Subgroups — C₂×C₁₀₉×#1834 #42138752

Order 27904.

1182 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#384×C₁₀₉	C₂	1	2	yes	maximal
3	C₁₀₉×#1834	C₂	1	8	yes	maximal, direct product factor
4	C₂×#340×C₁₀₉	C₂	1	4	yes	maximal
5	C₂³×C₈×C₁₀₉	C₂²	1	2	yes	—
6	C₂²×C₄⋊C₄×C₁₀₉	—	2	2	no	—
7	C₂²×C₄⋊C₄×C₁₀₉	—	2	2	no	—
8	C₂²×C₄⋊C₄×C₁₀₉	C₂²	1	4	yes	—
9	C₂²×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₁₀₉	C₄	1	8	yes	complemented
11	C₂×C₄×C₈×C₁₀₉	C₂²	1	4	yes	—
12	#384×C₁₀₉	—	2	4	no	—
13	#384×C₁₀₉	C₂²	1	8	yes	—
14	#340×C₁₀₉	C₂²	1	16	yes	—
15	C₂³×C₄×C₁₀₉	—	2	4	no	—
16	C₂³×C₄×C₁₀₉	D₄	1	1	yes	characteristic
17	C₂³×C₄×C₁₀₉	D₄	1	1	yes	characteristic
18	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
19	C₂²×C₈×C₁₀₉	—	2	8	no	—
20	C₂²×C₈×C₁₀₉	C₂×C₄	1	8	yes	—
21	C₂²×C₈×C₁₀₉	C₂³	1	4	yes	—
22	C₂×C₄⋊C₄×C₁₀₉	—	2	16	no	—
23	C₂×C₄⋊C₄×C₁₀₉	—	2	8	no	—
24	C₂×C₄⋊C₄×C₁₀₉	—	2	8	no	—
25	C₂×C₄⋊C₄×C₁₀₉	—	2	4	no	—
26	C₂×C₄⋊C₄×C₁₀₉	—	4	8	no	—
27	C₂×C₄⋊C₄×C₁₀₉	—	4	8	no	—
28	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	8	yes	—
29	C₂×C₄²×C₁₀₉	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₁₀₉	D₄	1	2	yes	—
31	C₂×C₄²×C₁₀₉	Q₈	1	4	yes	—
32	C₂×C₄²×C₁₀₉	C₂³	1	2	yes	—
33	C₄×C₈×C₁₀₉	C₂×C₄	1	16	yes	complemented
34	C₂⁴×C₁₀₉	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₀₉	—	2	8	no	—
36	C₂²×C₄×C₁₀₉	—	4	8	no	—
37	C₂²×C₄×C₁₀₉	—	4	8	no	—
38	C₂²×C₄×C₁₀₉	—	4	8	no	—
39	C₂²×C₄×C₁₀₉	C₄⋊C₄	1	4	yes	—
40	C₂²×C₄×C₁₀₉	C₄⋊C₄	1	4	yes	—
41	C₂²×C₄×C₁₀₉	D₈	1	4	yes	—
42	C₂²×C₄×C₁₀₉	SD₁₆	1	8	yes	—
43	C₂²×C₄×C₁₀₉	Q₁₆	1	4	yes	—
44	C₂²×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
45	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
47	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
48	C₂²×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
49	C₂²×C₄×C₁₀₉	#41	1	8	yes	—
50	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
51	C₂×C₈×C₁₀₉	—	2	16	no	—
52	C₂×C₈×C₁₀₉	—	2	8	no	—
53	C₂×C₈×C₁₀₉	C₂²×C₄	1	8	yes	—
54	C₄⋊C₄×C₁₀₉	—	2	16	no	—
55	C₄⋊C₄×C₁₀₉	—	4	8	no	—
56	C₄⋊C₄×C₁₀₉	—	4	16	no	—
57	C₄⋊C₄×C₁₀₉	—	4	16	no	—
58	C₄²×C₁₀₉	C₄⋊C₄	1	8	yes	—
59	C₄²×C₁₀₉	C₂²×C₄	1	4	yes	—
60	C₄²×C₁₀₉	C₂×Q₈	1	4	yes	—
61	C₂³×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
62	C₂³×C₁₀₉	#124	1	4	yes	—
63	C₂³×C₁₀₉	C₂×D₈	1	2	yes	—
64	C₂³×C₁₀₉	C₂×SD₁₆	1	4	yes	—
65	C₂³×C₁₀₉	C₂×Q₁₆	1	2	yes	—
66	C₂³×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
67	C₂×C₄×C₁₀₉	—	4	8	no	—
68	C₂×C₄×C₁₀₉	—	4	8	no	—
69	C₂×C₄×C₁₀₉	—	4	8	no	—
70	C₂×C₄×C₁₀₉	—	8	16	no	—
71	C₂×C₄×C₁₀₉	#106	1	16	yes	—
72	C₂×C₄×C₁₀₉	#107	1	16	yes	—
73	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
74	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	4	yes	—
75	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	4	yes	—
76	C₂×C₄×C₁₀₉	#117	1	8	yes	—
77	C₂×C₄×C₁₀₉	#124	1	4	yes	—
78	C₂×C₄×C₁₀₉	#125	1	8	yes	—
79	C₂×C₄×C₁₀₉	C₂×D₈	1	4	yes	—
80	C₂×C₄×C₁₀₉	C₂×SD₁₆	1	8	yes	—
81	C₂×C₄×C₁₀₉	C₂×Q₁₆	1	4	yes	—
82	C₂×C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
83	C₂×C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
84	C₂×C₄×C₁₀₉	C₂×#41	1	4	yes	—
85	C₈×C₁₀₉	—	2	8	no	—
86	C₂²×C₁₀₉	#382	1	4	yes	—
87	C₂²×C₁₀₉	C₂×#106	1	4	yes	—
88	C₂²×C₁₀₉	C₂×#107	1	4	yes	—
89	C₂²×C₁₀₉	#486	1	8	yes	—
90	C₂²×C₁₀₉	#487	1	8	yes	—
91	C₂²×C₁₀₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
92	C₂²×C₁₀₉	C₂×#124	1	2	yes	—
93	C₂²×C₁₀₉	C₂²×D₈	1	1	yes	characteristic
94	C₂²×C₁₀₉	C₂²×SD₁₆	1	2	yes	—
95	C₂²×C₁₀₉	C₂²×Q₁₆	1	1	yes	characteristic
96	C₄×C₁₀₉	—	8	8	no	—
97	C₄×C₁₀₉	#382	1	4	yes	—
98	C₄×C₁₀₉	C₂×#106	1	8	yes	—
99	C₄×C₁₀₉	C₂×#107	1	8	yes	—
100	C₄×C₁₀₉	C₂×#117	1	2	yes	—
101	C₄×C₁₀₉	C₂×#125	1	2	yes	—
102	C₂×#1834	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₂×C₁₀₉	#1834	1	8	yes	direct product factor
104	C₂×C₁₀₉	C₂×#382	1	1	yes	characteristic
105	C₂×C₁₀₉	C₂²×#106	1	1	yes	characteristic
106	C₂×C₁₀₉	C₂²×#107	1	1	yes	characteristic
107	C₂×C₁₀₉	C₂×#486	1	2	yes	—
108	C₂×C₁₀₉	C₂×#487	1	2	yes	—
109	C₂²×C₄×C₈	C₂×C₁₀₉	1	1	yes	characteristic
110	C₂×#384	C₂×C₁₀₉	1	2	yes	—
111	#1834	C₂×C₁₀₉	1	8	yes	direct product factor
112	C₂×#340	C₂×C₁₀₉	1	4	yes	—
113	C₁₀₉	C₂×#1834	1	1	yes	characteristic, direct product factor, Sylow
114	C₂³×C₈	C₂²×C₁₀₉	1	2	yes	—
115	C₂²×C₄⋊C₄	—	2	2	no	—
116	C₂²×C₄⋊C₄	—	2	2	no	—
117	C₂²×C₄⋊C₄	C₂²×C₁₀₉	1	4	yes	—
118	C₂²×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
119	C₂×C₄×C₈	C₄×C₁₀₉	1	8	yes	complemented
120	C₂×C₄×C₈	C₂²×C₁₀₉	1	4	yes	—
121	#384	—	2	4	no	—
122	#384	C₂²×C₁₀₉	1	8	yes	—
123	#340	C₂²×C₁₀₉	1	16	yes	—
124	C₂³×C₄	—	2	4	no	—
125	C₂³×C₄	D₄×C₁₀₉	1	1	yes	characteristic
126	C₂³×C₄	D₄×C₁₀₉	1	1	yes	characteristic
127	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
128	C₂²×C₈	—	2	8	no	—
129	C₂²×C₈	C₂×C₄×C₁₀₉	1	8	yes	—
130	C₂²×C₈	C₂³×C₁₀₉	1	4	yes	—
131	C₂×C₄⋊C₄	—	2	16	no	—
132	C₂×C₄⋊C₄	—	2	8	no	—
133	C₂×C₄⋊C₄	—	2	8	no	—
134	C₂×C₄⋊C₄	—	2	4	no	—
135	C₂×C₄⋊C₄	—	4	8	no	—
136	C₂×C₄⋊C₄	—	4	8	no	—
137	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	8	yes	—
138	C₂×C₄²	C₂×C₄×C₁₀₉	1	4	yes	—
139	C₂×C₄²	D₄×C₁₀₉	1	2	yes	—
140	C₂×C₄²	Q₈×C₁₀₉	1	4	yes	—
141	C₂×C₄²	C₂³×C₁₀₉	1	2	yes	—
142	C₄×C₈	C₂×C₄×C₁₀₉	1	16	yes	complemented
143	C₂⁴	C₂×D₄×C₁₀₉	1	1	yes	characteristic
144	C₂²×C₄	—	2	8	no	—
145	C₂²×C₄	—	4	8	no	—
146	C₂²×C₄	—	4	8	no	—
147	C₂²×C₄	—	4	8	no	—
148	C₂²×C₄	C₄⋊C₄×C₁₀₉	1	4	yes	—
149	C₂²×C₄	C₄⋊C₄×C₁₀₉	1	4	yes	—
150	C₂²×C₄	D₈×C₁₀₉	1	4	yes	—
151	C₂²×C₄	SD₁₆×C₁₀₉	1	8	yes	—
152	C₂²×C₄	Q₁₆×C₁₀₉	1	4	yes	—
153	C₂²×C₄	C₂²×C₄×C₁₀₉	1	2	yes	—
154	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
155	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
156	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
157	C₂²×C₄	C₂×Q₈×C₁₀₉	1	2	yes	—
158	C₂²×C₄	#41×C₁₀₉	1	8	yes	—
159	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
160	C₂×C₈	—	2	16	no	—
161	C₂×C₈	—	2	8	no	—
162	C₂×C₈	C₂²×C₄×C₁₀₉	1	8	yes	—
163	C₄⋊C₄	—	2	16	no	—
164	C₄⋊C₄	—	4	8	no	—
165	C₄⋊C₄	—	4	16	no	—
166	C₄⋊C₄	—	4	16	no	—
167	C₄²	C₄⋊C₄×C₁₀₉	1	8	yes	—
168	C₄²	C₂²×C₄×C₁₀₉	1	4	yes	—
169	C₄²	C₂×Q₈×C₁₀₉	1	4	yes	—
170	C₂³	C₂×C₄⋊C₄×C₁₀₉	1	2	yes	—
171	C₂³	#124×C₁₀₉	1	4	yes	—
172	C₂³	C₂×D₈×C₁₀₉	1	2	yes	—
173	C₂³	C₂×SD₁₆×C₁₀₉	1	4	yes	—
174	C₂³	C₂×Q₁₆×C₁₀₉	1	2	yes	—
175	C₂³	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
176	C₂×C₄	—	4	8	no	—
177	C₂×C₄	—	4	8	no	—
178	C₂×C₄	—	4	8	no	—
179	C₂×C₄	—	8	16	no	—
180	C₂×C₄	#106×C₁₀₉	1	16	yes	—
181	C₂×C₄	#107×C₁₀₉	1	16	yes	—
182	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	2	yes	—
183	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	4	yes	—
184	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	4	yes	—
185	C₂×C₄	#117×C₁₀₉	1	8	yes	—
186	C₂×C₄	#124×C₁₀₉	1	4	yes	—
187	C₂×C₄	#125×C₁₀₉	1	8	yes	—
188	C₂×C₄	C₂×D₈×C₁₀₉	1	4	yes	—
189	C₂×C₄	C₂×SD₁₆×C₁₀₉	1	8	yes	—
190	C₂×C₄	C₂×Q₁₆×C₁₀₉	1	4	yes	—
191	C₂×C₄	C₂³×C₄×C₁₀₉	1	1	yes	characteristic
192	C₂×C₄	C₂²×Q₈×C₁₀₉	1	1	yes	characteristic
193	C₂×C₄	C₂×#41×C₁₀₉	1	4	yes	—
194	C₈	—	2	8	no	—
195	C₂²	#382×C₁₀₉	1	4	yes	—
196	C₂²	C₂×#106×C₁₀₉	1	4	yes	—
197	C₂²	C₂×#107×C₁₀₉	1	4	yes	—
198	C₂²	#486×C₁₀₉	1	8	yes	—
199	C₂²	#487×C₁₀₉	1	8	yes	—
200	C₂²	C₂²×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
201	C₂²	C₂×#124×C₁₀₉	1	2	yes	—
202	C₂²	C₂²×D₈×C₁₀₉	1	1	yes	characteristic
203	C₂²	C₂²×SD₁₆×C₁₀₉	1	2	yes	—
204	C₂²	C₂²×Q₁₆×C₁₀₉	1	1	yes	characteristic
205	C₄	—	8	8	no	—
206	C₄	#382×C₁₀₉	1	4	yes	—
207	C₄	C₂×#106×C₁₀₉	1	8	yes	—
208	C₄	C₂×#107×C₁₀₉	1	8	yes	—
209	C₄	C₂×#117×C₁₀₉	1	2	yes	—
210	C₄	C₂×#125×C₁₀₉	1	2	yes	—
211	C₂	C₁₀₉×#1834	1	8	yes	direct product factor
212	C₂	C₂×#382×C₁₀₉	1	1	yes	characteristic
213	C₂	C₂²×#106×C₁₀₉	1	1	yes	characteristic
214	C₂	C₂²×#107×C₁₀₉	1	1	yes	characteristic
215	C₂	C₂×#486×C₁₀₉	1	2	yes	—
216	C₂	C₂×#487×C₁₀₉	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₁₀₉×#1834	C₂	direct (N × H)
10. C₂×C₄×C₈×C₁₀₉	C₄	semidirect
33. C₄×C₈×C₁₀₉	C₂×C₄	semidirect
102. C₂×#1834	C₁₀₉	direct (N × H)
103. C₂×C₁₀₉	#1834	direct (N × H)
111. #1834	C₂×C₁₀₉	direct (N × H)
113. C₁₀₉	C₂×#1834	direct (N × H)
119. C₂×C₄×C₈	C₄×C₁₀₉	semidirect
142. C₄×C₈	C₂×C₄×C₁₀₉	semidirect
211. C₂	C₁₀₉×#1834	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×C₄×C₈×C₁₀₉	2
C₂×#384×C₁₀₉	2
C₁₀₉×#1834	2
C₂×#340×C₁₀₉	2
C₂×#1834	109

Subgroups — C2×C109×#1834 #42138752

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₀₉×#1834 #42138752