Subgroups — C₁₀₉×#11602 #42129596

Order 27904.

380 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#443×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#364×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₀₉×#1450	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₀₉×#1335	C₂	1	4	yes	maximal, complemented
5	C₂×#135×C₁₀₉	C₄	1	1	yes	characteristic
6	C₂×#131×C₁₀₉	—	2	1	no	—
7	C₂×M₃₂×C₁₀₉	—	2	1	no	—
8	#443×C₁₀₉	—	2	1	no	—
9	#443×C₁₀₉	—	2	1	no	—
10	#443×C₁₀₉	C₄	1	2	yes	—
11	#443×C₁₀₉	C₂²	1	2	yes	—
12	#431×C₁₀₉	—	2	1	no	—
13	C₂×#108×C₁₀₉	C₄	1	1	yes	characteristic
14	C₂×#104×C₁₀₉	—	2	1	no	—
15	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
16	#364×C₁₀₉	—	2	1	no	—
17	#364×C₁₀₉	C₂²	1	2	yes	complemented
18	#322×C₁₀₉	—	2	1	no	—
19	C₈²×C₁₀₉	C₂²	1	2	yes	complemented
20	C₂×#41×C₁₀₉	—	2	1	no	—
21	#135×C₁₀₉	—	2	1	no	—
22	#135×C₁₀₉	—	2	1	no	—
23	#135×C₁₀₉	C₈	1	2	yes	complemented
24	#135×C₁₀₉	C₂×C₄	1	2	yes	—
25	C₂×Q₁₆×C₁₀₉	C₈	1	1	yes	characteristic, complemented
26	C₂×SD₁₆×C₁₀₉	—	2	1	no	—
27	C₂×D₈×C₁₀₉	C₈	1	1	yes	characteristic, complemented
28	#131×C₁₀₉	—	2	2	no	—
29	#131×C₁₀₉	—	2	2	no	—
30	#131×C₁₀₉	—	4	1	no	—
31	#131×C₁₀₉	—	4	1	no	—
32	C₂×M₁₆×C₁₀₉	—	2	1	no	—
33	C₂×M₁₆×C₁₀₉	—	4	1	no	—
34	C₂²×C₈×C₁₀₉	—	4	1	no	—
35	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₁₀₉	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₀₉	D₄	1	1	yes	characteristic
38	M₃₂×C₁₀₉	—	2	2	no	—
39	M₃₂×C₁₀₉	—	4	1	no	—
40	C₂×C₁₆×C₁₀₉	—	4	1	no	—
41	#108×C₁₀₉	—	2	1	no	—
42	#108×C₁₀₉	C₂×C₄	1	2	yes	—
43	C₄⋊C₈×C₁₀₉	—	2	2	no	—
44	#104×C₁₀₉	—	2	2	no	—
45	#104×C₁₀₉	—	2	2	no	—
46	C₂²⋊C₈×C₁₀₉	—	2	2	no	—
47	C₄×C₈×C₁₀₉	—	2	2	no	—
48	C₄×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₁₀₉	D₄	1	1	yes	characteristic
50	C₄×C₈×C₁₀₉	D₄	1	1	yes	characteristic
51	C₄×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
52	#41×C₁₀₉	—	2	2	no	—
53	#41×C₁₀₉	—	2	2	no	—
54	#41×C₁₀₉	—	4	1	no	—
55	#41×C₁₀₉	—	4	1	no	—
56	C₂×Q₈×C₁₀₉	—	2	1	no	—
57	C₂×D₄×C₁₀₉	—	2	1	no	—
58	C₂×D₄×C₁₀₉	—	4	1	no	—
59	C₂²×C₄×C₁₀₉	—	2	1	no	—
60	C₂²×C₄×C₁₀₉	—	4	1	no	—
61	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
62	Q₁₆×C₁₀₉	—	2	1	no	—
63	Q₁₆×C₁₀₉	C₂×C₈	1	2	yes	—
64	SD₁₆×C₁₀₉	—	2	2	no	—
65	SD₁₆×C₁₀₉	—	2	2	no	—
66	D₈×C₁₀₉	—	2	1	no	—
67	D₈×C₁₀₉	C₂×C₈	1	2	yes	—
68	M₁₆×C₁₀₉	—	2	2	no	—
69	M₁₆×C₁₀₉	—	4	1	no	—
70	M₁₆×C₁₀₉	—	4	2	no	—
71	M₁₆×C₁₀₉	—	4	2	no	—
72	C₂×C₈×C₁₀₉	—	2	4	no	—
73	C₂×C₈×C₁₀₉	—	4	1	no	—
74	C₂×C₈×C₁₀₉	—	4	2	no	—
75	C₂×C₈×C₁₀₉	—	4	2	no	—
76	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₁₀₉	C₂×C₈	1	1	yes	characteristic
79	C₂×C₈×C₁₀₉	C₂×C₈	1	1	yes	characteristic
80	C₂×C₈×C₁₀₉	M₁₆	1	1	yes	characteristic
81	C₂×C₈×C₁₀₉	M₁₆	1	1	yes	characteristic
82	C₂×C₈×C₁₀₉	D₈	1	1	yes	characteristic
83	C₂×C₈×C₁₀₉	D₈	1	1	yes	characteristic
84	C₂×C₈×C₁₀₉	SD₁₆	1	1	yes	characteristic
85	C₂×C₈×C₁₀₉	SD₁₆	1	1	yes	characteristic
86	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₁₀₉	—	2	1	no	—
89	C₄²×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
90	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
91	C₁₆×C₁₀₉	—	4	2	no	—
92	C₂³×C₁₀₉	—	4	1	no	—
93	C₂³×C₁₀₉	#104	1	1	yes	characteristic
94	Q₈×C₁₀₉	—	2	2	no	—
95	Q₈×C₁₀₉	—	4	1	no	—
96	D₄×C₁₀₉	—	2	2	no	—
97	D₄×C₁₀₉	—	4	2	no	—
98	D₄×C₁₀₉	—	4	2	no	—
99	D₄×C₁₀₉	—	4	1	no	—
100	C₂×C₄×C₁₀₉	—	2	1	no	—
101	C₂×C₄×C₁₀₉	—	2	1	no	—
102	C₂×C₄×C₁₀₉	—	4	2	no	—
103	C₂×C₄×C₁₀₉	—	4	1	no	—
104	C₂×C₄×C₁₀₉	—	4	2	no	—
105	C₂×C₄×C₁₀₉	—	4	2	no	—
106	C₂×C₄×C₁₀₉	C₂²⋊C₈	1	1	yes	characteristic
107	C₂×C₄×C₁₀₉	C₂²⋊C₈	1	1	yes	characteristic
108	C₂×C₄×C₁₀₉	#102	1	1	yes	characteristic
109	C₂×C₄×C₁₀₉	#102	1	1	yes	characteristic
110	C₂×C₄×C₁₀₉	#104	1	1	yes	characteristic
111	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₈×C₁₀₉	—	4	4	no	—
113	C₈×C₁₀₉	—	4	2	no	—
114	C₈×C₁₀₉	C₂²⋊C₈	1	1	yes	characteristic
115	C₈×C₁₀₉	C₂²⋊C₈	1	1	yes	characteristic
116	C₈×C₁₀₉	#102	1	1	yes	characteristic
117	C₈×C₁₀₉	#102	1	1	yes	characteristic
118	C₈×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
119	C₈×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
120	C₈×C₁₀₉	C₂×D₈	1	1	yes	characteristic
121	C₈×C₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
122	C₂²×C₁₀₉	—	4	1	no	—
123	C₂²×C₁₀₉	—	4	2	no	—
124	C₂²×C₁₀₉	—	8	2	no	—
125	C₂²×C₁₀₉	D₄⋊C₈	1	1	yes	characteristic
126	C₂²×C₁₀₉	D₄⋊C₈	1	1	yes	characteristic
127	C₂²×C₁₀₉	C₂×#104	1	1	yes	characteristic
128	C₄×C₁₀₉	—	2	2	no	—
129	C₄×C₁₀₉	—	4	1	no	—
130	C₄×C₁₀₉	—	4	2	no	—
131	C₄×C₁₀₉	D₄⋊C₈	1	1	yes	characteristic
132	C₄×C₁₀₉	D₄⋊C₈	1	1	yes	characteristic
133	C₄×C₁₀₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
134	C₄×C₁₀₉	C₂×#102	1	1	yes	characteristic
135	#11602	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
136	C₂×C₁₀₉	—	2	1	no	—
137	C₂×C₁₀₉	—	2	1	no	—
138	C₂×C₁₀₉	—	2	1	no	—
139	C₂×C₁₀₉	—	8	2	no	—
140	C₂×C₁₀₉	C₂×D₄⋊C₈	1	1	yes	characteristic
141	C₂×#443	C₂×C₁₀₉	1	1	yes	characteristic
142	C₂×#364	C₂×C₁₀₉	1	1	yes	characteristic, complemented
143	#1450	C₂×C₁₀₉	1	1	yes	characteristic, complemented
144	#1335	C₂×C₁₀₉	1	4	yes	complemented
145	C₁₀₉	#11602	1	1	yes	characteristic, direct product factor, Sylow
146	C₂×#135	C₄×C₁₀₉	1	1	yes	characteristic
147	C₂×#131	—	2	1	no	—
148	C₂×M₃₂	—	2	1	no	—
149	#443	—	2	1	no	—
150	#443	—	2	1	no	—
151	#443	C₄×C₁₀₉	1	2	yes	—
152	#443	C₂²×C₁₀₉	1	2	yes	—
153	#431	—	2	1	no	—
154	C₂×#108	C₄×C₁₀₉	1	1	yes	characteristic
155	C₂×#104	—	2	1	no	—
156	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
157	#364	—	2	1	no	—
158	#364	C₂²×C₁₀₉	1	2	yes	complemented
159	#322	—	2	1	no	—
160	C₈²	C₂²×C₁₀₉	1	2	yes	complemented
161	C₂×#41	—	2	1	no	—
162	#135	—	2	1	no	—
163	#135	—	2	1	no	—
164	#135	C₈×C₁₀₉	1	2	yes	complemented
165	#135	C₂×C₄×C₁₀₉	1	2	yes	—
166	C₂×Q₁₆	C₈×C₁₀₉	1	1	yes	characteristic, complemented
167	C₂×SD₁₆	—	2	1	no	—
168	C₂×D₈	C₈×C₁₀₉	1	1	yes	characteristic, complemented
169	#131	—	2	2	no	—
170	#131	—	2	2	no	—
171	#131	—	4	1	no	—
172	#131	—	4	1	no	—
173	C₂×M₁₆	—	2	1	no	—
174	C₂×M₁₆	—	4	1	no	—
175	C₂²×C₈	—	4	1	no	—
176	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
177	C₂²×C₈	D₄×C₁₀₉	1	1	yes	characteristic
178	C₂×C₄²	D₄×C₁₀₉	1	1	yes	characteristic
179	M₃₂	—	2	2	no	—
180	M₃₂	—	4	1	no	—
181	C₂×C₁₆	—	4	1	no	—
182	#108	—	2	1	no	—
183	#108	C₂×C₄×C₁₀₉	1	2	yes	—
184	C₄⋊C₈	—	2	2	no	—
185	#104	—	2	2	no	—
186	#104	—	2	2	no	—
187	C₂²⋊C₈	—	2	2	no	—
188	C₄×C₈	—	2	2	no	—
189	C₄×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
190	C₄×C₈	D₄×C₁₀₉	1	1	yes	characteristic
191	C₄×C₈	D₄×C₁₀₉	1	1	yes	characteristic
192	C₄×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
193	#41	—	2	2	no	—
194	#41	—	2	2	no	—
195	#41	—	4	1	no	—
196	#41	—	4	1	no	—
197	C₂×Q₈	—	2	1	no	—
198	C₂×D₄	—	2	1	no	—
199	C₂×D₄	—	4	1	no	—
200	C₂²×C₄	—	2	1	no	—
201	C₂²×C₄	—	4	1	no	—
202	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
203	Q₁₆	—	2	1	no	—
204	Q₁₆	C₂×C₈×C₁₀₉	1	2	yes	—
205	SD₁₆	—	2	2	no	—
206	SD₁₆	—	2	2	no	—
207	D₈	—	2	1	no	—
208	D₈	C₂×C₈×C₁₀₉	1	2	yes	—
209	M₁₆	—	2	2	no	—
210	M₁₆	—	4	1	no	—
211	M₁₆	—	4	2	no	—
212	M₁₆	—	4	2	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	4	1	no	—
215	C₂×C₈	—	4	2	no	—
216	C₂×C₈	—	4	2	no	—
217	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
218	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
219	C₂×C₈	C₂×C₈×C₁₀₉	1	1	yes	characteristic
220	C₂×C₈	C₂×C₈×C₁₀₉	1	1	yes	characteristic
221	C₂×C₈	M₁₆×C₁₀₉	1	1	yes	characteristic
222	C₂×C₈	M₁₆×C₁₀₉	1	1	yes	characteristic
223	C₂×C₈	D₈×C₁₀₉	1	1	yes	characteristic
224	C₂×C₈	D₈×C₁₀₉	1	1	yes	characteristic
225	C₂×C₈	SD₁₆×C₁₀₉	1	1	yes	characteristic
226	C₂×C₈	SD₁₆×C₁₀₉	1	1	yes	characteristic
227	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
228	C₂×C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
229	C₄²	—	2	1	no	—
230	C₄²	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
231	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
232	C₁₆	—	4	2	no	—
233	C₂³	—	4	1	no	—
234	C₂³	#104×C₁₀₉	1	1	yes	characteristic
235	Q₈	—	2	2	no	—
236	Q₈	—	4	1	no	—
237	D₄	—	2	2	no	—
238	D₄	—	4	2	no	—
239	D₄	—	4	2	no	—
240	D₄	—	4	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	1	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	C₂²⋊C₈×C₁₀₉	1	1	yes	characteristic
248	C₂×C₄	C₂²⋊C₈×C₁₀₉	1	1	yes	characteristic
249	C₂×C₄	#102×C₁₀₉	1	1	yes	characteristic
250	C₂×C₄	#102×C₁₀₉	1	1	yes	characteristic
251	C₂×C₄	#104×C₁₀₉	1	1	yes	characteristic
252	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
253	C₈	—	4	4	no	—
254	C₈	—	4	2	no	—
255	C₈	C₂²⋊C₈×C₁₀₉	1	1	yes	characteristic
256	C₈	C₂²⋊C₈×C₁₀₉	1	1	yes	characteristic
257	C₈	#102×C₁₀₉	1	1	yes	characteristic
258	C₈	#102×C₁₀₉	1	1	yes	characteristic
259	C₈	C₂²×C₈×C₁₀₉	1	1	yes	characteristic
260	C₈	C₂×M₁₆×C₁₀₉	1	1	yes	characteristic
261	C₈	C₂×D₈×C₁₀₉	1	1	yes	characteristic
262	C₈	C₂×SD₁₆×C₁₀₉	1	1	yes	characteristic
263	C₂²	—	4	1	no	—
264	C₂²	—	4	2	no	—
265	C₂²	—	8	2	no	—
266	C₂²	D₄⋊C₈×C₁₀₉	1	1	yes	characteristic
267	C₂²	D₄⋊C₈×C₁₀₉	1	1	yes	characteristic
268	C₂²	C₂×#104×C₁₀₉	1	1	yes	characteristic
269	C₄	—	2	2	no	—
270	C₄	—	4	1	no	—
271	C₄	—	4	2	no	—
272	C₄	D₄⋊C₈×C₁₀₉	1	1	yes	characteristic
273	C₄	D₄⋊C₈×C₁₀₉	1	1	yes	characteristic
274	C₄	C₂×C₂²⋊C₈×C₁₀₉	1	1	yes	characteristic
275	C₄	C₂×#102×C₁₀₉	1	1	yes	characteristic
276	C₂	—	2	1	no	—
277	C₂	—	2	1	no	—
278	C₂	—	2	1	no	—
279	C₂	—	8	2	no	—
280	C₂	C₂×D₄⋊C₈×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₁₀₉	C₂×D₈
commutator	C₈	C₂²×C₈×C₁₀₉
Frattini, cosocle	C₄×C₈	C₂³×C₁₀₉
socle	C₂×C₁₀₉	C₂×D₄⋊C₈
prime radical	C₂×#135×C₁₀₉	C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₁₀₉ ◁ C₂²×C₈×C₁₀₉ ◁ C₂×C₄×C₈×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#364×C₁₀₉	C₂	semidirect
3. C₁₀₉×#1450	C₂	semidirect
4. C₁₀₉×#1335	C₂	semidirect
17. #364×C₁₀₉	C₂²	semidirect
19. C₈²×C₁₀₉	C₂²	semidirect
23. #135×C₁₀₉	C₈	semidirect
25. C₂×Q₁₆×C₁₀₉	C₈	semidirect
27. C₂×D₈×C₁₀₉	C₈	semidirect
135. #11602	C₁₀₉	direct (N × H)
142. C₂×#364	C₂×C₁₀₉	semidirect
143. #1450	C₂×C₁₀₉	semidirect
144. #1335	C₂×C₁₀₉	semidirect
145. C₁₀₉	#11602	direct (N × H)
158. #364	C₂²×C₁₀₉	semidirect
160. C₈²	C₂²×C₁₀₉	semidirect
164. #135	C₈×C₁₀₉	semidirect
166. C₂×Q₁₆	C₈×C₁₀₉	semidirect
168. C₂×D₈	C₈×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#443×C₁₀₉	2
C₂×#364×C₁₀₉	2
C₁₀₉×#1450	2
C₁₀₉×#1335	2
#11602	109

Subgroups — C109×#11602 #42129596

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#11602 #42129596