Subgroups — C₁₀₉×#10276 #42128270

Order 27904.

574 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#494×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#475×C₁₀₉	C₂	1	2	yes	maximal
3	C₁₀₉×#1984	C₂	1	1	yes	maximal, characteristic
4	C₂×#332×C₁₀₉	C₂	1	2	yes	maximal
5	C₁₀₉×#1277	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₁₆×C₁₀₉	—	2	2	no	—
7	C₂×C₄⋊Q₈×C₁₀₉	C₂²	1	2	yes	—
8	C₂×C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
9	#494×C₁₀₉	—	2	4	no	—
10	#475×C₁₀₉	—	2	4	no	—
11	#475×C₁₀₉	—	2	4	no	—
12	C₂×#106×C₁₀₉	—	2	2	no	—
13	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
15	C₂×#103×C₁₀₉	—	2	2	no	—
16	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
17	#336×C₁₀₉	—	2	2	no	—
18	#332×C₁₀₉	—	2	4	no	—
19	C₂²×Q₈×C₁₀₉	—	2	2	no	—
20	C₂²×Q₈×C₁₀₉	D₄	1	1	yes	characteristic
21	C₂×Q₁₆×C₁₀₉	—	2	8	no	—
22	C₂×Q₁₆×C₁₀₉	—	4	4	no	—
23	C₂×Q₁₆×C₁₀₉	—	4	4	no	—
24	C₂²×C₈×C₁₀₉	—	2	2	no	—
25	C₂²×C₈×C₁₀₉	—	2	1	no	—
26	C₂²×C₈×C₁₀₉	D₄	1	2	yes	—
27	C₄⋊Q₈×C₁₀₉	—	2	2	no	—
28	C₄⋊Q₈×C₁₀₉	—	2	2	no	—
29	C₄⋊Q₈×C₁₀₉	—	2	2	no	—
30	C₄⋊Q₈×C₁₀₉	D₄	1	4	yes	—
31	C₄×Q₈×C₁₀₉	—	2	2	no	—
32	C₄×Q₈×C₁₀₉	—	2	2	no	—
33	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
34	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
35	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
36	C₂×C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₀₉	—	2	1	no	—
38	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
39	#106×C₁₀₉	—	2	8	no	—
40	C₄⋊C₈×C₁₀₉	—	2	2	no	—
41	C₄⋊C₈×C₁₀₉	—	2	2	no	—
42	C₄⋊C₈×C₁₀₉	D₄	1	4	yes	—
43	#103×C₁₀₉	—	2	8	no	—
44	C₄×C₈×C₁₀₉	—	2	2	no	—
45	C₂×Q₈×C₁₀₉	—	2	1	no	—
46	C₂×Q₈×C₁₀₉	—	2	2	no	—
47	C₂×Q₈×C₁₀₉	—	2	4	no	—
48	C₂×Q₈×C₁₀₉	—	2	2	no	—
49	C₂×Q₈×C₁₀₉	—	4	2	no	—
50	C₂×Q₈×C₁₀₉	—	4	4	no	—
51	C₂×Q₈×C₁₀₉	—	4	4	no	—
52	C₂×Q₈×C₁₀₉	Q₁₆	1	2	yes	complemented
53	C₂²×C₄×C₁₀₉	—	2	1	no	—
54	C₂²×C₄×C₁₀₉	—	2	1	no	—
55	C₂²×C₄×C₁₀₉	—	4	2	no	—
56	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₀₉	#41	1	1	yes	characteristic
59	Q₁₆×C₁₀₉	—	4	16	no	—
60	C₂×C₈×C₁₀₉	—	2	2	no	—
61	C₂×C₈×C₁₀₉	—	2	2	no	—
62	C₂×C₈×C₁₀₉	—	2	4	no	—
63	C₂×C₈×C₁₀₉	—	4	2	no	—
64	C₂×C₈×C₁₀₉	—	4	2	no	—
65	C₂×C₈×C₁₀₉	—	4	1	no	—
66	C₂×C₈×C₁₀₉	—	4	2	no	—
67	C₂×C₈×C₁₀₉	SD₁₆	1	4	yes	—
68	C₄⋊C₄×C₁₀₉	—	2	4	no	—
69	C₄⋊C₄×C₁₀₉	—	2	2	no	—
70	C₄⋊C₄×C₁₀₉	—	2	4	no	—
71	C₄⋊C₄×C₁₀₉	—	4	2	no	—
72	C₄⋊C₄×C₁₀₉	—	4	2	no	—
73	C₄⋊C₄×C₁₀₉	—	4	2	no	—
74	C₄²×C₁₀₉	—	4	1	no	—
75	C₄²×C₁₀₉	—	4	1	no	—
76	C₄²×C₁₀₉	C₂×D₄	1	2	yes	—
77	C₄²×C₁₀₉	C₂×D₄	1	2	yes	—
78	C₂³×C₁₀₉	#121	1	1	yes	characteristic
79	Q₈×C₁₀₉	—	2	4	no	—
80	Q₈×C₁₀₉	—	4	2	no	—
81	Q₈×C₁₀₉	—	4	8	no	—
82	Q₈×C₁₀₉	—	8	2	no	—
83	Q₈×C₁₀₉	—	8	2	no	—
84	C₂×C₄×C₁₀₉	—	2	2	no	—
85	C₂×C₄×C₁₀₉	—	2	2	no	—
86	C₂×C₄×C₁₀₉	—	4	1	no	—
87	C₂×C₄×C₁₀₉	—	4	1	no	—
88	C₂×C₄×C₁₀₉	—	4	1	no	—
89	C₂×C₄×C₁₀₉	—	4	4	no	—
90	C₂×C₄×C₁₀₉	—	4	2	no	—
91	C₂×C₄×C₁₀₉	—	8	4	no	—
92	C₂×C₄×C₁₀₉	#120	1	1	yes	characteristic
93	C₂×C₄×C₁₀₉	#120	1	2	yes	—
94	C₂×C₄×C₁₀₉	#121	1	2	yes	—
95	C₂×C₄×C₁₀₉	#127	1	1	yes	characteristic
96	C₂×C₄×C₁₀₉	C₂×SD₁₆	1	2	yes	—
97	C₂×C₄×C₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
98	C₂×C₄×C₁₀₉	#135	1	1	yes	characteristic
99	C₂×C₄×C₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
100	C₂×C₄×C₁₀₉	#137	1	2	yes	—
101	C₈×C₁₀₉	—	2	4	no	—
102	C₈×C₁₀₉	—	4	4	no	—
103	C₈×C₁₀₉	—	8	1	no	—
104	C₂²×C₁₀₉	#392	1	1	yes	characteristic
105	C₂²×C₁₀₉	#460	1	2	yes	—
106	C₂²×C₁₀₉	#467	1	1	yes	characteristic
107	C₂²×C₁₀₉	#470	1	1	yes	characteristic
108	C₂²×C₁₀₉	#473	1	2	yes	—
109	C₄×C₁₀₉	—	2	2	no	—
110	C₄×C₁₀₉	—	4	2	no	—
111	C₄×C₁₀₉	—	8	4	no	—
112	C₄×C₁₀₉	—	8	1	no	—
113	C₄×C₁₀₉	#450	1	2	yes	—
114	C₄×C₁₀₉	#451	1	2	yes	—
115	C₄×C₁₀₉	#452	1	2	yes	—
116	C₄×C₁₀₉	#492	1	1	yes	characteristic
117	C₄×C₁₀₉	#496	1	1	yes	characteristic
118	#10276	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
119	C₂×C₁₀₉	#1670	1	2	yes	—
120	C₂×C₁₀₉	#1690	1	2	yes	—
121	C₂×C₁₀₉	#2001	1	1	yes	characteristic
122	C₂×C₁₀₉	#2014	1	1	yes	characteristic
123	C₂×C₁₀₉	#2020	1	1	yes	characteristic
124	C₂×#494	C₂×C₁₀₉	1	1	yes	characteristic
125	C₂×#475	C₂×C₁₀₉	1	2	yes	—
126	#1984	C₂×C₁₀₉	1	1	yes	characteristic
127	C₂×#332	C₂×C₁₀₉	1	2	yes	—
128	#1277	C₂×C₁₀₉	1	1	yes	characteristic
129	C₁₀₉	#10276	1	1	yes	characteristic, direct product factor, Sylow
130	C₂²×Q₁₆	—	2	2	no	—
131	C₂×C₄⋊Q₈	C₂²×C₁₀₉	1	2	yes	—
132	C₂×C₄×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
133	#494	—	2	4	no	—
134	#475	—	2	4	no	—
135	#475	—	2	4	no	—
136	C₂×#106	—	2	2	no	—
137	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
138	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
139	C₂×#103	—	2	2	no	—
140	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
141	#336	—	2	2	no	—
142	#332	—	2	4	no	—
143	C₂²×Q₈	—	2	2	no	—
144	C₂²×Q₈	D₄×C₁₀₉	1	1	yes	characteristic
145	C₂×Q₁₆	—	2	8	no	—
146	C₂×Q₁₆	—	4	4	no	—
147	C₂×Q₁₆	—	4	4	no	—
148	C₂²×C₈	—	2	2	no	—
149	C₂²×C₈	—	2	1	no	—
150	C₂²×C₈	D₄×C₁₀₉	1	2	yes	—
151	C₄⋊Q₈	—	2	2	no	—
152	C₄⋊Q₈	—	2	2	no	—
153	C₄⋊Q₈	—	2	2	no	—
154	C₄⋊Q₈	D₄×C₁₀₉	1	4	yes	—
155	C₄×Q₈	—	2	2	no	—
156	C₄×Q₈	—	2	2	no	—
157	C₂×C₄⋊C₄	—	2	2	no	—
158	C₂×C₄⋊C₄	—	2	2	no	—
159	C₂×C₄⋊C₄	—	2	1	no	—
160	C₂×C₄⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
161	C₂×C₄²	—	2	1	no	—
162	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
163	#106	—	2	8	no	—
164	C₄⋊C₈	—	2	2	no	—
165	C₄⋊C₈	—	2	2	no	—
166	C₄⋊C₈	D₄×C₁₀₉	1	4	yes	—
167	#103	—	2	8	no	—
168	C₄×C₈	—	2	2	no	—
169	C₂×Q₈	—	2	1	no	—
170	C₂×Q₈	—	2	2	no	—
171	C₂×Q₈	—	2	4	no	—
172	C₂×Q₈	—	2	2	no	—
173	C₂×Q₈	—	4	2	no	—
174	C₂×Q₈	—	4	4	no	—
175	C₂×Q₈	—	4	4	no	—
176	C₂×Q₈	Q₁₆×C₁₀₉	1	2	yes	complemented
177	C₂²×C₄	—	2	1	no	—
178	C₂²×C₄	—	2	1	no	—
179	C₂²×C₄	—	4	2	no	—
180	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
181	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
182	C₂²×C₄	#41×C₁₀₉	1	1	yes	characteristic
183	Q₁₆	—	4	16	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	4	no	—
187	C₂×C₈	—	4	2	no	—
188	C₂×C₈	—	4	2	no	—
189	C₂×C₈	—	4	1	no	—
190	C₂×C₈	—	4	2	no	—
191	C₂×C₈	SD₁₆×C₁₀₉	1	4	yes	—
192	C₄⋊C₄	—	2	4	no	—
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	2	4	no	—
195	C₄⋊C₄	—	4	2	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₄⋊C₄	—	4	2	no	—
198	C₄²	—	4	1	no	—
199	C₄²	—	4	1	no	—
200	C₄²	C₂×D₄×C₁₀₉	1	2	yes	—
201	C₄²	C₂×D₄×C₁₀₉	1	2	yes	—
202	C₂³	#121×C₁₀₉	1	1	yes	characteristic
203	Q₈	—	2	4	no	—
204	Q₈	—	4	2	no	—
205	Q₈	—	4	8	no	—
206	Q₈	—	8	2	no	—
207	Q₈	—	8	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	4	1	no	—
211	C₂×C₄	—	4	1	no	—
212	C₂×C₄	—	4	1	no	—
213	C₂×C₄	—	4	4	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	8	4	no	—
216	C₂×C₄	#120×C₁₀₉	1	1	yes	characteristic
217	C₂×C₄	#120×C₁₀₉	1	2	yes	—
218	C₂×C₄	#121×C₁₀₉	1	2	yes	—
219	C₂×C₄	#127×C₁₀₉	1	1	yes	characteristic
220	C₂×C₄	C₂×SD₁₆×C₁₀₉	1	2	yes	—
221	C₂×C₄	C₂×Q₁₆×C₁₀₉	1	1	yes	characteristic
222	C₂×C₄	#135×C₁₀₉	1	1	yes	characteristic
223	C₂×C₄	AGL(1,ℤ/8)×C₁₀₉	1	2	yes	—
224	C₂×C₄	#137×C₁₀₉	1	2	yes	—
225	C₈	—	2	4	no	—
226	C₈	—	4	4	no	—
227	C₈	—	8	1	no	—
228	C₂²	#392×C₁₀₉	1	1	yes	characteristic
229	C₂²	#460×C₁₀₉	1	2	yes	—
230	C₂²	#467×C₁₀₉	1	1	yes	characteristic
231	C₂²	#470×C₁₀₉	1	1	yes	characteristic
232	C₂²	#473×C₁₀₉	1	2	yes	—
233	C₄	—	2	2	no	—
234	C₄	—	4	2	no	—
235	C₄	—	8	4	no	—
236	C₄	—	8	1	no	—
237	C₄	#450×C₁₀₉	1	2	yes	—
238	C₄	#451×C₁₀₉	1	2	yes	—
239	C₄	#452×C₁₀₉	1	2	yes	—
240	C₄	#492×C₁₀₉	1	1	yes	characteristic
241	C₄	#496×C₁₀₉	1	1	yes	characteristic
242	C₂	C₁₀₉×#1670	1	2	yes	—
243	C₂	C₁₀₉×#1690	1	2	yes	—
244	C₂	C₁₀₉×#2001	1	1	yes	characteristic
245	C₂	C₁₀₉×#2014	1	1	yes	characteristic
246	C₂	C₁₀₉×#2020	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₀₉	#121
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₉
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
52. C₂×Q₈×C₁₀₉	Q₁₆	semidirect
118. #10276	C₁₀₉	direct (N × H)
129. C₁₀₉	#10276	direct (N × H)
176. C₂×Q₈	Q₁₆×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#494×C₁₀₉	2
C₂×#475×C₁₀₉	2
C₁₀₉×#1984	2
C₂×#332×C₁₀₉	2
C₁₀₉×#1277	2
#10276	109

Subgroups — C109×#10276 #42128270

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#10276 #42128270