Subgroups — C₁₀₉×#12745 #42130739

Order 27904.

890 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₉×#3584	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#2145	C₂	1	2	yes	maximal, complemented
3	C₁₀₉×#2138	C₂	1	2	yes	maximal, complemented
4	C₁₀₉×#2113	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₀₉×#1812	C₂	1	1	yes	maximal, characteristic, complemented
6	#585×C₁₀₉	—	2	1	no	—
7	#578×C₁₀₉	—	2	2	no	—
8	#577×C₁₀₉	—	2	4	no	—
9	#576×C₁₀₉	—	2	2	no	—
10	#575×C₁₀₉	—	2	1	no	—
11	#575×C₁₀₉	C₄	1	4	yes	complemented
12	C₂×#135×C₁₀₉	—	2	2	no	—
13	C₂×#135×C₁₀₉	C₂²	1	2	yes	—
14	C₂×#131×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂×M₃₂×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
16	C₂²×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
17	#427×C₁₀₉	C₂²	1	2	yes	complemented
18	#410×C₁₀₉	—	2	1	no	—
19	#360×C₁₀₉	—	2	4	no	—
20	#358×C₁₀₉	—	2	2	no	—
21	#357×C₁₀₉	—	2	2	no	—
22	C₂²⋊C₁₆×C₁₀₉	—	2	2	no	—
23	H(3,ℤ/4)×C₁₀₉	—	2	1	no	—
24	2₋¹⁺⁴×C₁₀₉	—	2	4	no	—
25	2₋¹⁺⁴×C₁₀₉	—	4	1	no	—
26	H(5,𝔽₂)×C₁₀₉	—	2	4	no	—
27	H(5,𝔽₂)×C₁₀₉	—	4	2	no	—
28	H(5,𝔽₂)×C₁₀₉	—	4	1	no	—
29	C₂×#41×C₁₀₉	—	2	4	no	—
30	C₂×#41×C₁₀₉	—	2	2	no	—
31	C₂×#41×C₁₀₉	—	4	4	no	—
32	C₂×#41×C₁₀₉	D₄	1	1	yes	characteristic
33	#137×C₁₀₉	—	2	4	no	—
34	#137×C₁₀₉	—	2	8	no	—
35	AGL(1,ℤ/8)×C₁₀₉	—	2	4	no	—
36	AGL(1,ℤ/8)×C₁₀₉	—	2	8	no	—
37	#135×C₁₀₉	—	2	4	no	—
38	#135×C₁₀₉	—	2	4	no	—
39	#135×C₁₀₉	—	2	4	no	—
40	#135×C₁₀₉	—	2	2	no	—
41	#135×C₁₀₉	—	2	2	no	—
42	#135×C₁₀₉	C₂×C₄	1	4	yes	—
43	#135×C₁₀₉	D₄	1	4	yes	—
44	C₂×Q₁₆×C₁₀₉	—	2	2	no	—
45	C₂×Q₁₆×C₁₀₉	D₄	1	2	yes	—
46	C₂×SD₁₆×C₁₀₉	—	2	4	no	—
47	C₂×SD₁₆×C₁₀₉	—	2	2	no	—
48	C₂×D₈×C₁₀₉	—	2	2	no	—
49	C₂×D₈×C₁₀₉	D₄	1	2	yes	—
50	#131×C₁₀₉	—	2	2	no	—
51	#131×C₁₀₉	—	2	2	no	—
52	C₂×M₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
53	C₂×M₁₆×C₁₀₉	D₄	1	1	yes	characteristic
54	C₂²×C₈×C₁₀₉	D₄	1	2	yes	—
55	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
56	#117×C₁₀₉	—	2	2	no	—
57	M₃₂×C₁₀₉	—	2	2	no	—
58	C₂×C₁₆×C₁₀₉	—	2	1	no	—
59	C₂×C₁₆×C₁₀₉	—	2	2	no	—
60	C₂×C₁₆×C₁₀₉	—	2	1	no	—
61	#107×C₁₀₉	—	2	2	no	—
62	#106×C₁₀₉	—	2	2	no	—
63	#99×C₁₀₉	—	4	2	no	—
64	#41×C₁₀₉	—	2	2	no	—
65	#41×C₁₀₉	—	2	2	no	—
66	#41×C₁₀₉	—	2	8	no	—
67	#41×C₁₀₉	—	2	4	no	—
68	#41×C₁₀₉	—	2	4	no	—
69	#41×C₁₀₉	—	4	4	no	—
70	#41×C₁₀₉	—	4	8	no	—
71	#41×C₁₀₉	—	4	4	no	—
72	#41×C₁₀₉	—	4	4	no	—
73	#41×C₁₀₉	—	4	4	no	—
74	#41×C₁₀₉	—	4	2	no	—
75	#41×C₁₀₉	—	4	2	no	—
76	C₂×Q₈×C₁₀₉	—	2	4	no	—
77	C₂×Q₈×C₁₀₉	—	2	2	no	—
78	C₂×Q₈×C₁₀₉	—	4	4	no	—
79	C₂×Q₈×C₁₀₉	SD₁₆	1	1	yes	characteristic
80	C₂×D₄×C₁₀₉	—	2	4	no	—
81	C₂×D₄×C₁₀₉	—	2	2	no	—
82	C₂×D₄×C₁₀₉	—	4	8	no	—
83	C₂×D₄×C₁₀₉	—	4	4	no	—
84	C₂×D₄×C₁₀₉	—	4	4	no	—
85	C₂×D₄×C₁₀₉	—	4	2	no	—
86	C₂×D₄×C₁₀₉	D₈	1	2	yes	—
87	C₂×D₄×C₁₀₉	SD₁₆	1	1	yes	characteristic
88	C₂²×C₄×C₁₀₉	—	4	4	no	—
89	C₂²×C₄×C₁₀₉	—	4	2	no	—
90	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
91	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
92	Q₁₆×C₁₀₉	—	2	4	no	—
93	Q₁₆×C₁₀₉	—	2	2	no	—
94	Q₁₆×C₁₀₉	C₂²⋊C₄	1	4	yes	—
95	SD₁₆×C₁₀₉	—	2	8	no	—
96	SD₁₆×C₁₀₉	—	2	4	no	—
97	SD₁₆×C₁₀₉	—	2	4	no	—
98	D₈×C₁₀₉	—	2	4	no	—
99	D₈×C₁₀₉	—	2	2	no	—
100	D₈×C₁₀₉	C₂²⋊C₄	1	4	yes	—
101	M₁₆×C₁₀₉	—	2	2	no	—
102	M₁₆×C₁₀₉	—	2	4	no	—
103	C₂×C₈×C₁₀₉	—	2	2	no	—
104	C₂×C₈×C₁₀₉	—	2	2	no	—
105	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	2	yes	—
106	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₁₀₉	C₂×D₄	1	2	yes	—
108	C₂×C₈×C₁₀₉	C₂×D₄	1	2	yes	—
109	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
110	C₄⋊C₄×C₁₀₉	—	4	2	no	—
111	C₂²⋊C₄×C₁₀₉	—	4	2	no	—
112	C₄²×C₁₀₉	—	4	2	no	—
113	C₁₆×C₁₀₉	—	4	1	no	—
114	C₁₆×C₁₀₉	—	4	1	no	—
115	C₂³×C₁₀₉	—	4	4	no	—
116	C₂³×C₁₀₉	—	4	2	no	—
117	C₂³×C₁₀₉	#102	1	2	yes	—
118	C₂³×C₁₀₉	C₂×D₈	1	1	yes	characteristic
119	Q₈×C₁₀₉	—	2	2	no	—
120	Q₈×C₁₀₉	—	2	4	no	—
121	Q₈×C₁₀₉	—	4	4	no	—
122	Q₈×C₁₀₉	—	4	2	no	—
123	D₄×C₁₀₉	—	2	4	no	—
124	D₄×C₁₀₉	—	2	4	no	—
125	D₄×C₁₀₉	—	2	2	no	—
126	D₄×C₁₀₉	—	4	4	no	—
127	D₄×C₁₀₉	—	4	8	no	—
128	D₄×C₁₀₉	—	4	4	no	—
129	D₄×C₁₀₉	—	4	4	no	—
130	D₄×C₁₀₉	—	4	2	no	—
131	C₂×C₄×C₁₀₉	—	2	2	no	—
132	C₂×C₄×C₁₀₉	—	2	2	no	—
133	C₂×C₄×C₁₀₉	—	4	8	no	—
134	C₂×C₄×C₁₀₉	—	4	4	no	—
135	C₂×C₄×C₁₀₉	—	4	2	no	—
136	C₂×C₄×C₁₀₉	—	4	4	no	—
137	C₂×C₄×C₁₀₉	—	4	4	no	—
138	C₂×C₄×C₁₀₉	—	8	2	no	—
139	C₂×C₄×C₁₀₉	#102	1	2	yes	—
140	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
141	C₂×C₄×C₁₀₉	#120	1	2	yes	—
142	C₂×C₄×C₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
143	C₂×C₄×C₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
144	C₈×C₁₀₉	—	2	2	no	—
145	C₈×C₁₀₉	C₂×C₂²⋊C₄	1	2	yes	—
146	C₈×C₁₀₉	#120	1	1	yes	characteristic
147	C₈×C₁₀₉	#120	1	1	yes	characteristic
148	C₂²×C₁₀₉	—	2	2	no	—
149	C₂²×C₁₀₉	—	4	1	no	—
150	C₂²×C₁₀₉	—	4	2	no	—
151	C₂²×C₁₀₉	—	4	4	no	—
152	C₂²×C₁₀₉	—	16	2	no	—
153	C₂²×C₁₀₉	—	16	4	no	—
154	C₂²×C₁₀₉	C₂×#102	1	1	yes	characteristic
155	C₂²×C₁₀₉	#447	1	2	yes	—
156	C₄×C₁₀₉	—	2	2	no	—
157	C₄×C₁₀₉	—	4	4	no	—
158	C₄×C₁₀₉	—	16	2	no	—
159	C₄×C₁₀₉	#379	1	1	yes	characteristic
160	C₄×C₁₀₉	#418	1	1	yes	characteristic
161	C₄×C₁₀₉	#450	1	2	yes	—
162	#12745	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
163	C₂×C₁₀₉	—	2	2	no	—
164	C₂×C₁₀₉	—	2	1	no	—
165	C₂×C₁₀₉	—	4	2	no	—
166	C₂×C₁₀₉	—	8	4	no	—
167	C₂×C₁₀₉	#1786	1	1	yes	characteristic
168	#3584	C₂×C₁₀₉	1	1	yes	characteristic
169	#2145	C₂×C₁₀₉	1	2	yes	complemented
170	#2138	C₂×C₁₀₉	1	2	yes	complemented
171	#2113	C₂×C₁₀₉	1	1	yes	characteristic, complemented
172	#1812	C₂×C₁₀₉	1	1	yes	characteristic, complemented
173	C₁₀₉	#12745	1	1	yes	characteristic, direct product factor, Sylow
174	#585	—	2	1	no	—
175	#578	—	2	2	no	—
176	#577	—	2	4	no	—
177	#576	—	2	2	no	—
178	#575	—	2	1	no	—
179	#575	C₄×C₁₀₉	1	4	yes	complemented
180	C₂×#135	—	2	2	no	—
181	C₂×#135	C₂²×C₁₀₉	1	2	yes	—
182	C₂×#131	C₂²×C₁₀₉	1	1	yes	characteristic
183	C₂×M₃₂	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
184	C₂²×C₁₆	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
185	#427	C₂²×C₁₀₉	1	2	yes	complemented
186	#410	—	2	1	no	—
187	#360	—	2	4	no	—
188	#358	—	2	2	no	—
189	#357	—	2	2	no	—
190	C₂²⋊C₁₆	—	2	2	no	—
191	H(3,ℤ/4)	—	2	1	no	—
192	2₋¹⁺⁴	—	2	4	no	—
193	2₋¹⁺⁴	—	4	1	no	—
194	H(5,𝔽₂)	—	2	4	no	—
195	H(5,𝔽₂)	—	4	2	no	—
196	H(5,𝔽₂)	—	4	1	no	—
197	C₂×#41	—	2	4	no	—
198	C₂×#41	—	2	2	no	—
199	C₂×#41	—	4	4	no	—
200	C₂×#41	D₄×C₁₀₉	1	1	yes	characteristic
201	#137	—	2	4	no	—
202	#137	—	2	8	no	—
203	AGL(1,ℤ/8)	—	2	4	no	—
204	AGL(1,ℤ/8)	—	2	8	no	—
205	#135	—	2	4	no	—
206	#135	—	2	4	no	—
207	#135	—	2	4	no	—
208	#135	—	2	2	no	—
209	#135	—	2	2	no	—
210	#135	C₂×C₄×C₁₀₉	1	4	yes	—
211	#135	D₄×C₁₀₉	1	4	yes	—
212	C₂×Q₁₆	—	2	2	no	—
213	C₂×Q₁₆	D₄×C₁₀₉	1	2	yes	—
214	C₂×SD₁₆	—	2	4	no	—
215	C₂×SD₁₆	—	2	2	no	—
216	C₂×D₈	—	2	2	no	—
217	C₂×D₈	D₄×C₁₀₉	1	2	yes	—
218	#131	—	2	2	no	—
219	#131	—	2	2	no	—
220	C₂×M₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
221	C₂×M₁₆	D₄×C₁₀₉	1	1	yes	characteristic
222	C₂²×C₈	D₄×C₁₀₉	1	2	yes	—
223	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
224	#117	—	2	2	no	—
225	M₃₂	—	2	2	no	—
226	C₂×C₁₆	—	2	1	no	—
227	C₂×C₁₆	—	2	1	no	—
228	C₂×C₁₆	—	2	2	no	—
229	#107	—	2	2	no	—
230	#106	—	2	2	no	—
231	#99	—	4	2	no	—
232	#41	—	2	2	no	—
233	#41	—	2	2	no	—
234	#41	—	2	8	no	—
235	#41	—	2	4	no	—
236	#41	—	2	4	no	—
237	#41	—	4	4	no	—
238	#41	—	4	8	no	—
239	#41	—	4	4	no	—
240	#41	—	4	4	no	—
241	#41	—	4	4	no	—
242	#41	—	4	2	no	—
243	#41	—	4	2	no	—
244	C₂×Q₈	—	2	4	no	—
245	C₂×Q₈	—	2	2	no	—
246	C₂×Q₈	—	4	4	no	—
247	C₂×Q₈	SD₁₆×C₁₀₉	1	1	yes	characteristic
248	C₂×D₄	—	2	4	no	—
249	C₂×D₄	—	2	2	no	—
250	C₂×D₄	—	4	8	no	—
251	C₂×D₄	—	4	4	no	—
252	C₂×D₄	—	4	4	no	—
253	C₂×D₄	—	4	2	no	—
254	C₂×D₄	D₈×C₁₀₉	1	2	yes	—
255	C₂×D₄	SD₁₆×C₁₀₉	1	1	yes	characteristic
256	C₂²×C₄	—	4	4	no	—
257	C₂²×C₄	—	4	2	no	—
258	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	2	yes	—
259	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
260	Q₁₆	—	2	4	no	—
261	Q₁₆	—	2	2	no	—
262	Q₁₆	C₂²⋊C₄×C₁₀₉	1	4	yes	—
263	SD₁₆	—	2	8	no	—
264	SD₁₆	—	2	4	no	—
265	SD₁₆	—	2	4	no	—
266	D₈	—	2	4	no	—
267	D₈	—	2	2	no	—
268	D₈	C₂²⋊C₄×C₁₀₉	1	4	yes	—
269	M₁₆	—	2	2	no	—
270	M₁₆	—	2	4	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	2	yes	—
274	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
275	C₂×C₈	C₂×D₄×C₁₀₉	1	2	yes	—
276	C₂×C₈	C₂×D₄×C₁₀₉	1	2	yes	—
277	C₂×C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
278	C₄⋊C₄	—	4	2	no	—
279	C₂²⋊C₄	—	4	2	no	—
280	C₄²	—	4	2	no	—
281	C₁₆	—	4	1	no	—
282	C₁₆	—	4	1	no	—
283	C₂³	—	4	4	no	—
284	C₂³	—	4	2	no	—
285	C₂³	#102×C₁₀₉	1	2	yes	—
286	C₂³	C₂×D₈×C₁₀₉	1	1	yes	characteristic
287	Q₈	—	2	2	no	—
288	Q₈	—	2	4	no	—
289	Q₈	—	4	4	no	—
290	Q₈	—	4	2	no	—
291	D₄	—	2	4	no	—
292	D₄	—	2	4	no	—
293	D₄	—	2	2	no	—
294	D₄	—	4	4	no	—
295	D₄	—	4	8	no	—
296	D₄	—	4	4	no	—
297	D₄	—	4	4	no	—
298	D₄	—	4	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	4	8	no	—
302	C₂×C₄	—	4	4	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	4	no	—
305	C₂×C₄	—	4	4	no	—
306	C₂×C₄	—	8	2	no	—
307	C₂×C₄	#102×C₁₀₉	1	2	yes	—
308	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
309	C₂×C₄	#120×C₁₀₉	1	2	yes	—
310	C₂×C₄	C₂×SD₁₆×C₁₀₉	1	1	yes	characteristic
311	C₂×C₄	AGL(1,ℤ/8)×C₁₀₉	1	2	yes	—
312	C₈	—	2	2	no	—
313	C₈	C₂×C₂²⋊C₄×C₁₀₉	1	2	yes	—
314	C₈	#120×C₁₀₉	1	1	yes	characteristic
315	C₈	#120×C₁₀₉	1	1	yes	characteristic
316	C₂²	—	2	2	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	4	2	no	—
319	C₂²	—	4	4	no	—
320	C₂²	—	16	2	no	—
321	C₂²	—	16	4	no	—
322	C₂²	C₂×#102×C₁₀₉	1	1	yes	characteristic
323	C₂²	#447×C₁₀₉	1	2	yes	—
324	C₄	—	2	2	no	—
325	C₄	—	4	4	no	—
326	C₄	—	16	2	no	—
327	C₄	#379×C₁₀₉	1	1	yes	characteristic
328	C₄	#418×C₁₀₉	1	1	yes	characteristic
329	C₄	#450×C₁₀₉	1	2	yes	—
330	C₂	—	2	2	no	—
331	C₂	—	2	1	no	—
332	C₂	—	4	2	no	—
333	C₂	—	8	4	no	—
334	C₂	C₁₀₉×#1786	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₀₉	#418
commutator	C₂×C₈	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₀₉
socle	C₂×C₁₀₉	#1786
prime radical	C₁₀₉×#3584	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₀₉ ◁ C₂²×C₄×C₁₀₉ ◁ C₂×#131×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₀₉×#2145	C₂	semidirect
3. C₁₀₉×#2138	C₂	semidirect
4. C₁₀₉×#2113	C₂	semidirect
5. C₁₀₉×#1812	C₂	semidirect
11. #575×C₁₀₉	C₄	semidirect
15. C₂×M₃₂×C₁₀₉	C₂²	semidirect
16. C₂²×C₁₆×C₁₀₉	C₂²	semidirect
17. #427×C₁₀₉	C₂²	semidirect
162. #12745	C₁₀₉	direct (N × H)
169. #2145	C₂×C₁₀₉	semidirect
170. #2138	C₂×C₁₀₉	semidirect
171. #2113	C₂×C₁₀₉	semidirect
172. #1812	C₂×C₁₀₉	semidirect
173. C₁₀₉	#12745	direct (N × H)
179. #575	C₄×C₁₀₉	semidirect
183. C₂×M₃₂	C₂²×C₁₀₉	semidirect
184. C₂²×C₁₆	C₂²×C₁₀₉	semidirect
185. #427	C₂²×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₁₀₉×#3584	2
C₁₀₉×#2145	2
C₁₀₉×#2138	2
C₁₀₉×#2113	2
C₁₀₉×#1812	2
#12745	109

Subgroups — C109×#12745 #42130739

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#12745 #42130739