Subgroups — C₁₀₉×#10278 #42128272

Order 27904.

562 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#500×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#476×C₁₀₉	C₂	1	2	yes	maximal
3	C₁₀₉×#1954	C₂	1	1	yes	maximal, characteristic
4	C₂×#333×C₁₀₉	C₂	1	2	yes	maximal, complemented
5	C₁₀₉×#1277	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄⋊Q₈×C₁₀₉	C₂²	1	2	yes	—
7	C₂×C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
8	#500×C₁₀₉	—	2	4	no	—
9	#476×C₁₀₉	—	2	4	no	—
10	#476×C₁₀₉	—	2	4	no	—
11	C₂×#107×C₁₀₉	—	2	2	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×#102×C₁₀₉	—	2	2	no	—
16	C₂×C₂²⋊C₈×C₁₀₉	—	2	2	no	—
17	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
18	#333×C₁₀₉	—	2	4	no	—
19	C₂²×Q₈×C₁₀₉	—	2	2	no	—
20	C₂²×D₄×C₁₀₉	D₄	1	1	yes	characteristic
21	C₂³×C₄×C₁₀₉	—	2	1	no	—
22	C₂²×C₈×C₁₀₉	—	2	2	no	—
23	C₂²×C₈×C₁₀₉	—	2	1	no	—
24	C₂²×C₈×C₁₀₉	D₄	1	2	yes	—
25	C₄⋊Q₈×C₁₀₉	—	2	2	no	—
26	C₄⋊Q₈×C₁₀₉	—	2	2	no	—
27	C₄⋊Q₈×C₁₀₉	—	2	2	no	—
28	C₄⋊Q₈×C₁₀₉	D₄	1	4	yes	—
29	C₄×D₄×C₁₀₉	—	2	2	no	—
30	C₄×D₄×C₁₀₉	—	2	2	no	—
31	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
32	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
33	C₂×C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
34	C₂×C₂²⋊C₄×C₁₀₉	—	2	1	no	—
35	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
36	#107×C₁₀₉	—	2	8	no	—
37	#106×C₁₀₉	—	2	8	no	—
38	C₄⋊C₈×C₁₀₉	—	2	2	no	—
39	C₄⋊C₈×C₁₀₉	—	2	2	no	—
40	C₄⋊C₈×C₁₀₉	D₄	1	4	yes	—
41	#102×C₁₀₉	—	2	8	no	—
42	C₂²⋊C₈×C₁₀₉	—	4	4	no	—
43	C₄×C₈×C₁₀₉	—	2	2	no	—
44	C₂⁴×C₁₀₉	—	2	1	no	—
45	C₂×Q₈×C₁₀₉	—	2	4	no	—
46	C₂×Q₈×C₁₀₉	—	4	2	no	—
47	C₂×Q₈×C₁₀₉	—	8	4	no	—
48	C₂×D₄×C₁₀₉	—	2	2	no	—
49	C₂×D₄×C₁₀₉	—	2	1	no	—
50	C₂×D₄×C₁₀₉	—	2	2	no	—
51	C₂×D₄×C₁₀₉	Q₁₆	1	2	yes	—
52	C₂²×C₄×C₁₀₉	—	2	1	no	—
53	C₂²×C₄×C₁₀₉	—	4	2	no	—
54	C₂²×C₄×C₁₀₉	—	4	1	no	—
55	C₂²×C₄×C₁₀₉	—	4	1	no	—
56	C₂²×C₄×C₁₀₉	—	4	2	no	—
57	C₂²×C₄×C₁₀₉	—	4	2	no	—
58	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₀₉	#41	1	1	yes	characteristic
61	C₂×C₈×C₁₀₉	—	2	2	no	—
62	C₂×C₈×C₁₀₉	—	2	2	no	—
63	C₂×C₈×C₁₀₉	—	2	4	no	—
64	C₂×C₈×C₁₀₉	—	4	2	no	—
65	C₂×C₈×C₁₀₉	—	4	2	no	—
66	C₂×C₈×C₁₀₉	—	4	1	no	—
67	C₂×C₈×C₁₀₉	—	4	2	no	—
68	C₂×C₈×C₁₀₉	SD₁₆	1	4	yes	—
69	C₄⋊C₄×C₁₀₉	—	2	4	no	—
70	C₄⋊C₄×C₁₀₉	—	2	2	no	—
71	C₄⋊C₄×C₁₀₉	—	2	4	no	—
72	C₄⋊C₄×C₁₀₉	—	4	2	no	—
73	C₄⋊C₄×C₁₀₉	—	4	2	no	—
74	C₂²⋊C₄×C₁₀₉	—	4	2	no	—
75	C₄²×C₁₀₉	C₂×D₄	1	2	yes	—
76	C₄²×C₁₀₉	C₂×D₄	1	2	yes	—
77	C₂³×C₁₀₉	—	2	2	no	—
78	C₂³×C₁₀₉	—	4	1	no	—
79	C₂³×C₁₀₉	—	4	2	no	—
80	C₂³×C₁₀₉	—	4	1	no	—
81	C₂³×C₁₀₉	—	4	2	no	—
82	C₂³×C₁₀₉	#121	1	1	yes	characteristic
83	Q₈×C₁₀₉	—	8	4	no	—
84	Q₈×C₁₀₉	—	8	4	no	—
85	D₄×C₁₀₉	—	2	4	no	—
86	D₄×C₁₀₉	—	4	2	no	—
87	C₂×C₄×C₁₀₉	—	2	2	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	—	8	2	no	—
93	C₂×C₄×C₁₀₉	—	8	1	no	—
94	C₂×C₄×C₁₀₉	—	8	1	no	—
95	C₂×C₄×C₁₀₉	#120	1	1	yes	characteristic
96	C₂×C₄×C₁₀₉	#120	1	2	yes	—
97	C₂×C₄×C₁₀₉	#121	1	2	yes	—
98	C₂×C₄×C₁₀₉	#127	1	1	yes	characteristic
99	C₂×C₄×C₁₀₉	C₂×SD₁₆	1	2	yes	—
100	C₂×C₄×C₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
101	C₂×C₄×C₁₀₉	#135	1	1	yes	characteristic
102	C₂×C₄×C₁₀₉	#137	1	2	yes	—
103	C₂×C₄×C₁₀₉	#137	1	2	yes	—
104	C₈×C₁₀₉	—	2	4	no	—
105	C₈×C₁₀₉	—	4	4	no	—
106	C₈×C₁₀₉	—	8	1	no	—
107	C₂²×C₁₀₉	—	4	4	no	—
108	C₂²×C₁₀₉	—	4	2	no	—
109	C₂²×C₁₀₉	—	8	2	no	—
110	C₂²×C₁₀₉	—	8	1	no	—
111	C₂²×C₁₀₉	—	8	1	no	—
112	C₂²×C₁₀₉	#392	1	1	yes	characteristic
113	C₂²×C₁₀₉	#461	1	2	yes	—
114	C₂²×C₁₀₉	#467	1	1	yes	characteristic
115	C₂²×C₁₀₉	#470	1	1	yes	characteristic
116	C₂²×C₁₀₉	#472	1	2	yes	—
117	C₄×C₁₀₉	—	2	2	no	—
118	C₄×C₁₀₉	—	8	1	no	—
119	C₄×C₁₀₉	—	8	4	no	—
120	C₄×C₁₀₉	#448	1	2	yes	—
121	C₄×C₁₀₉	#451	1	2	yes	—
122	C₄×C₁₀₉	#452	1	2	yes	—
123	C₄×C₁₀₉	#492	1	1	yes	characteristic
124	C₄×C₁₀₉	#497	1	1	yes	characteristic
125	#10278	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
126	C₂×C₁₀₉	—	8	2	no	—
127	C₂×C₁₀₉	#1675	1	2	yes	—
128	C₂×C₁₀₉	#1695	1	2	yes	—
129	C₂×C₁₀₉	#2001	1	1	yes	characteristic
130	C₂×C₁₀₉	#2013	1	1	yes	characteristic
131	C₂×C₁₀₉	#2018	1	1	yes	characteristic
132	C₂×#500	C₂×C₁₀₉	1	1	yes	characteristic, complemented
133	C₂×#476	C₂×C₁₀₉	1	2	yes	—
134	#1954	C₂×C₁₀₉	1	1	yes	characteristic
135	C₂×#333	C₂×C₁₀₉	1	2	yes	complemented
136	#1277	C₂×C₁₀₉	1	1	yes	characteristic, complemented
137	C₁₀₉	#10278	1	1	yes	characteristic, direct product factor, Sylow
138	C₂×C₄⋊Q₈	C₂²×C₁₀₉	1	2	yes	—
139	C₂×C₄×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
140	#500	—	2	4	no	—
141	#476	—	2	4	no	—
142	#476	—	2	4	no	—
143	C₂×#107	—	2	2	no	—
144	C₂×#106	—	2	2	no	—
145	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
146	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
147	C₂×#102	—	2	2	no	—
148	C₂×C₂²⋊C₈	—	2	2	no	—
149	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
150	#333	—	2	4	no	—
151	C₂²×Q₈	—	2	2	no	—
152	C₂²×D₄	D₄×C₁₀₉	1	1	yes	characteristic
153	C₂³×C₄	—	2	1	no	—
154	C₂²×C₈	—	2	2	no	—
155	C₂²×C₈	—	2	1	no	—
156	C₂²×C₈	D₄×C₁₀₉	1	2	yes	—
157	C₄⋊Q₈	—	2	2	no	—
158	C₄⋊Q₈	—	2	2	no	—
159	C₄⋊Q₈	—	2	2	no	—
160	C₄⋊Q₈	D₄×C₁₀₉	1	4	yes	—
161	C₄×D₄	—	2	2	no	—
162	C₄×D₄	—	2	2	no	—
163	C₂×C₄⋊C₄	—	2	2	no	—
164	C₂×C₄⋊C₄	—	2	2	no	—
165	C₂×C₄⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
166	C₂×C₂²⋊C₄	—	2	1	no	—
167	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
168	#107	—	2	8	no	—
169	#106	—	2	8	no	—
170	C₄⋊C₈	—	2	2	no	—
171	C₄⋊C₈	—	2	2	no	—
172	C₄⋊C₈	D₄×C₁₀₉	1	4	yes	—
173	#102	—	2	8	no	—
174	C₂²⋊C₈	—	4	4	no	—
175	C₄×C₈	—	2	2	no	—
176	C₂⁴	—	2	1	no	—
177	C₂×Q₈	—	2	4	no	—
178	C₂×Q₈	—	4	2	no	—
179	C₂×Q₈	—	8	4	no	—
180	C₂×D₄	—	2	2	no	—
181	C₂×D₄	—	2	1	no	—
182	C₂×D₄	—	2	2	no	—
183	C₂×D₄	Q₁₆×C₁₀₉	1	2	yes	—
184	C₂²×C₄	—	2	1	no	—
185	C₂²×C₄	—	4	2	no	—
186	C₂²×C₄	—	4	1	no	—
187	C₂²×C₄	—	4	1	no	—
188	C₂²×C₄	—	4	2	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
191	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
192	C₂²×C₄	#41×C₁₀₉	1	1	yes	characteristic
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	4	no	—
196	C₂×C₈	—	4	2	no	—
197	C₂×C₈	—	4	2	no	—
198	C₂×C₈	—	4	1	no	—
199	C₂×C₈	—	4	2	no	—
200	C₂×C₈	SD₁₆×C₁₀₉	1	4	yes	—
201	C₄⋊C₄	—	2	4	no	—
202	C₄⋊C₄	—	2	2	no	—
203	C₄⋊C₄	—	2	4	no	—
204	C₄⋊C₄	—	4	2	no	—
205	C₄⋊C₄	—	4	2	no	—
206	C₂²⋊C₄	—	4	2	no	—
207	C₄²	C₂×D₄×C₁₀₉	1	2	yes	—
208	C₄²	C₂×D₄×C₁₀₉	1	2	yes	—
209	C₂³	—	2	2	no	—
210	C₂³	—	4	1	no	—
211	C₂³	—	4	2	no	—
212	C₂³	—	4	1	no	—
213	C₂³	—	4	2	no	—
214	C₂³	#121×C₁₀₉	1	1	yes	characteristic
215	Q₈	—	8	4	no	—
216	Q₈	—	8	4	no	—
217	D₄	—	2	4	no	—
218	D₄	—	4	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	4	1	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	8	4	no	—
224	C₂×C₄	—	8	2	no	—
225	C₂×C₄	—	8	1	no	—
226	C₂×C₄	—	8	1	no	—
227	C₂×C₄	#120×C₁₀₉	1	1	yes	characteristic
228	C₂×C₄	#120×C₁₀₉	1	2	yes	—
229	C₂×C₄	#121×C₁₀₉	1	2	yes	—
230	C₂×C₄	#127×C₁₀₉	1	1	yes	characteristic
231	C₂×C₄	C₂×SD₁₆×C₁₀₉	1	2	yes	—
232	C₂×C₄	C₂×Q₁₆×C₁₀₉	1	1	yes	characteristic
233	C₂×C₄	#135×C₁₀₉	1	1	yes	characteristic
234	C₂×C₄	#137×C₁₀₉	1	2	yes	—
235	C₂×C₄	#137×C₁₀₉	1	2	yes	—
236	C₈	—	2	4	no	—
237	C₈	—	4	4	no	—
238	C₈	—	8	1	no	—
239	C₂²	—	4	4	no	—
240	C₂²	—	4	2	no	—
241	C₂²	—	8	2	no	—
242	C₂²	—	8	1	no	—
243	C₂²	—	8	1	no	—
244	C₂²	#392×C₁₀₉	1	1	yes	characteristic
245	C₂²	#461×C₁₀₉	1	2	yes	—
246	C₂²	#467×C₁₀₉	1	1	yes	characteristic
247	C₂²	#470×C₁₀₉	1	1	yes	characteristic
248	C₂²	#472×C₁₀₉	1	2	yes	—
249	C₄	—	2	2	no	—
250	C₄	—	8	1	no	—
251	C₄	—	8	4	no	—
252	C₄	#448×C₁₀₉	1	2	yes	—
253	C₄	#451×C₁₀₉	1	2	yes	—
254	C₄	#452×C₁₀₉	1	2	yes	—
255	C₄	#492×C₁₀₉	1	1	yes	characteristic
256	C₄	#497×C₁₀₉	1	1	yes	characteristic
257	C₂	—	8	2	no	—
258	C₂	C₁₀₉×#1675	1	2	yes	—
259	C₂	C₁₀₉×#1695	1	2	yes	—
260	C₂	C₁₀₉×#2001	1	1	yes	characteristic
261	C₂	C₁₀₉×#2013	1	1	yes	characteristic
262	C₂	C₁₀₉×#2018	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₀₉	#121
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₉
prime radical	C₂²×D₄×C₁₀₉	D₄
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
1. C₂×#500×C₁₀₉	C₂	semidirect
4. C₂×#333×C₁₀₉	C₂	semidirect
5. C₁₀₉×#1277	C₂	semidirect
125. #10278	C₁₀₉	direct (N × H)
132. C₂×#500	C₂×C₁₀₉	semidirect
135. C₂×#333	C₂×C₁₀₉	semidirect
136. #1277	C₂×C₁₀₉	semidirect
137. C₁₀₉	#10278	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×#500×C₁₀₉	2
C₂×#476×C₁₀₉	2
C₁₀₉×#1954	2
C₂×#333×C₁₀₉	2
C₁₀₉×#1277	2
#10278	109

Subgroups — C109×#10278 #42128272

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#10278 #42128272