Subgroups — C₁₀₉×#3388 #28360642

Order 13952.

334 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#557×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
2	#554×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	#501×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#501×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#499×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#479×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
7	#479×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#478×C₁₀₉	C₂	1	1	yes	maximal, characteristic
9	#478×C₁₀₉	C₂	1	1	yes	maximal, characteristic
10	#476×C₁₀₉	C₂	1	1	yes	maximal, characteristic
11	#475×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#446×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
13	#440×C₁₀₉	C₂	1	1	yes	maximal, characteristic
14	#440×C₁₀₉	C₂	1	1	yes	maximal, characteristic
15	C₄×SD₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
16	C₂×SD₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₁₀₉	—	2	1	no	—
18	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
20	#125×C₁₀₉	—	2	1	no	—
21	#125×C₁₀₉	—	2	1	no	—
22	#125×C₁₀₉	C₂²	1	1	yes	characteristic
23	#125×C₁₀₉	C₂²	1	1	yes	characteristic
24	#125×C₁₀₉	C₂²	1	1	yes	characteristic
25	#125×C₁₀₉	C₂²	1	1	yes	characteristic
26	C₂²⋊Q₈×C₁₀₉	—	2	1	no	—
27	C₂²⋊Q₈×C₁₀₉	—	2	1	no	—
28	C₂²⋊Q₈×C₁₀₉	—	2	1	no	—
29	C₄×Q₈×C₁₀₉	—	2	1	no	—
30	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
31	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
32	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
33	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
34	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
35	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
36	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
37	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
38	#107×C₁₀₉	C₂²	1	1	yes	characteristic
39	#107×C₁₀₉	C₂²	1	1	yes	characteristic
40	#107×C₁₀₉	C₂²	1	1	yes	characteristic
41	#107×C₁₀₉	C₂²	1	1	yes	characteristic
42	#106×C₁₀₉	C₂²	1	1	yes	characteristic
43	#106×C₁₀₉	C₂²	1	1	yes	characteristic
44	#106×C₁₀₉	C₂²	1	1	yes	characteristic
45	#106×C₁₀₉	C₂²	1	1	yes	characteristic
46	#106×C₁₀₉	C₂²	1	1	yes	characteristic
47	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
48	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
49	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
50	#103×C₁₀₉	C₂²	1	1	yes	characteristic
51	#103×C₁₀₉	C₂²	1	1	yes	characteristic
52	#103×C₁₀₉	C₂²	1	1	yes	characteristic
53	#102×C₁₀₉	C₂²	1	1	yes	characteristic
54	#102×C₁₀₉	C₂²	1	1	yes	characteristic
55	#102×C₁₀₉	C₂²	1	1	yes	characteristic
56	#97×C₁₀₉	C₂²	1	1	yes	characteristic
57	#97×C₁₀₉	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₀₉	—	2	1	no	—
60	C₂×Q₈×C₁₀₉	—	2	1	no	—
61	C₂×Q₈×C₁₀₉	D₄	1	1	yes	characteristic
62	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
63	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₁₀₉	—	2	1	no	—
65	C₂²×C₄×C₁₀₉	—	2	1	no	—
66	C₂²×C₄×C₁₀₉	—	2	1	no	—
67	SD₁₆×C₁₀₉	Q₈	1	4	yes	complemented
68	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
69	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
70	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
72	C₄⋊C₄×C₁₀₉	—	2	1	no	—
73	C₄⋊C₄×C₁₀₉	—	2	1	no	—
74	C₄⋊C₄×C₁₀₉	—	2	1	no	—
75	C₄⋊C₄×C₁₀₉	—	2	1	no	—
76	C₄⋊C₄×C₁₀₉	—	2	1	no	—
77	C₄⋊C₄×C₁₀₉	—	2	1	no	—
78	C₄⋊C₄×C₁₀₉	—	2	1	no	—
79	C₄⋊C₄×C₁₀₉	—	2	1	no	—
80	C₄⋊C₄×C₁₀₉	—	2	1	no	—
81	C₄⋊C₄×C₁₀₉	—	2	1	no	—
82	C₄⋊C₄×C₁₀₉	—	2	1	no	—
83	C₄⋊C₄×C₁₀₉	—	2	1	no	—
84	C₄⋊C₄×C₁₀₉	—	2	1	no	—
85	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
86	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
87	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
88	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
89	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
90	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
91	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
92	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
93	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
94	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
95	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
96	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
97	C₄²×C₁₀₉	—	2	1	no	—
98	C₄²×C₁₀₉	—	2	1	no	—
99	C₄²×C₁₀₉	—	2	1	no	—
100	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
101	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
102	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
103	C₂³×C₁₀₉	—	2	1	no	—
104	Q₈×C₁₀₉	—	2	1	no	—
105	Q₈×C₁₀₉	—	4	1	no	—
106	Q₈×C₁₀₉	—	4	2	no	—
107	Q₈×C₁₀₉	—	8	1	no	—
108	Q₈×C₁₀₉	C₂×Q₈	1	2	yes	—
109	D₄×C₁₀₉	—	2	1	no	—
110	D₄×C₁₀₉	C₂×Q₈	1	2	yes	—
111	C₂×C₄×C₁₀₉	—	2	1	no	—
112	C₂×C₄×C₁₀₉	—	2	1	no	—
113	C₂×C₄×C₁₀₉	—	2	1	no	—
114	C₂×C₄×C₁₀₉	—	2	1	no	—
115	C₂×C₄×C₁₀₉	—	2	1	no	—
116	C₂×C₄×C₁₀₉	—	2	1	no	—
117	C₂×C₄×C₁₀₉	—	2	1	no	—
118	C₂×C₄×C₁₀₉	—	4	2	no	—
119	C₂×C₄×C₁₀₉	—	8	1	no	—
120	C₂×C₄×C₁₀₉	—	8	1	no	—
121	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
128	C₈×C₁₀₉	—	2	1	no	—
129	C₈×C₁₀₉	—	2	1	no	—
130	C₈×C₁₀₉	—	2	1	no	—
131	C₈×C₁₀₉	C₂×Q₈	1	2	yes	—
132	C₂²×C₁₀₉	—	2	2	no	—
133	C₂²×C₁₀₉	—	4	1	no	—
134	C₂²×C₁₀₉	—	4	1	no	—
135	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
136	C₄×C₁₀₉	—	2	2	no	—
137	C₄×C₁₀₉	—	2	1	no	—
138	C₄×C₁₀₉	—	2	1	no	—
139	C₄×C₁₀₉	—	2	1	no	—
140	C₄×C₁₀₉	—	2	1	no	—
141	C₄×C₁₀₉	—	2	1	no	—
142	C₄×C₁₀₉	—	2	1	no	—
143	C₄×C₁₀₉	—	4	1	no	—
144	C₄×C₁₀₉	—	4	1	no	—
145	C₄×C₁₀₉	—	4	1	no	—
146	C₄×C₁₀₉	—	4	1	no	—
147	C₄×C₁₀₉	—	4	1	no	—
148	C₄×C₁₀₉	—	4	1	no	—
149	C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
150	C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
151	C₂×C₁₀₉	—	4	2	no	—
152	C₂×C₁₀₉	D₄×Q₈	1	1	yes	characteristic
153	C₂×C₁₀₉	#575	1	1	yes	characteristic
154	C₂×C₁₀₉	#577	1	1	yes	characteristic
155	#3388	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
156	C₁₀₉	#3388	1	1	yes	characteristic, direct product factor, Sylow
157	#557	C₂×C₁₀₉	1	1	yes	characteristic, complemented
158	#554	C₂×C₁₀₉	1	1	yes	characteristic
159	#501	C₂×C₁₀₉	1	1	yes	characteristic, complemented
160	#501	C₂×C₁₀₉	1	1	yes	characteristic, complemented
161	#499	C₂×C₁₀₉	1	1	yes	characteristic, complemented
162	#479	C₂×C₁₀₉	1	1	yes	characteristic, complemented
163	#479	C₂×C₁₀₉	1	1	yes	characteristic, complemented
164	#478	C₂×C₁₀₉	1	1	yes	characteristic
165	#478	C₂×C₁₀₉	1	1	yes	characteristic
166	#476	C₂×C₁₀₉	1	1	yes	characteristic
167	#475	C₂×C₁₀₉	1	1	yes	characteristic, complemented
168	#446	C₂×C₁₀₉	1	1	yes	characteristic, complemented
169	#440	C₂×C₁₀₉	1	1	yes	characteristic
170	#440	C₂×C₁₀₉	1	1	yes	characteristic
171	C₄×SD₁₆	C₂×C₁₀₉	1	1	yes	characteristic
172	C₂×SD₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
173	C₄⋊Q₈	—	2	1	no	—
174	C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
175	C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
176	#125	—	2	1	no	—
177	#125	—	2	1	no	—
178	#125	C₂²×C₁₀₉	1	1	yes	characteristic
179	#125	C₂²×C₁₀₉	1	1	yes	characteristic
180	#125	C₂²×C₁₀₉	1	1	yes	characteristic
181	#125	C₂²×C₁₀₉	1	1	yes	characteristic
182	C₂²⋊Q₈	—	2	1	no	—
183	C₂²⋊Q₈	—	2	1	no	—
184	C₂²⋊Q₈	—	2	1	no	—
185	C₄×Q₈	—	2	1	no	—
186	C₄×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
187	C₄×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
188	C₄×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
189	C₄×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
190	C₄×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
191	C₄×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
192	C₄×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
193	C₂×C₄⋊C₄	—	2	1	no	—
194	#107	C₂²×C₁₀₉	1	1	yes	characteristic
195	#107	C₂²×C₁₀₉	1	1	yes	characteristic
196	#107	C₂²×C₁₀₉	1	1	yes	characteristic
197	#107	C₂²×C₁₀₉	1	1	yes	characteristic
198	#106	C₂²×C₁₀₉	1	1	yes	characteristic
199	#106	C₂²×C₁₀₉	1	1	yes	characteristic
200	#106	C₂²×C₁₀₉	1	1	yes	characteristic
201	#106	C₂²×C₁₀₉	1	1	yes	characteristic
202	#106	C₂²×C₁₀₉	1	1	yes	characteristic
203	C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
204	C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
205	C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
206	#103	C₂²×C₁₀₉	1	1	yes	characteristic
207	#103	C₂²×C₁₀₉	1	1	yes	characteristic
208	#103	C₂²×C₁₀₉	1	1	yes	characteristic
209	#102	C₂²×C₁₀₉	1	1	yes	characteristic
210	#102	C₂²×C₁₀₉	1	1	yes	characteristic
211	#102	C₂²×C₁₀₉	1	1	yes	characteristic
212	#97	C₂²×C₁₀₉	1	1	yes	characteristic
213	#97	C₂²×C₁₀₉	1	1	yes	characteristic
214	C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
215	C₂×Q₈	—	2	1	no	—
216	C₂×Q₈	—	2	1	no	—
217	C₂×Q₈	D₄×C₁₀₉	1	1	yes	characteristic
218	C₂×Q₈	C₂³×C₁₀₉	1	1	yes	characteristic
219	C₂×D₄	C₂³×C₁₀₉	1	1	yes	characteristic
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	2	1	no	—
223	SD₁₆	Q₈×C₁₀₉	1	4	yes	complemented
224	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
225	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
226	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
227	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
228	C₄⋊C₄	—	2	1	no	—
229	C₄⋊C₄	—	2	1	no	—
230	C₄⋊C₄	—	2	1	no	—
231	C₄⋊C₄	—	2	1	no	—
232	C₄⋊C₄	—	2	1	no	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	2	1	no	—
235	C₄⋊C₄	—	2	1	no	—
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	—	2	1	no	—
238	C₄⋊C₄	—	2	1	no	—
239	C₄⋊C₄	—	2	1	no	—
240	C₄⋊C₄	—	2	1	no	—
241	C₄⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
242	C₄⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
243	C₄⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
244	C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
245	C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
246	C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
247	C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
248	C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
249	C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
250	C₂²⋊C₄	—	2	1	no	—
251	C₂²⋊C₄	—	2	1	no	—
252	C₂²⋊C₄	—	2	1	no	—
253	C₄²	—	2	1	no	—
254	C₄²	—	2	1	no	—
255	C₄²	—	2	1	no	—
256	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
257	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
258	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
259	C₂³	—	2	1	no	—
260	Q₈	—	2	1	no	—
261	Q₈	—	4	1	no	—
262	Q₈	—	4	2	no	—
263	Q₈	—	8	1	no	—
264	Q₈	C₂×Q₈×C₁₀₉	1	2	yes	—
265	D₄	—	2	1	no	—
266	D₄	C₂×Q₈×C₁₀₉	1	2	yes	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	8	1	no	—
276	C₂×C₄	—	8	1	no	—
277	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
278	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
279	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
280	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
281	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
282	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
283	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
284	C₈	—	2	1	no	—
285	C₈	—	2	1	no	—
286	C₈	—	2	1	no	—
287	C₈	C₂×Q₈×C₁₀₉	1	2	yes	—
288	C₂²	—	2	2	no	—
289	C₂²	—	4	1	no	—
290	C₂²	—	4	1	no	—
291	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
292	C₄	—	2	2	no	—
293	C₄	—	2	1	no	—
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	4	1	no	—
300	C₄	—	4	1	no	—
301	C₄	—	4	1	no	—
302	C₄	—	4	1	no	—
303	C₄	—	4	1	no	—
304	C₄	—	4	1	no	—
305	C₄	C₂²×Q₈×C₁₀₉	1	1	yes	characteristic
306	C₄	2₋¹⁺⁴×C₁₀₉	1	1	yes	characteristic
307	C₂	—	4	2	no	—
308	C₂	D₄×Q₈×C₁₀₉	1	1	yes	characteristic
309	C₂	#575×C₁₀₉	1	1	yes	characteristic
310	C₂	#577×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₀₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₀₉
prime radical	C₂×D₄×C₁₀₉	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₄×Q₈×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. #557×C₁₀₉	C₂	semidirect
3. #501×C₁₀₉	C₂	semidirect
4. #501×C₁₀₉	C₂	semidirect
5. #499×C₁₀₉	C₂	semidirect
6. #479×C₁₀₉	C₂	semidirect
7. #479×C₁₀₉	C₂	semidirect
11. #475×C₁₀₉	C₂	semidirect
12. #446×C₁₀₉	C₂	semidirect
67. SD₁₆×C₁₀₉	Q₈	semidirect
155. #3388	C₁₀₉	direct (N × H)
156. C₁₀₉	#3388	direct (N × H)
157. #557	C₂×C₁₀₉	semidirect
159. #501	C₂×C₁₀₉	semidirect
160. #501	C₂×C₁₀₉	semidirect
161. #499	C₂×C₁₀₉	semidirect
162. #479	C₂×C₁₀₉	semidirect
163. #479	C₂×C₁₀₉	semidirect
167. #475	C₂×C₁₀₉	semidirect
168. #446	C₂×C₁₀₉	semidirect
223. SD₁₆	Q₈×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#557×C₁₀₉	2
#554×C₁₀₉	2
#501×C₁₀₉	2
#499×C₁₀₉	2
#479×C₁₀₉	2
#478×C₁₀₉	2
#476×C₁₀₉	2
#475×C₁₀₉	2
#446×C₁₀₉	2
#440×C₁₀₉	2
C₄×SD₁₆×C₁₀₉	2
#3388	109

Subgroups — C109×#3388 #28360642

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#3388 #28360642