Subgroups — C₁₀₉×#12638 #42130632

Order 27904.

462 conjugacy classes of subgroups.

The subgroup lattice has 236 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×#346×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	complemented
17	#346×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	1	no	—
79	C₂×C₄×C₁₀₉	—	2	1	no	—
80	C₂×C₄×C₁₀₉	—	2	2	no	—
81	C₂×C₄×C₁₀₉	—	4	2	no	—
82	C₂×C₄×C₁₀₉	—	4	4	no	—
83	C₂×C₄×C₁₀₉	—	4	2	no	—
84	C₂×C₄×C₁₀₉	—	4	4	no	—
85	C₂×C₄×C₁₀₉	—	8	2	no	—
86	C₂×C₄×C₁₀₉	—	8	1	no	—
87	C₂×C₄×C₁₀₉	—	8	1	no	—
88	C₂×C₄×C₁₀₉	—	8	2	no	—
89	C₂×C₄×C₁₀₉	#102	1	2	yes	—
90	C₂×C₄×C₁₀₉	#102	1	2	yes	—
91	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₁₀₉	C₄×D₄	1	2	yes	—
93	C₂×C₄×C₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
94	C₂×C₄×C₁₀₉	#135	1	2	yes	—
95	C₈×C₁₀₉	—	2	2	no	—
96	C₈×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
97	C₈×C₁₀₉	#117	1	1	yes	characteristic
98	C₈×C₁₀₉	C₄×D₄	1	2	yes	—
99	C₂²×C₁₀₉	C₂×#102	1	1	yes	characteristic
100	C₂²×C₁₀₉	C₄×D₈	1	2	yes	—
101	C₂²×C₁₀₉	#509	1	1	yes	characteristic
102	C₂²×C₁₀₉	#510	1	1	yes	characteristic
103	C₂²×C₁₀₉	#510	1	2	yes	—
104	C₄×C₁₀₉	—	2	2	no	—
105	C₄×C₁₀₉	—	2	2	no	—
106	C₄×C₁₀₉	—	4	4	no	—
107	C₄×C₁₀₉	—	8	2	no	—
108	C₄×C₁₀₉	—	8	4	no	—
109	C₄×C₁₀₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
110	C₄×C₁₀₉	#416	1	1	yes	characteristic
111	C₄×C₁₀₉	C₄×SD₁₆	1	2	yes	—
112	#12638	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂×C₁₀₉	C₄×#102	1	1	yes	characteristic
114	C₂×C₁₀₉	#2139	1	1	yes	characteristic
115	C₂×C₁₀₉	#2141	1	1	yes	characteristic
116	C₂×C₁₀₉	#2175	1	2	yes	—
117	C₂×C₁₀₉	#2176	1	2	yes	—
118	C₂×C₄×Q₁₆	C₂×C₁₀₉	1	1	yes	characteristic
119	C₂×#358	C₂×C₁₀₉	1	2	yes	—
120	C₂×#346	C₂×C₁₀₉	1	1	yes	characteristic
121	C₄×#107	C₂×C₁₀₉	1	1	yes	characteristic
122	#1380	C₂×C₁₀₉	1	2	yes	—
123	C₁₀₉	#12638	1	1	yes	characteristic, direct product factor, Sylow
124	C₂²×Q₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
125	C₂×C₄×Q₈	—	2	1	no	—
126	C₂²×C₁₆	C₂²×C₁₀₉	1	2	yes	—
127	C₄×Q₁₆	—	2	2	no	—
128	C₄×Q₁₆	C₄×C₁₀₉	1	4	yes	complemented
129	C₂×#107	C₂²×C₁₀₉	1	2	yes	—
130	C₂×#107	C₂²×C₁₀₉	1	1	yes	characteristic
131	C₂×#103	—	2	1	no	—
132	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
133	C₄×C₄⋊C₄	—	2	1	no	—
134	#358	C₄×C₁₀₉	1	8	yes	complemented
135	#346	—	2	2	no	—
136	#340	—	2	1	no	—
137	C₂²×Q₈	—	2	1	no	—
138	C₂×Q₁₆	—	2	1	no	—
139	C₂×Q₁₆	—	2	2	no	—
140	C₂×Q₁₆	C₂×C₄×C₁₀₉	1	4	yes	—
141	C₂×Q₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
142	C₂²×C₈	D₄×C₁₀₉	1	1	yes	characteristic
143	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
144	C₄×Q₈	—	2	4	no	—
145	C₄×Q₈	—	4	2	no	—
146	C₂×C₄⋊C₄	—	2	1	no	—
147	C₂×C₄⋊C₄	—	2	2	no	—
148	C₂×C₄⋊C₄	—	4	1	no	—
149	C₂×C₄²	—	4	1	no	—
150	C₂×C₄²	—	4	1	no	—
151	C₂×C₄²	D₄×C₁₀₉	1	1	yes	characteristic
152	C₂×C₁₆	—	2	2	no	—
153	C₂×C₁₆	—	2	2	no	—
154	C₂×C₁₆	C₂×C₄×C₁₀₉	1	4	yes	—
155	#107	—	2	1	no	—
156	#107	—	2	2	no	—
157	#107	C₂×C₄×C₁₀₉	1	4	yes	—
158	#107	C₂×C₄×C₁₀₉	1	2	yes	—
159	#103	—	2	4	no	—
160	C₄×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
161	C₄×C₈	D₄×C₁₀₉	1	2	yes	—
162	#95	—	4	1	no	—
163	C₂×Q₈	—	2	2	no	—
164	C₂×Q₈	—	2	4	no	—
165	C₂×Q₈	—	4	1	no	—
166	C₂×Q₈	—	4	2	no	—
167	C₂²×C₄	—	4	1	no	—
168	C₂²×C₄	—	4	2	no	—
169	C₂²×C₄	—	4	1	no	—
170	C₂²×C₄	D₈×C₁₀₉	1	1	yes	characteristic
171	C₂²×C₄	D₈×C₁₀₉	1	1	yes	characteristic
172	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
173	Q₁₆	—	2	2	no	—
174	Q₁₆	—	2	4	no	—
175	Q₁₆	C₄²×C₁₀₉	1	4	yes	complemented
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	2	yes	—
178	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
179	C₂×C₈	C₂²×C₄×C₁₀₉	1	2	yes	—
180	C₂×C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
181	C₂×C₈	#41×C₁₀₉	1	2	yes	—
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	—	2	4	no	—
184	C₄⋊C₄	—	4	4	no	—
185	C₄⋊C₄	—	4	1	no	—
186	C₄⋊C₄	—	4	2	no	—
187	C₄²	—	4	4	no	—
188	C₄²	—	8	2	no	—
189	C₄²	C₂²⋊C₄×C₁₀₉	1	2	yes	—
190	C₄²	SD₁₆×C₁₀₉	1	2	yes	—
191	C₁₆	—	2	4	no	—
192	C₂³	C₂×D₈×C₁₀₉	1	1	yes	characteristic
193	Q₈	—	2	4	no	—
194	Q₈	—	4	2	no	—
195	Q₈	—	4	4	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	—	4	4	no	—
201	C₂×C₄	—	4	2	no	—
202	C₂×C₄	—	4	4	no	—
203	C₂×C₄	—	8	2	no	—
204	C₂×C₄	—	8	1	no	—
205	C₂×C₄	—	8	2	no	—
206	C₂×C₄	—	8	1	no	—
207	C₂×C₄	#102×C₁₀₉	1	2	yes	—
208	C₂×C₄	#102×C₁₀₉	1	2	yes	—
209	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
210	C₂×C₄	C₄×D₄×C₁₀₉	1	2	yes	—
211	C₂×C₄	C₂×SD₁₆×C₁₀₉	1	1	yes	characteristic
212	C₂×C₄	#135×C₁₀₉	1	2	yes	—
213	C₈	—	2	2	no	—
214	C₈	C₂×C₄²×C₁₀₉	1	1	yes	characteristic
215	C₈	#117×C₁₀₉	1	1	yes	characteristic
216	C₈	C₄×D₄×C₁₀₉	1	2	yes	—
217	C₂²	C₂×#102×C₁₀₉	1	1	yes	characteristic
218	C₂²	C₄×D₈×C₁₀₉	1	2	yes	—
219	C₂²	#509×C₁₀₉	1	1	yes	characteristic
220	C₂²	#510×C₁₀₉	1	2	yes	—
221	C₂²	#510×C₁₀₉	1	1	yes	characteristic
222	C₄	—	2	2	no	—
223	C₄	—	2	2	no	—
224	C₄	—	4	4	no	—
225	C₄	—	8	4	no	—
226	C₄	—	8	2	no	—
227	C₄	C₄×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
228	C₄	#416×C₁₀₉	1	1	yes	characteristic
229	C₄	C₄×SD₁₆×C₁₀₉	1	2	yes	—
230	C₂	C₄×#102×C₁₀₉	1	1	yes	characteristic
231	C₂	C₁₀₉×#2139	1	1	yes	characteristic
232	C₂	C₁₀₉×#2141	1	1	yes	characteristic
233	C₂	C₁₀₉×#2175	1	2	yes	—
234	C₂	C₁₀₉×#2176	1	2	yes	—

	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
10. C₄×Q₁₆×C₁₀₉	C₄	semidirect
16. #358×C₁₀₉	C₄	semidirect
57. Q₁₆×C₁₀₉	C₄²	semidirect
112. #12638	C₁₀₉	direct (N × H)
123. C₁₀₉	#12638	direct (N × H)
128. C₄×Q₁₆	C₄×C₁₀₉	semidirect
134. #358	C₄×C₁₀₉	semidirect
175. Q₁₆	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×C₄×Q₁₆×C₁₀₉	2
C₂×#358×C₁₀₉	2
C₂×#346×C₁₀₉	2
C₄×#107×C₁₀₉	2
C₁₀₉×#1380	2
#12638	109

Subgroups — C109×#12638 #42130632

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#12638 #42130632