Subgroups — C₁₀₉×#12651 #42130645

Order 27904.

412 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#444×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#2148	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#2143	C₂	1	1	yes	maximal, characteristic
4	C₂×#347×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₀₉×#1778	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₀₉×#1388	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₀₉×#1385	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#135×C₁₀₉	C₂²	1	1	yes	characteristic
9	C₂×#131×C₁₀₉	—	2	1	no	—
10	C₂×M₃₂×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₂×M₃₂×C₁₀₉	C₂²	1	1	yes	characteristic
12	#444×C₁₀₉	—	2	2	no	—
13	#444×C₁₀₉	C₄	1	4	yes	—
14	C₂×#108×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂×#108×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₂×#108×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₂×#104×C₁₀₉	—	2	1	no	—
18	C₄×M₁₆×C₁₀₉	—	2	1	no	—
19	C₂×#97×C₁₀₉	C₂²	1	1	yes	characteristic
20	#362×C₁₀₉	C₄	1	2	yes	complemented
21	#361×C₁₀₉	C₄	1	2	yes	complemented
22	#359×C₁₀₉	C₄	1	4	yes	complemented
23	#347×C₁₀₉	—	2	2	no	—
24	#341×C₁₀₉	—	2	1	no	—
25	C₂×#41×C₁₀₉	—	2	1	no	—
26	#135×C₁₀₉	—	2	1	no	—
27	#135×C₁₀₉	—	2	1	no	—
28	#135×C₁₀₉	C₂×C₄	1	2	yes	—
29	#135×C₁₀₉	C₂×C₄	1	2	yes	—
30	C₂×Q₁₆×C₁₀₉	C₂×C₄	1	1	yes	characteristic
31	C₂×SD₁₆×C₁₀₉	—	2	1	no	—
32	C₂×D₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
33	#131×C₁₀₉	—	2	4	no	—
34	#131×C₁₀₉	—	4	2	no	—
35	C₂×M₁₆×C₁₀₉	—	2	1	no	—
36	C₂×M₁₆×C₁₀₉	—	2	1	no	—
37	C₂×M₁₆×C₁₀₉	—	2	1	no	—
38	C₂×M₁₆×C₁₀₉	—	4	1	no	—
39	C₂²×C₈×C₁₀₉	—	4	1	no	—
40	C₂²×C₈×C₁₀₉	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₁₀₉	D₄	1	1	yes	characteristic
43	M₃₂×C₁₀₉	—	2	1	no	—
44	M₃₂×C₁₀₉	—	2	1	no	—
45	M₃₂×C₁₀₉	—	2	1	no	—
46	M₃₂×C₁₀₉	C₂×C₄	1	2	yes	—
47	C₂×C₁₆×C₁₀₉	—	2	1	no	—
48	C₂×C₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
49	#108×C₁₀₉	—	2	1	no	—
50	#108×C₁₀₉	—	2	1	no	—
51	#108×C₁₀₉	—	2	1	no	—
52	#108×C₁₀₉	C₂×C₄	1	2	yes	—
53	#108×C₁₀₉	C₂×C₄	1	2	yes	—
54	#108×C₁₀₉	C₂×C₄	1	2	yes	—
55	#104×C₁₀₉	—	2	4	no	—
56	#97×C₁₀₉	—	4	1	no	—
57	#97×C₁₀₉	C₂×C₄	1	2	yes	—
58	#97×C₁₀₉	D₄	1	2	yes	—
59	C₄×C₈×C₁₀₉	—	4	1	no	—
60	#41×C₁₀₉	—	2	2	no	—
61	#41×C₁₀₉	—	2	2	no	—
62	#41×C₁₀₉	—	4	1	no	—
63	#41×C₁₀₉	—	4	1	no	—
64	C₂×Q₈×C₁₀₉	—	2	1	no	—
65	C₂×D₄×C₁₀₉	—	2	1	no	—
66	C₂×D₄×C₁₀₉	—	4	1	no	—
67	C₂²×C₄×C₁₀₉	—	2	1	no	—
68	C₂²×C₄×C₁₀₉	—	4	1	no	—
69	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
70	Q₁₆×C₁₀₉	—	2	1	no	—
71	Q₁₆×C₁₀₉	C₄²	1	2	yes	—
72	SD₁₆×C₁₀₉	—	2	2	no	—
73	SD₁₆×C₁₀₉	—	2	2	no	—
74	D₈×C₁₀₉	—	2	1	no	—
75	D₈×C₁₀₉	C₄²	1	2	yes	—
76	M₁₆×C₁₀₉	—	2	2	no	—
77	M₁₆×C₁₀₉	—	2	2	no	—
78	M₁₆×C₁₀₉	—	2	2	no	—
79	M₁₆×C₁₀₉	—	4	4	no	—
80	M₁₆×C₁₀₉	—	4	1	no	—
81	M₁₆×C₁₀₉	—	4	1	no	—
82	M₁₆×C₁₀₉	—	4	1	no	—
83	C₂×C₈×C₁₀₉	—	2	1	no	—
84	C₂×C₈×C₁₀₉	—	2	1	no	—
85	C₂×C₈×C₁₀₉	—	4	4	no	—
86	C₂×C₈×C₁₀₉	—	4	1	no	—
87	C₂×C₈×C₁₀₉	—	4	1	no	—
88	C₂×C₈×C₁₀₉	—	4	1	no	—
89	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	2	yes	—
90	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₁₀₉	#41	1	1	yes	characteristic
95	C₂×C₈×C₁₀₉	#41	1	1	yes	characteristic
96	C₄²×C₁₀₉	—	2	1	no	—
97	C₄²×C₁₀₉	C₂²⋊C₄	1	2	yes	—
98	C₁₆×C₁₀₉	—	2	2	no	—
99	C₁₆×C₁₀₉	—	2	2	no	—
100	C₂³×C₁₀₉	—	4	1	no	—
101	C₂³×C₁₀₉	#137	1	1	yes	characteristic
102	Q₈×C₁₀₉	—	2	2	no	—
103	Q₈×C₁₀₉	—	4	1	no	—
104	D₄×C₁₀₉	—	2	2	no	—
105	D₄×C₁₀₉	—	4	2	no	—
106	D₄×C₁₀₉	—	4	2	no	—
107	D₄×C₁₀₉	—	4	1	no	—
108	C₂×C₄×C₁₀₉	—	2	2	no	—
109	C₂×C₄×C₁₀₉	—	4	1	no	—
110	C₂×C₄×C₁₀₉	—	4	2	no	—
111	C₂×C₄×C₁₀₉	—	4	2	no	—
112	C₂×C₄×C₁₀₉	—	8	1	no	—
113	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₀₉	AGL(1,ℤ/8)	1	1	yes	characteristic
117	C₂×C₄×C₁₀₉	AGL(1,ℤ/8)	1	1	yes	characteristic
118	C₂×C₄×C₁₀₉	AGL(1,ℤ/8)	1	1	yes	characteristic
119	C₈×C₁₀₉	—	2	2	no	—
120	C₈×C₁₀₉	—	4	2	no	—
121	C₈×C₁₀₉	—	4	2	no	—
122	C₈×C₁₀₉	—	4	2	no	—
123	C₈×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
124	C₈×C₁₀₉	#117	1	1	yes	characteristic
125	C₈×C₁₀₉	C₄×D₄	1	1	yes	characteristic
126	C₈×C₁₀₉	C₄×D₄	1	1	yes	characteristic
127	C₂²×C₁₀₉	—	4	1	no	—
128	C₂²×C₁₀₉	—	4	2	no	—
129	C₂²×C₁₀₉	—	8	2	no	—
130	C₂²×C₁₀₉	#417	1	1	yes	characteristic
131	C₂²×C₁₀₉	#440	1	1	yes	characteristic
132	C₂²×C₁₀₉	#440	1	1	yes	characteristic
133	C₄×C₁₀₉	—	4	2	no	—
134	C₄×C₁₀₉	—	4	2	no	—
135	C₄×C₁₀₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
136	C₄×C₁₀₉	#418	1	1	yes	characteristic
137	C₄×C₁₀₉	#442	1	1	yes	characteristic
138	C₄×C₁₀₉	#442	1	1	yes	characteristic
139	#12651	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
140	C₂×C₁₀₉	—	2	1	no	—
141	C₂×C₁₀₉	—	2	1	no	—
142	C₂×C₁₀₉	—	2	1	no	—
143	C₂×C₁₀₉	—	8	2	no	—
144	C₂×C₁₀₉	#1762	1	1	yes	characteristic
145	C₂×#444	C₂×C₁₀₉	1	1	yes	characteristic
146	#2148	C₂×C₁₀₉	1	1	yes	characteristic
147	#2143	C₂×C₁₀₉	1	1	yes	characteristic
148	C₂×#347	C₂×C₁₀₉	1	1	yes	characteristic, complemented
149	#1778	C₂×C₁₀₉	1	1	yes	characteristic, complemented
150	#1388	C₂×C₁₀₉	1	1	yes	characteristic, complemented
151	#1385	C₂×C₁₀₉	1	1	yes	characteristic, complemented
152	C₁₀₉	#12651	1	1	yes	characteristic, direct product factor, Sylow
153	C₂×#135	C₂²×C₁₀₉	1	1	yes	characteristic
154	C₂×#131	—	2	1	no	—
155	C₂×M₃₂	C₂²×C₁₀₉	1	1	yes	characteristic
156	C₂×M₃₂	C₂²×C₁₀₉	1	1	yes	characteristic
157	#444	—	2	2	no	—
158	#444	C₄×C₁₀₉	1	4	yes	—
159	C₂×#108	C₂²×C₁₀₉	1	1	yes	characteristic
160	C₂×#108	C₂²×C₁₀₉	1	1	yes	characteristic
161	C₂×#108	C₂²×C₁₀₉	1	1	yes	characteristic
162	C₂×#104	—	2	1	no	—
163	C₄×M₁₆	—	2	1	no	—
164	C₂×#97	C₂²×C₁₀₉	1	1	yes	characteristic
165	#362	C₄×C₁₀₉	1	2	yes	complemented
166	#361	C₄×C₁₀₉	1	2	yes	complemented
167	#359	C₄×C₁₀₉	1	4	yes	complemented
168	#347	—	2	2	no	—
169	#341	—	2	1	no	—
170	C₂×#41	—	2	1	no	—
171	#135	—	2	1	no	—
172	#135	—	2	1	no	—
173	#135	C₂×C₄×C₁₀₉	1	2	yes	—
174	#135	C₂×C₄×C₁₀₉	1	2	yes	—
175	C₂×Q₁₆	C₂×C₄×C₁₀₉	1	1	yes	characteristic
176	C₂×SD₁₆	—	2	1	no	—
177	C₂×D₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
178	#131	—	2	4	no	—
179	#131	—	4	2	no	—
180	C₂×M₁₆	—	2	1	no	—
181	C₂×M₁₆	—	2	1	no	—
182	C₂×M₁₆	—	2	1	no	—
183	C₂×M₁₆	—	4	1	no	—
184	C₂²×C₈	—	4	1	no	—
185	C₂²×C₈	D₄×C₁₀₉	1	1	yes	characteristic
186	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
187	C₂×C₄²	D₄×C₁₀₉	1	1	yes	characteristic
188	M₃₂	—	2	1	no	—
189	M₃₂	—	2	1	no	—
190	M₃₂	—	2	1	no	—
191	M₃₂	C₂×C₄×C₁₀₉	1	2	yes	—
192	C₂×C₁₆	—	2	1	no	—
193	C₂×C₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
194	#108	—	2	1	no	—
195	#108	—	2	1	no	—
196	#108	—	2	1	no	—
197	#108	C₂×C₄×C₁₀₉	1	2	yes	—
198	#108	C₂×C₄×C₁₀₉	1	2	yes	—
199	#108	C₂×C₄×C₁₀₉	1	2	yes	—
200	#104	—	2	4	no	—
201	#97	—	4	1	no	—
202	#97	C₂×C₄×C₁₀₉	1	2	yes	—
203	#97	D₄×C₁₀₉	1	2	yes	—
204	C₄×C₈	—	4	1	no	—
205	#41	—	2	2	no	—
206	#41	—	2	2	no	—
207	#41	—	4	1	no	—
208	#41	—	4	1	no	—
209	C₂×Q₈	—	2	1	no	—
210	C₂×D₄	—	2	1	no	—
211	C₂×D₄	—	4	1	no	—
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	4	1	no	—
214	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
215	Q₁₆	—	2	1	no	—
216	Q₁₆	C₄²×C₁₀₉	1	2	yes	—
217	SD₁₆	—	2	2	no	—
218	SD₁₆	—	2	2	no	—
219	D₈	—	2	1	no	—
220	D₈	C₄²×C₁₀₉	1	2	yes	—
221	M₁₆	—	2	2	no	—
222	M₁₆	—	2	2	no	—
223	M₁₆	—	2	2	no	—
224	M₁₆	—	4	4	no	—
225	M₁₆	—	4	1	no	—
226	M₁₆	—	4	1	no	—
227	M₁₆	—	4	1	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	2	1	no	—
230	C₂×C₈	—	4	4	no	—
231	C₂×C₈	—	4	1	no	—
232	C₂×C₈	—	4	1	no	—
233	C₂×C₈	—	4	1	no	—
234	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	2	yes	—
235	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
236	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
237	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
238	C₂×C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
239	C₂×C₈	#41×C₁₀₉	1	1	yes	characteristic
240	C₂×C₈	#41×C₁₀₉	1	1	yes	characteristic
241	C₄²	—	2	1	no	—
242	C₄²	C₂²⋊C₄×C₁₀₉	1	2	yes	—
243	C₁₆	—	2	2	no	—
244	C₁₆	—	2	2	no	—
245	C₂³	—	4	1	no	—
246	C₂³	#137×C₁₀₉	1	1	yes	characteristic
247	Q₈	—	2	2	no	—
248	Q₈	—	4	1	no	—
249	D₄	—	2	2	no	—
250	D₄	—	4	2	no	—
251	D₄	—	4	2	no	—
252	D₄	—	4	1	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	4	1	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	8	1	no	—
258	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
259	C₂×C₄	C₄×D₄×C₁₀₉	1	1	yes	characteristic
260	C₂×C₄	C₄×D₄×C₁₀₉	1	1	yes	characteristic
261	C₂×C₄	AGL(1,ℤ/8)×C₁₀₉	1	1	yes	characteristic
262	C₂×C₄	AGL(1,ℤ/8)×C₁₀₉	1	1	yes	characteristic
263	C₂×C₄	AGL(1,ℤ/8)×C₁₀₉	1	1	yes	characteristic
264	C₈	—	2	2	no	—
265	C₈	—	4	2	no	—
266	C₈	—	4	2	no	—
267	C₈	—	4	2	no	—
268	C₈	C₂×C₄²×C₁₀₉	1	1	yes	characteristic
269	C₈	#117×C₁₀₉	1	1	yes	characteristic
270	C₈	C₄×D₄×C₁₀₉	1	1	yes	characteristic
271	C₈	C₄×D₄×C₁₀₉	1	1	yes	characteristic
272	C₂²	—	4	1	no	—
273	C₂²	—	4	2	no	—
274	C₂²	—	8	2	no	—
275	C₂²	#417×C₁₀₉	1	1	yes	characteristic
276	C₂²	#440×C₁₀₉	1	1	yes	characteristic
277	C₂²	#440×C₁₀₉	1	1	yes	characteristic
278	C₄	—	4	2	no	—
279	C₄	—	4	2	no	—
280	C₄	C₄×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
281	C₄	#418×C₁₀₉	1	1	yes	characteristic
282	C₄	#442×C₁₀₉	1	1	yes	characteristic
283	C₄	#442×C₁₀₉	1	1	yes	characteristic
284	C₂	—	2	1	no	—
285	C₂	—	2	1	no	—
286	C₂	—	2	1	no	—
287	C₂	—	8	2	no	—
288	C₂	C₁₀₉×#1762	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₀₉	#442
commutator	C₈	C₂×C₄²×C₁₀₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₀₉
socle	C₂×C₁₀₉	#1762
prime radical	C₂×#135×C₁₀₉	C₂²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₀₉ ◁ C₂²×C₄×C₁₀₉ ◁ C₂×#97×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂×#347×C₁₀₉	C₂	semidirect
5. C₁₀₉×#1778	C₂	semidirect
6. C₁₀₉×#1388	C₂	semidirect
7. C₁₀₉×#1385	C₂	semidirect
20. #362×C₁₀₉	C₄	semidirect
21. #361×C₁₀₉	C₄	semidirect
22. #359×C₁₀₉	C₄	semidirect
139. #12651	C₁₀₉	direct (N × H)
148. C₂×#347	C₂×C₁₀₉	semidirect
149. #1778	C₂×C₁₀₉	semidirect
150. #1388	C₂×C₁₀₉	semidirect
151. #1385	C₂×C₁₀₉	semidirect
152. C₁₀₉	#12651	direct (N × H)
165. #362	C₄×C₁₀₉	semidirect
166. #361	C₄×C₁₀₉	semidirect
167. #359	C₄×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#444×C₁₀₉	2
C₁₀₉×#2148	2
C₁₀₉×#2143	2
C₂×#347×C₁₀₉	2
C₁₀₉×#1778	2
C₁₀₉×#1388	2
C₁₀₉×#1385	2
#12651	109

Subgroups — C109×#12651 #42130645

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#12651 #42130645