Subgroups — C₁₀₉×#19344 #42137338

Order 27904.

612 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₉×#3254	C₂	1	2	yes	maximal, complemented
2	C₁₀₉×#3169	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#2960	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₀₉×#1698	C₂	1	4	yes	maximal, complemented
5	C₁₀₉×#1676	C₂	1	2	yes	maximal, complemented
6	C₁₀₉×#1668	C₂	1	2	yes	maximal, complemented
7	C₁₀₉×#1567	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₀₉×#1560	C₂	1	2	yes	maximal, complemented
9	#533×C₁₀₉	C₂²	1	1	yes	characteristic
10	#532×C₁₀₉	C₂²	1	2	yes	—
11	#500×C₁₀₉	C₂²	1	1	yes	characteristic
12	#493×C₁₀₉	C₂²	1	1	yes	characteristic
13	#486×C₁₀₉	C₂²	1	2	yes	—
14	#477×C₁₀₉	C₂²	1	4	yes	—
15	#474×C₁₀₉	—	2	2	no	—
16	#468×C₁₀₉	—	2	4	no	—
17	#466×C₁₀₉	—	2	1	no	—
18	#466×C₁₀₉	—	2	1	no	—
19	#461×C₁₀₉	—	2	2	no	—
20	#460×C₁₀₉	C₂²	1	4	yes	—
21	#446×C₁₀₉	—	2	1	no	—
22	#446×C₁₀₉	C₂²	1	2	yes	—
23	C₈×Q₈×C₁₀₉	—	2	1	no	—
24	C₈×Q₈×C₁₀₉	C₂²	1	2	yes	—
25	C₄×D₈×C₁₀₉	—	2	1	no	—
26	C₄×D₈×C₁₀₉	—	2	1	no	—
27	#424×C₁₀₉	—	2	1	no	—
28	#423×C₁₀₉	C₂²	1	1	yes	characteristic
29	#423×C₁₀₉	C₂²	1	2	yes	—
30	#405×C₁₀₉	—	2	1	no	—
31	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
32	#335×C₁₀₉	C₂²	1	2	yes	—
33	#335×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	C₂×SD₁₆×C₁₀₉	—	2	4	no	—
38	C₂×D₈×C₁₀₉	—	2	1	no	—
39	C₂×D₈×C₁₀₉	—	2	1	no	—
40	C₂×M₁₆×C₁₀₉	—	2	1	no	—
41	C₂×M₁₆×C₁₀₉	—	2	2	no	—
42	C₂²×C₈×C₁₀₉	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₁₀₉	D₄	1	1	yes	characteristic
44	C₄⋊Q₈×C₁₀₉	D₄	1	1	yes	characteristic
45	C₄⋊Q₈×C₁₀₉	D₄	1	1	yes	characteristic
46	C₄⋊Q₈×C₁₀₉	C₂³	1	2	yes	—
47	#127×C₁₀₉	C₂³	1	2	yes	—
48	#125×C₁₀₉	—	2	1	no	—
49	#124×C₁₀₉	—	2	2	no	—
50	C₂²⋊Q₈×C₁₀₉	—	2	1	no	—
51	C₂²⋊Q₈×C₁₀₉	D₄	1	2	yes	—
52	#121×C₁₀₉	—	2	2	no	—
53	#121×C₁₀₉	—	2	2	no	—
54	C₄×Q₈×C₁₀₉	—	2	1	no	—
55	C₄×Q₈×C₁₀₉	C₂³	1	2	yes	—
56	C₄×D₄×C₁₀₉	—	2	2	no	—
57	C₄×D₄×C₁₀₉	—	2	2	no	—
58	#117×C₁₀₉	—	2	1	no	—
59	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
60	#107×C₁₀₉	—	2	1	no	—
61	#107×C₁₀₉	—	2	1	no	—
62	#107×C₁₀₉	—	2	4	no	—
63	C₄⋊C₈×C₁₀₉	—	2	2	no	—
64	C₄⋊C₈×C₁₀₉	—	2	2	no	—
65	C₄⋊C₈×C₁₀₉	—	2	1	no	—
66	C₄⋊C₈×C₁₀₉	—	2	2	no	—
67	C₄⋊C₈×C₁₀₉	C₂³	1	2	yes	—
68	C₄⋊C₈×C₁₀₉	C₂³	1	4	yes	—
69	#103×C₁₀₉	—	2	4	no	—
70	#102×C₁₀₉	—	2	2	no	—
71	#102×C₁₀₉	—	2	2	no	—
72	#102×C₁₀₉	—	2	4	no	—
73	#97×C₁₀₉	—	2	2	no	—
74	C₄×C₈×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	C₂×D₄	1	2	yes	—
83	C₂×D₄×C₁₀₉	—	2	2	no	—
84	C₂×D₄×C₁₀₉	—	2	2	no	—
85	C₂×D₄×C₁₀₉	—	2	2	no	—
86	C₂×D₄×C₁₀₉	—	4	2	no	—
87	C₂²×C₄×C₁₀₉	—	4	2	no	—
88	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₁₀₉	#41	1	1	yes	characteristic
91	SD₁₆×C₁₀₉	—	8	4	no	—
92	D₈×C₁₀₉	—	4	2	no	—
93	D₈×C₁₀₉	—	4	2	no	—
94	M₁₆×C₁₀₉	—	4	2	no	—
95	M₁₆×C₁₀₉	—	8	2	no	—
96	C₂×C₈×C₁₀₉	—	2	2	no	—
97	C₂×C₈×C₁₀₉	—	2	2	no	—
98	C₂×C₈×C₁₀₉	—	2	4	no	—
99	C₂×C₈×C₁₀₉	—	4	1	no	—
100	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
104	C₄⋊C₄×C₁₀₉	—	2	1	no	—
105	C₄⋊C₄×C₁₀₉	—	2	1	no	—
106	C₄⋊C₄×C₁₀₉	—	2	1	no	—
107	C₄⋊C₄×C₁₀₉	—	2	2	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	—	2	2	no	—
113	C₂²⋊C₄×C₁₀₉	—	4	2	no	—
114	C₂²⋊C₄×C₁₀₉	—	4	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	2	no	—
123	Q₈×C₁₀₉	—	2	2	no	—
124	Q₈×C₁₀₉	—	4	2	no	—
125	Q₈×C₁₀₉	—	8	2	no	—
126	D₄×C₁₀₉	—	4	2	no	—
127	D₄×C₁₀₉	—	4	2	no	—
128	D₄×C₁₀₉	—	4	2	no	—
129	D₄×C₁₀₉	—	4	2	no	—
130	D₄×C₁₀₉	—	8	2	no	—
131	D₄×C₁₀₉	—	8	4	no	—
132	C₂×C₄×C₁₀₉	—	2	1	no	—
133	C₂×C₄×C₁₀₉	—	2	1	no	—
134	C₂×C₄×C₁₀₉	—	2	1	no	—
135	C₂×C₄×C₁₀₉	—	2	1	no	—
136	C₂×C₄×C₁₀₉	—	2	2	no	—
137	C₂×C₄×C₁₀₉	—	2	2	no	—
138	C₂×C₄×C₁₀₉	—	2	2	no	—
139	C₂×C₄×C₁₀₉	—	4	2	no	—
140	C₂×C₄×C₁₀₉	—	8	4	no	—
141	C₂×C₄×C₁₀₉	#121	1	1	yes	characteristic
142	C₂×C₄×C₁₀₉	#121	1	1	yes	characteristic
143	C₂×C₄×C₁₀₉	#121	1	1	yes	characteristic
144	C₂×C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
147	C₈×C₁₀₉	—	2	1	no	—
148	C₈×C₁₀₉	—	2	1	no	—
149	C₈×C₁₀₉	—	2	1	no	—
150	C₈×C₁₀₉	—	4	2	no	—
151	C₈×C₁₀₉	—	4	4	no	—
152	C₈×C₁₀₉	#137	1	2	yes	—
153	C₂²×C₁₀₉	—	2	1	no	—
154	C₂²×C₁₀₉	—	2	1	no	—
155	C₂²×C₁₀₉	—	2	1	no	—
156	C₂²×C₁₀₉	—	8	2	no	—
157	C₂²×C₁₀₉	—	8	2	no	—
158	C₂²×C₁₀₉	—	8	2	no	—
159	C₂²×C₁₀₉	C₂×#121	1	1	yes	characteristic
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	#472	1	2	yes	—
167	C₄×C₁₀₉	C₂×#137	1	1	yes	characteristic
168	C₄×C₁₀₉	#575	1	1	yes	characteristic
169	C₄×C₁₀₉	#576	1	1	yes	characteristic
170	C₄×C₁₀₉	#578	1	1	yes	characteristic
171	#19344	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
172	C₂×C₁₀₉	—	4	1	no	—
173	C₂×C₁₀₉	—	16	2	no	—
174	C₂×C₁₀₉	#3037	1	1	yes	characteristic
175	C₂×C₁₀₉	#3061	1	1	yes	characteristic
176	C₂×C₁₀₉	C₂×#472	1	1	yes	characteristic
177	#3254	C₂×C₁₀₉	1	2	yes	complemented
178	#3169	C₂×C₁₀₉	1	1	yes	characteristic
179	#2960	C₂×C₁₀₉	1	1	yes	characteristic, complemented
180	#1698	C₂×C₁₀₉	1	4	yes	complemented
181	#1676	C₂×C₁₀₉	1	2	yes	complemented
182	#1668	C₂×C₁₀₉	1	2	yes	complemented
183	#1567	C₂×C₁₀₉	1	1	yes	characteristic, complemented
184	#1560	C₂×C₁₀₉	1	2	yes	complemented
185	C₁₀₉	#19344	1	1	yes	characteristic, direct product factor, Sylow
186	#533	C₂²×C₁₀₉	1	1	yes	characteristic
187	#532	C₂²×C₁₀₉	1	2	yes	—
188	#500	C₂²×C₁₀₉	1	1	yes	characteristic
189	#493	C₂²×C₁₀₉	1	1	yes	characteristic
190	#486	C₂²×C₁₀₉	1	2	yes	—
191	#477	C₂²×C₁₀₉	1	4	yes	—
192	#474	—	2	2	no	—
193	#468	—	2	4	no	—
194	#466	—	2	1	no	—
195	#466	—	2	1	no	—
196	#461	—	2	2	no	—
197	#460	C₂²×C₁₀₉	1	4	yes	—
198	#446	—	2	1	no	—
199	#446	C₂²×C₁₀₉	1	2	yes	—
200	C₈×Q₈	—	2	1	no	—
201	C₈×Q₈	C₂²×C₁₀₉	1	2	yes	—
202	C₄×D₈	—	2	1	no	—
203	C₄×D₈	—	2	1	no	—
204	#424	—	2	1	no	—
205	#423	C₂²×C₁₀₉	1	1	yes	characteristic
206	#423	C₂²×C₁₀₉	1	2	yes	—
207	#405	—	2	1	no	—
208	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
209	#335	C₂²×C₁₀₉	1	2	yes	—
210	#335	C₂²×C₁₀₉	1	2	yes	—
211	#333	C₂²×C₁₀₉	1	4	yes	—
212	#331	C₂²×C₁₀₉	1	4	yes	—
213	C₂×#41	—	2	2	no	—
214	C₂×SD₁₆	—	2	4	no	—
215	C₂×D₈	—	2	1	no	—
216	C₂×D₈	—	2	1	no	—
217	C₂×M₁₆	—	2	1	no	—
218	C₂×M₁₆	—	2	2	no	—
219	C₂²×C₈	D₄×C₁₀₉	1	1	yes	characteristic
220	C₂²×C₈	D₄×C₁₀₉	1	1	yes	characteristic
221	C₄⋊Q₈	D₄×C₁₀₉	1	1	yes	characteristic
222	C₄⋊Q₈	D₄×C₁₀₉	1	1	yes	characteristic
223	C₄⋊Q₈	C₂³×C₁₀₉	1	2	yes	—
224	#127	C₂³×C₁₀₉	1	2	yes	—
225	#125	—	2	1	no	—
226	#124	—	2	2	no	—
227	C₂²⋊Q₈	—	2	1	no	—
228	C₂²⋊Q₈	D₄×C₁₀₉	1	2	yes	—
229	#121	—	2	2	no	—
230	#121	—	2	2	no	—
231	C₄×Q₈	—	2	1	no	—
232	C₄×Q₈	C₂³×C₁₀₉	1	2	yes	—
233	C₄×D₄	—	2	2	no	—
234	C₄×D₄	—	2	2	no	—
235	#117	—	2	1	no	—
236	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
237	#107	—	2	1	no	—
238	#107	—	2	1	no	—
239	#107	—	2	4	no	—
240	C₄⋊C₈	—	2	2	no	—
241	C₄⋊C₈	—	2	2	no	—
242	C₄⋊C₈	—	2	1	no	—
243	C₄⋊C₈	—	2	2	no	—
244	C₄⋊C₈	C₂³×C₁₀₉	1	2	yes	—
245	C₄⋊C₈	C₂³×C₁₀₉	1	4	yes	—
246	#103	—	2	4	no	—
247	#102	—	2	2	no	—
248	#102	—	2	2	no	—
249	#102	—	2	4	no	—
250	#97	—	2	2	no	—
251	C₄×C₈	—	2	2	no	—
252	C₄×C₈	D₄×C₁₀₉	1	1	yes	characteristic
253	C₄×C₈	D₄×C₁₀₉	1	1	yes	characteristic
254	C₄×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
255	C₄×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
256	#41	—	8	4	no	—
257	C₂×Q₈	—	2	1	no	—
258	C₂×Q₈	—	2	2	no	—
259	C₂×Q₈	C₂×D₄×C₁₀₉	1	2	yes	—
260	C₂×D₄	—	2	2	no	—
261	C₂×D₄	—	2	2	no	—
262	C₂×D₄	—	2	2	no	—
263	C₂×D₄	—	4	2	no	—
264	C₂²×C₄	—	4	2	no	—
265	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
266	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
267	C₂²×C₄	#41×C₁₀₉	1	1	yes	characteristic
268	SD₁₆	—	8	4	no	—
269	D₈	—	4	2	no	—
270	D₈	—	4	2	no	—
271	M₁₆	—	4	2	no	—
272	M₁₆	—	8	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	2	4	no	—
276	C₂×C₈	—	4	1	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	1	no	—
282	C₄⋊C₄	—	2	1	no	—
283	C₄⋊C₄	—	2	1	no	—
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	2	2	no	—
286	C₄⋊C₄	—	2	2	no	—
287	C₄⋊C₄	—	2	2	no	—
288	C₄⋊C₄	C₂×D₄×C₁₀₉	1	2	yes	—
289	C₂²⋊C₄	—	2	2	no	—
290	C₂²⋊C₄	—	4	2	no	—
291	C₂²⋊C₄	—	4	2	no	—
292	C₄²	—	2	2	no	—
293	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
294	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
295	C₄²	#41×C₁₀₉	1	1	yes	characteristic
296	C₄²	C₂⁴×C₁₀₉	1	1	yes	characteristic
297	C₂³	—	4	2	no	—
298	C₂³	#121×C₁₀₉	1	1	yes	characteristic
299	Q₈	—	2	2	no	—
300	Q₈	—	2	2	no	—
301	Q₈	—	4	2	no	—
302	Q₈	—	8	2	no	—
303	D₄	—	4	2	no	—
304	D₄	—	4	2	no	—
305	D₄	—	4	2	no	—
306	D₄	—	4	2	no	—
307	D₄	—	8	2	no	—
308	D₄	—	8	4	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	8	4	no	—
318	C₂×C₄	#121×C₁₀₉	1	1	yes	characteristic
319	C₂×C₄	#121×C₁₀₉	1	1	yes	characteristic
320	C₂×C₄	#121×C₁₀₉	1	1	yes	characteristic
321	C₂×C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
322	C₂×C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
323	C₂×C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
324	C₈	—	2	1	no	—
325	C₈	—	2	1	no	—
326	C₈	—	2	1	no	—
327	C₈	—	4	2	no	—
328	C₈	—	4	4	no	—
329	C₈	#137×C₁₀₉	1	2	yes	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	8	2	no	—
334	C₂²	—	8	2	no	—
335	C₂²	—	8	2	no	—
336	C₂²	C₂×#121×C₁₀₉	1	1	yes	characteristic
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	4	2	no	—
341	C₄	—	4	2	no	—
342	C₄	—	8	2	no	—
343	C₄	#472×C₁₀₉	1	2	yes	—
344	C₄	C₂×#137×C₁₀₉	1	1	yes	characteristic
345	C₄	#575×C₁₀₉	1	1	yes	characteristic
346	C₄	#576×C₁₀₉	1	1	yes	characteristic
347	C₄	#578×C₁₀₉	1	1	yes	characteristic
348	C₂	—	4	1	no	—
349	C₂	—	16	2	no	—
350	C₂	C₁₀₉×#3037	1	1	yes	characteristic
351	C₂	C₁₀₉×#3061	1	1	yes	characteristic
352	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
1. C₁₀₉×#3254	C₂	semidirect
3. C₁₀₉×#2960	C₂	semidirect
4. C₁₀₉×#1698	C₂	semidirect
5. C₁₀₉×#1676	C₂	semidirect
6. C₁₀₉×#1668	C₂	semidirect
7. C₁₀₉×#1567	C₂	semidirect
8. C₁₀₉×#1560	C₂	semidirect
171. #19344	C₁₀₉	direct (N × H)
177. #3254	C₂×C₁₀₉	semidirect
179. #2960	C₂×C₁₀₉	semidirect
180. #1698	C₂×C₁₀₉	semidirect
181. #1676	C₂×C₁₀₉	semidirect
182. #1668	C₂×C₁₀₉	semidirect
183. #1567	C₂×C₁₀₉	semidirect
184. #1560	C₂×C₁₀₉	semidirect
185. C₁₀₉	#19344	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×#3254	2
C₁₀₉×#3169	2
C₁₀₉×#2960	2
C₁₀₉×#1698	2
C₁₀₉×#1676	2
C₁₀₉×#1668	2
C₁₀₉×#1567	2
C₁₀₉×#1560	2
#19344	109

Subgroups — C109×#19344 #42137338

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#19344 #42137338