Subgroups — C₁₀₉×#11272 #42129266

Order 27904.

574 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#108×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#344×C₁₀₉	C₂	1	2	yes	maximal
3	C₁₀₉×#1738	C₂	1	2	yes	maximal
4	C₂×#341×C₁₀₉	C₂	1	1	yes	maximal, characteristic
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	C₂²×M₁₆×C₁₀₉	C₂²	1	2	yes	—
9	C₂³×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
10	C₂×#117×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₂×#108×C₁₀₉	—	2	4	no	—
12	C₂×#108×C₁₀₉	—	2	2	no	—
13	#344×C₁₀₉	—	2	4	no	—
14	#341×C₁₀₉	—	2	4	no	—
15	#341×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	1	no	—
20	C₂×M₁₆×C₁₀₉	—	2	2	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	—	2	2	no	—
28	C₂×M₁₆×C₁₀₉	—	2	2	no	—
29	C₂×M₁₆×C₁₀₉	—	2	2	no	—
30	C₂×M₁₆×C₁₀₉	—	2	2	no	—
31	C₂×M₁₆×C₁₀₉	C₂×C₄	1	4	yes	—
32	C₂×M₁₆×C₁₀₉	D₄	1	2	yes	—
33	C₂×M₁₆×C₁₀₉	D₄	1	4	yes	—
34	C₂²×C₈×C₁₀₉	—	2	2	no	—
35	C₂²×C₈×C₁₀₉	—	2	2	no	—
36	C₂²×C₈×C₁₀₉	—	2	1	no	—
37	C₂²×C₈×C₁₀₉	—	2	2	no	—
38	C₂²×C₈×C₁₀₉	—	2	2	no	—
39	C₂²×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
40	C₂²×C₈×C₁₀₉	D₄	1	2	yes	—
41	C₂²×C₈×C₁₀₉	Q₈	1	2	yes	—
42	#117×C₁₀₉	—	2	2	no	—
43	#117×C₁₀₉	—	2	2	no	—
44	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
45	C₂×C₂²⋊C₄×C₁₀₉	Q₈	1	2	yes	—
46	C₂×C₄²×C₁₀₉	—	2	1	no	—
47	#108×C₁₀₉	—	2	8	no	—
48	C₂⁴×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
49	C₂²×C₄×C₁₀₉	—	2	2	no	—
50	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
53	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
55	C₂²×C₄×C₁₀₉	#41	1	2	yes	—
56	C₂²×C₄×C₁₀₉	#41	1	2	yes	—
57	C₂²×C₄×C₁₀₉	#41	1	2	yes	—
58	C₂²×C₄×C₁₀₉	#41	1	1	yes	characteristic
59	C₂²×C₄×C₁₀₉	#41	1	1	yes	characteristic
60	M₁₆×C₁₀₉	—	2	4	no	—
61	M₁₆×C₁₀₉	—	2	4	no	—
62	M₁₆×C₁₀₉	—	2	4	no	—
63	M₁₆×C₁₀₉	—	2	4	no	—
64	M₁₆×C₁₀₉	—	2	4	no	—
65	M₁₆×C₁₀₉	—	4	2	no	—
66	M₁₆×C₁₀₉	—	4	2	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	—	2	4	no	—
77	C₂×C₈×C₁₀₉	—	4	2	no	—
78	C₂×C₈×C₁₀₉	—	4	2	no	—
79	C₂×C₈×C₁₀₉	—	4	2	no	—
80	C₂×C₈×C₁₀₉	—	4	2	no	—
81	C₂×C₈×C₁₀₉	—	4	1	no	—
82	C₂×C₈×C₁₀₉	—	4	1	no	—
83	C₂×C₈×C₁₀₉	C₄⋊C₄	1	4	yes	—
84	C₄⋊C₄×C₁₀₉	—	4	2	no	—
85	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
86	C₄²×C₁₀₉	—	4	1	no	—
87	C₄²×C₁₀₉	—	4	1	no	—
88	C₂³×C₁₀₉	—	4	2	no	—
89	C₂³×C₁₀₉	C₄×Q₈	1	1	yes	characteristic
90	C₂³×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
91	C₂³×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
92	C₂³×C₁₀₉	#125	1	2	yes	—
93	C₂³×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
94	C₂×C₄×C₁₀₉	—	4	4	no	—
95	C₂×C₄×C₁₀₉	—	4	2	no	—
96	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₁₀₉	#117	1	2	yes	—
98	C₂×C₄×C₁₀₉	C₄×D₄	1	2	yes	—
99	C₂×C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
100	C₂×C₄×C₁₀₉	#120	1	1	yes	characteristic
101	C₂×C₄×C₁₀₉	#121	1	2	yes	—
102	C₂×C₄×C₁₀₉	#121	1	1	yes	characteristic
103	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
104	C₂×C₄×C₁₀₉	#123	1	1	yes	characteristic
105	C₂×C₄×C₁₀₉	#123	1	2	yes	—
106	C₂×C₄×C₁₀₉	#123	1	2	yes	—
107	C₂×C₄×C₁₀₉	#124	1	1	yes	characteristic
108	C₂×C₄×C₁₀₉	#124	1	1	yes	characteristic
109	C₂×C₄×C₁₀₉	#124	1	2	yes	—
110	C₂×C₄×C₁₀₉	#125	1	2	yes	—
111	C₂×C₄×C₁₀₉	#125	1	1	yes	characteristic
112	C₂×C₄×C₁₀₉	#126	1	2	yes	—
113	C₂×C₄×C₁₀₉	#126	1	2	yes	—
114	C₈×C₁₀₉	—	2	4	no	—
115	C₈×C₁₀₉	—	4	4	no	—
116	C₈×C₁₀₉	—	4	4	no	—
117	C₈×C₁₀₉	—	4	2	no	—
118	C₂²×C₁₀₉	—	2	2	no	—
119	C₂²×C₁₀₉	—	2	2	no	—
120	C₂²×C₁₀₉	—	2	2	no	—
121	C₂²×C₁₀₉	—	4	2	no	—
122	C₂²×C₁₀₉	—	4	2	no	—
123	C₂²×C₁₀₉	#387	1	2	yes	—
124	C₂²×C₁₀₉	#389	1	1	yes	characteristic
125	C₂²×C₁₀₉	#393	1	1	yes	characteristic
126	C₂²×C₁₀₉	#398	1	2	yes	—
127	C₂²×C₁₀₉	#400	1	1	yes	characteristic
128	C₄×C₁₀₉	—	8	2	no	—
129	C₄×C₁₀₉	#382	1	1	yes	characteristic
130	C₄×C₁₀₉	#385	1	1	yes	characteristic
131	C₄×C₁₀₉	#388	1	2	yes	—
132	C₄×C₁₀₉	#394	1	1	yes	characteristic
133	C₄×C₁₀₉	#397	1	1	yes	characteristic
134	C₄×C₁₀₉	#400	1	2	yes	—
135	#11272	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
136	C₂×C₁₀₉	—	2	1	no	—
137	C₂×C₁₀₉	—	2	2	no	—
138	C₂×C₁₀₉	—	2	2	no	—
139	C₂×C₁₀₉	—	2	1	no	—
140	C₂×C₁₀₉	#1442	1	1	yes	characteristic
141	C₂×C₁₀₉	#1953	1	2	yes	—
142	C₂²×#108	C₂×C₁₀₉	1	1	yes	characteristic
143	C₂×#344	C₂×C₁₀₉	1	2	yes	—
144	#1738	C₂×C₁₀₉	1	2	yes	—
145	C₂×#341	C₂×C₁₀₉	1	1	yes	characteristic
146	#1729	C₂×C₁₀₉	1	1	yes	characteristic
147	C₁₀₉	#11272	1	1	yes	characteristic, direct product factor, Sylow
148	C₂²×M₁₆	C₂²×C₁₀₉	1	2	yes	—
149	C₂²×M₁₆	C₂²×C₁₀₉	1	2	yes	—
150	C₂²×M₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
151	C₂³×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
152	C₂×#117	C₂²×C₁₀₉	1	1	yes	characteristic
153	C₂×#108	—	2	4	no	—
154	C₂×#108	—	2	2	no	—
155	#344	—	2	4	no	—
156	#341	—	2	4	no	—
157	#341	C₄×C₁₀₉	1	4	yes	complemented
158	#336	—	2	2	no	—
159	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
160	C₂×M₁₆	—	2	2	no	—
161	C₂×M₁₆	—	2	1	no	—
162	C₂×M₁₆	—	2	2	no	—
163	C₂×M₁₆	—	2	2	no	—
164	C₂×M₁₆	—	2	2	no	—
165	C₂×M₁₆	—	2	2	no	—
166	C₂×M₁₆	—	2	2	no	—
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	C₂×C₄×C₁₀₉	1	4	yes	—
174	C₂×M₁₆	D₄×C₁₀₉	1	2	yes	—
175	C₂×M₁₆	D₄×C₁₀₉	1	4	yes	—
176	C₂²×C₈	—	2	2	no	—
177	C₂²×C₈	—	2	2	no	—
178	C₂²×C₈	—	2	1	no	—
179	C₂²×C₈	—	2	2	no	—
180	C₂²×C₈	—	2	2	no	—
181	C₂²×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
182	C₂²×C₈	D₄×C₁₀₉	1	2	yes	—
183	C₂²×C₈	Q₈×C₁₀₉	1	2	yes	—
184	#117	—	2	2	no	—
185	#117	—	2	2	no	—
186	C₂×C₄⋊C₄	—	2	1	no	—
187	C₂×C₂²⋊C₄	Q₈×C₁₀₉	1	2	yes	—
188	C₂×C₄²	—	2	1	no	—
189	#108	—	2	8	no	—
190	C₂⁴	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
193	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
194	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
195	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
196	C₂²×C₄	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
197	C₂²×C₄	#41×C₁₀₉	1	2	yes	—
198	C₂²×C₄	#41×C₁₀₉	1	2	yes	—
199	C₂²×C₄	#41×C₁₀₉	1	2	yes	—
200	C₂²×C₄	#41×C₁₀₉	1	1	yes	characteristic
201	C₂²×C₄	#41×C₁₀₉	1	1	yes	characteristic
202	M₁₆	—	2	4	no	—
203	M₁₆	—	2	4	no	—
204	M₁₆	—	2	4	no	—
205	M₁₆	—	2	4	no	—
206	M₁₆	—	2	4	no	—
207	M₁₆	—	4	2	no	—
208	M₁₆	—	4	2	no	—
209	M₁₆	—	4	2	no	—
210	M₁₆	—	4	2	no	—
211	M₁₆	—	4	2	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	4	2	no	—
220	C₂×C₈	—	4	2	no	—
221	C₂×C₈	—	4	2	no	—
222	C₂×C₈	—	4	2	no	—
223	C₂×C₈	—	4	1	no	—
224	C₂×C₈	—	4	1	no	—
225	C₂×C₈	C₄⋊C₄×C₁₀₉	1	4	yes	—
226	C₄⋊C₄	—	4	2	no	—
227	C₂²⋊C₄	—	2	4	no	—
228	C₄²	—	4	1	no	—
229	C₄²	—	4	1	no	—
230	C₂³	—	4	2	no	—
231	C₂³	C₄×Q₈×C₁₀₉	1	1	yes	characteristic
232	C₂³	C₂²⋊Q₈×C₁₀₉	1	1	yes	characteristic
233	C₂³	C₂²⋊Q₈×C₁₀₉	1	2	yes	—
234	C₂³	#125×C₁₀₉	1	2	yes	—
235	C₂³	C₄⋊Q₈×C₁₀₉	1	1	yes	characteristic
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
239	C₂×C₄	#117×C₁₀₉	1	2	yes	—
240	C₂×C₄	C₄×D₄×C₁₀₉	1	2	yes	—
241	C₂×C₄	C₄×D₄×C₁₀₉	1	1	yes	characteristic
242	C₂×C₄	#120×C₁₀₉	1	1	yes	characteristic
243	C₂×C₄	#121×C₁₀₉	1	2	yes	—
244	C₂×C₄	#121×C₁₀₉	1	1	yes	characteristic
245	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	2	yes	—
246	C₂×C₄	#123×C₁₀₉	1	1	yes	characteristic
247	C₂×C₄	#123×C₁₀₉	1	2	yes	—
248	C₂×C₄	#123×C₁₀₉	1	2	yes	—
249	C₂×C₄	#124×C₁₀₉	1	1	yes	characteristic
250	C₂×C₄	#124×C₁₀₉	1	1	yes	characteristic
251	C₂×C₄	#124×C₁₀₉	1	2	yes	—
252	C₂×C₄	#125×C₁₀₉	1	2	yes	—
253	C₂×C₄	#125×C₁₀₉	1	1	yes	characteristic
254	C₂×C₄	#126×C₁₀₉	1	2	yes	—
255	C₂×C₄	#126×C₁₀₉	1	2	yes	—
256	C₈	—	2	4	no	—
257	C₈	—	4	4	no	—
258	C₈	—	4	4	no	—
259	C₈	—	4	2	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	4	2	no	—
264	C₂²	—	4	2	no	—
265	C₂²	#387×C₁₀₉	1	2	yes	—
266	C₂²	#389×C₁₀₉	1	1	yes	characteristic
267	C₂²	#393×C₁₀₉	1	1	yes	characteristic
268	C₂²	#398×C₁₀₉	1	2	yes	—
269	C₂²	#400×C₁₀₉	1	1	yes	characteristic
270	C₄	—	8	2	no	—
271	C₄	#382×C₁₀₉	1	1	yes	characteristic
272	C₄	#385×C₁₀₉	1	1	yes	characteristic
273	C₄	#388×C₁₀₉	1	2	yes	—
274	C₄	#394×C₁₀₉	1	1	yes	characteristic
275	C₄	#397×C₁₀₉	1	1	yes	characteristic
276	C₄	#400×C₁₀₉	1	2	yes	—
277	C₂	—	2	1	no	—
278	C₂	—	2	2	no	—
279	C₂	—	2	2	no	—
280	C₂	—	2	1	no	—
281	C₂	C₁₀₉×#1442	1	1	yes	characteristic
282	C₂	C₁₀₉×#1953	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₀₉	#121
commutator	C₂²×C₄	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂³×C₄	C₂³×C₁₀₉
socle	C₂²×C₁₀₉	#389
prime radical	C₂⁴×C₁₀₉	C₂×Q₈
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. #341×C₁₀₉	C₄	semidirect
135. #11272	C₁₀₉	direct (N × H)
147. C₁₀₉	#11272	direct (N × H)
157. #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×#108×C₁₀₉	2
C₂×#344×C₁₀₉	2
C₁₀₉×#1738	2
C₂×#341×C₁₀₉	2
C₁₀₉×#1729	2
#11272	109

Subgroups — C109×#11272 #42129266

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#11272 #42129266