Subgroups — C₄×#358×C₁₀₉ #42130630

Order 27904.

470 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#358×C₁₀₉	C₂	1	2	yes	maximal
3	C₂×C₄×C₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₄×#107×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₁₀₉×#1380	C₂	1	2	yes	maximal
6	C₂²×Q₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
7	C₂×C₄×Q₈×C₁₀₉	—	2	1	no	—
8	C₂²×C₁₆×C₁₀₉	C₂²	1	2	yes	—
9	C₄×Q₁₆×C₁₀₉	—	2	2	no	—
10	C₄×Q₁₆×C₁₀₉	C₄	1	4	yes	complemented
11	C₂×#107×C₁₀₉	C₂²	1	1	yes	characteristic
12	C₂×#107×C₁₀₉	C₂²	1	2	yes	—
13	C₂×#103×C₁₀₉	—	2	1	no	—
14	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₁₀₉	—	2	1	no	—
16	#358×C₁₀₉	C₄	1	8	yes	direct product factor
17	C₄×C₁₆×C₁₀₉	—	2	2	no	—
18	#340×C₁₀₉	—	2	1	no	—
19	C₂²×Q₈×C₁₀₉	—	2	1	no	—
20	C₂×Q₁₆×C₁₀₉	—	2	1	no	—
21	C₂×Q₁₆×C₁₀₉	—	2	2	no	—
22	C₂×Q₁₆×C₁₀₉	C₂×C₄	1	4	yes	—
23	C₂×Q₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₁₀₉	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
26	C₄×Q₈×C₁₀₉	—	2	4	no	—
27	C₄×Q₈×C₁₀₉	—	4	2	no	—
28	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
29	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
30	C₂×C₄⋊C₄×C₁₀₉	—	4	1	no	—
31	C₂×C₄²×C₁₀₉	—	4	1	no	—
32	C₂×C₄²×C₁₀₉	—	4	1	no	—
33	C₂×C₄²×C₁₀₉	D₄	1	1	yes	characteristic
34	C₂×C₁₆×C₁₀₉	—	2	2	no	—
35	C₂×C₁₆×C₁₀₉	—	2	2	no	—
36	C₂×C₁₆×C₁₀₉	C₂×C₄	1	4	yes	—
37	#107×C₁₀₉	—	2	1	no	—
38	#107×C₁₀₉	—	2	2	no	—
39	#107×C₁₀₉	C₂×C₄	1	4	yes	—
40	#107×C₁₀₉	C₂×C₄	1	2	yes	—
41	#103×C₁₀₉	—	2	4	no	—
42	C₄×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
43	C₄×C₈×C₁₀₉	D₄	1	2	yes	—
44	#95×C₁₀₉	—	4	1	no	—
45	C₂×Q₈×C₁₀₉	—	2	2	no	—
46	C₂×Q₈×C₁₀₉	—	2	4	no	—
47	C₂×Q₈×C₁₀₉	—	4	1	no	—
48	C₂×Q₈×C₁₀₉	—	4	2	no	—
49	C₂²×C₄×C₁₀₉	—	4	1	no	—
50	C₂²×C₄×C₁₀₉	—	4	2	no	—
51	C₂²×C₄×C₁₀₉	—	4	1	no	—
52	C₂²×C₄×C₁₀₉	D₈	1	1	yes	characteristic
53	C₂²×C₄×C₁₀₉	D₈	1	1	yes	characteristic
54	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
55	Q₁₆×C₁₀₉	—	2	2	no	—
56	Q₁₆×C₁₀₉	—	2	4	no	—
57	Q₁₆×C₁₀₉	C₄²	1	4	yes	complemented
58	C₂×C₈×C₁₀₉	—	2	2	no	—
59	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	2	yes	—
60	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
62	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₁₀₉	#41	1	2	yes	—
64	C₄⋊C₄×C₁₀₉	—	2	2	no	—
65	C₄⋊C₄×C₁₀₉	—	2	4	no	—
66	C₄⋊C₄×C₁₀₉	—	4	4	no	—
67	C₄⋊C₄×C₁₀₉	—	4	1	no	—
68	C₄⋊C₄×C₁₀₉	—	4	2	no	—
69	C₄²×C₁₀₉	—	4	4	no	—
70	C₄²×C₁₀₉	—	8	2	no	—
71	C₄²×C₁₀₉	C₂²⋊C₄	1	2	yes	—
72	C₄²×C₁₀₉	SD₁₆	1	2	yes	—
73	C₁₆×C₁₀₉	—	2	4	no	—
74	C₂³×C₁₀₉	C₂×D₈	1	1	yes	characteristic
75	Q₈×C₁₀₉	—	2	4	no	—
76	Q₈×C₁₀₉	—	4	2	no	—
77	Q₈×C₁₀₉	—	4	4	no	—
78	C₂×C₄×C₁₀₉	—	2	2	no	—
79	C₂×C₄×C₁₀₉	—	4	2	no	—
80	C₂×C₄×C₁₀₉	—	4	4	no	—
81	C₂×C₄×C₁₀₉	—	4	2	no	—
82	C₂×C₄×C₁₀₉	—	4	4	no	—
83	C₂×C₄×C₁₀₉	—	8	2	no	—
84	C₂×C₄×C₁₀₉	—	8	2	no	—
85	C₂×C₄×C₁₀₉	—	8	2	no	—
86	C₂×C₄×C₁₀₉	#102	1	2	yes	—
87	C₂×C₄×C₁₀₉	#102	1	2	yes	—
88	C₂×C₄×C₁₀₉	SD₃₂	1	2	yes	—
89	C₂×C₄×C₁₀₉	Q₃₂	1	2	yes	—
90	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₁₀₉	C₄×D₄	1	2	yes	—
92	C₂×C₄×C₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
93	C₂×C₄×C₁₀₉	#135	1	2	yes	—
94	C₈×C₁₀₉	—	2	2	no	—
95	C₈×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
96	C₈×C₁₀₉	#117	1	1	yes	characteristic
97	C₈×C₁₀₉	C₄×D₄	1	2	yes	—
98	C₂²×C₁₀₉	C₂×#102	1	1	yes	characteristic
99	C₂²×C₁₀₉	C₄×D₈	1	2	yes	—
100	C₂²×C₁₀₉	C₂×SD₃₂	1	1	yes	characteristic
101	C₂²×C₁₀₉	C₂×Q₃₂	1	1	yes	characteristic
102	C₂²×C₁₀₉	#508	1	2	yes	—
103	C₄×C₁₀₉	—	2	2	no	—
104	C₄×C₁₀₉	—	4	4	no	—
105	C₄×C₁₀₉	—	8	2	no	—
106	C₄×C₁₀₉	—	8	4	no	—
107	C₄×C₁₀₉	#358	1	4	yes	direct product factor
108	C₄×C₁₀₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
109	C₄×C₁₀₉	#416	1	1	yes	characteristic
110	C₄×C₁₀₉	C₄×SD₁₆	1	2	yes	—
111	C₄×#358	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
112	C₂×C₁₀₉	C₄×#102	1	1	yes	characteristic
113	C₂×C₁₀₉	C₂×#358	1	1	yes	characteristic
114	C₂×C₁₀₉	#2138	1	1	yes	characteristic
115	C₂×C₁₀₉	C₄×SD₃₂	1	2	yes	—
116	C₂×C₁₀₉	C₄×Q₃₂	1	2	yes	—
117	C₂×C₄×Q₁₆	C₂×C₁₀₉	1	1	yes	characteristic
118	C₂×#358	C₂×C₁₀₉	1	2	yes	—
119	C₂×C₄×C₁₆	C₂×C₁₀₉	1	1	yes	characteristic
120	C₄×#107	C₂×C₁₀₉	1	1	yes	characteristic
121	#1380	C₂×C₁₀₉	1	2	yes	—
122	C₁₀₉	C₄×#358	1	1	yes	characteristic, direct product factor, Sylow
123	C₂²×Q₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
124	C₂×C₄×Q₈	—	2	1	no	—
125	C₂²×C₁₆	C₂²×C₁₀₉	1	2	yes	—
126	C₄×Q₁₆	—	2	2	no	—
127	C₄×Q₁₆	C₄×C₁₀₉	1	4	yes	complemented
128	C₂×#107	C₂²×C₁₀₉	1	2	yes	—
129	C₂×#107	C₂²×C₁₀₉	1	1	yes	characteristic
130	C₂×#103	—	2	1	no	—
131	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
132	C₄×C₄⋊C₄	—	2	1	no	—
133	#358	C₄×C₁₀₉	1	8	yes	direct product factor
134	C₄×C₁₆	—	2	2	no	—
135	#340	—	2	1	no	—
136	C₂²×Q₈	—	2	1	no	—
137	C₂×Q₁₆	—	2	1	no	—
138	C₂×Q₁₆	—	2	2	no	—
139	C₂×Q₁₆	C₂×C₄×C₁₀₉	1	4	yes	—
140	C₂×Q₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
141	C₂²×C₈	D₄×C₁₀₉	1	1	yes	characteristic
142	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
143	C₄×Q₈	—	2	4	no	—
144	C₄×Q₈	—	4	2	no	—
145	C₂×C₄⋊C₄	—	2	1	no	—
146	C₂×C₄⋊C₄	—	2	2	no	—
147	C₂×C₄⋊C₄	—	4	1	no	—
148	C₂×C₄²	—	4	1	no	—
149	C₂×C₄²	—	4	1	no	—
150	C₂×C₄²	D₄×C₁₀₉	1	1	yes	characteristic
151	C₂×C₁₆	—	2	2	no	—
152	C₂×C₁₆	—	2	2	no	—
153	C₂×C₁₆	C₂×C₄×C₁₀₉	1	4	yes	—
154	#107	—	2	1	no	—
155	#107	—	2	2	no	—
156	#107	C₂×C₄×C₁₀₉	1	4	yes	—
157	#107	C₂×C₄×C₁₀₉	1	2	yes	—
158	#103	—	2	4	no	—
159	C₄×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
160	C₄×C₈	D₄×C₁₀₉	1	2	yes	—
161	#95	—	4	1	no	—
162	C₂×Q₈	—	2	2	no	—
163	C₂×Q₈	—	2	4	no	—
164	C₂×Q₈	—	4	1	no	—
165	C₂×Q₈	—	4	2	no	—
166	C₂²×C₄	—	4	1	no	—
167	C₂²×C₄	—	4	2	no	—
168	C₂²×C₄	—	4	1	no	—
169	C₂²×C₄	D₈×C₁₀₉	1	1	yes	characteristic
170	C₂²×C₄	D₈×C₁₀₉	1	1	yes	characteristic
171	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
172	Q₁₆	—	2	2	no	—
173	Q₁₆	—	2	4	no	—
174	Q₁₆	C₄²×C₁₀₉	1	4	yes	complemented
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	2	yes	—
177	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
178	C₂×C₈	C₂²×C₄×C₁₀₉	1	2	yes	—
179	C₂×C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
180	C₂×C₈	#41×C₁₀₉	1	2	yes	—
181	C₄⋊C₄	—	2	2	no	—
182	C₄⋊C₄	—	2	4	no	—
183	C₄⋊C₄	—	4	4	no	—
184	C₄⋊C₄	—	4	1	no	—
185	C₄⋊C₄	—	4	2	no	—
186	C₄²	—	4	4	no	—
187	C₄²	—	8	2	no	—
188	C₄²	C₂²⋊C₄×C₁₀₉	1	2	yes	—
189	C₄²	SD₁₆×C₁₀₉	1	2	yes	—
190	C₁₆	—	2	4	no	—
191	C₂³	C₂×D₈×C₁₀₉	1	1	yes	characteristic
192	Q₈	—	2	4	no	—
193	Q₈	—	4	2	no	—
194	Q₈	—	4	4	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	—	4	4	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	—	4	4	no	—
200	C₂×C₄	—	8	2	no	—
201	C₂×C₄	—	8	2	no	—
202	C₂×C₄	—	8	2	no	—
203	C₂×C₄	#102×C₁₀₉	1	2	yes	—
204	C₂×C₄	#102×C₁₀₉	1	2	yes	—
205	C₂×C₄	SD₃₂×C₁₀₉	1	2	yes	—
206	C₂×C₄	Q₃₂×C₁₀₉	1	2	yes	—
207	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
208	C₂×C₄	C₄×D₄×C₁₀₉	1	2	yes	—
209	C₂×C₄	C₂×SD₁₆×C₁₀₉	1	1	yes	characteristic
210	C₂×C₄	#135×C₁₀₉	1	2	yes	—
211	C₈	—	2	2	no	—
212	C₈	C₂×C₄²×C₁₀₉	1	1	yes	characteristic
213	C₈	#117×C₁₀₉	1	1	yes	characteristic
214	C₈	C₄×D₄×C₁₀₉	1	2	yes	—
215	C₂²	C₂×#102×C₁₀₉	1	1	yes	characteristic
216	C₂²	C₄×D₈×C₁₀₉	1	2	yes	—
217	C₂²	C₂×SD₃₂×C₁₀₉	1	1	yes	characteristic
218	C₂²	C₂×Q₃₂×C₁₀₉	1	1	yes	characteristic
219	C₂²	#508×C₁₀₉	1	2	yes	—
220	C₄	—	2	2	no	—
221	C₄	—	4	4	no	—
222	C₄	—	8	4	no	—
223	C₄	—	8	2	no	—
224	C₄	#358×C₁₀₉	1	4	yes	direct product factor
225	C₄	C₄×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
226	C₄	#416×C₁₀₉	1	1	yes	characteristic
227	C₄	C₄×SD₁₆×C₁₀₉	1	2	yes	—
228	C₂	C₄×#102×C₁₀₉	1	1	yes	characteristic
229	C₂	C₂×#358×C₁₀₉	1	1	yes	characteristic
230	C₂	C₁₀₉×#2138	1	1	yes	characteristic
231	C₂	C₄×SD₃₂×C₁₀₉	1	2	yes	—
232	C₂	C₄×Q₃₂×C₁₀₉	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₀₉	D₈
commutator	C₈	C₂×C₄²×C₁₀₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₀₉
socle, prime radical	C₂³×C₁₀₉	C₂×D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×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
10. C₄×Q₁₆×C₁₀₉	C₄	semidirect
16. #358×C₁₀₉	C₄	direct (N × H)
57. Q₁₆×C₁₀₉	C₄²	semidirect
107. C₄×C₁₀₉	#358	direct (N × H)
111. C₄×#358	C₁₀₉	direct (N × H)
122. C₁₀₉	C₄×#358	direct (N × H)
127. C₄×Q₁₆	C₄×C₁₀₉	semidirect
133. #358	C₄×C₁₀₉	direct (N × H)
174. Q₁₆	C₄²×C₁₀₉	semidirect
224. C₄	#358×C₁₀₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×Q₁₆×C₁₀₉	2
C₂×#358×C₁₀₉	2
C₂×C₄×C₁₆×C₁₀₉	2
C₄×#107×C₁₀₉	2
C₁₀₉×#1380	2
C₄×#358	109

Subgroups — C4×#358×C109 #42130630

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#358×C₁₀₉ #42130630