Subgroups — C₁₀₉×#11270 #42129264

Order 27904.

634 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#107×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#343×C₁₀₉	C₂	1	2	yes	maximal
3	C₂×#341×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₁₀₉×#1736	C₂	1	2	yes	maximal
5	C₁₀₉×#1729	C₂	1	1	yes	maximal, characteristic
6	C₂²×M₁₆×C₁₀₉	C₂²	1	2	yes	—
7	C₂²×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
9	C₂×#117×C₁₀₉	C₂²	1	1	yes	characteristic
10	C₂²×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
11	C₂×#107×C₁₀₉	—	2	2	no	—
12	C₂×#107×C₁₀₉	—	2	4	no	—
13	#343×C₁₀₉	—	2	4	no	—
14	#341×C₁₀₉	C₄	1	4	yes	complemented
15	#340×C₁₀₉	—	2	4	no	—
16	#336×C₁₀₉	—	2	2	no	—
17	C₂³×C₄×C₁₀₉	—	2	2	no	—
18	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
19	C₂×M₁₆×C₁₀₉	—	2	2	no	—
20	C₂×M₁₆×C₁₀₉	—	2	1	no	—
21	C₂×M₁₆×C₁₀₉	—	2	2	no	—
22	C₂×M₁₆×C₁₀₉	—	2	2	no	—
23	C₂×M₁₆×C₁₀₉	—	2	2	no	—
24	C₂×M₁₆×C₁₀₉	—	2	2	no	—
25	C₂×M₁₆×C₁₀₉	—	2	2	no	—
26	C₂×M₁₆×C₁₀₉	—	2	2	no	—
27	C₂×M₁₆×C₁₀₉	C₂×C₄	1	4	yes	—
28	C₂×M₁₆×C₁₀₉	Q₈	1	2	yes	—
29	C₂²×C₈×C₁₀₉	—	2	2	no	—
30	C₂²×C₈×C₁₀₉	—	2	2	no	—
31	C₂²×C₈×C₁₀₉	—	2	1	no	—
32	C₂²×C₈×C₁₀₉	—	2	2	no	—
33	C₂²×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₁₀₉	D₄	1	2	yes	—
35	C₂²×C₈×C₁₀₉	Q₈	1	2	yes	—
36	#117×C₁₀₉	—	2	2	no	—
37	#117×C₁₀₉	—	2	2	no	—
38	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
39	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
40	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
41	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
42	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
43	C₂×C₄⋊C₄×C₁₀₉	D₄	1	4	yes	—
44	C₂×C₄⋊C₄×C₁₀₉	D₄	1	2	yes	—
45	C₂×C₂²⋊C₄×C₁₀₉	—	2	1	no	—
46	C₂×C₄²×C₁₀₉	—	2	1	no	—
47	#107×C₁₀₉	—	2	8	no	—
48	C₂⁴×C₁₀₉	#41	1	1	yes	characteristic
49	C₂²×C₄×C₁₀₉	—	2	2	no	—
50	C₂²×C₄×C₁₀₉	—	2	4	no	—
51	C₂²×C₄×C₁₀₉	—	4	2	no	—
52	C₂²×C₄×C₁₀₉	—	4	2	no	—
53	C₂²×C₄×C₁₀₉	—	8	4	no	—
54	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
60	C₂²×C₄×C₁₀₉	#41	1	2	yes	—
61	C₂²×C₄×C₁₀₉	#41	1	2	yes	—
62	C₂²×C₄×C₁₀₉	#41	1	2	yes	—
63	C₂²×C₄×C₁₀₉	#41	1	1	yes	characteristic
64	M₁₆×C₁₀₉	—	2	4	no	—
65	M₁₆×C₁₀₉	—	2	4	no	—
66	M₁₆×C₁₀₉	—	2	4	no	—
67	M₁₆×C₁₀₉	—	4	2	no	—
68	M₁₆×C₁₀₉	—	4	2	no	—
69	M₁₆×C₁₀₉	—	4	2	no	—
70	C₂×C₈×C₁₀₉	—	2	4	no	—
71	C₂×C₈×C₁₀₉	—	2	2	no	—
72	C₂×C₈×C₁₀₉	—	2	4	no	—
73	C₂×C₈×C₁₀₉	—	2	2	no	—
74	C₂×C₈×C₁₀₉	—	2	4	no	—
75	C₂×C₈×C₁₀₉	—	2	2	no	—
76	C₂×C₈×C₁₀₉	—	4	1	no	—
77	C₂×C₈×C₁₀₉	—	4	2	no	—
78	C₂×C₈×C₁₀₉	—	4	2	no	—
79	C₂×C₈×C₁₀₉	—	4	1	no	—
80	C₂×C₈×C₁₀₉	C₄⋊C₄	1	4	yes	—
81	C₄⋊C₄×C₁₀₉	—	2	4	no	—
82	C₄⋊C₄×C₁₀₉	—	2	4	no	—
83	C₄⋊C₄×C₁₀₉	—	2	4	no	—
84	C₄⋊C₄×C₁₀₉	—	4	2	no	—
85	C₄⋊C₄×C₁₀₉	—	4	2	no	—
86	C₂²⋊C₄×C₁₀₉	—	4	2	no	—
87	C₄²×C₁₀₉	—	4	1	no	—
88	C₄²×C₁₀₉	—	4	1	no	—
89	C₂³×C₁₀₉	—	2	2	no	—
90	C₂³×C₁₀₉	—	2	2	no	—
91	C₂³×C₁₀₉	C₄×Q₈	1	1	yes	characteristic
92	C₂³×C₁₀₉	#121	1	1	yes	characteristic
93	C₂³×C₁₀₉	#123	1	2	yes	—
94	C₂³×C₁₀₉	#125	1	1	yes	characteristic
95	C₂³×C₁₀₉	#126	1	2	yes	—
96	C₂×C₄×C₁₀₉	—	4	4	no	—
97	C₂×C₄×C₁₀₉	—	4	2	no	—
98	C₂×C₄×C₁₀₉	—	4	2	no	—
99	C₂×C₄×C₁₀₉	—	4	2	no	—
100	C₂×C₄×C₁₀₉	—	8	4	no	—
101	C₂×C₄×C₁₀₉	—	8	4	no	—
102	C₂×C₄×C₁₀₉	—	8	4	no	—
103	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₁₀₉	#117	1	2	yes	—
105	C₂×C₄×C₁₀₉	C₄×D₄	1	2	yes	—
106	C₂×C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₁₀₉	#120	1	1	yes	characteristic
108	C₂×C₄×C₁₀₉	#121	1	2	yes	—
109	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
110	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
111	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
112	C₂×C₄×C₁₀₉	#123	1	1	yes	characteristic
113	C₂×C₄×C₁₀₉	#123	1	2	yes	—
114	C₂×C₄×C₁₀₉	#124	1	1	yes	characteristic
115	C₂×C₄×C₁₀₉	#124	1	1	yes	characteristic
116	C₂×C₄×C₁₀₉	#124	1	2	yes	—
117	C₂×C₄×C₁₀₉	#125	1	2	yes	—
118	C₂×C₄×C₁₀₉	#125	1	2	yes	—
119	C₂×C₄×C₁₀₉	#126	1	2	yes	—
120	C₂×C₄×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
121	C₈×C₁₀₉	—	2	4	no	—
122	C₈×C₁₀₉	—	4	4	no	—
123	C₈×C₁₀₉	—	4	2	no	—
124	C₂²×C₁₀₉	—	2	4	no	—
125	C₂²×C₁₀₉	—	2	2	no	—
126	C₂²×C₁₀₉	—	2	2	no	—
127	C₂²×C₁₀₉	—	2	2	no	—
128	C₂²×C₁₀₉	—	2	2	no	—
129	C₂²×C₁₀₉	#387	1	2	yes	—
130	C₂²×C₁₀₉	#389	1	1	yes	characteristic
131	C₂²×C₁₀₉	#394	1	1	yes	characteristic
132	C₂²×C₁₀₉	#397	1	1	yes	characteristic
133	C₂²×C₁₀₉	#400	1	2	yes	—
134	C₂²×C₁₀₉	#472	1	2	yes	—
135	C₂²×C₁₀₉	#473	1	2	yes	—
136	C₄×C₁₀₉	—	8	2	no	—
137	C₄×C₁₀₉	—	8	4	no	—
138	C₄×C₁₀₉	#382	1	1	yes	characteristic
139	C₄×C₁₀₉	#385	1	1	yes	characteristic
140	C₄×C₁₀₉	#388	1	2	yes	—
141	C₄×C₁₀₉	#393	1	1	yes	characteristic
142	C₄×C₁₀₉	#398	1	2	yes	—
143	C₄×C₁₀₉	#400	1	1	yes	characteristic
144	#11270	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
145	C₂×C₁₀₉	—	2	2	no	—
146	C₂×C₁₀₉	—	2	2	no	—
147	C₂×C₁₀₉	#1442	1	1	yes	characteristic
148	C₂×C₁₀₉	#1952	1	2	yes	—
149	C₂×C₁₀₉	#2045	1	1	yes	characteristic
150	C₂×C₁₀₉	#2047	1	1	yes	characteristic
151	C₂×C₁₀₉	#2078	1	2	yes	—
152	C₂²×#107	C₂×C₁₀₉	1	1	yes	characteristic
153	C₂×#343	C₂×C₁₀₉	1	2	yes	—
154	C₂×#341	C₂×C₁₀₉	1	1	yes	characteristic
155	#1736	C₂×C₁₀₉	1	2	yes	—
156	#1729	C₂×C₁₀₉	1	1	yes	characteristic
157	C₁₀₉	#11270	1	1	yes	characteristic, direct product factor, Sylow
158	C₂²×M₁₆	C₂²×C₁₀₉	1	2	yes	—
159	C₂²×M₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
160	C₂³×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
161	C₂×#117	C₂²×C₁₀₉	1	1	yes	characteristic
162	C₂²×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
163	C₂×#107	—	2	2	no	—
164	C₂×#107	—	2	4	no	—
165	#343	—	2	4	no	—
166	#341	C₄×C₁₀₉	1	4	yes	complemented
167	#340	—	2	4	no	—
168	#336	—	2	2	no	—
169	C₂³×C₄	—	2	2	no	—
170	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	1	no	—
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	C₂×C₄×C₁₀₉	1	4	yes	—
180	C₂×M₁₆	Q₈×C₁₀₉	1	2	yes	—
181	C₂²×C₈	—	2	2	no	—
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	—	2	1	no	—
184	C₂²×C₈	—	2	2	no	—
185	C₂²×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
186	C₂²×C₈	D₄×C₁₀₉	1	2	yes	—
187	C₂²×C₈	Q₈×C₁₀₉	1	2	yes	—
188	#117	—	2	2	no	—
189	#117	—	2	2	no	—
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₄⋊C₄	—	2	2	no	—
192	C₂×C₄⋊C₄	—	2	2	no	—
193	C₂×C₄⋊C₄	—	2	2	no	—
194	C₂×C₄⋊C₄	—	2	2	no	—
195	C₂×C₄⋊C₄	D₄×C₁₀₉	1	4	yes	—
196	C₂×C₄⋊C₄	D₄×C₁₀₉	1	2	yes	—
197	C₂×C₂²⋊C₄	—	2	1	no	—
198	C₂×C₄²	—	2	1	no	—
199	#107	—	2	8	no	—
200	C₂⁴	#41×C₁₀₉	1	1	yes	characteristic
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	—	4	2	no	—
204	C₂²×C₄	—	4	2	no	—
205	C₂²×C₄	—	8	4	no	—
206	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
207	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
208	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
209	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
210	C₂²×C₄	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
211	C₂²×C₄	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
212	C₂²×C₄	#41×C₁₀₉	1	2	yes	—
213	C₂²×C₄	#41×C₁₀₉	1	2	yes	—
214	C₂²×C₄	#41×C₁₀₉	1	2	yes	—
215	C₂²×C₄	#41×C₁₀₉	1	1	yes	characteristic
216	M₁₆	—	2	4	no	—
217	M₁₆	—	2	4	no	—
218	M₁₆	—	2	4	no	—
219	M₁₆	—	4	2	no	—
220	M₁₆	—	4	2	no	—
221	M₁₆	—	4	2	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	4	1	no	—
229	C₂×C₈	—	4	2	no	—
230	C₂×C₈	—	4	2	no	—
231	C₂×C₈	—	4	1	no	—
232	C₂×C₈	C₄⋊C₄×C₁₀₉	1	4	yes	—
233	C₄⋊C₄	—	2	4	no	—
234	C₄⋊C₄	—	2	4	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	4	2	no	—
237	C₄⋊C₄	—	4	2	no	—
238	C₂²⋊C₄	—	4	2	no	—
239	C₄²	—	4	1	no	—
240	C₄²	—	4	1	no	—
241	C₂³	—	2	2	no	—
242	C₂³	—	2	2	no	—
243	C₂³	C₄×Q₈×C₁₀₉	1	1	yes	characteristic
244	C₂³	#121×C₁₀₉	1	1	yes	characteristic
245	C₂³	#123×C₁₀₉	1	2	yes	—
246	C₂³	#125×C₁₀₉	1	1	yes	characteristic
247	C₂³	#126×C₁₀₉	1	2	yes	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	8	4	no	—
253	C₂×C₄	—	8	4	no	—
254	C₂×C₄	—	8	4	no	—
255	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
256	C₂×C₄	#117×C₁₀₉	1	2	yes	—
257	C₂×C₄	C₄×D₄×C₁₀₉	1	2	yes	—
258	C₂×C₄	C₄×D₄×C₁₀₉	1	1	yes	characteristic
259	C₂×C₄	#120×C₁₀₉	1	1	yes	characteristic
260	C₂×C₄	#121×C₁₀₉	1	2	yes	—
261	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	2	yes	—
262	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	2	yes	—
263	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	1	yes	characteristic
264	C₂×C₄	#123×C₁₀₉	1	1	yes	characteristic
265	C₂×C₄	#123×C₁₀₉	1	2	yes	—
266	C₂×C₄	#124×C₁₀₉	1	1	yes	characteristic
267	C₂×C₄	#124×C₁₀₉	1	1	yes	characteristic
268	C₂×C₄	#124×C₁₀₉	1	2	yes	—
269	C₂×C₄	#125×C₁₀₉	1	2	yes	—
270	C₂×C₄	#125×C₁₀₉	1	2	yes	—
271	C₂×C₄	#126×C₁₀₉	1	2	yes	—
272	C₂×C₄	C₄⋊Q₈×C₁₀₉	1	1	yes	characteristic
273	C₈	—	2	4	no	—
274	C₈	—	4	4	no	—
275	C₈	—	4	2	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	2	4	no	—
281	C₂²	#387×C₁₀₉	1	2	yes	—
282	C₂²	#389×C₁₀₉	1	1	yes	characteristic
283	C₂²	#394×C₁₀₉	1	1	yes	characteristic
284	C₂²	#397×C₁₀₉	1	1	yes	characteristic
285	C₂²	#400×C₁₀₉	1	2	yes	—
286	C₂²	#472×C₁₀₉	1	2	yes	—
287	C₂²	#473×C₁₀₉	1	2	yes	—
288	C₄	—	8	4	no	—
289	C₄	—	8	2	no	—
290	C₄	#382×C₁₀₉	1	1	yes	characteristic
291	C₄	#385×C₁₀₉	1	1	yes	characteristic
292	C₄	#388×C₁₀₉	1	2	yes	—
293	C₄	#393×C₁₀₉	1	1	yes	characteristic
294	C₄	#398×C₁₀₉	1	2	yes	—
295	C₄	#400×C₁₀₉	1	1	yes	characteristic
296	C₂	—	2	2	no	—
297	C₂	—	2	2	no	—
298	C₂	C₁₀₉×#1442	1	1	yes	characteristic
299	C₂	C₁₀₉×#1952	1	2	yes	—
300	C₂	C₁₀₉×#2045	1	1	yes	characteristic
301	C₂	C₁₀₉×#2047	1	1	yes	characteristic
302	C₂	C₁₀₉×#2078	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₀₉	#121
commutator	C₂²×C₄	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂³×C₄	C₂³×C₁₀₉
prime radical	C₂⁴×C₁₀₉	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₉ ◁ C₂³×C₄×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
14. #341×C₁₀₉	C₄	semidirect
144. #11270	C₁₀₉	direct (N × H)
157. C₁₀₉	#11270	direct (N × H)
166. #341	C₄×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#107×C₁₀₉	2
C₂×#343×C₁₀₉	2
C₂×#341×C₁₀₉	2
C₁₀₉×#1736	2
C₁₀₉×#1729	2
#11270	109

Subgroups — C109×#11270 #42129264

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#11270 #42129264