Subgroups — C₁₀₉×#8600 #42126594

Order 27904.

478 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#431×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#1797	C₂	1	2	yes	maximal
3	C₂×#335×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#334×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₀₉×#1277	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
7	#431×C₁₀₉	—	2	2	no	—
8	#431×C₁₀₉	—	2	2	no	—
9	C₂×C₄⋊C₈×C₁₀₉	C₄	1	2	yes	—
10	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
11	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
12	C₂×C₂²⋊C₈×C₁₀₉	—	2	2	no	—
13	C₂×C₂²⋊C₈×C₁₀₉	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
16	#336×C₁₀₉	—	2	2	no	—
17	#335×C₁₀₉	—	2	2	no	—
18	#334×C₁₀₉	—	2	2	no	—
19	C₂³×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
20	C₂³×C₄×C₁₀₉	D₄	1	1	yes	characteristic
21	C₂³×C₄×C₁₀₉	Q₈	1	1	yes	characteristic
22	C₂²×C₈×C₁₀₉	—	2	2	no	—
23	C₂²×C₈×C₁₀₉	—	2	2	no	—
24	C₂²×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₁₀₉	—	2	1	no	—
27	C₂×C₄²×C₁₀₉	—	2	1	no	—
28	C₂×C₄²×C₁₀₉	—	2	1	no	—
29	C₂×C₄²×C₁₀₉	—	2	1	no	—
30	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₀₉	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₀₉	Q₈	1	1	yes	characteristic
33	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₁₀₉	—	2	4	no	—
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	C₂²⋊C₈×C₁₀₉	—	2	4	no	—
40	C₂²⋊C₈×C₁₀₉	—	4	4	no	—
41	C₄×C₈×C₁₀₉	—	2	1	no	—
42	C₄×C₈×C₁₀₉	—	2	1	no	—
43	C₄×C₈×C₁₀₉	D₄	1	2	yes	—
44	C₄×C₈×C₁₀₉	D₄	1	2	yes	—
45	C₂⁴×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₀₉	—	2	2	no	—
47	C₂²×C₄×C₁₀₉	—	2	2	no	—
48	C₂²×C₄×C₁₀₉	—	2	1	no	—
49	C₂²×C₄×C₁₀₉	—	2	1	no	—
50	C₂²×C₄×C₁₀₉	—	2	1	no	—
51	C₂²×C₄×C₁₀₉	—	2	1	no	—
52	C₂²×C₄×C₁₀₉	—	2	1	no	—
53	C₂²×C₄×C₁₀₉	—	2	1	no	—
54	C₂²×C₄×C₁₀₉	—	2	4	no	—
55	C₂²×C₄×C₁₀₉	—	2	2	no	—
56	C₂²×C₄×C₁₀₉	—	2	2	no	—
57	C₂²×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₈×C₁₀₉	—	2	4	no	—
64	C₂×C₈×C₁₀₉	—	2	2	no	—
65	C₂×C₈×C₁₀₉	—	2	4	no	—
66	C₂×C₈×C₁₀₉	—	4	2	no	—
67	C₂×C₈×C₁₀₉	—	4	2	no	—
68	C₂×C₈×C₁₀₉	—	4	2	no	—
69	C₂×C₈×C₁₀₉	—	4	2	no	—
70	C₂×C₈×C₁₀₉	—	4	2	no	—
71	C₂×C₈×C₁₀₉	—	4	2	no	—
72	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	4	yes	—
73	C₄²×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	—	2	1	no	—
78	C₄²×C₁₀₉	—	2	1	no	—
79	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₁₀₉	#41	1	1	yes	characteristic
82	C₄²×C₁₀₉	#41	1	1	yes	characteristic
83	C₂³×C₁₀₉	—	2	1	no	—
84	C₂³×C₁₀₉	—	2	2	no	—
85	C₂³×C₁₀₉	—	2	2	no	—
86	C₂³×C₁₀₉	—	2	1	no	—
87	C₂³×C₁₀₉	#108	1	2	yes	—
88	C₂³×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₁₀₉	—	2	4	no	—
90	C₂×C₄×C₁₀₉	—	2	4	no	—
91	C₂×C₄×C₁₀₉	—	2	1	no	—
92	C₂×C₄×C₁₀₉	—	2	1	no	—
93	C₂×C₄×C₁₀₉	—	2	1	no	—
94	C₂×C₄×C₁₀₉	—	2	1	no	—
95	C₂×C₄×C₁₀₉	—	2	1	no	—
96	C₂×C₄×C₁₀₉	—	2	1	no	—
97	C₂×C₄×C₁₀₉	—	2	2	no	—
98	C₂×C₄×C₁₀₉	—	2	2	no	—
99	C₂×C₄×C₁₀₉	—	4	2	no	—
100	C₂×C₄×C₁₀₉	—	4	2	no	—
101	C₂×C₄×C₁₀₉	—	4	2	no	—
102	C₂×C₄×C₁₀₉	—	4	1	no	—
103	C₂×C₄×C₁₀₉	—	4	1	no	—
104	C₂×C₄×C₁₀₉	#108	1	2	yes	—
105	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₀₉	#117	1	1	yes	characteristic
107	C₂×C₄×C₁₀₉	#121	1	1	yes	characteristic
108	C₂×C₄×C₁₀₉	#121	1	1	yes	characteristic
109	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
110	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
111	C₂×C₄×C₁₀₉	#131	1	4	yes	—
112	C₂×C₄×C₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
113	C₂×C₄×C₁₀₉	#137	1	2	yes	—
114	C₈×C₁₀₉	—	2	4	no	—
115	C₈×C₁₀₉	—	4	2	no	—
116	C₈×C₁₀₉	—	8	2	no	—
117	C₈×C₁₀₉	—	8	2	no	—
118	C₂²×C₁₀₉	—	2	2	no	—
119	C₂²×C₁₀₉	—	2	4	no	—
120	C₂²×C₁₀₉	—	4	1	no	—
121	C₂²×C₁₀₉	—	4	1	no	—
122	C₂²×C₁₀₉	—	4	2	no	—
123	C₂²×C₁₀₉	#380	1	1	yes	characteristic
124	C₂²×C₁₀₉	#424	1	2	yes	—
125	C₂²×C₁₀₉	#428	1	2	yes	—
126	C₂²×C₁₀₉	C₂×#108	1	1	yes	characteristic
127	C₂²×C₁₀₉	C₂×#108	1	1	yes	characteristic
128	C₄×C₁₀₉	—	2	2	no	—
129	C₄×C₁₀₉	—	2	1	no	—
130	C₄×C₁₀₉	—	2	1	no	—
131	C₄×C₁₀₉	—	4	1	no	—
132	C₄×C₁₀₉	—	4	1	no	—
133	C₄×C₁₀₉	#408	1	2	yes	—
134	C₄×C₁₀₉	#433	1	2	yes	—
135	C₄×C₁₀₉	#468	1	2	yes	—
136	C₄×C₁₀₉	#469	1	1	yes	characteristic
137	C₄×C₁₀₉	#470	1	1	yes	characteristic
138	#8600	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
139	C₂×C₁₀₉	—	4	2	no	—
140	C₂×C₁₀₉	#1573	1	2	yes	—
141	C₂×C₁₀₉	#1574	1	2	yes	—
142	C₂×C₁₀₉	#1798	1	1	yes	characteristic
143	C₂×C₁₀₉	#1809	1	1	yes	characteristic
144	C₂×C₁₀₉	#1810	1	1	yes	characteristic
145	C₂×#431	C₂×C₁₀₉	1	1	yes	characteristic
146	#1797	C₂×C₁₀₉	1	2	yes	—
147	C₂×#335	C₂×C₁₀₉	1	1	yes	characteristic, complemented
148	C₂×#334	C₂×C₁₀₉	1	1	yes	characteristic, complemented
149	#1277	C₂×C₁₀₉	1	2	yes	complemented
150	C₁₀₉	#8600	1	1	yes	characteristic, direct product factor, Sylow
151	C₂²×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
152	#431	—	2	2	no	—
153	#431	—	2	2	no	—
154	C₂×C₄⋊C₈	C₄×C₁₀₉	1	2	yes	—
155	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
156	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
157	C₂×C₂²⋊C₈	—	2	2	no	—
158	C₂×C₂²⋊C₈	C₄×C₁₀₉	1	2	yes	—
159	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
160	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
161	#336	—	2	2	no	—
162	#335	—	2	2	no	—
163	#334	—	2	2	no	—
164	C₂³×C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
165	C₂³×C₄	D₄×C₁₀₉	1	1	yes	characteristic
166	C₂³×C₄	Q₈×C₁₀₉	1	1	yes	characteristic
167	C₂²×C₈	—	2	2	no	—
168	C₂²×C₈	—	2	2	no	—
169	C₂²×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
170	C₂²×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
171	C₂×C₄²	—	2	1	no	—
172	C₂×C₄²	—	2	1	no	—
173	C₂×C₄²	—	2	1	no	—
174	C₂×C₄²	—	2	1	no	—
175	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
176	C₂×C₄²	D₄×C₁₀₉	1	1	yes	characteristic
177	C₂×C₄²	Q₈×C₁₀₉	1	1	yes	characteristic
178	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
179	C₄⋊C₈	—	2	4	no	—
180	C₄⋊C₈	—	2	2	no	—
181	C₄⋊C₈	—	2	2	no	—
182	C₄⋊C₈	—	2	2	no	—
183	C₄⋊C₈	—	2	2	no	—
184	C₂²⋊C₈	—	2	4	no	—
185	C₂²⋊C₈	—	4	4	no	—
186	C₄×C₈	—	2	1	no	—
187	C₄×C₈	—	2	1	no	—
188	C₄×C₈	D₄×C₁₀₉	1	2	yes	—
189	C₄×C₈	D₄×C₁₀₉	1	2	yes	—
190	C₂⁴	C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	2	1	no	—
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	—	2	1	no	—
196	C₂²×C₄	—	2	1	no	—
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	—	2	1	no	—
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
203	C₂²×C₄	C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
204	C₂²×C₄	C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
205	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
206	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
207	C₂²×C₄	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	4	2	no	—
212	C₂×C₈	—	4	2	no	—
213	C₂×C₈	—	4	2	no	—
214	C₂×C₈	—	4	2	no	—
215	C₂×C₈	—	4	2	no	—
216	C₂×C₈	—	4	2	no	—
217	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	4	yes	—
218	C₄²	—	2	1	no	—
219	C₄²	—	2	1	no	—
220	C₄²	—	2	1	no	—
221	C₄²	—	2	1	no	—
222	C₄²	—	2	1	no	—
223	C₄²	—	2	1	no	—
224	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
225	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
226	C₄²	#41×C₁₀₉	1	1	yes	characteristic
227	C₄²	#41×C₁₀₉	1	1	yes	characteristic
228	C₂³	—	2	1	no	—
229	C₂³	—	2	2	no	—
230	C₂³	—	2	2	no	—
231	C₂³	—	2	1	no	—
232	C₂³	#108×C₁₀₉	1	2	yes	—
233	C₂³	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	4	2	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	#108×C₁₀₉	1	2	yes	—
250	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
251	C₂×C₄	#117×C₁₀₉	1	1	yes	characteristic
252	C₂×C₄	#121×C₁₀₉	1	1	yes	characteristic
253	C₂×C₄	#121×C₁₀₉	1	1	yes	characteristic
254	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	1	yes	characteristic
255	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	1	yes	characteristic
256	C₂×C₄	#131×C₁₀₉	1	4	yes	—
257	C₂×C₄	AGL(1,ℤ/8)×C₁₀₉	1	2	yes	—
258	C₂×C₄	#137×C₁₀₉	1	2	yes	—
259	C₈	—	2	4	no	—
260	C₈	—	4	2	no	—
261	C₈	—	8	2	no	—
262	C₈	—	8	2	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	4	no	—
265	C₂²	—	4	1	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	4	2	no	—
268	C₂²	#380×C₁₀₉	1	1	yes	characteristic
269	C₂²	#424×C₁₀₉	1	2	yes	—
270	C₂²	#428×C₁₀₉	1	2	yes	—
271	C₂²	C₂×#108×C₁₀₉	1	1	yes	characteristic
272	C₂²	C₂×#108×C₁₀₉	1	1	yes	characteristic
273	C₄	—	2	2	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	4	1	no	—
277	C₄	—	4	1	no	—
278	C₄	#408×C₁₀₉	1	2	yes	—
279	C₄	#433×C₁₀₉	1	2	yes	—
280	C₄	#468×C₁₀₉	1	2	yes	—
281	C₄	#469×C₁₀₉	1	1	yes	characteristic
282	C₄	#470×C₁₀₉	1	1	yes	characteristic
283	C₂	—	4	2	no	—
284	C₂	C₁₀₉×#1573	1	2	yes	—
285	C₂	C₁₀₉×#1574	1	2	yes	—
286	C₂	C₁₀₉×#1798	1	1	yes	characteristic
287	C₂	C₁₀₉×#1809	1	1	yes	characteristic
288	C₂	C₁₀₉×#1810	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₀₉	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₉
socle	C₂³×C₁₀₉	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₀₉	C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₀₉ ◁ C₂²×C₄²×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×#335×C₁₀₉	C₂	semidirect
4. C₂×#334×C₁₀₉	C₂	semidirect
5. C₁₀₉×#1277	C₂	semidirect
138. #8600	C₁₀₉	direct (N × H)
147. C₂×#335	C₂×C₁₀₉	semidirect
148. C₂×#334	C₂×C₁₀₉	semidirect
149. #1277	C₂×C₁₀₉	semidirect
150. C₁₀₉	#8600	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#431×C₁₀₉	2
C₁₀₉×#1797	2
C₂×#335×C₁₀₉	2
C₂×#334×C₁₀₉	2
C₁₀₉×#1277	2
#8600	109

Subgroups — C109×#8600 #42126594

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#8600 #42126594