Subgroups — C₁₀₉×#8607 #42126601

Order 27904.

372 conjugacy classes of subgroups.

The subgroup lattice has 256 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×#1564	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#334×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₀₉×#1477	C₂	1	2	yes	maximal
5	C₁₀₉×#1283	C₂	1	2	yes	maximal, complemented
6	C₄×#41×C₁₀₉	C₂²	1	1	yes	characteristic
7	#433×C₁₀₉	—	2	1	no	—
8	#433×C₁₀₉	—	2	1	no	—
9	#431×C₁₀₉	—	2	2	no	—
10	#431×C₁₀₉	C₂²	1	2	yes	—
11	#424×C₁₀₉	C₄	1	2	yes	—
12	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
13	#408×C₁₀₉	C₄	1	2	yes	—
14	C₄×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
16	#335×C₁₀₉	—	2	1	no	—
17	#334×C₁₀₉	—	2	2	no	—
18	#334×C₁₀₉	—	2	1	no	—
19	Q₈⋊C₈×C₁₀₉	—	2	2	no	—
20	D₄⋊C₈×C₁₀₉	—	2	2	no	—
21	C₂×#41×C₁₀₉	C₂×C₄	1	1	yes	characteristic
22	C₂×M₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₁₀₉	—	2	2	no	—
24	C₂²×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
25	C₄×Q₈×C₁₀₉	D₄	1	1	yes	characteristic
26	C₄×Q₈×C₁₀₉	Q₈	1	1	yes	characteristic
27	C₄×D₄×C₁₀₉	—	2	1	no	—
28	C₄×D₄×C₁₀₉	—	2	1	no	—
29	C₄×D₄×C₁₀₉	D₄	1	1	yes	characteristic
30	C₄×D₄×C₁₀₉	Q₈	1	1	yes	characteristic
31	#117×C₁₀₉	—	2	1	no	—
32	#117×C₁₀₉	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₀₉	—	2	1	no	—
34	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
35	C₄⋊C₈×C₁₀₉	—	2	2	no	—
36	C₄⋊C₈×C₁₀₉	—	2	2	no	—
37	C₄⋊C₈×C₁₀₉	—	2	2	no	—
38	C₄⋊C₈×C₁₀₉	—	2	2	no	—
39	C₄⋊C₈×C₁₀₉	—	2	2	no	—
40	C₂²⋊C₈×C₁₀₉	—	2	2	no	—
41	C₂²⋊C₈×C₁₀₉	—	2	2	no	—
42	C₂²⋊C₈×C₁₀₉	—	2	2	no	—
43	#97×C₁₀₉	—	2	1	no	—
44	C₄×C₈×C₁₀₉	—	2	1	no	—
45	C₄×C₈×C₁₀₉	—	2	2	no	—
46	C₄×C₈×C₁₀₉	D₄	1	2	yes	—
47	C₄×C₈×C₁₀₉	D₄	1	1	yes	characteristic
48	C₄×C₈×C₁₀₉	D₄	1	1	yes	characteristic
49	#41×C₁₀₉	—	2	4	no	—
50	C₂×Q₈×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
51	C₂×D₄×C₁₀₉	—	2	1	no	—
52	C₂×D₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₀₉	—	2	1	no	—
54	C₂²×C₄×C₁₀₉	—	2	1	no	—
55	C₂²×C₄×C₁₀₉	—	2	1	no	—
56	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₀₉	#41	1	1	yes	characteristic
58	C₂²×C₄×C₁₀₉	#41	1	1	yes	characteristic
59	M₁₆×C₁₀₉	—	2	4	no	—
60	C₂×C₈×C₁₀₉	—	2	4	no	—
61	C₂×C₈×C₁₀₉	—	2	2	no	—
62	C₂×C₈×C₁₀₉	—	4	2	no	—
63	C₂×C₈×C₁₀₉	—	4	2	no	—
64	C₂×C₈×C₁₀₉	—	8	2	no	—
65	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	2	yes	—
66	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	2	yes	—
67	C₄⋊C₄×C₁₀₉	—	2	1	no	—
68	C₄⋊C₄×C₁₀₉	—	2	1	no	—
69	C₄⋊C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
70	C₄⋊C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
71	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
72	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
73	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
74	C₄²×C₁₀₉	—	2	1	no	—
75	C₄²×C₁₀₉	—	2	1	no	—
76	C₄²×C₁₀₉	—	2	1	no	—
77	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
81	C₂³×C₁₀₉	—	2	1	no	—
82	C₂³×C₁₀₉	#117	1	1	yes	characteristic
83	Q₈×C₁₀₉	—	2	2	no	—
84	D₄×C₁₀₉	—	2	2	no	—
85	D₄×C₁₀₉	—	4	2	no	—
86	C₂×C₄×C₁₀₉	—	2	2	no	—
87	C₂×C₄×C₁₀₉	—	2	1	no	—
88	C₂×C₄×C₁₀₉	—	2	1	no	—
89	C₂×C₄×C₁₀₉	—	2	1	no	—
90	C₂×C₄×C₁₀₉	—	4	1	no	—
91	C₂×C₄×C₁₀₉	—	4	2	no	—
92	C₂×C₄×C₁₀₉	—	4	2	no	—
93	C₂×C₄×C₁₀₉	—	4	2	no	—
94	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₁₀₉	#121	1	1	yes	characteristic
97	C₂×C₄×C₁₀₉	#121	1	1	yes	characteristic
98	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
99	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
100	C₂×C₄×C₁₀₉	#131	1	4	yes	—
101	C₈×C₁₀₉	—	2	2	no	—
102	C₈×C₁₀₉	—	2	2	no	—
103	C₈×C₁₀₉	—	4	2	no	—
104	C₈×C₁₀₉	—	8	2	no	—
105	C₈×C₁₀₉	—	8	2	no	—
106	C₂²×C₁₀₉	—	2	1	no	—
107	C₂²×C₁₀₉	—	2	1	no	—
108	C₂²×C₁₀₉	—	4	1	no	—
109	C₂²×C₁₀₉	—	4	1	no	—
110	C₂²×C₁₀₉	—	4	1	no	—
111	C₂²×C₁₀₉	#380	1	1	yes	characteristic
112	C₂²×C₁₀₉	#433	1	2	yes	—
113	C₄×C₁₀₉	—	2	1	no	—
114	C₄×C₁₀₉	—	2	1	no	—
115	C₄×C₁₀₉	—	2	1	no	—
116	C₄×C₁₀₉	—	4	1	no	—
117	C₄×C₁₀₉	—	4	1	no	—
118	C₄×C₁₀₉	—	4	1	no	—
119	C₄×C₁₀₉	#408	1	2	yes	—
120	C₄×C₁₀₉	#424	1	2	yes	—
121	C₄×C₁₀₉	#471	1	2	yes	—
122	#8607	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
123	C₂×C₁₀₉	—	2	2	no	—
124	C₂×C₁₀₉	—	8	1	no	—
125	C₂×C₁₀₉	#1798	1	1	yes	characteristic
126	C₂×C₁₀₉	#1814	1	1	yes	characteristic
127	C₂×C₁₀₉	#1815	1	1	yes	characteristic
128	#2922	C₂×C₁₀₉	1	1	yes	characteristic
129	#1564	C₂×C₁₀₉	1	1	yes	characteristic, complemented
130	C₂×#334	C₂×C₁₀₉	1	1	yes	characteristic, complemented
131	#1477	C₂×C₁₀₉	1	2	yes	—
132	#1283	C₂×C₁₀₉	1	2	yes	complemented
133	C₁₀₉	#8607	1	1	yes	characteristic, direct product factor, Sylow
134	C₄×#41	C₂²×C₁₀₉	1	1	yes	characteristic
135	#433	—	2	1	no	—
136	#433	—	2	1	no	—
137	#431	—	2	2	no	—
138	#431	C₂²×C₁₀₉	1	2	yes	—
139	#424	C₄×C₁₀₉	1	2	yes	—
140	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
141	#408	C₄×C₁₀₉	1	2	yes	—
142	C₄×M₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
143	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
144	#335	—	2	1	no	—
145	#334	—	2	2	no	—
146	#334	—	2	1	no	—
147	Q₈⋊C₈	—	2	2	no	—
148	D₄⋊C₈	—	2	2	no	—
149	C₂×#41	C₂×C₄×C₁₀₉	1	1	yes	characteristic
150	C₂×M₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
151	C₂²×C₈	—	2	2	no	—
152	C₂²×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
153	C₄×Q₈	D₄×C₁₀₉	1	1	yes	characteristic
154	C₄×Q₈	Q₈×C₁₀₉	1	1	yes	characteristic
155	C₄×D₄	—	2	1	no	—
156	C₄×D₄	—	2	1	no	—
157	C₄×D₄	D₄×C₁₀₉	1	1	yes	characteristic
158	C₄×D₄	Q₈×C₁₀₉	1	1	yes	characteristic
159	#117	—	2	1	no	—
160	#117	C₂×C₄×C₁₀₉	1	1	yes	characteristic
161	C₂×C₄²	—	2	1	no	—
162	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
163	C₄⋊C₈	—	2	2	no	—
164	C₄⋊C₈	—	2	2	no	—
165	C₄⋊C₈	—	2	2	no	—
166	C₄⋊C₈	—	2	2	no	—
167	C₄⋊C₈	—	2	2	no	—
168	C₂²⋊C₈	—	2	2	no	—
169	C₂²⋊C₈	—	2	2	no	—
170	C₂²⋊C₈	—	2	2	no	—
171	#97	—	2	1	no	—
172	C₄×C₈	—	2	1	no	—
173	C₄×C₈	—	2	2	no	—
174	C₄×C₈	D₄×C₁₀₉	1	2	yes	—
175	C₄×C₈	D₄×C₁₀₉	1	1	yes	characteristic
176	C₄×C₈	D₄×C₁₀₉	1	1	yes	characteristic
177	#41	—	2	4	no	—
178	C₂×Q₈	C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
179	C₂×D₄	—	2	1	no	—
180	C₂×D₄	C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
185	C₂²×C₄	#41×C₁₀₉	1	1	yes	characteristic
186	C₂²×C₄	#41×C₁₀₉	1	1	yes	characteristic
187	M₁₆	—	2	4	no	—
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	4	2	no	—
191	C₂×C₈	—	4	2	no	—
192	C₂×C₈	—	8	2	no	—
193	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	2	yes	—
194	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	2	yes	—
195	C₄⋊C₄	—	2	1	no	—
196	C₄⋊C₄	—	2	1	no	—
197	C₄⋊C₄	C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
198	C₄⋊C₄	C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
199	C₂²⋊C₄	—	2	1	no	—
200	C₂²⋊C₄	—	2	1	no	—
201	C₂²⋊C₄	—	2	1	no	—
202	C₄²	—	2	1	no	—
203	C₄²	—	2	1	no	—
204	C₄²	—	2	1	no	—
205	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
206	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
207	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
208	C₄²	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
209	C₂³	—	2	1	no	—
210	C₂³	#117×C₁₀₉	1	1	yes	characteristic
211	Q₈	—	2	2	no	—
212	D₄	—	2	2	no	—
213	D₄	—	4	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	4	1	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
223	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
224	C₂×C₄	#121×C₁₀₉	1	1	yes	characteristic
225	C₂×C₄	#121×C₁₀₉	1	1	yes	characteristic
226	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	1	yes	characteristic
227	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	1	yes	characteristic
228	C₂×C₄	#131×C₁₀₉	1	4	yes	—
229	C₈	—	2	2	no	—
230	C₈	—	2	2	no	—
231	C₈	—	4	2	no	—
232	C₈	—	8	2	no	—
233	C₈	—	8	2	no	—
234	C₂²	—	2	1	no	—
235	C₂²	—	2	1	no	—
236	C₂²	—	4	1	no	—
237	C₂²	—	4	1	no	—
238	C₂²	—	4	1	no	—
239	C₂²	#380×C₁₀₉	1	1	yes	characteristic
240	C₂²	#433×C₁₀₉	1	2	yes	—
241	C₄	—	2	1	no	—
242	C₄	—	2	1	no	—
243	C₄	—	2	1	no	—
244	C₄	—	4	1	no	—
245	C₄	—	4	1	no	—
246	C₄	—	4	1	no	—
247	C₄	#408×C₁₀₉	1	2	yes	—
248	C₄	#424×C₁₀₉	1	2	yes	—
249	C₄	#471×C₁₀₉	1	2	yes	—
250	C₂	—	2	2	no	—
251	C₂	—	8	1	no	—
252	C₂	C₁₀₉×#1798	1	1	yes	characteristic
253	C₂	C₁₀₉×#1814	1	1	yes	characteristic
254	C₂	C₁₀₉×#1815	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₀₉	#121
commutator	C₂²×C₄	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₉
socle	C₂²×C₁₀₉	#380
prime radical	C₂×#41×C₁₀₉	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₀₉×#1564	C₂	semidirect
3. C₂×#334×C₁₀₉	C₂	semidirect
5. C₁₀₉×#1283	C₂	semidirect
122. #8607	C₁₀₉	direct (N × H)
129. #1564	C₂×C₁₀₉	semidirect
130. C₂×#334	C₂×C₁₀₉	semidirect
132. #1283	C₂×C₁₀₉	semidirect
133. C₁₀₉	#8607	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×#1564	2
C₂×#334×C₁₀₉	2
C₁₀₉×#1477	2
C₁₀₉×#1283	2
#8607	109

Subgroups — C109×#8607 #42126601

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#8607 #42126601