Subgroups — C₁₀₉×#3010 #28360264

Order 13952.

732 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×2₋¹⁺⁴×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×AGL(1,ℤ/8)×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×SD₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#535×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#449×C₁₀₉	C₂	1	4	yes	maximal, complemented
6	#448×C₁₀₉	C₂	1	4	yes	maximal, complemented
7	#419×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#416×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
9	#408×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
10	2₋¹⁺⁴×C₁₀₉	—	2	4	no	—
11	2₋¹⁺⁴×C₁₀₉	—	2	4	no	—
12	C₂×#41×C₁₀₉	—	2	2	no	—
13	C₂×#41×C₁₀₉	—	2	1	no	—
14	C₂×#41×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂×#41×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₂×#41×C₁₀₉	C₂²	1	2	yes	complemented
17	C₂²×Q₈×C₁₀₉	—	2	1	no	—
18	C₂²×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₂²×Q₈×C₁₀₉	C₂²	1	2	yes	complemented
20	C₂²×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
21	AGL(1,ℤ/8)×C₁₀₉	—	2	4	no	—
22	C₂×SD₁₆×C₁₀₉	—	2	4	no	—
23	C₂×SD₁₆×C₁₀₉	C₂²	1	4	yes	—
24	C₂×SD₁₆×C₁₀₉	C₂²	1	2	yes	—
25	C₂×D₈×C₁₀₉	C₂²	1	2	yes	—
26	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
27	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
28	#127×C₁₀₉	—	2	1	no	—
29	#124×C₁₀₉	—	2	1	no	—
30	#121×C₁₀₉	C₂²	1	4	yes	—
31	#120×C₁₀₉	—	2	2	no	—
32	#117×C₁₀₉	C₂²	1	1	yes	characteristic
33	#103×C₁₀₉	C₂²	1	2	yes	complemented
34	#103×C₁₀₉	C₂²	1	4	yes	complemented
35	#102×C₁₀₉	C₂²	1	2	yes	complemented
36	C₂²⋊C₈×C₁₀₉	C₂²	1	4	yes	complemented
37	C₂⁴×C₁₀₉	—	2	1	no	—
38	#41×C₁₀₉	—	2	16	no	—
39	#41×C₁₀₉	—	2	8	no	—
40	#41×C₁₀₉	—	2	4	no	—
41	#41×C₁₀₉	—	2	4	no	—
42	#41×C₁₀₉	—	2	2	no	—
43	#41×C₁₀₉	—	2	2	no	—
44	#41×C₁₀₉	—	2	2	no	—
45	#41×C₁₀₉	D₄	1	4	yes	complemented
46	C₂×Q₈×C₁₀₉	—	2	8	no	—
47	C₂×Q₈×C₁₀₉	—	2	4	no	—
48	C₂×Q₈×C₁₀₉	—	2	4	no	—
49	C₂×Q₈×C₁₀₉	—	2	1	no	—
50	C₂×Q₈×C₁₀₉	—	2	2	no	—
51	C₂×Q₈×C₁₀₉	—	2	2	no	—
52	C₂×Q₈×C₁₀₉	D₄	1	1	yes	characteristic
53	C₂×Q₈×C₁₀₉	D₄	1	4	yes	complemented
54	C₂×Q₈×C₁₀₉	C₂³	1	2	yes	—
55	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
56	C₂×D₄×C₁₀₉	—	2	1	no	—
57	C₂×D₄×C₁₀₉	—	2	2	no	—
58	C₂×D₄×C₁₀₉	—	2	4	no	—
59	C₂×D₄×C₁₀₉	—	2	2	no	—
60	C₂×D₄×C₁₀₉	—	2	2	no	—
61	C₂×D₄×C₁₀₉	—	2	1	no	—
62	C₂×D₄×C₁₀₉	D₄	1	2	yes	—
63	C₂×D₄×C₁₀₉	D₄	1	1	yes	characteristic
64	C₂×D₄×C₁₀₉	C₂³	1	2	yes	—
65	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
66	C₂²×C₄×C₁₀₉	—	2	1	no	—
67	C₂²×C₄×C₁₀₉	—	2	2	no	—
68	C₂²×C₄×C₁₀₉	—	2	1	no	—
69	C₂²×C₄×C₁₀₉	—	2	2	no	—
70	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
71	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
72	SD₁₆×C₁₀₉	—	2	8	no	—
73	SD₁₆×C₁₀₉	—	2	4	no	—
74	D₈×C₁₀₉	—	2	4	no	—
75	M₁₆×C₁₀₉	—	2	2	no	—
76	C₂×C₈×C₁₀₉	—	2	2	no	—
77	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
78	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
79	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
80	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
81	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
82	C₄²×C₁₀₉	—	2	1	no	—
83	C₂³×C₁₀₉	—	2	1	no	—
84	C₂³×C₁₀₉	—	2	2	no	—
85	C₂³×C₁₀₉	—	4	2	no	—
86	C₂³×C₁₀₉	—	8	2	no	—
87	C₂³×C₁₀₉	C₂×D₄	1	2	yes	—
88	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
89	Q₈×C₁₀₉	—	2	8	no	—
90	Q₈×C₁₀₉	—	2	2	no	—
91	Q₈×C₁₀₉	—	2	4	no	—
92	Q₈×C₁₀₉	—	2	1	no	—
93	Q₈×C₁₀₉	—	2	2	no	—
94	Q₈×C₁₀₉	C₂×D₄	1	2	yes	—
95	Q₈×C₁₀₉	C₂×D₄	1	4	yes	—
96	D₄×C₁₀₉	—	2	8	no	—
97	D₄×C₁₀₉	—	2	4	no	—
98	D₄×C₁₀₉	—	2	4	no	—
99	D₄×C₁₀₉	—	2	2	no	—
100	D₄×C₁₀₉	—	2	1	no	—
101	D₄×C₁₀₉	—	2	2	no	—
102	D₄×C₁₀₉	—	2	2	no	—
103	D₄×C₁₀₉	—	4	2	no	—
104	D₄×C₁₀₉	—	8	4	no	—
105	D₄×C₁₀₉	C₂×D₄	1	2	yes	—
106	C₂×C₄×C₁₀₉	—	2	8	no	—
107	C₂×C₄×C₁₀₉	—	2	8	no	—
108	C₂×C₄×C₁₀₉	—	2	4	no	—
109	C₂×C₄×C₁₀₉	—	2	2	no	—
110	C₂×C₄×C₁₀₉	—	2	1	no	—
111	C₂×C₄×C₁₀₉	—	2	4	no	—
112	C₂×C₄×C₁₀₉	—	2	2	no	—
113	C₂×C₄×C₁₀₉	—	2	2	no	—
114	C₂×C₄×C₁₀₉	—	2	2	no	—
115	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
116	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
118	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
119	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
120	C₈×C₁₀₉	—	2	2	no	—
121	C₈×C₁₀₉	—	2	2	no	—
122	C₂²×C₁₀₉	—	2	2	no	—
123	C₂²×C₁₀₉	—	2	2	no	—
124	C₂²×C₁₀₉	—	2	2	no	—
125	C₂²×C₁₀₉	—	2	4	no	—
126	C₂²×C₁₀₉	—	4	2	no	—
127	C₂²×C₁₀₉	—	4	2	no	—
128	C₂²×C₁₀₉	—	4	4	no	—
129	C₂²×C₁₀₉	—	8	4	no	—
130	C₂²×C₁₀₉	#120	1	2	yes	—
131	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
132	C₄×C₁₀₉	—	2	4	no	—
133	C₄×C₁₀₉	—	2	2	no	—
134	C₄×C₁₀₉	—	2	2	no	—
135	C₄×C₁₀₉	—	4	2	no	—
136	C₄×C₁₀₉	#120	1	2	yes	—
137	C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
138	C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₁₀₉	—	2	2	no	—
140	C₂×C₁₀₉	—	4	2	no	—
141	C₂×C₁₀₉	—	4	2	no	—
142	C₂×C₁₀₉	—	8	2	no	—
143	C₂×C₁₀₉	C₂×#120	1	1	yes	characteristic
144	C₂×C₁₀₉	#577	1	2	yes	—
145	#3010	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
146	C₁₀₉	#3010	1	1	yes	characteristic, direct product factor, Sylow
147	C₂×2₋¹⁺⁴	C₂×C₁₀₉	1	1	yes	characteristic, complemented
148	C₂×AGL(1,ℤ/8)	C₂×C₁₀₉	1	1	yes	characteristic, complemented
149	C₂²×SD₁₆	C₂×C₁₀₉	1	1	yes	characteristic, complemented
150	#535	C₂×C₁₀₉	1	1	yes	characteristic, complemented
151	#449	C₂×C₁₀₉	1	4	yes	complemented
152	#448	C₂×C₁₀₉	1	4	yes	complemented
153	#419	C₂×C₁₀₉	1	1	yes	characteristic, complemented
154	#416	C₂×C₁₀₉	1	1	yes	characteristic, complemented
155	#408	C₂×C₁₀₉	1	1	yes	characteristic, complemented
156	2₋¹⁺⁴	—	2	4	no	—
157	2₋¹⁺⁴	—	2	4	no	—
158	C₂×#41	—	2	2	no	—
159	C₂×#41	—	2	1	no	—
160	C₂×#41	C₂²×C₁₀₉	1	1	yes	characteristic
161	C₂×#41	C₂²×C₁₀₉	1	1	yes	characteristic
162	C₂×#41	C₂²×C₁₀₉	1	2	yes	complemented
163	C₂²×Q₈	—	2	1	no	—
164	C₂²×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
165	C₂²×Q₈	C₂²×C₁₀₉	1	2	yes	complemented
166	C₂²×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
167	AGL(1,ℤ/8)	—	2	4	no	—
168	C₂×SD₁₆	—	2	4	no	—
169	C₂×SD₁₆	C₂²×C₁₀₉	1	4	yes	—
170	C₂×SD₁₆	C₂²×C₁₀₉	1	2	yes	—
171	C₂×D₈	C₂²×C₁₀₉	1	2	yes	—
172	C₂×M₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
173	C₂²×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
174	#127	—	2	1	no	—
175	#124	—	2	1	no	—
176	#121	C₂²×C₁₀₉	1	4	yes	—
177	#120	—	2	2	no	—
178	#117	C₂²×C₁₀₉	1	1	yes	characteristic
179	#103	C₂²×C₁₀₉	1	2	yes	complemented
180	#103	C₂²×C₁₀₉	1	4	yes	complemented
181	#102	C₂²×C₁₀₉	1	2	yes	complemented
182	C₂²⋊C₈	C₂²×C₁₀₉	1	4	yes	complemented
183	C₂⁴	—	2	1	no	—
184	#41	—	2	16	no	—
185	#41	—	2	8	no	—
186	#41	—	2	4	no	—
187	#41	—	2	4	no	—
188	#41	—	2	2	no	—
189	#41	—	2	2	no	—
190	#41	—	2	2	no	—
191	#41	D₄×C₁₀₉	1	4	yes	complemented
192	C₂×Q₈	—	2	8	no	—
193	C₂×Q₈	—	2	4	no	—
194	C₂×Q₈	—	2	4	no	—
195	C₂×Q₈	—	2	1	no	—
196	C₂×Q₈	—	2	2	no	—
197	C₂×Q₈	—	2	2	no	—
198	C₂×Q₈	D₄×C₁₀₉	1	1	yes	characteristic
199	C₂×Q₈	D₄×C₁₀₉	1	4	yes	complemented
200	C₂×Q₈	C₂³×C₁₀₉	1	2	yes	—
201	C₂×Q₈	C₂³×C₁₀₉	1	1	yes	characteristic
202	C₂×D₄	—	2	1	no	—
203	C₂×D₄	—	2	2	no	—
204	C₂×D₄	—	2	4	no	—
205	C₂×D₄	—	2	2	no	—
206	C₂×D₄	—	2	2	no	—
207	C₂×D₄	—	2	1	no	—
208	C₂×D₄	D₄×C₁₀₉	1	2	yes	—
209	C₂×D₄	D₄×C₁₀₉	1	1	yes	characteristic
210	C₂×D₄	C₂³×C₁₀₉	1	2	yes	—
211	C₂×D₄	C₂³×C₁₀₉	1	1	yes	characteristic
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
217	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
218	SD₁₆	—	2	8	no	—
219	SD₁₆	—	2	4	no	—
220	D₈	—	2	4	no	—
221	M₁₆	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
224	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
225	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
226	C₂²⋊C₄	—	2	1	no	—
227	C₂²⋊C₄	—	2	4	no	—
228	C₄²	—	2	1	no	—
229	C₂³	—	2	1	no	—
230	C₂³	—	2	2	no	—
231	C₂³	—	4	2	no	—
232	C₂³	—	8	2	no	—
233	C₂³	C₂×D₄×C₁₀₉	1	2	yes	—
234	C₂³	C₂×D₄×C₁₀₉	1	1	yes	characteristic
235	Q₈	—	2	8	no	—
236	Q₈	—	2	2	no	—
237	Q₈	—	2	4	no	—
238	Q₈	—	2	1	no	—
239	Q₈	—	2	2	no	—
240	Q₈	C₂×D₄×C₁₀₉	1	2	yes	—
241	Q₈	C₂×D₄×C₁₀₉	1	4	yes	—
242	D₄	—	2	8	no	—
243	D₄	—	2	4	no	—
244	D₄	—	2	4	no	—
245	D₄	—	2	2	no	—
246	D₄	—	2	1	no	—
247	D₄	—	2	2	no	—
248	D₄	—	2	2	no	—
249	D₄	—	4	2	no	—
250	D₄	—	8	4	no	—
251	D₄	C₂×D₄×C₁₀₉	1	2	yes	—
252	C₂×C₄	—	2	8	no	—
253	C₂×C₄	—	2	8	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
262	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
263	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
264	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
265	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
266	C₈	—	2	2	no	—
267	C₈	—	2	2	no	—
268	C₂²	—	2	2	no	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	2	4	no	—
272	C₂²	—	4	2	no	—
273	C₂²	—	4	2	no	—
274	C₂²	—	4	4	no	—
275	C₂²	—	8	4	no	—
276	C₂²	#120×C₁₀₉	1	2	yes	—
277	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
278	C₄	—	2	4	no	—
279	C₄	—	2	2	no	—
280	C₄	—	2	2	no	—
281	C₄	—	4	2	no	—
282	C₄	#120×C₁₀₉	1	2	yes	—
283	C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
284	C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
285	C₂	—	2	2	no	—
286	C₂	—	4	2	no	—
287	C₂	—	4	2	no	—
288	C₂	—	8	2	no	—
289	C₂	C₂×#120×C₁₀₉	1	1	yes	characteristic
290	C₂	#577×C₁₀₉	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₀₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₀₉
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₂×#41×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×2₋¹⁺⁴×C₁₀₉	C₂	semidirect
2. C₂×AGL(1,ℤ/8)×C₁₀₉	C₂	semidirect
3. C₂²×SD₁₆×C₁₀₉	C₂	semidirect
4. #535×C₁₀₉	C₂	semidirect
5. #449×C₁₀₉	C₂	semidirect
6. #448×C₁₀₉	C₂	semidirect
7. #419×C₁₀₉	C₂	semidirect
8. #416×C₁₀₉	C₂	semidirect
9. #408×C₁₀₉	C₂	semidirect
16. C₂×#41×C₁₀₉	C₂²	semidirect
19. C₂²×Q₈×C₁₀₉	C₂²	semidirect
33. #103×C₁₀₉	C₂²	semidirect
34. #103×C₁₀₉	C₂²	semidirect
35. #102×C₁₀₉	C₂²	semidirect
36. C₂²⋊C₈×C₁₀₉	C₂²	semidirect
45. #41×C₁₀₉	D₄	semidirect
53. C₂×Q₈×C₁₀₉	D₄	semidirect
145. #3010	C₁₀₉	direct (N × H)
146. C₁₀₉	#3010	direct (N × H)
147. C₂×2₋¹⁺⁴	C₂×C₁₀₉	semidirect
148. C₂×AGL(1,ℤ/8)	C₂×C₁₀₉	semidirect
149. C₂²×SD₁₆	C₂×C₁₀₉	semidirect
150. #535	C₂×C₁₀₉	semidirect
151. #449	C₂×C₁₀₉	semidirect
152. #448	C₂×C₁₀₉	semidirect
153. #419	C₂×C₁₀₉	semidirect
154. #416	C₂×C₁₀₉	semidirect
155. #408	C₂×C₁₀₉	semidirect
162. C₂×#41	C₂²×C₁₀₉	semidirect
165. C₂²×Q₈	C₂²×C₁₀₉	semidirect
179. #103	C₂²×C₁₀₉	semidirect
180. #103	C₂²×C₁₀₉	semidirect
181. #102	C₂²×C₁₀₉	semidirect
182. C₂²⋊C₈	C₂²×C₁₀₉	semidirect
191. #41	D₄×C₁₀₉	semidirect
199. C₂×Q₈	D₄×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×2₋¹⁺⁴×C₁₀₉	2
C₂×AGL(1,ℤ/8)×C₁₀₉	2
C₂²×SD₁₆×C₁₀₉	2
#535×C₁₀₉	2
#449×C₁₀₉	2
#448×C₁₀₉	2
#419×C₁₀₉	2
#416×C₁₀₉	2
#408×C₁₀₉	2
#3010	109

Subgroups — C109×#3010 #28360264

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#3010 #28360264