Subgroups — C₁₀₉×#12998 #42130992

Order 27904.

500 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₉×#3054	C₂	1	1	yes	maximal, characteristic
2	C₂×#430×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₀₉×#2143	C₂	1	2	yes	maximal, complemented
4	C₂×#349×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₀₉×#1389	C₂	1	2	yes	maximal, complemented
6	C₂×#135×C₁₀₉	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₁₀₉	—	2	1	no	—
8	C₂²×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
9	#525×C₁₀₉	—	2	1	no	—
10	C₂×M₃₂×C₁₀₉	C₂²	1	2	yes	—
11	#470×C₁₀₉	C₄	1	2	yes	—
12	#469×C₁₀₉	C₄	1	2	yes	—
13	#468×C₁₀₉	—	2	2	no	—
14	#430×C₁₀₉	—	2	4	no	—
15	C₂×#108×C₁₀₉	—	2	1	no	—
16	C₂×#108×C₁₀₉	C₂²	1	2	yes	—
17	#427×C₁₀₉	C₂²	1	1	yes	characteristic
18	#416×C₁₀₉	—	2	1	no	—
19	#359×C₁₀₉	—	2	4	no	—
20	#349×C₁₀₉	—	2	2	no	—
21	C₂×#41×C₁₀₉	—	2	1	no	—
22	C₂³×C₄×C₁₀₉	D₄	1	1	yes	characteristic
23	#135×C₁₀₉	—	2	2	no	—
24	#135×C₁₀₉	—	2	2	no	—
25	C₂×Q₁₆×C₁₀₉	C₂×C₄	1	1	yes	characteristic
26	C₂×SD₁₆×C₁₀₉	—	2	1	no	—
27	C₂×D₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
28	C₂×M₁₆×C₁₀₉	—	2	2	no	—
29	C₂×M₁₆×C₁₀₉	—	2	2	no	—
30	C₂×M₁₆×C₁₀₉	—	2	4	no	—
31	C₂×M₁₆×C₁₀₉	—	2	2	no	—
32	C₂×M₁₆×C₁₀₉	—	4	1	no	—
33	C₂×M₁₆×C₁₀₉	—	4	2	no	—
34	C₂×M₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
35	C₂×M₁₆×C₁₀₉	D₄	1	2	yes	—
36	C₂²×C₈×C₁₀₉	—	4	1	no	—
37	C₂²×C₈×C₁₀₉	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
39	#123×C₁₀₉	—	4	1	no	—
40	C₂²⋊Q₈×C₁₀₉	—	2	2	no	—
41	#121×C₁₀₉	—	2	2	no	—
42	#120×C₁₀₉	—	4	1	no	—
43	C₄×D₄×C₁₀₉	—	4	2	no	—
44	#117×C₁₀₉	—	2	1	no	—
45	M₃₂×C₁₀₉	—	2	2	no	—
46	M₃₂×C₁₀₉	—	2	2	no	—
47	C₂×C₁₆×C₁₀₉	—	2	2	no	—
48	#108×C₁₀₉	—	2	4	no	—
49	#108×C₁₀₉	—	2	2	no	—
50	#108×C₁₀₉	D₄	1	4	yes	complemented
51	#107×C₁₀₉	—	2	1	no	—
52	#106×C₁₀₉	C₂×C₄	1	2	yes	—
53	#103×C₁₀₉	—	2	2	no	—
54	#102×C₁₀₉	—	2	2	no	—
55	C₂⁴×C₁₀₉	SD₁₆	1	1	yes	characteristic
56	#41×C₁₀₉	—	4	1	no	—
57	#41×C₁₀₉	—	4	1	no	—
58	#41×C₁₀₉	—	4	1	no	—
59	#41×C₁₀₉	—	4	1	no	—
60	C₂×Q₈×C₁₀₉	—	2	1	no	—
61	C₂×D₄×C₁₀₉	—	2	1	no	—
62	C₂×D₄×C₁₀₉	—	4	2	no	—
63	C₂×D₄×C₁₀₉	—	4	1	no	—
64	C₂²×C₄×C₁₀₉	—	2	1	no	—
65	C₂²×C₄×C₁₀₉	—	2	1	no	—
66	C₂²×C₄×C₁₀₉	—	2	1	no	—
67	C₂²×C₄×C₁₀₉	—	2	1	no	—
68	C₂²×C₄×C₁₀₉	—	4	1	no	—
69	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
70	C₂²×C₄×C₁₀₉	D₈	1	2	yes	—
71	C₂²×C₄×C₁₀₉	SD₁₆	1	1	yes	characteristic
72	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
73	Q₁₆×C₁₀₉	—	2	2	no	—
74	SD₁₆×C₁₀₉	—	4	2	no	—
75	D₈×C₁₀₉	—	2	2	no	—
76	M₁₆×C₁₀₉	—	2	4	no	—
77	M₁₆×C₁₀₉	—	2	4	no	—
78	M₁₆×C₁₀₉	—	2	4	no	—
79	M₁₆×C₁₀₉	—	4	4	no	—
80	M₁₆×C₁₀₉	—	4	2	no	—
81	C₂×C₈×C₁₀₉	—	2	1	no	—
82	C₂×C₈×C₁₀₉	—	2	1	no	—
83	C₂×C₈×C₁₀₉	—	4	4	no	—
84	C₂×C₈×C₁₀₉	—	4	2	no	—
85	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	2	yes	—
86	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₁₀₉	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₁₀₉	#41	1	2	yes	—
90	C₄⋊C₄×C₁₀₉	—	2	2	no	—
91	C₄⋊C₄×C₁₀₉	—	4	2	no	—
92	C₂²⋊C₄×C₁₀₉	—	4	2	no	—
93	C₂²⋊C₄×C₁₀₉	—	4	1	no	—
94	C₂²⋊C₄×C₁₀₉	—	4	2	no	—
95	C₄²×C₁₀₉	—	4	1	no	—
96	C₁₆×C₁₀₉	—	4	2	no	—
97	C₂³×C₁₀₉	—	2	1	no	—
98	C₂³×C₁₀₉	—	2	1	no	—
99	C₂³×C₁₀₉	—	4	1	no	—
100	C₂³×C₁₀₉	—	8	1	no	—
101	C₂³×C₁₀₉	#102	1	2	yes	—
102	C₂³×C₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
103	Q₈×C₁₀₉	—	4	1	no	—
104	Q₈×C₁₀₉	—	4	1	no	—
105	D₄×C₁₀₉	—	4	1	no	—
106	D₄×C₁₀₉	—	4	1	no	—
107	D₄×C₁₀₉	—	8	4	no	—
108	D₄×C₁₀₉	—	8	1	no	—
109	D₄×C₁₀₉	—	8	1	no	—
110	C₂×C₄×C₁₀₉	—	2	1	no	—
111	C₂×C₄×C₁₀₉	—	2	1	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	—	2	2	no	—
116	C₂×C₄×C₁₀₉	—	4	2	no	—
117	C₂×C₄×C₁₀₉	—	4	1	no	—
118	C₂×C₄×C₁₀₉	—	8	1	no	—
119	C₂×C₄×C₁₀₉	—	8	1	no	—
120	C₂×C₄×C₁₀₉	#102	1	2	yes	—
121	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₁₀₉	#120	1	1	yes	characteristic
123	C₂×C₄×C₁₀₉	#123	1	1	yes	characteristic
124	C₂×C₄×C₁₀₉	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₁₀₉	AGL(1,ℤ/8)	1	1	yes	characteristic
126	C₂×C₄×C₁₀₉	#137	1	1	yes	characteristic
127	C₈×C₁₀₉	—	2	2	no	—
128	C₈×C₁₀₉	—	4	4	no	—
129	C₈×C₁₀₉	C₄×D₄	1	2	yes	—
130	C₈×C₁₀₉	#121	1	2	yes	—
131	C₂²×C₁₀₉	—	2	2	no	—
132	C₂²×C₁₀₉	—	4	1	no	—
133	C₂²×C₁₀₉	—	4	2	no	—
134	C₂²×C₁₀₉	—	4	2	no	—
135	C₂²×C₁₀₉	—	8	1	no	—
136	C₂²×C₁₀₉	—	8	2	no	—
137	C₂²×C₁₀₉	—	8	1	no	—
138	C₂²×C₁₀₉	C₂×#102	1	1	yes	characteristic
139	C₂²×C₁₀₉	#448	1	1	yes	characteristic
140	C₂²×C₁₀₉	#481	1	1	yes	characteristic
141	C₄×C₁₀₉	—	2	2	no	—
142	C₄×C₁₀₉	—	8	2	no	—
143	C₄×C₁₀₉	—	8	1	no	—
144	C₄×C₁₀₉	#386	1	1	yes	characteristic
145	C₄×C₁₀₉	#417	1	1	yes	characteristic
146	C₄×C₁₀₉	#447	1	1	yes	characteristic
147	C₄×C₁₀₉	#480	1	1	yes	characteristic
148	#12998	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
149	C₂×C₁₀₉	—	2	1	no	—
150	C₂×C₁₀₉	—	2	1	no	—
151	C₂×C₁₀₉	—	2	1	no	—
152	C₂×C₁₀₉	—	4	2	no	—
153	C₂×C₁₀₉	—	16	1	no	—
154	C₂×C₁₀₉	#1874	1	1	yes	characteristic
155	#3054	C₂×C₁₀₉	1	1	yes	characteristic
156	C₂×#430	C₂×C₁₀₉	1	1	yes	characteristic, complemented
157	#2143	C₂×C₁₀₉	1	2	yes	complemented
158	C₂×#349	C₂×C₁₀₉	1	1	yes	characteristic, complemented
159	#1389	C₂×C₁₀₉	1	2	yes	complemented
160	C₁₀₉	#12998	1	1	yes	characteristic, direct product factor, Sylow
161	C₂×#135	C₂²×C₁₀₉	1	1	yes	characteristic
162	C₂²×M₁₆	—	2	1	no	—
163	C₂²×M₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
164	#525	—	2	1	no	—
165	C₂×M₃₂	C₂²×C₁₀₉	1	2	yes	—
166	#470	C₄×C₁₀₉	1	2	yes	—
167	#469	C₄×C₁₀₉	1	2	yes	—
168	#468	—	2	2	no	—
169	#430	—	2	4	no	—
170	C₂×#108	—	2	1	no	—
171	C₂×#108	C₂²×C₁₀₉	1	2	yes	—
172	#427	C₂²×C₁₀₉	1	1	yes	characteristic
173	#416	—	2	1	no	—
174	#359	—	2	4	no	—
175	#349	—	2	2	no	—
176	C₂×#41	—	2	1	no	—
177	C₂³×C₄	D₄×C₁₀₉	1	1	yes	characteristic
178	#135	—	2	2	no	—
179	#135	—	2	2	no	—
180	C₂×Q₁₆	C₂×C₄×C₁₀₉	1	1	yes	characteristic
181	C₂×SD₁₆	—	2	1	no	—
182	C₂×D₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
183	C₂×M₁₆	—	2	2	no	—
184	C₂×M₁₆	—	2	2	no	—
185	C₂×M₁₆	—	2	4	no	—
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	—	4	1	no	—
188	C₂×M₁₆	—	4	2	no	—
189	C₂×M₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
190	C₂×M₁₆	D₄×C₁₀₉	1	2	yes	—
191	C₂²×C₈	—	4	1	no	—
192	C₂²×C₈	D₄×C₁₀₉	1	1	yes	characteristic
193	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
194	#123	—	4	1	no	—
195	C₂²⋊Q₈	—	2	2	no	—
196	#121	—	2	2	no	—
197	#120	—	4	1	no	—
198	C₄×D₄	—	4	2	no	—
199	#117	—	2	1	no	—
200	M₃₂	—	2	2	no	—
201	M₃₂	—	2	2	no	—
202	C₂×C₁₆	—	2	2	no	—
203	#108	—	2	4	no	—
204	#108	—	2	2	no	—
205	#108	D₄×C₁₀₉	1	4	yes	complemented
206	#107	—	2	1	no	—
207	#106	C₂×C₄×C₁₀₉	1	2	yes	—
208	#103	—	2	2	no	—
209	#102	—	2	2	no	—
210	C₂⁴	SD₁₆×C₁₀₉	1	1	yes	characteristic
211	#41	—	4	1	no	—
212	#41	—	4	1	no	—
213	#41	—	4	1	no	—
214	#41	—	4	1	no	—
215	C₂×Q₈	—	2	1	no	—
216	C₂×D₄	—	2	1	no	—
217	C₂×D₄	—	4	2	no	—
218	C₂×D₄	—	4	1	no	—
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	2	1	no	—
223	C₂²×C₄	—	4	1	no	—
224	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	2	yes	—
225	C₂²×C₄	D₈×C₁₀₉	1	2	yes	—
226	C₂²×C₄	SD₁₆×C₁₀₉	1	1	yes	characteristic
227	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
228	Q₁₆	—	2	2	no	—
229	SD₁₆	—	4	2	no	—
230	D₈	—	2	2	no	—
231	M₁₆	—	2	4	no	—
232	M₁₆	—	2	4	no	—
233	M₁₆	—	2	4	no	—
234	M₁₆	—	4	4	no	—
235	M₁₆	—	4	2	no	—
236	C₂×C₈	—	2	1	no	—
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	4	4	no	—
239	C₂×C₈	—	4	2	no	—
240	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	2	yes	—
241	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
242	C₂×C₈	C₂×D₄×C₁₀₉	1	2	yes	—
243	C₂×C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
244	C₂×C₈	#41×C₁₀₉	1	2	yes	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	4	2	no	—
247	C₂²⋊C₄	—	4	2	no	—
248	C₂²⋊C₄	—	4	1	no	—
249	C₂²⋊C₄	—	4	2	no	—
250	C₄²	—	4	1	no	—
251	C₁₆	—	4	2	no	—
252	C₂³	—	2	1	no	—
253	C₂³	—	2	1	no	—
254	C₂³	—	4	1	no	—
255	C₂³	—	8	1	no	—
256	C₂³	#102×C₁₀₉	1	2	yes	—
257	C₂³	C₂×SD₁₆×C₁₀₉	1	1	yes	characteristic
258	Q₈	—	4	1	no	—
259	Q₈	—	4	1	no	—
260	D₄	—	4	1	no	—
261	D₄	—	4	1	no	—
262	D₄	—	8	4	no	—
263	D₄	—	8	1	no	—
264	D₄	—	8	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	1	no	—
273	C₂×C₄	—	8	1	no	—
274	C₂×C₄	—	8	1	no	—
275	C₂×C₄	#102×C₁₀₉	1	2	yes	—
276	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
277	C₂×C₄	#120×C₁₀₉	1	1	yes	characteristic
278	C₂×C₄	#123×C₁₀₉	1	1	yes	characteristic
279	C₂×C₄	C₂×D₈×C₁₀₉	1	1	yes	characteristic
280	C₂×C₄	AGL(1,ℤ/8)×C₁₀₉	1	1	yes	characteristic
281	C₂×C₄	#137×C₁₀₉	1	1	yes	characteristic
282	C₈	—	2	2	no	—
283	C₈	—	4	4	no	—
284	C₈	C₄×D₄×C₁₀₉	1	2	yes	—
285	C₈	#121×C₁₀₉	1	2	yes	—
286	C₂²	—	2	2	no	—
287	C₂²	—	4	1	no	—
288	C₂²	—	4	2	no	—
289	C₂²	—	4	2	no	—
290	C₂²	—	8	1	no	—
291	C₂²	—	8	2	no	—
292	C₂²	—	8	1	no	—
293	C₂²	C₂×#102×C₁₀₉	1	1	yes	characteristic
294	C₂²	#448×C₁₀₉	1	1	yes	characteristic
295	C₂²	#481×C₁₀₉	1	1	yes	characteristic
296	C₄	—	2	2	no	—
297	C₄	—	8	2	no	—
298	C₄	—	8	1	no	—
299	C₄	#386×C₁₀₉	1	1	yes	characteristic
300	C₄	#417×C₁₀₉	1	1	yes	characteristic
301	C₄	#447×C₁₀₉	1	1	yes	characteristic
302	C₄	#480×C₁₀₉	1	1	yes	characteristic
303	C₂	—	2	1	no	—
304	C₂	—	2	1	no	—
305	C₂	—	2	1	no	—
306	C₂	—	4	2	no	—
307	C₂	—	16	1	no	—
308	C₂	C₁₀₉×#1874	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₀₉	#447
commutator	C₂×C₈	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₀₉
socle	C₂×C₁₀₉	#1874
prime radical	C₁₀₉×#3054	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₀₉ ◁ C₂²×C₄×C₁₀₉ ◁ C₂²×M₁₆×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#430×C₁₀₉	C₂	semidirect
3. C₁₀₉×#2143	C₂	semidirect
4. C₂×#349×C₁₀₉	C₂	semidirect
5. C₁₀₉×#1389	C₂	semidirect
50. #108×C₁₀₉	D₄	semidirect
148. #12998	C₁₀₉	direct (N × H)
156. C₂×#430	C₂×C₁₀₉	semidirect
157. #2143	C₂×C₁₀₉	semidirect
158. C₂×#349	C₂×C₁₀₉	semidirect
159. #1389	C₂×C₁₀₉	semidirect
160. C₁₀₉	#12998	direct (N × H)
205. #108	D₄×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₁₀₉×#3054	2
C₂×#430×C₁₀₉	2
C₁₀₉×#2143	2
C₂×#349×C₁₀₉	2
C₁₀₉×#1389	2
#12998	109

Subgroups — C109×#12998 #42130992

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#12998 #42130992