Subgroups — C₁₀₉×#11288 #42129282

Order 27904.

602 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₉×#2905	C₂	1	1	yes	maximal, characteristic
2	C₂×#344×C₁₀₉	C₂	1	2	yes	maximal
3	C₁₀₉×#1740	C₂	1	1	yes	maximal, characteristic
4	C₁₀₉×#1729	C₂	1	2	yes	maximal
5	C₁₀₉×#1729	C₂	1	1	yes	maximal, characteristic
6	C₂²×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₁₀₉	C₂²	1	2	yes	—
8	C₂³×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
9	C₂×#117×C₁₀₉	C₂²	1	2	yes	—
10	C₂×#117×C₁₀₉	C₂²	1	1	yes	characteristic
11	#424×C₁₀₉	—	2	4	no	—
12	#423×C₁₀₉	—	2	2	no	—
13	#344×C₁₀₉	—	2	4	no	—
14	#342×C₁₀₉	C₄	1	4	yes	—
15	#336×C₁₀₉	—	2	4	no	—
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	—	2	2	no	—
21	C₂×M₁₆×C₁₀₉	—	2	1	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	D₄	1	2	yes	—
28	C₂²×C₈×C₁₀₉	—	2	2	no	—
29	C₂²×C₈×C₁₀₉	—	2	1	no	—
30	C₂²×C₈×C₁₀₉	—	2	1	no	—
31	C₂²×C₈×C₁₀₉	—	2	2	no	—
32	C₂²×C₈×C₁₀₉	—	2	1	no	—
33	C₂²×C₈×C₁₀₉	D₄	1	4	yes	—
34	C₂²×C₈×C₁₀₉	Q₈	1	2	yes	—
35	#117×C₁₀₉	—	2	2	no	—
36	#117×C₁₀₉	—	2	2	no	—
37	#117×C₁₀₉	—	2	2	no	—
38	#117×C₁₀₉	—	2	2	no	—
39	#117×C₁₀₉	—	2	2	no	—
40	#117×C₁₀₉	—	2	2	no	—
41	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
42	C₂×C₄⋊C₄×C₁₀₉	Q₈	1	2	yes	—
43	C₂×C₂²⋊C₄×C₁₀₉	C₂×C₄	1	4	yes	—
44	C₂×C₂²⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₁₀₉	—	2	2	no	—
46	C₂×C₄²×C₁₀₉	D₄	1	2	yes	—
47	C₄⋊C₈×C₁₀₉	—	2	8	no	—
48	C₂⁴×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₀₉	—	2	4	no	—
50	C₂²×C₄×C₁₀₉	C₄⋊C₄	1	4	yes	—
51	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
52	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
55	C₂²×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
56	C₂²×C₄×C₁₀₉	#41	1	1	yes	characteristic
57	C₂²×C₄×C₁₀₉	#41	1	2	yes	—
58	C₂²×C₄×C₁₀₉	#41	1	2	yes	—
59	C₂²×C₄×C₁₀₉	#41	1	2	yes	—
60	C₂²×C₄×C₁₀₉	#41	1	1	yes	characteristic
61	M₁₆×C₁₀₉	—	2	4	no	—
62	M₁₆×C₁₀₉	—	4	2	no	—
63	M₁₆×C₁₀₉	—	4	2	no	—
64	M₁₆×C₁₀₉	—	4	2	no	—
65	M₁₆×C₁₀₉	—	4	2	no	—
66	M₁₆×C₁₀₉	—	4	2	no	—
67	C₂×C₈×C₁₀₉	—	2	4	no	—
68	C₂×C₈×C₁₀₉	—	2	2	no	—
69	C₂×C₈×C₁₀₉	—	2	4	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	—	4	1	no	—
74	C₂×C₈×C₁₀₉	—	4	1	no	—
75	C₂×C₈×C₁₀₉	—	4	1	no	—
76	C₂×C₈×C₁₀₉	—	4	2	no	—
77	C₂×C₈×C₁₀₉	—	4	1	no	—
78	C₄⋊C₄×C₁₀₉	—	2	4	no	—
79	C₄⋊C₄×C₁₀₉	—	4	2	no	—
80	C₄⋊C₄×C₁₀₉	—	4	2	no	—
81	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
82	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
83	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
84	C₄²×C₁₀₉	—	2	2	no	—
85	C₄²×C₁₀₉	—	2	2	no	—
86	C₄²×C₁₀₉	—	4	2	no	—
87	C₄²×C₁₀₉	—	4	2	no	—
88	C₂³×C₁₀₉	—	4	2	no	—
89	C₂³×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂³×C₁₀₉	#117	1	2	yes	—
91	C₂³×C₁₀₉	C₄×D₄	1	2	yes	—
92	C₂³×C₁₀₉	C₄×D₄	1	1	yes	characteristic
93	C₂³×C₁₀₉	C₄×Q₈	1	1	yes	characteristic
94	C₂×C₄×C₁₀₉	—	2	4	no	—
95	C₂×C₄×C₁₀₉	—	4	4	no	—
96	C₂×C₄×C₁₀₉	—	4	8	no	—
97	C₂×C₄×C₁₀₉	—	4	4	no	—
98	C₂×C₄×C₁₀₉	#120	1	1	yes	characteristic
99	C₂×C₄×C₁₀₉	#121	1	1	yes	characteristic
100	C₂×C₄×C₁₀₉	#121	1	2	yes	—
101	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
102	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
103	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
104	C₂×C₄×C₁₀₉	#123	1	2	yes	—
105	C₂×C₄×C₁₀₉	#123	1	2	yes	—
106	C₂×C₄×C₁₀₉	#123	1	1	yes	characteristic
107	C₂×C₄×C₁₀₉	#124	1	1	yes	characteristic
108	C₂×C₄×C₁₀₉	#124	1	2	yes	—
109	C₂×C₄×C₁₀₉	#124	1	1	yes	characteristic
110	C₂×C₄×C₁₀₉	#125	1	2	yes	—
111	C₂×C₄×C₁₀₉	#125	1	2	yes	—
112	C₂×C₄×C₁₀₉	#125	1	1	yes	characteristic
113	C₂×C₄×C₁₀₉	#126	1	2	yes	—
114	C₂×C₄×C₁₀₉	#126	1	2	yes	—
115	C₂×C₄×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
116	C₈×C₁₀₉	—	4	2	no	—
117	C₈×C₁₀₉	—	4	4	no	—
118	C₈×C₁₀₉	—	4	2	no	—
119	C₂²×C₁₀₉	—	2	2	no	—
120	C₂²×C₁₀₉	—	2	4	no	—
121	C₂²×C₁₀₉	—	4	2	no	—
122	C₂²×C₁₀₉	—	4	1	no	—
123	C₂²×C₁₀₉	—	4	1	no	—
124	C₂²×C₁₀₉	#382	1	1	yes	characteristic
125	C₂²×C₁₀₉	#385	1	1	yes	characteristic
126	C₂²×C₁₀₉	#387	1	2	yes	—
127	C₂²×C₁₀₉	#388	1	2	yes	—
128	C₂²×C₁₀₉	#389	1	1	yes	characteristic
129	C₄×C₁₀₉	—	4	4	no	—
130	C₄×C₁₀₉	—	8	4	no	—
131	C₄×C₁₀₉	#393	1	1	yes	characteristic
132	C₄×C₁₀₉	#394	1	1	yes	characteristic
133	C₄×C₁₀₉	#397	1	1	yes	characteristic
134	C₄×C₁₀₉	#398	1	2	yes	—
135	C₄×C₁₀₉	#400	1	1	yes	characteristic
136	C₄×C₁₀₉	#400	1	2	yes	—
137	#11288	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
138	C₂×C₁₀₉	—	2	2	no	—
139	C₂×C₁₀₉	—	2	1	no	—
140	C₂×C₁₀₉	—	2	2	no	—
141	C₂×C₁₀₉	—	2	1	no	—
142	C₂×C₁₀₉	#1442	1	1	yes	characteristic
143	C₂×C₁₀₉	#1868	1	2	yes	—
144	#2905	C₂×C₁₀₉	1	1	yes	characteristic
145	C₂×#344	C₂×C₁₀₉	1	2	yes	—
146	#1740	C₂×C₁₀₉	1	1	yes	characteristic
147	#1729	C₂×C₁₀₉	1	2	yes	—
148	#1729	C₂×C₁₀₉	1	1	yes	characteristic
149	C₁₀₉	#11288	1	1	yes	characteristic, direct product factor, Sylow
150	C₂²×M₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
151	C₂²×M₁₆	C₂²×C₁₀₉	1	2	yes	—
152	C₂³×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
153	C₂×#117	C₂²×C₁₀₉	1	2	yes	—
154	C₂×#117	C₂²×C₁₀₉	1	1	yes	characteristic
155	#424	—	2	4	no	—
156	#423	—	2	2	no	—
157	#344	—	2	4	no	—
158	#342	C₄×C₁₀₉	1	4	yes	—
159	#336	—	2	4	no	—
160	#336	—	2	2	no	—
161	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
162	C₂×M₁₆	—	2	2	no	—
163	C₂×M₁₆	—	2	2	no	—
164	C₂×M₁₆	—	2	2	no	—
165	C₂×M₁₆	—	2	1	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	D₄×C₁₀₉	1	2	yes	—
172	C₂²×C₈	—	2	2	no	—
173	C₂²×C₈	—	2	1	no	—
174	C₂²×C₈	—	2	1	no	—
175	C₂²×C₈	—	2	2	no	—
176	C₂²×C₈	—	2	1	no	—
177	C₂²×C₈	D₄×C₁₀₉	1	4	yes	—
178	C₂²×C₈	Q₈×C₁₀₉	1	2	yes	—
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	—	2	2	no	—
185	C₂×C₄⋊C₄	—	2	2	no	—
186	C₂×C₄⋊C₄	Q₈×C₁₀₉	1	2	yes	—
187	C₂×C₂²⋊C₄	C₂×C₄×C₁₀₉	1	4	yes	—
188	C₂×C₂²⋊C₄	C₂×C₄×C₁₀₉	1	2	yes	—
189	C₂×C₄²	—	2	2	no	—
190	C₂×C₄²	D₄×C₁₀₉	1	2	yes	—
191	C₄⋊C₈	—	2	8	no	—
192	C₂⁴	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	C₄⋊C₄×C₁₀₉	1	4	yes	—
195	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
196	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
197	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
198	C₂²×C₄	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
199	C₂²×C₄	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
200	C₂²×C₄	#41×C₁₀₉	1	1	yes	characteristic
201	C₂²×C₄	#41×C₁₀₉	1	2	yes	—
202	C₂²×C₄	#41×C₁₀₉	1	2	yes	—
203	C₂²×C₄	#41×C₁₀₉	1	2	yes	—
204	C₂²×C₄	#41×C₁₀₉	1	1	yes	characteristic
205	M₁₆	—	2	4	no	—
206	M₁₆	—	4	2	no	—
207	M₁₆	—	4	2	no	—
208	M₁₆	—	4	2	no	—
209	M₁₆	—	4	2	no	—
210	M₁₆	—	4	2	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	4	1	no	—
218	C₂×C₈	—	4	1	no	—
219	C₂×C₈	—	4	1	no	—
220	C₂×C₈	—	4	2	no	—
221	C₂×C₈	—	4	1	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	—	2	4	no	—
228	C₄²	—	2	2	no	—
229	C₄²	—	2	2	no	—
230	C₄²	—	4	2	no	—
231	C₄²	—	4	2	no	—
232	C₂³	—	4	2	no	—
233	C₂³	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
234	C₂³	#117×C₁₀₉	1	2	yes	—
235	C₂³	C₄×D₄×C₁₀₉	1	2	yes	—
236	C₂³	C₄×D₄×C₁₀₉	1	1	yes	characteristic
237	C₂³	C₄×Q₈×C₁₀₉	1	1	yes	characteristic
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	—	4	8	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	#120×C₁₀₉	1	1	yes	characteristic
243	C₂×C₄	#121×C₁₀₉	1	1	yes	characteristic
244	C₂×C₄	#121×C₁₀₉	1	2	yes	—
245	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	2	yes	—
246	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	1	yes	characteristic
247	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	2	yes	—
248	C₂×C₄	#123×C₁₀₉	1	2	yes	—
249	C₂×C₄	#123×C₁₀₉	1	2	yes	—
250	C₂×C₄	#123×C₁₀₉	1	1	yes	characteristic
251	C₂×C₄	#124×C₁₀₉	1	1	yes	characteristic
252	C₂×C₄	#124×C₁₀₉	1	2	yes	—
253	C₂×C₄	#124×C₁₀₉	1	1	yes	characteristic
254	C₂×C₄	#125×C₁₀₉	1	2	yes	—
255	C₂×C₄	#125×C₁₀₉	1	2	yes	—
256	C₂×C₄	#125×C₁₀₉	1	1	yes	characteristic
257	C₂×C₄	#126×C₁₀₉	1	2	yes	—
258	C₂×C₄	#126×C₁₀₉	1	2	yes	—
259	C₂×C₄	C₄⋊Q₈×C₁₀₉	1	1	yes	characteristic
260	C₈	—	4	2	no	—
261	C₈	—	4	4	no	—
262	C₈	—	4	2	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	4	no	—
265	C₂²	—	4	2	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	4	1	no	—
268	C₂²	#382×C₁₀₉	1	1	yes	characteristic
269	C₂²	#385×C₁₀₉	1	1	yes	characteristic
270	C₂²	#387×C₁₀₉	1	2	yes	—
271	C₂²	#388×C₁₀₉	1	2	yes	—
272	C₂²	#389×C₁₀₉	1	1	yes	characteristic
273	C₄	—	4	4	no	—
274	C₄	—	8	4	no	—
275	C₄	#393×C₁₀₉	1	1	yes	characteristic
276	C₄	#394×C₁₀₉	1	1	yes	characteristic
277	C₄	#397×C₁₀₉	1	1	yes	characteristic
278	C₄	#398×C₁₀₉	1	2	yes	—
279	C₄	#400×C₁₀₉	1	1	yes	characteristic
280	C₄	#400×C₁₀₉	1	2	yes	—
281	C₂	—	2	2	no	—
282	C₂	—	2	1	no	—
283	C₂	—	2	2	no	—
284	C₂	—	2	1	no	—
285	C₂	C₁₀₉×#1442	1	1	yes	characteristic
286	C₂	C₁₀₉×#1868	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₀₉	#120
commutator	C₂⁴	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂³×C₄	C₂³×C₁₀₉
socle	C₂²×C₁₀₉	#385
prime radical	C₂⁴×C₁₀₉	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ 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
137. #11288	C₁₀₉	direct (N × H)
149. C₁₀₉	#11288	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×#2905	2
C₂×#344×C₁₀₉	2
C₁₀₉×#1740	2
C₁₀₉×#1729	2
#11288	109

Subgroups — C109×#11288 #42129282

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#11288 #42129282