Subgroups — C₁₀₉×#11285 #42129279

Order 27904.

606 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₉×#2911	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#1740	C₂	1	2	yes	maximal
3	C₁₀₉×#1735	C₂	1	2	yes	maximal
4	C₂×H(3,ℤ/4)×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₁₀₉×#1729	C₂	1	1	yes	maximal, characteristic
6	C₂²×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
8	C₂×#117×C₁₀₉	C₂²	1	2	yes	—
9	C₂×#117×C₁₀₉	C₂²	1	2	yes	—
10	C₂×#117×C₁₀₉	C₂²	1	1	yes	characteristic
11	#428×C₁₀₉	—	2	4	no	—
12	#427×C₁₀₉	—	2	2	no	—
13	#342×C₁₀₉	—	2	4	no	—
14	#340×C₁₀₉	—	2	4	no	—
15	H(3,ℤ/4)×C₁₀₉	C₄	1	4	yes	complemented
16	#336×C₁₀₉	—	2	2	no	—
17	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
18	C₂×M₁₆×C₁₀₉	—	2	2	no	—
19	C₂×M₁₆×C₁₀₉	—	2	2	no	—
20	C₂×M₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
21	C₂×M₁₆×C₁₀₉	Q₈	1	2	yes	—
22	C₂²×C₈×C₁₀₉	—	2	1	no	—
23	C₂²×C₈×C₁₀₉	—	2	2	no	—
24	C₂²×C₈×C₁₀₉	—	2	2	no	—
25	C₂²×C₈×C₁₀₉	—	2	1	no	—
26	C₂²×C₈×C₁₀₉	D₄	1	2	yes	—
27	C₂²×C₈×C₁₀₉	Q₈	1	2	yes	—
28	#117×C₁₀₉	—	2	2	no	—
29	#117×C₁₀₉	—	2	2	no	—
30	#117×C₁₀₉	—	2	2	no	—
31	#117×C₁₀₉	—	2	2	no	—
32	#117×C₁₀₉	—	2	2	no	—
33	#117×C₁₀₉	—	2	2	no	—
34	#117×C₁₀₉	—	2	2	no	—
35	#117×C₁₀₉	—	2	2	no	—
36	#117×C₁₀₉	—	2	2	no	—
37	#117×C₁₀₉	C₂×C₄	1	4	yes	—
38	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
39	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
40	C₂×C₄⋊C₄×C₁₀₉	D₄	1	4	yes	—
41	C₂×C₂²⋊C₄×C₁₀₉	—	2	2	no	—
42	C₂×C₂²⋊C₄×C₁₀₉	—	2	2	no	—
43	C₂×C₂²⋊C₄×C₁₀₉	D₄	1	2	yes	—
44	C₂×C₄²×C₁₀₉	—	2	2	no	—
45	C₂×C₄²×C₁₀₉	—	2	1	no	—
46	C₂×C₄²×C₁₀₉	—	2	2	no	—
47	#107×C₁₀₉	—	2	4	no	—
48	#106×C₁₀₉	—	2	4	no	—
49	C₂⁴×C₁₀₉	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₀₉	—	2	2	no	—
51	C₂²×C₄×C₁₀₉	—	2	4	no	—
52	C₂²×C₄×C₁₀₉	—	4	2	no	—
53	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
57	C₂²×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
58	C₂²×C₄×C₁₀₉	#41	1	2	yes	—
59	C₂²×C₄×C₁₀₉	#41	1	2	yes	—
60	C₂²×C₄×C₁₀₉	#41	1	2	yes	—
61	C₂²×C₄×C₁₀₉	#41	1	1	yes	characteristic
62	C₂²×C₄×C₁₀₉	#41	1	1	yes	characteristic
63	M₁₆×C₁₀₉	—	2	4	no	—
64	M₁₆×C₁₀₉	—	2	4	no	—
65	C₂×C₈×C₁₀₉	—	2	2	no	—
66	C₂×C₈×C₁₀₉	—	2	2	no	—
67	C₂×C₈×C₁₀₉	—	2	4	no	—
68	C₂×C₈×C₁₀₉	—	2	2	no	—
69	C₂×C₈×C₁₀₉	—	4	1	no	—
70	C₂×C₈×C₁₀₉	—	4	1	no	—
71	C₂×C₈×C₁₀₉	—	4	2	no	—
72	C₂×C₈×C₁₀₉	C₄⋊C₄	1	4	yes	—
73	C₄⋊C₄×C₁₀₉	—	2	4	no	—
74	C₄⋊C₄×C₁₀₉	—	2	4	no	—
75	C₄⋊C₄×C₁₀₉	—	2	4	no	—
76	C₄⋊C₄×C₁₀₉	—	4	2	no	—
77	C₄⋊C₄×C₁₀₉	—	4	2	no	—
78	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
79	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
80	C₂²⋊C₄×C₁₀₉	—	4	2	no	—
81	C₂²⋊C₄×C₁₀₉	—	4	2	no	—
82	C₂²⋊C₄×C₁₀₉	—	4	2	no	—
83	C₄²×C₁₀₉	—	2	4	no	—
84	C₄²×C₁₀₉	—	4	2	no	—
85	C₄²×C₁₀₉	—	4	1	no	—
86	C₄²×C₁₀₉	—	4	1	no	—
87	C₄²×C₁₀₉	—	4	2	no	—
88	C₄²×C₁₀₉	—	4	2	no	—
89	C₂³×C₁₀₉	—	4	2	no	—
90	C₂³×C₁₀₉	C₄×D₄	1	1	yes	characteristic
91	C₂³×C₁₀₉	#121	1	2	yes	—
92	C₂³×C₁₀₉	#121	1	1	yes	characteristic
93	C₂³×C₁₀₉	#123	1	2	yes	—
94	C₂³×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
95	C₂×C₄×C₁₀₉	—	4	4	no	—
96	C₂×C₄×C₁₀₉	—	4	4	no	—
97	C₂×C₄×C₁₀₉	—	4	4	no	—
98	C₂×C₄×C₁₀₉	—	4	2	no	—
99	C₂×C₄×C₁₀₉	—	8	4	no	—
100	C₂×C₄×C₁₀₉	—	8	4	no	—
101	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₁₀₉	#117	1	2	yes	—
103	C₂×C₄×C₁₀₉	C₄×D₄	1	2	yes	—
104	C₂×C₄×C₁₀₉	C₄×Q₈	1	1	yes	characteristic
105	C₂×C₄×C₁₀₉	#120	1	1	yes	characteristic
106	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
107	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
108	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
109	C₂×C₄×C₁₀₉	#123	1	1	yes	characteristic
110	C₂×C₄×C₁₀₉	#123	1	2	yes	—
111	C₂×C₄×C₁₀₉	#124	1	1	yes	characteristic
112	C₂×C₄×C₁₀₉	#124	1	2	yes	—
113	C₂×C₄×C₁₀₉	#124	1	1	yes	characteristic
114	C₂×C₄×C₁₀₉	#125	1	2	yes	—
115	C₂×C₄×C₁₀₉	#125	1	2	yes	—
116	C₂×C₄×C₁₀₉	#125	1	1	yes	characteristic
117	C₂×C₄×C₁₀₉	#126	1	2	yes	—
118	C₂×C₄×C₁₀₉	#126	1	2	yes	—
119	C₈×C₁₀₉	—	2	4	no	—
120	C₈×C₁₀₉	—	4	2	no	—
121	C₂²×C₁₀₉	—	2	2	no	—
122	C₂²×C₁₀₉	—	2	4	no	—
123	C₂²×C₁₀₉	—	4	1	no	—
124	C₂²×C₁₀₉	—	4	2	no	—
125	C₂²×C₁₀₉	—	4	1	no	—
126	C₂²×C₁₀₉	#388	1	2	yes	—
127	C₂²×C₁₀₉	#389	1	1	yes	characteristic
128	C₂²×C₁₀₉	#394	1	1	yes	characteristic
129	C₂²×C₁₀₉	#397	1	1	yes	characteristic
130	C₂²×C₁₀₉	#398	1	2	yes	—
131	C₄×C₁₀₉	—	8	2	no	—
132	C₄×C₁₀₉	—	8	4	no	—
133	C₄×C₁₀₉	—	8	4	no	—
134	C₄×C₁₀₉	#382	1	1	yes	characteristic
135	C₄×C₁₀₉	#385	1	1	yes	characteristic
136	C₄×C₁₀₉	#387	1	2	yes	—
137	C₄×C₁₀₉	#393	1	1	yes	characteristic
138	C₄×C₁₀₉	#400	1	2	yes	—
139	C₄×C₁₀₉	#400	1	1	yes	characteristic
140	#11285	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
141	C₂×C₁₀₉	—	2	2	no	—
142	C₂×C₁₀₉	—	2	1	no	—
143	C₂×C₁₀₉	—	2	1	no	—
144	C₂×C₁₀₉	—	2	2	no	—
145	C₂×C₁₀₉	#1442	1	1	yes	characteristic
146	C₂×C₁₀₉	#2053	1	2	yes	—
147	#2911	C₂×C₁₀₉	1	1	yes	characteristic
148	#1740	C₂×C₁₀₉	1	2	yes	—
149	#1735	C₂×C₁₀₉	1	2	yes	—
150	C₂×H(3,ℤ/4)	C₂×C₁₀₉	1	1	yes	characteristic
151	#1729	C₂×C₁₀₉	1	1	yes	characteristic
152	C₁₀₉	#11285	1	1	yes	characteristic, direct product factor, Sylow
153	C₂²×M₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
154	C₂³×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
155	C₂×#117	C₂²×C₁₀₉	1	2	yes	—
156	C₂×#117	C₂²×C₁₀₉	1	2	yes	—
157	C₂×#117	C₂²×C₁₀₉	1	1	yes	characteristic
158	#428	—	2	4	no	—
159	#427	—	2	2	no	—
160	#342	—	2	4	no	—
161	#340	—	2	4	no	—
162	H(3,ℤ/4)	C₄×C₁₀₉	1	4	yes	complemented
163	#336	—	2	2	no	—
164	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
165	C₂×M₁₆	—	2	2	no	—
166	C₂×M₁₆	—	2	2	no	—
167	C₂×M₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
168	C₂×M₁₆	Q₈×C₁₀₉	1	2	yes	—
169	C₂²×C₈	—	2	1	no	—
170	C₂²×C₈	—	2	2	no	—
171	C₂²×C₈	—	2	2	no	—
172	C₂²×C₈	—	2	1	no	—
173	C₂²×C₈	D₄×C₁₀₉	1	2	yes	—
174	C₂²×C₈	Q₈×C₁₀₉	1	2	yes	—
175	#117	—	2	2	no	—
176	#117	—	2	2	no	—
177	#117	—	2	2	no	—
178	#117	—	2	2	no	—
179	#117	—	2	2	no	—
180	#117	—	2	2	no	—
181	#117	—	2	2	no	—
182	#117	—	2	2	no	—
183	#117	—	2	2	no	—
184	#117	C₂×C₄×C₁₀₉	1	4	yes	—
185	C₂×C₄⋊C₄	—	2	2	no	—
186	C₂×C₄⋊C₄	—	2	1	no	—
187	C₂×C₄⋊C₄	D₄×C₁₀₉	1	4	yes	—
188	C₂×C₂²⋊C₄	—	2	2	no	—
189	C₂×C₂²⋊C₄	—	2	2	no	—
190	C₂×C₂²⋊C₄	D₄×C₁₀₉	1	2	yes	—
191	C₂×C₄²	—	2	2	no	—
192	C₂×C₄²	—	2	1	no	—
193	C₂×C₄²	—	2	2	no	—
194	#107	—	2	4	no	—
195	#106	—	2	4	no	—
196	C₂⁴	C₂×D₄×C₁₀₉	1	1	yes	characteristic
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	—	4	2	no	—
200	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
201	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
202	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
203	C₂²×C₄	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
204	C₂²×C₄	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
205	C₂²×C₄	#41×C₁₀₉	1	2	yes	—
206	C₂²×C₄	#41×C₁₀₉	1	2	yes	—
207	C₂²×C₄	#41×C₁₀₉	1	2	yes	—
208	C₂²×C₄	#41×C₁₀₉	1	1	yes	characteristic
209	C₂²×C₄	#41×C₁₀₉	1	1	yes	characteristic
210	M₁₆	—	2	4	no	—
211	M₁₆	—	2	4	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	4	1	no	—
217	C₂×C₈	—	4	1	no	—
218	C₂×C₈	—	4	2	no	—
219	C₂×C₈	C₄⋊C₄×C₁₀₉	1	4	yes	—
220	C₄⋊C₄	—	2	4	no	—
221	C₄⋊C₄	—	2	4	no	—
222	C₄⋊C₄	—	2	4	no	—
223	C₄⋊C₄	—	4	2	no	—
224	C₄⋊C₄	—	4	2	no	—
225	C₂²⋊C₄	—	2	4	no	—
226	C₂²⋊C₄	—	2	4	no	—
227	C₂²⋊C₄	—	4	2	no	—
228	C₂²⋊C₄	—	4	2	no	—
229	C₂²⋊C₄	—	4	2	no	—
230	C₄²	—	2	4	no	—
231	C₄²	—	4	2	no	—
232	C₄²	—	4	1	no	—
233	C₄²	—	4	2	no	—
234	C₄²	—	4	1	no	—
235	C₄²	—	4	2	no	—
236	C₂³	—	4	2	no	—
237	C₂³	C₄×D₄×C₁₀₉	1	1	yes	characteristic
238	C₂³	#121×C₁₀₉	1	2	yes	—
239	C₂³	#121×C₁₀₉	1	1	yes	characteristic
240	C₂³	#123×C₁₀₉	1	2	yes	—
241	C₂³	C₄⋊Q₈×C₁₀₉	1	1	yes	characteristic
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	8	4	no	—
247	C₂×C₄	—	8	4	no	—
248	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
249	C₂×C₄	#117×C₁₀₉	1	2	yes	—
250	C₂×C₄	C₄×D₄×C₁₀₉	1	2	yes	—
251	C₂×C₄	C₄×Q₈×C₁₀₉	1	1	yes	characteristic
252	C₂×C₄	#120×C₁₀₉	1	1	yes	characteristic
253	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	2	yes	—
254	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	2	yes	—
255	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	1	yes	characteristic
256	C₂×C₄	#123×C₁₀₉	1	1	yes	characteristic
257	C₂×C₄	#123×C₁₀₉	1	2	yes	—
258	C₂×C₄	#124×C₁₀₉	1	1	yes	characteristic
259	C₂×C₄	#124×C₁₀₉	1	1	yes	characteristic
260	C₂×C₄	#124×C₁₀₉	1	2	yes	—
261	C₂×C₄	#125×C₁₀₉	1	2	yes	—
262	C₂×C₄	#125×C₁₀₉	1	2	yes	—
263	C₂×C₄	#125×C₁₀₉	1	1	yes	characteristic
264	C₂×C₄	#126×C₁₀₉	1	2	yes	—
265	C₂×C₄	#126×C₁₀₉	1	2	yes	—
266	C₈	—	2	4	no	—
267	C₈	—	4	2	no	—
268	C₂²	—	2	2	no	—
269	C₂²	—	2	4	no	—
270	C₂²	—	4	1	no	—
271	C₂²	—	4	2	no	—
272	C₂²	—	4	1	no	—
273	C₂²	#388×C₁₀₉	1	2	yes	—
274	C₂²	#389×C₁₀₉	1	1	yes	characteristic
275	C₂²	#394×C₁₀₉	1	1	yes	characteristic
276	C₂²	#397×C₁₀₉	1	1	yes	characteristic
277	C₂²	#398×C₁₀₉	1	2	yes	—
278	C₄	—	8	2	no	—
279	C₄	—	8	4	no	—
280	C₄	—	8	4	no	—
281	C₄	#382×C₁₀₉	1	1	yes	characteristic
282	C₄	#385×C₁₀₉	1	1	yes	characteristic
283	C₄	#387×C₁₀₉	1	2	yes	—
284	C₄	#393×C₁₀₉	1	1	yes	characteristic
285	C₄	#400×C₁₀₉	1	2	yes	—
286	C₄	#400×C₁₀₉	1	1	yes	characteristic
287	C₂	—	2	2	no	—
288	C₂	—	2	1	no	—
289	C₂	—	2	1	no	—
290	C₂	—	2	2	no	—
291	C₂	C₁₀₉×#1442	1	1	yes	characteristic
292	C₂	C₁₀₉×#2053	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₀₉	#124
commutator	C₂²×C₄	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂³×C₄	C₂³×C₁₀₉
socle	C₂²×C₁₀₉	#394
prime radical	C₂⁴×C₁₀₉	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₉ ◁ C₂³×C₄×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
15. H(3,ℤ/4)×C₁₀₉	C₄	semidirect
140. #11285	C₁₀₉	direct (N × H)
152. C₁₀₉	#11285	direct (N × H)
162. H(3,ℤ/4)	C₄×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₁₀₉×#2911	2
C₁₀₉×#1740	2
C₁₀₉×#1735	2
C₂×H(3,ℤ/4)×C₁₀₉	2
C₁₀₉×#1729	2
#11285	109

Subgroups — C109×#11285 #42129279

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#11285 #42129279