Subgroups — C₁₀₉×#19289 #42137283

Order 27904.

630 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₉×#3169	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#3112	C₂	1	2	yes	maximal, complemented
3	C₂×#446×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₀₉×#1698	C₂	1	4	yes	maximal, complemented
5	C₁₀₉×#1696	C₂	1	2	yes	maximal, complemented
6	C₁₀₉×#1688	C₂	1	2	yes	maximal, complemented
7	C₁₀₉×#1568	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₀₉×#1551	C₂	1	2	yes	maximal, complemented
9	#532×C₁₀₉	C₂²	1	2	yes	—
10	C₂×C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
11	#500×C₁₀₉	C₂²	1	1	yes	characteristic
12	#493×C₁₀₉	C₂²	1	1	yes	characteristic
13	#486×C₁₀₉	C₂²	1	2	yes	—
14	#480×C₁₀₉	—	2	2	no	—
15	#477×C₁₀₉	C₂²	1	4	yes	—
16	#466×C₁₀₉	—	2	1	no	—
17	#466×C₁₀₉	—	2	1	no	—
18	#460×C₁₀₉	C₂²	1	4	yes	—
19	#448×C₁₀₉	—	2	2	no	—
20	#446×C₁₀₉	—	2	1	no	—
21	#446×C₁₀₉	—	2	1	no	—
22	#446×C₁₀₉	C₂²	1	2	yes	—
23	#446×C₁₀₉	C₂²	1	2	yes	—
24	#440×C₁₀₉	—	2	4	no	—
25	C₄×D₈×C₁₀₉	—	2	1	no	—
26	C₄×D₈×C₁₀₉	—	2	1	no	—
27	#432×C₁₀₉	C₂²	1	2	yes	—
28	C₂×C₄⋊C₈×C₁₀₉	—	2	1	no	—
29	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
30	C₂×#97×C₁₀₉	—	2	1	no	—
31	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
32	#335×C₁₀₉	C₂²	1	2	yes	—
33	#334×C₁₀₉	C₂²	1	2	yes	—
34	#333×C₁₀₉	C₂²	1	4	yes	—
35	#331×C₁₀₉	C₂²	1	4	yes	—
36	C₂×#41×C₁₀₉	—	2	2	no	—
37	C₂²×Q₈×C₁₀₉	D₄	1	1	yes	characteristic
38	C₂×SD₁₆×C₁₀₉	—	2	4	no	—
39	C₂×D₈×C₁₀₉	—	2	1	no	—
40	C₂×D₈×C₁₀₉	—	2	1	no	—
41	C₂²×C₈×C₁₀₉	—	2	1	no	—
42	C₂²×C₈×C₁₀₉	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₁₀₉	D₄	1	1	yes	characteristic
44	C₄⋊Q₈×C₁₀₉	C₂³	1	2	yes	—
45	#127×C₁₀₉	C₂³	1	2	yes	—
46	#124×C₁₀₉	—	2	2	no	—
47	#121×C₁₀₉	—	2	2	no	—
48	#121×C₁₀₉	—	2	2	no	—
49	C₄×Q₈×C₁₀₉	—	2	1	no	—
50	C₄×Q₈×C₁₀₉	—	2	1	no	—
51	C₄×Q₈×C₁₀₉	D₄	1	2	yes	—
52	C₄×Q₈×C₁₀₉	C₂³	1	2	yes	—
53	C₄×D₄×C₁₀₉	—	2	2	no	—
54	C₄×D₄×C₁₀₉	—	2	2	no	—
55	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
56	C₂×C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
57	C₂×C₄²×C₁₀₉	—	2	1	no	—
58	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
59	#107×C₁₀₉	—	2	1	no	—
60	#107×C₁₀₉	—	2	1	no	—
61	#107×C₁₀₉	—	2	4	no	—
62	C₄⋊C₈×C₁₀₉	—	2	2	no	—
63	C₄⋊C₈×C₁₀₉	—	2	2	no	—
64	C₄⋊C₈×C₁₀₉	—	2	1	no	—
65	C₄⋊C₈×C₁₀₉	C₂³	1	2	yes	—
66	C₄⋊C₈×C₁₀₉	C₂³	1	4	yes	—
67	#103×C₁₀₉	—	2	4	no	—
68	#102×C₁₀₉	—	2	2	no	—
69	#102×C₁₀₉	—	2	2	no	—
70	#102×C₁₀₉	—	2	4	no	—
71	C₂²⋊C₈×C₁₀₉	—	2	2	no	—
72	#97×C₁₀₉	—	2	2	no	—
73	#97×C₁₀₉	—	2	2	no	—
74	#97×C₁₀₉	—	2	2	no	—
75	C₄×C₈×C₁₀₉	D₄	1	1	yes	characteristic
76	C₄×C₈×C₁₀₉	D₄	1	1	yes	characteristic
77	C₄×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
79	#41×C₁₀₉	—	8	4	no	—
80	C₂×Q₈×C₁₀₉	—	2	1	no	—
81	C₂×Q₈×C₁₀₉	—	2	2	no	—
82	C₂×Q₈×C₁₀₉	—	4	2	no	—
83	C₂×Q₈×C₁₀₉	C₂×D₄	1	2	yes	—
84	C₂×D₄×C₁₀₉	—	2	2	no	—
85	C₂×D₄×C₁₀₉	—	2	2	no	—
86	C₂×D₄×C₁₀₉	—	2	2	no	—
87	C₂×D₄×C₁₀₉	—	4	2	no	—
88	C₂²×C₄×C₁₀₉	—	2	1	no	—
89	C₂²×C₄×C₁₀₉	—	2	1	no	—
90	C₂²×C₄×C₁₀₉	—	4	2	no	—
91	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₁₀₉	#41	1	1	yes	characteristic
94	SD₁₆×C₁₀₉	—	4	8	no	—
95	D₈×C₁₀₉	—	4	2	no	—
96	D₈×C₁₀₉	—	4	2	no	—
97	C₂×C₈×C₁₀₉	—	2	2	no	—
98	C₂×C₈×C₁₀₉	—	2	2	no	—
99	C₂×C₈×C₁₀₉	—	2	4	no	—
100	C₂×C₈×C₁₀₉	—	4	1	no	—
101	C₂×C₈×C₁₀₉	—	4	2	no	—
102	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
106	C₄⋊C₄×C₁₀₉	—	2	2	no	—
107	C₄⋊C₄×C₁₀₉	—	2	1	no	—
108	C₄⋊C₄×C₁₀₉	—	2	2	no	—
109	C₄⋊C₄×C₁₀₉	—	2	2	no	—
110	C₄⋊C₄×C₁₀₉	—	2	2	no	—
111	C₄⋊C₄×C₁₀₉	C₂×D₄	1	2	yes	—
112	C₂²⋊C₄×C₁₀₉	—	4	2	no	—
113	C₂²⋊C₄×C₁₀₉	—	4	2	no	—
114	C₄²×C₁₀₉	—	2	2	no	—
115	C₄²×C₁₀₉	—	2	2	no	—
116	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₁₀₉	#41	1	1	yes	characteristic
119	C₄²×C₁₀₉	C₂⁴	1	1	yes	characteristic
120	C₂³×C₁₀₉	—	4	2	no	—
121	C₂³×C₁₀₉	#121	1	1	yes	characteristic
122	Q₈×C₁₀₉	—	2	4	no	—
123	Q₈×C₁₀₉	—	4	2	no	—
124	Q₈×C₁₀₉	—	8	2	no	—
125	D₄×C₁₀₉	—	4	2	no	—
126	D₄×C₁₀₉	—	4	2	no	—
127	D₄×C₁₀₉	—	4	4	no	—
128	D₄×C₁₀₉	—	8	2	no	—
129	D₄×C₁₀₉	—	8	4	no	—
130	C₂×C₄×C₁₀₉	—	2	2	no	—
131	C₂×C₄×C₁₀₉	—	2	1	no	—
132	C₂×C₄×C₁₀₉	—	2	1	no	—
133	C₂×C₄×C₁₀₉	—	2	2	no	—
134	C₂×C₄×C₁₀₉	—	2	2	no	—
135	C₂×C₄×C₁₀₉	—	2	2	no	—
136	C₂×C₄×C₁₀₉	—	4	2	no	—
137	C₂×C₄×C₁₀₉	—	4	2	no	—
138	C₂×C₄×C₁₀₉	—	4	2	no	—
139	C₂×C₄×C₁₀₉	—	8	4	no	—
140	C₂×C₄×C₁₀₉	#121	1	1	yes	characteristic
141	C₂×C₄×C₁₀₉	#121	1	1	yes	characteristic
142	C₂×C₄×C₁₀₉	#121	1	1	yes	characteristic
143	C₂×C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
146	C₈×C₁₀₉	—	2	1	no	—
147	C₈×C₁₀₉	—	2	1	no	—
148	C₈×C₁₀₉	—	2	1	no	—
149	C₈×C₁₀₉	—	4	2	no	—
150	C₈×C₁₀₉	—	4	4	no	—
151	C₈×C₁₀₉	#137	1	2	yes	—
152	C₂²×C₁₀₉	—	2	1	no	—
153	C₂²×C₁₀₉	—	2	1	no	—
154	C₂²×C₁₀₉	—	8	2	no	—
155	C₂²×C₁₀₉	—	8	2	no	—
156	C₂²×C₁₀₉	—	8	2	no	—
157	C₂²×C₁₀₉	#472	1	2	yes	—
158	C₂²×C₁₀₉	C₂×#121	1	1	yes	characteristic
159	C₄×C₁₀₉	—	2	1	no	—
160	C₄×C₁₀₉	—	2	1	no	—
161	C₄×C₁₀₉	—	2	1	no	—
162	C₄×C₁₀₉	—	2	1	no	—
163	C₄×C₁₀₉	—	4	2	no	—
164	C₄×C₁₀₉	—	4	2	no	—
165	C₄×C₁₀₉	—	8	2	no	—
166	C₄×C₁₀₉	C₂×#137	1	1	yes	characteristic
167	C₄×C₁₀₉	#575	1	1	yes	characteristic
168	C₄×C₁₀₉	#575	1	2	yes	—
169	#19289	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
170	C₂×C₁₀₉	—	2	2	no	—
171	C₂×C₁₀₉	—	16	2	no	—
172	C₂×C₁₀₉	#3037	1	1	yes	characteristic
173	C₂×C₁₀₉	#3054	1	1	yes	characteristic
174	C₂×C₁₀₉	C₂×#472	1	1	yes	characteristic
175	#3169	C₂×C₁₀₉	1	1	yes	characteristic
176	#3112	C₂×C₁₀₉	1	2	yes	complemented
177	C₂×#446	C₂×C₁₀₉	1	1	yes	characteristic, complemented
178	#1698	C₂×C₁₀₉	1	4	yes	complemented
179	#1696	C₂×C₁₀₉	1	2	yes	complemented
180	#1688	C₂×C₁₀₉	1	2	yes	complemented
181	#1568	C₂×C₁₀₉	1	1	yes	characteristic, complemented
182	#1551	C₂×C₁₀₉	1	2	yes	complemented
183	C₁₀₉	#19289	1	1	yes	characteristic, direct product factor, Sylow
184	#532	C₂²×C₁₀₉	1	2	yes	—
185	C₂×C₄×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
186	#500	C₂²×C₁₀₉	1	1	yes	characteristic
187	#493	C₂²×C₁₀₉	1	1	yes	characteristic
188	#486	C₂²×C₁₀₉	1	2	yes	—
189	#480	—	2	2	no	—
190	#477	C₂²×C₁₀₉	1	4	yes	—
191	#466	—	2	1	no	—
192	#466	—	2	1	no	—
193	#460	C₂²×C₁₀₉	1	4	yes	—
194	#448	—	2	2	no	—
195	#446	—	2	1	no	—
196	#446	—	2	1	no	—
197	#446	C₂²×C₁₀₉	1	2	yes	—
198	#446	C₂²×C₁₀₉	1	2	yes	—
199	#440	—	2	4	no	—
200	C₄×D₈	—	2	1	no	—
201	C₄×D₈	—	2	1	no	—
202	#432	C₂²×C₁₀₉	1	2	yes	—
203	C₂×C₄⋊C₈	—	2	1	no	—
204	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
205	C₂×#97	—	2	1	no	—
206	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
207	#335	C₂²×C₁₀₉	1	2	yes	—
208	#334	C₂²×C₁₀₉	1	2	yes	—
209	#333	C₂²×C₁₀₉	1	4	yes	—
210	#331	C₂²×C₁₀₉	1	4	yes	—
211	C₂×#41	—	2	2	no	—
212	C₂²×Q₈	D₄×C₁₀₉	1	1	yes	characteristic
213	C₂×SD₁₆	—	2	4	no	—
214	C₂×D₈	—	2	1	no	—
215	C₂×D₈	—	2	1	no	—
216	C₂²×C₈	—	2	1	no	—
217	C₂²×C₈	D₄×C₁₀₉	1	1	yes	characteristic
218	C₂²×C₈	D₄×C₁₀₉	1	1	yes	characteristic
219	C₄⋊Q₈	C₂³×C₁₀₉	1	2	yes	—
220	#127	C₂³×C₁₀₉	1	2	yes	—
221	#124	—	2	2	no	—
222	#121	—	2	2	no	—
223	#121	—	2	2	no	—
224	C₄×Q₈	—	2	1	no	—
225	C₄×Q₈	—	2	1	no	—
226	C₄×Q₈	D₄×C₁₀₉	1	2	yes	—
227	C₄×Q₈	C₂³×C₁₀₉	1	2	yes	—
228	C₄×D₄	—	2	2	no	—
229	C₄×D₄	—	2	2	no	—
230	C₂×C₄⋊C₄	—	2	1	no	—
231	C₂×C₄⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
232	C₂×C₄²	—	2	1	no	—
233	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
234	#107	—	2	1	no	—
235	#107	—	2	1	no	—
236	#107	—	2	4	no	—
237	C₄⋊C₈	—	2	2	no	—
238	C₄⋊C₈	—	2	2	no	—
239	C₄⋊C₈	—	2	1	no	—
240	C₄⋊C₈	C₂³×C₁₀₉	1	2	yes	—
241	C₄⋊C₈	C₂³×C₁₀₉	1	4	yes	—
242	#103	—	2	4	no	—
243	#102	—	2	2	no	—
244	#102	—	2	2	no	—
245	#102	—	2	4	no	—
246	C₂²⋊C₈	—	2	2	no	—
247	#97	—	2	2	no	—
248	#97	—	2	2	no	—
249	#97	—	2	2	no	—
250	C₄×C₈	D₄×C₁₀₉	1	1	yes	characteristic
251	C₄×C₈	D₄×C₁₀₉	1	1	yes	characteristic
252	C₄×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
253	C₄×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
254	#41	—	8	4	no	—
255	C₂×Q₈	—	2	1	no	—
256	C₂×Q₈	—	2	2	no	—
257	C₂×Q₈	—	4	2	no	—
258	C₂×Q₈	C₂×D₄×C₁₀₉	1	2	yes	—
259	C₂×D₄	—	2	2	no	—
260	C₂×D₄	—	2	2	no	—
261	C₂×D₄	—	2	2	no	—
262	C₂×D₄	—	4	2	no	—
263	C₂²×C₄	—	2	1	no	—
264	C₂²×C₄	—	2	1	no	—
265	C₂²×C₄	—	4	2	no	—
266	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
267	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
268	C₂²×C₄	#41×C₁₀₉	1	1	yes	characteristic
269	SD₁₆	—	4	8	no	—
270	D₈	—	4	2	no	—
271	D₈	—	4	2	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	2	4	no	—
275	C₂×C₈	—	4	1	no	—
276	C₂×C₈	—	4	2	no	—
277	C₂×C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
278	C₂×C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
279	C₂×C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
280	C₂×C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
281	C₄⋊C₄	—	2	2	no	—
282	C₄⋊C₄	—	2	1	no	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	2	2	no	—
286	C₄⋊C₄	C₂×D₄×C₁₀₉	1	2	yes	—
287	C₂²⋊C₄	—	4	2	no	—
288	C₂²⋊C₄	—	4	2	no	—
289	C₄²	—	2	2	no	—
290	C₄²	—	2	2	no	—
291	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
292	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
293	C₄²	#41×C₁₀₉	1	1	yes	characteristic
294	C₄²	C₂⁴×C₁₀₉	1	1	yes	characteristic
295	C₂³	—	4	2	no	—
296	C₂³	#121×C₁₀₉	1	1	yes	characteristic
297	Q₈	—	2	4	no	—
298	Q₈	—	4	2	no	—
299	Q₈	—	8	2	no	—
300	D₄	—	4	2	no	—
301	D₄	—	4	2	no	—
302	D₄	—	4	4	no	—
303	D₄	—	8	2	no	—
304	D₄	—	8	4	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	8	4	no	—
315	C₂×C₄	#121×C₁₀₉	1	1	yes	characteristic
316	C₂×C₄	#121×C₁₀₉	1	1	yes	characteristic
317	C₂×C₄	#121×C₁₀₉	1	1	yes	characteristic
318	C₂×C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
319	C₂×C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
320	C₂×C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
321	C₈	—	2	1	no	—
322	C₈	—	2	1	no	—
323	C₈	—	2	1	no	—
324	C₈	—	4	2	no	—
325	C₈	—	4	4	no	—
326	C₈	#137×C₁₀₉	1	2	yes	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	8	2	no	—
330	C₂²	—	8	2	no	—
331	C₂²	—	8	2	no	—
332	C₂²	#472×C₁₀₉	1	2	yes	—
333	C₂²	C₂×#121×C₁₀₉	1	1	yes	characteristic
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	4	2	no	—
339	C₄	—	4	2	no	—
340	C₄	—	8	2	no	—
341	C₄	C₂×#137×C₁₀₉	1	1	yes	characteristic
342	C₄	#575×C₁₀₉	1	1	yes	characteristic
343	C₄	#575×C₁₀₉	1	2	yes	—
344	C₂	—	2	2	no	—
345	C₂	—	16	2	no	—
346	C₂	C₁₀₉×#3037	1	1	yes	characteristic
347	C₂	C₁₀₉×#3054	1	1	yes	characteristic
348	C₂	C₂×#472×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₀₉	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₀₉
prime radical	C₁₀₉×#3169	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₀₉×#3112	C₂	semidirect
3. C₂×#446×C₁₀₉	C₂	semidirect
4. C₁₀₉×#1698	C₂	semidirect
5. C₁₀₉×#1696	C₂	semidirect
6. C₁₀₉×#1688	C₂	semidirect
7. C₁₀₉×#1568	C₂	semidirect
8. C₁₀₉×#1551	C₂	semidirect
169. #19289	C₁₀₉	direct (N × H)
176. #3112	C₂×C₁₀₉	semidirect
177. C₂×#446	C₂×C₁₀₉	semidirect
178. #1698	C₂×C₁₀₉	semidirect
179. #1696	C₂×C₁₀₉	semidirect
180. #1688	C₂×C₁₀₉	semidirect
181. #1568	C₂×C₁₀₉	semidirect
182. #1551	C₂×C₁₀₉	semidirect
183. C₁₀₉	#19289	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₀₉×#3169	2
C₁₀₉×#3112	2
C₂×#446×C₁₀₉	2
C₁₀₉×#1698	2
C₁₀₉×#1696	2
C₁₀₉×#1688	2
C₁₀₉×#1568	2
C₁₀₉×#1551	2
#19289	109

Subgroups — C109×#19289 #42137283

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#19289 #42137283