Subgroups — C₁₀₉×#11464 #42129458

Order 27904.

310 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	C₁₀₉×#2922	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#2163	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₀₉×#2162	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₀₉×#1323	C₂	1	2	yes	maximal, complemented
5	C₁₀₉×#1315	C₂	1	2	yes	maximal
6	C₄×#41×C₁₀₉	C₄	1	1	yes	characteristic
7	#433×C₁₀₉	—	2	1	no	—
8	#433×C₁₀₉	—	2	1	no	—
9	#431×C₁₀₉	—	2	1	no	—
10	#431×C₁₀₉	—	2	1	no	—
11	#424×C₁₀₉	C₄	1	1	yes	characteristic
12	#424×C₁₀₉	C₂²	1	1	yes	characteristic
13	#408×C₁₀₉	C₄	1	1	yes	characteristic
14	#408×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₄×M₁₆×C₁₀₉	C₄	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₄⋊C₁₆×C₁₀₉	—	2	1	no	—
18	C₄⋊C₁₆×C₁₀₉	—	2	1	no	—
19	C₂²⋊C₁₆×C₁₀₉	C₂²	1	2	yes	—
20	C₂²⋊C₁₆×C₁₀₉	C₂²	1	2	yes	—
21	#346×C₁₀₉	—	2	1	no	—
22	C₄×C₁₆×C₁₀₉	—	2	1	no	—
23	C₂×#41×C₁₀₉	C₂×C₄	1	1	yes	characteristic
24	C₂×M₁₆×C₁₀₉	C₂×C₄	1	1	yes	characteristic
25	C₂×M₁₆×C₁₀₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
28	C₄×Q₈×C₁₀₉	C₈	1	1	yes	characteristic
29	C₄×Q₈×C₁₀₉	C₈	1	1	yes	characteristic
30	C₄×D₄×C₁₀₉	—	2	1	no	—
31	C₄×D₄×C₁₀₉	—	2	1	no	—
32	C₄×D₄×C₁₀₉	C₈	1	1	yes	characteristic
33	C₄×D₄×C₁₀₉	C₈	1	1	yes	characteristic
34	#117×C₁₀₉	—	2	1	no	—
35	#117×C₁₀₉	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₀₉	—	2	1	no	—
37	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
38	C₂×C₁₆×C₁₀₉	—	2	2	no	—
39	C₂×C₁₆×C₁₀₉	—	2	2	no	—
40	C₄⋊C₈×C₁₀₉	—	2	1	no	—
41	C₄⋊C₈×C₁₀₉	—	2	1	no	—
42	C₄⋊C₈×C₁₀₉	—	2	1	no	—
43	C₄⋊C₈×C₁₀₉	—	2	1	no	—
44	C₂²⋊C₈×C₁₀₉	—	2	1	no	—
45	C₂²⋊C₈×C₁₀₉	—	2	1	no	—
46	C₂²⋊C₈×C₁₀₉	—	2	1	no	—
47	C₂²⋊C₈×C₁₀₉	—	2	1	no	—
48	#97×C₁₀₉	—	2	1	no	—
49	C₄×C₈×C₁₀₉	—	2	1	no	—
50	C₄×C₈×C₁₀₉	D₄	1	1	yes	characteristic
51	C₄×C₈×C₁₀₉	D₄	1	1	yes	characteristic
52	C₄×C₈×C₁₀₉	D₄	1	1	yes	characteristic
53	C₄×C₈×C₁₀₉	D₄	1	1	yes	characteristic
54	#41×C₁₀₉	—	4	1	no	—
55	#41×C₁₀₉	—	4	1	no	—
56	C₂×Q₈×C₁₀₉	C₂×C₈	1	1	yes	characteristic
57	C₂×D₄×C₁₀₉	—	2	1	no	—
58	C₂×D₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
59	C₂²×C₄×C₁₀₉	—	2	1	no	—
60	C₂²×C₄×C₁₀₉	—	2	1	no	—
61	C₂²×C₄×C₁₀₉	—	2	1	no	—
62	C₂²×C₄×C₁₀₉	M₁₆	1	1	yes	characteristic
63	C₂²×C₄×C₁₀₉	M₁₆	1	1	yes	characteristic
64	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
65	M₁₆×C₁₀₉	—	4	1	no	—
66	M₁₆×C₁₀₉	—	4	1	no	—
67	C₂×C₈×C₁₀₉	—	2	1	no	—
68	C₂×C₈×C₁₀₉	—	2	1	no	—
69	C₂×C₈×C₁₀₉	—	4	1	no	—
70	C₂×C₈×C₁₀₉	—	4	1	no	—
71	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
75	C₄⋊C₄×C₁₀₉	—	2	1	no	—
76	C₄⋊C₄×C₁₀₉	—	2	1	no	—
77	C₄⋊C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
78	C₄⋊C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
79	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
80	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
81	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
82	C₄²×C₁₀₉	—	2	1	no	—
83	C₄²×C₁₀₉	—	2	1	no	—
84	C₄²×C₁₀₉	—	2	1	no	—
85	C₄²×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
86	C₄²×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
87	C₄²×C₁₀₉	C₂×C₈	1	1	yes	characteristic
88	C₄²×C₁₀₉	C₂×C₈	1	1	yes	characteristic
89	C₁₆×C₁₀₉	—	4	2	no	—
90	C₁₆×C₁₀₉	—	4	2	no	—
91	C₂³×C₁₀₉	—	2	1	no	—
92	C₂³×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
93	Q₈×C₁₀₉	—	4	1	no	—
94	D₄×C₁₀₉	—	4	1	no	—
95	D₄×C₁₀₉	—	4	2	no	—
96	C₂×C₄×C₁₀₉	—	2	2	no	—
97	C₂×C₄×C₁₀₉	—	2	1	no	—
98	C₂×C₄×C₁₀₉	—	2	1	no	—
99	C₂×C₄×C₁₀₉	—	2	1	no	—
100	C₂×C₄×C₁₀₉	—	2	1	no	—
101	C₂×C₄×C₁₀₉	—	2	1	no	—
102	C₂×C₄×C₁₀₉	—	4	1	no	—
103	C₂×C₄×C₁₀₉	—	4	2	no	—
104	C₂×C₄×C₁₀₉	—	4	2	no	—
105	C₂×C₄×C₁₀₉	—	4	1	no	—
106	C₂×C₄×C₁₀₉	—	4	1	no	—
107	C₂×C₄×C₁₀₉	C₂²⋊C₈	1	1	yes	characteristic
108	C₂×C₄×C₁₀₉	C₂²⋊C₈	1	1	yes	characteristic
109	C₂×C₄×C₁₀₉	C₂²⋊C₈	1	1	yes	characteristic
110	C₂×C₄×C₁₀₉	C₂²⋊C₈	1	1	yes	characteristic
111	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
113	C₈×C₁₀₉	—	2	1	no	—
114	C₈×C₁₀₉	—	2	1	no	—
115	C₈×C₁₀₉	—	2	1	no	—
116	C₈×C₁₀₉	—	2	1	no	—
117	C₈×C₁₀₉	—	4	1	no	—
118	C₈×C₁₀₉	—	4	1	no	—
119	C₂²×C₁₀₉	—	2	1	no	—
120	C₂²×C₁₀₉	—	2	1	no	—
121	C₂²×C₁₀₉	—	2	1	no	—
122	C₂²×C₁₀₉	—	4	1	no	—
123	C₂²×C₁₀₉	—	4	1	no	—
124	C₂²×C₁₀₉	—	4	1	no	—
125	C₂²×C₁₀₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
126	C₄×C₁₀₉	—	2	1	no	—
127	C₄×C₁₀₉	—	2	1	no	—
128	C₄×C₁₀₉	—	2	1	no	—
129	C₄×C₁₀₉	—	2	1	no	—
130	C₄×C₁₀₉	—	4	1	no	—
131	C₄×C₁₀₉	—	4	1	no	—
132	C₄×C₁₀₉	—	4	1	no	—
133	C₄×C₁₀₉	#350	1	2	yes	—
134	C₄×C₁₀₉	#410	1	1	yes	characteristic
135	C₄×C₁₀₉	#413	1	1	yes	characteristic
136	#11464	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
137	C₂×C₁₀₉	—	4	1	no	—
138	C₂×C₁₀₉	—	8	1	no	—
139	C₂×C₁₀₉	#1458	1	1	yes	characteristic
140	C₂×C₁₀₉	C₂×#350	1	1	yes	characteristic
141	C₂×C₁₀₉	#2117	1	1	yes	characteristic
142	#2922	C₂×C₁₀₉	1	1	yes	characteristic
143	#2163	C₂×C₁₀₉	1	1	yes	characteristic, complemented
144	#2162	C₂×C₁₀₉	1	1	yes	characteristic, complemented
145	#1323	C₂×C₁₀₉	1	2	yes	complemented
146	#1315	C₂×C₁₀₉	1	2	yes	—
147	C₁₀₉	#11464	1	1	yes	characteristic, direct product factor, Sylow
148	C₄×#41	C₄×C₁₀₉	1	1	yes	characteristic
149	#433	—	2	1	no	—
150	#433	—	2	1	no	—
151	#431	—	2	1	no	—
152	#431	—	2	1	no	—
153	#424	C₄×C₁₀₉	1	1	yes	characteristic
154	#424	C₂²×C₁₀₉	1	1	yes	characteristic
155	#408	C₄×C₁₀₉	1	1	yes	characteristic
156	#408	C₂²×C₁₀₉	1	1	yes	characteristic
157	C₄×M₁₆	C₄×C₁₀₉	1	1	yes	characteristic
158	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
159	C₄⋊C₁₆	—	2	1	no	—
160	C₄⋊C₁₆	—	2	1	no	—
161	C₂²⋊C₁₆	C₂²×C₁₀₉	1	2	yes	—
162	C₂²⋊C₁₆	C₂²×C₁₀₉	1	2	yes	—
163	#346	—	2	1	no	—
164	C₄×C₁₆	—	2	1	no	—
165	C₂×#41	C₂×C₄×C₁₀₉	1	1	yes	characteristic
166	C₂×M₁₆	C₂×C₄×C₁₀₉	1	1	yes	characteristic
167	C₂×M₁₆	C₂×C₄×C₁₀₉	1	1	yes	characteristic
168	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
169	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
170	C₄×Q₈	C₈×C₁₀₉	1	1	yes	characteristic
171	C₄×Q₈	C₈×C₁₀₉	1	1	yes	characteristic
172	C₄×D₄	—	2	1	no	—
173	C₄×D₄	—	2	1	no	—
174	C₄×D₄	C₈×C₁₀₉	1	1	yes	characteristic
175	C₄×D₄	C₈×C₁₀₉	1	1	yes	characteristic
176	#117	—	2	1	no	—
177	#117	C₂×C₄×C₁₀₉	1	1	yes	characteristic
178	C₂×C₄²	—	2	1	no	—
179	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
180	C₂×C₁₆	—	2	2	no	—
181	C₂×C₁₆	—	2	2	no	—
182	C₄⋊C₈	—	2	1	no	—
183	C₄⋊C₈	—	2	1	no	—
184	C₄⋊C₈	—	2	1	no	—
185	C₄⋊C₈	—	2	1	no	—
186	C₂²⋊C₈	—	2	1	no	—
187	C₂²⋊C₈	—	2	1	no	—
188	C₂²⋊C₈	—	2	1	no	—
189	C₂²⋊C₈	—	2	1	no	—
190	#97	—	2	1	no	—
191	C₄×C₈	—	2	1	no	—
192	C₄×C₈	D₄×C₁₀₉	1	1	yes	characteristic
193	C₄×C₈	D₄×C₁₀₉	1	1	yes	characteristic
194	C₄×C₈	D₄×C₁₀₉	1	1	yes	characteristic
195	C₄×C₈	D₄×C₁₀₉	1	1	yes	characteristic
196	#41	—	4	1	no	—
197	#41	—	4	1	no	—
198	C₂×Q₈	C₂×C₈×C₁₀₉	1	1	yes	characteristic
199	C₂×D₄	—	2	1	no	—
200	C₂×D₄	C₂×C₈×C₁₀₉	1	1	yes	characteristic
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	2	1	no	—
203	C₂²×C₄	—	2	1	no	—
204	C₂²×C₄	M₁₆×C₁₀₉	1	1	yes	characteristic
205	C₂²×C₄	M₁₆×C₁₀₉	1	1	yes	characteristic
206	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
207	M₁₆	—	4	1	no	—
208	M₁₆	—	4	1	no	—
209	C₂×C₈	—	2	1	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	4	1	no	—
212	C₂×C₈	—	4	1	no	—
213	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
214	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
215	C₂×C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
216	C₂×C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
217	C₄⋊C₄	—	2	1	no	—
218	C₄⋊C₄	—	2	1	no	—
219	C₄⋊C₄	C₂×C₈×C₁₀₉	1	1	yes	characteristic
220	C₄⋊C₄	C₂×C₈×C₁₀₉	1	1	yes	characteristic
221	C₂²⋊C₄	—	2	1	no	—
222	C₂²⋊C₄	—	2	1	no	—
223	C₂²⋊C₄	—	2	1	no	—
224	C₄²	—	2	1	no	—
225	C₄²	—	2	1	no	—
226	C₄²	—	2	1	no	—
227	C₄²	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
228	C₄²	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
229	C₄²	C₂×C₈×C₁₀₉	1	1	yes	characteristic
230	C₄²	C₂×C₈×C₁₀₉	1	1	yes	characteristic
231	C₁₆	—	4	2	no	—
232	C₁₆	—	4	2	no	—
233	C₂³	—	2	1	no	—
234	C₂³	C₂×M₁₆×C₁₀₉	1	1	yes	characteristic
235	Q₈	—	4	1	no	—
236	D₄	—	4	1	no	—
237	D₄	—	4	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	4	1	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	C₂²⋊C₈×C₁₀₉	1	1	yes	characteristic
250	C₂×C₄	C₂²⋊C₈×C₁₀₉	1	1	yes	characteristic
251	C₂×C₄	C₂²⋊C₈×C₁₀₉	1	1	yes	characteristic
252	C₂×C₄	C₂²⋊C₈×C₁₀₉	1	1	yes	characteristic
253	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
254	C₂×C₄	C₂²×C₈×C₁₀₉	1	1	yes	characteristic
255	C₈	—	2	1	no	—
256	C₈	—	2	1	no	—
257	C₈	—	2	1	no	—
258	C₈	—	2	1	no	—
259	C₈	—	4	1	no	—
260	C₈	—	4	1	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	4	1	no	—
265	C₂²	—	4	1	no	—
266	C₂²	—	4	1	no	—
267	C₂²	C₂×C₂²⋊C₈×C₁₀₉	1	1	yes	characteristic
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	2	1	no	—
272	C₄	—	4	1	no	—
273	C₄	—	4	1	no	—
274	C₄	—	4	1	no	—
275	C₄	#350×C₁₀₉	1	2	yes	—
276	C₄	#410×C₁₀₉	1	1	yes	characteristic
277	C₄	#413×C₁₀₉	1	1	yes	characteristic
278	C₂	—	4	1	no	—
279	C₂	—	8	1	no	—
280	C₂	C₁₀₉×#1458	1	1	yes	characteristic
281	C₂	C₂×#350×C₁₀₉	1	1	yes	characteristic
282	C₂	C₁₀₉×#2117	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₈×C₁₀₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₀₉
socle	C₂²×C₁₀₉	C₂×C₂²⋊C₈
prime radical	C₂×#41×C₁₀₉	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₉ ◁ C₂×C₄×C₈×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₀₉×#2163	C₂	semidirect
3. C₁₀₉×#2162	C₂	semidirect
4. C₁₀₉×#1323	C₂	semidirect
136. #11464	C₁₀₉	direct (N × H)
143. #2163	C₂×C₁₀₉	semidirect
144. #2162	C₂×C₁₀₉	semidirect
145. #1323	C₂×C₁₀₉	semidirect
147. C₁₀₉	#11464	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×#2922	2
C₁₀₉×#2163	2
C₁₀₉×#2162	2
C₁₀₉×#1323	2
C₁₀₉×#1315	2
#11464	109

Subgroups — C109×#11464 #42129458

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#11464 #42129458