—
88	C₂²×C₁₀₉	—	2	8	no	—
89	C₂²×C₁₀₉	—	2	4	no	—
90	C₂²×C₁₀₉	—	4	16	no	—
91	C₂²×C₁₀₉	—	4	8	no	—
92	C₂²×C₁₀₉	#381	1	4	yes	—
93	C₂²×C₁₀₉	#408	1	8	yes	—
94	C₂²×C₁₀₉	#433	1	16	yes	—
95	C₂²×C₁₀₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
96	C₂²×C₁₀₉	C₂×#117	1	2	yes	—
97	C₂²×C₁₀₉	C₂×#131	1	4	yes	—
98	C₄×C₁₀₉	—	2	4	no	—
99	C₄×C₁₀₉	—	4	8	no	—
100	C₄×C₁₀₉	#381	1	4	yes	—
101	C₄×C₁₀₉	C₂×#123	1	4	yes	—
102	C₂×#1818	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₂×C₁₀₉	—	4	4	no	—
104	C₂×C₁₀₉	#1818	1	8	yes	direct product factor
105	C₂×C₁₀₉	C₂×#381	1	1	yes	characteristic
106	C₂×C₁₀₉	C₂×#408	1	2	yes	—
107	C₂×C₁₀₉	C₂×#433	1	4	yes	—
108	C₂²×#117	C₂×C₁₀₉	1	1	yes	characteristic
109	C₂²×C₂²⋊C₈	C₂×C₁₀₉	1	2	yes	—
110	#1818	C₂×C₁₀₉	1	8	yes	direct product factor
111	C₂×#336	C₂×C₁₀₉	1	4	yes	complemented
112	C₁₀₉	C₂×#1818	1	1	yes	characteristic, direct product factor, Sylow
113	C₂⁴×C₄	C₂²×C₁₀₉	1	1	yes	characteristic
114	C₂³×C₈	C₂²×C₁₀₉	1	4	yes	—
115	C₂×#117	—	2	8	no	—
116	C₂×#117	C₄×C₁₀₉	1	4	yes	—
117	C₂×#117	C₂²×C₁₀₉	1	4	yes	—
118	C₂²×C₄⋊C₄	C₄×C₁₀₉	1	2	yes	—
119	C₂²×C₂²⋊C₄	C₄×C₁₀₉	1	2	yes	—
120	C₂²×C₄²	C₂²×C₁₀₉	1	2	yes	—
121	C₂×C₂²⋊C₈	—	2	8	no	—
122	C₂×C₂²⋊C₈	C₂²×C₁₀₉	1	8	yes	—
123	#336	C₂²×C₁₀₉	1	16	yes	complemented
124	C₂⁵	C₂×C₄×C₁₀₉	1	1	yes	characteristic
125	C₂³×C₄	—	2	8	no	—
126	C₂³×C₄	C₂×C₄×C₁₀₉	1	4	yes	—
127	C₂³×C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
128	C₂³×C₄	C₂×C₄×C₁₀₉	1	2	yes	—
129	C₂³×C₄	D₄×C₁₀₉	1	8	yes	—
130	C₂³×C₄	C₂³×C₁₀₉	1	2	yes	—
131	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
132	C₂²×C₈	—	2	8	no	—
133	C₂²×C₈	—	2	16	no	—
134	C₂²×C₈	C₂³×C₁₀₉	1	8	yes	—
135	#117	—	2	16	no	—
136	#117	—	2	16	no	—
137	C₂×C₄⋊C₄	—	2	8	no	—
138	C₂×C₄⋊C₄	C₂×C₄×C₁₀₉	1	8	yes	—
139	C₂×C₂²⋊C₄	—	2	8	no	—
140	C₂×C₂²⋊C₄	C₂×C₄×C₁₀₉	1	8	yes	—
141	C₂×C₄²	—	2	8	no	—
142	C₂×C₄²	C₂×C₄×C₁₀₉	1	4	yes	—
143	C₂×C₄²	C₂³×C₁₀₉	1	4	yes	—
144	C₂²⋊C₈	—	2	16	no	—
145	C₂⁴	—	2	4	no	—
146	C₂⁴	—	2	8	no	—
147	C₂⁴	C₂²⋊C₄×C₁₀₉	1	4	yes	—
148	C₂⁴	C₂²×C₄×C₁₀₉	1	2	yes	—
149	C₂⁴	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
150	C₂²×C₄	—	2	4	no	—
151	C₂²×C₄	—	2	8	no	—
152	C₂²×C₄	—	2	32	no	—
153	C₂²×C₄	—	2	8	no	—
154	C₂²×C₄	—	2	8	no	—
155	C₂²×C₄	—	2	16	no	—
156	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	4	yes	—
157	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	8	yes	—
158	C₂²×C₄	C₂²×C₄×C₁₀₉	1	8	yes	—
159	C₂²×C₄	C₂²×C₄×C₁₀₉	1	2	yes	—
160	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
161	C₂²×C₄	C₂×D₄×C₁₀₉	1	8	yes	—
162	C₂²×C₄	C₂×D₄×C₁₀₉	1	4	yes	—
163	C₂²×C₄	#41×C₁₀₉	1	8	yes	—
164	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
165	C₂×C₈	—	2	8	no	—
166	C₂×C₈	—	2	16	no	—
167	C₂×C₈	—	4	16	no	—
168	C₄⋊C₄	—	2	16	no	—
169	C₂²⋊C₄	—	2	16	no	—
170	C₄²	—	2	8	no	—
171	C₄²	—	2	8	no	—
172	C₂³	—	2	4	no	—
173	C₂³	—	2	8	no	—
174	C₂³	—	2	8	no	—
175	C₂³	—	2	16	no	—
176	C₂³	—	4	16	no	—
177	C₂³	C₂×C₂²⋊C₄×C₁₀₉	1	4	yes	—
178	C₂³	C₂×C₂²⋊C₄×C₁₀₉	1	2	yes	—
179	C₂³	#117×C₁₀₉	1	4	yes	—
180	C₂³	#131×C₁₀₉	1	8	yes	—
181	C₂³	C₂³×C₄×C₁₀₉	1	1	yes	characteristic
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	—	2	8	no	—
184	C₂×C₄	—	2	16	no	—
185	C₂×C₄	—	2	16	no	—
186	C₂×C₄	—	2	8	no	—
187	C₂×C₄	—	2	16	no	—
188	C₂×C₄	—	4	16	no	—
189	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	4	yes	—
190	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	2	yes	—
191	C₂×C₄	#117×C₁₀₉	1	4	yes	—
192	C₂×C₄	#123×C₁₀₉	1	16	yes	—
193	C₂×C₄	C₂²×D₄×C₁₀₉	1	2	yes	—
194	C₂×C₄	C₂×#41×C₁₀₉	1	4	yes	—
195	C₈	—	4	8	no	—
196	C₂²	—	2	4	no	—
197	C₂²	—	2	8	no	—
198	C₂²	—	4	16	no	—
199	C₂²	—	4	8	no	—
200	C₂²	#381×C₁₀₉	1	4	yes	—
201	C₂²	#408×C₁₀₉	1	8	yes	—
202	C₂²	#433×C₁₀₉	1	16	yes	—
203	C₂²	C₂²×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
204	C₂²	C₂×#117×C₁₀₉	1	2	yes	—
205	C₂²	C₂×#131×C₁₀₉	1	4	yes	—
206	C₄	—	2	4	no	—
207	C₄	—	4	8	no	—
208	C₄	#381×C₁₀₉	1	4	yes	—
209	C₄	C₂×#123×C₁₀₉	1	4	yes	—
210	C₂	—	4	4	no	—
211	C₂	C₁₀₉×#1818	1	8	yes	direct product factor
212	C₂	C₂×#381×C₁₀₉	1	1	yes	characteristic
213	C₂	C₂×#408×C₁₀₉	1	2	yes	—
214	C₂	C₂×#433×C₁₀₉	1	4	yes	—

	subgroup	G/H
center	C₂³×C₄×C₁₀₉	C₂³
commutator	C₂³	C₂³×C₄×C₁₀₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₀₉
socle	C₂⁴×C₁₀₉	C₂²×C₄
prime radical	C₂⁵×C₁₀₉	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₀₉×#1818	C₂	direct (N × H)
4. C₂×#336×C₁₀₉	C₂	semidirect
15. #336×C₁₀₉	C₂²	semidirect
102. C₂×#1818	C₁₀₉	direct (N × H)
104. C₂×C₁₀₉	#1818	direct (N × H)
110. #1818	C₂×C₁₀₉	direct (N × H)
111. C₂×#336	C₂×C₁₀₉	semidirect
112. C₁₀₉	C₂×#1818	direct (N × H)
123. #336	C₂²×C₁₀₉	semidirect
211. C₂	C₁₀₉×#1818	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#117×C₁₀₉	2
C₂²×C₂²⋊C₈×C₁₀₉	2
C₁₀₉×#1818	2
C₂×#336×C₁₀₉	2
C₂×#1818	109

Subgroups — C2×C109×#1818 #42138675

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₀₉×#1818 #42138675