Subgroups — C₁₀₉×#12999 #42130993

Order 27904.

564 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#469×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#2911	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₀₉×#2140	C₂	1	2	yes	maximal, complemented
4	C₂×#349×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₀₉×#1386	C₂	1	2	yes	maximal, complemented
6	C₂²×D₈×C₁₀₉	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
8	C₂×#121×C₁₀₉	—	2	1	no	—
9	C₂×#117×C₁₀₉	—	2	1	no	—
10	C₂×M₃₂×C₁₀₉	C₂²	1	2	yes	—
11	#469×C₁₀₉	—	2	2	no	—
12	#469×C₁₀₉	C₄	1	4	yes	complemented
13	#428×C₁₀₉	—	2	4	no	—
14	#427×C₁₀₉	—	2	1	no	—
15	#427×C₁₀₉	C₂²	1	2	yes	—
16	C₂×#106×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₂×#102×C₁₀₉	—	2	1	no	—
18	#357×C₁₀₉	—	2	4	no	—
19	#349×C₁₀₉	—	2	2	no	—
20	C₂²×D₄×C₁₀₉	—	2	1	no	—
21	C₂²×D₄×C₁₀₉	—	4	1	no	—
22	C₂³×C₄×C₁₀₉	D₄	1	1	yes	characteristic
23	C₂×D₈×C₁₀₉	—	2	2	no	—
24	C₂×D₈×C₁₀₉	—	2	1	no	—
25	C₂×D₈×C₁₀₉	—	2	2	no	—
26	C₂×D₈×C₁₀₉	C₂×C₄	1	2	yes	complemented
27	C₂×M₁₆×C₁₀₉	—	2	2	no	—
28	C₂×M₁₆×C₁₀₉	—	2	2	no	—
29	C₂×M₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
30	C₂×M₁₆×C₁₀₉	D₄	1	2	yes	—
31	C₂²×C₈×C₁₀₉	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
33	#121×C₁₀₉	—	2	4	no	—
34	#121×C₁₀₉	—	4	2	no	—
35	#117×C₁₀₉	—	2	2	no	—
36	#117×C₁₀₉	—	2	2	no	—
37	#117×C₁₀₉	—	4	1	no	—
38	#117×C₁₀₉	—	4	1	no	—
39	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
40	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
41	C₂×C₂²⋊C₄×C₁₀₉	—	4	1	no	—
42	C₂×C₂²⋊C₄×C₁₀₉	—	4	1	no	—
43	C₂×C₄²×C₁₀₉	—	4	1	no	—
44	M₃₂×C₁₀₉	—	2	2	no	—
45	M₃₂×C₁₀₉	—	2	2	no	—
46	C₂×C₁₆×C₁₀₉	—	2	2	no	—
47	#107×C₁₀₉	—	2	2	no	—
48	#107×C₁₀₉	D₄	1	4	yes	complemented
49	#106×C₁₀₉	—	2	2	no	—
50	#106×C₁₀₉	—	2	1	no	—
51	#106×C₁₀₉	—	2	2	no	—
52	#106×C₁₀₉	C₂×C₄	1	2	yes	complemented
53	#102×C₁₀₉	—	2	4	no	—
54	C₂⁴×C₁₀₉	—	4	1	no	—
55	C₂⁴×C₁₀₉	D₈	1	1	yes	characteristic
56	C₂×D₄×C₁₀₉	—	2	2	no	—
57	C₂×D₄×C₁₀₉	—	4	4	no	—
58	C₂×D₄×C₁₀₉	—	4	1	no	—
59	C₂×D₄×C₁₀₉	—	4	1	no	—
60	C₂×D₄×C₁₀₉	—	4	1	no	—
61	C₂×D₄×C₁₀₉	—	4	1	no	—
62	C₂×D₄×C₁₀₉	—	4	1	no	—
63	C₂×D₄×C₁₀₉	—	8	4	no	—
64	C₂²×C₄×C₁₀₉	—	2	1	no	—
65	C₂²×C₄×C₁₀₉	—	2	1	no	—
66	C₂²×C₄×C₁₀₉	—	2	1	no	—
67	C₂²×C₄×C₁₀₉	—	2	1	no	—
68	C₂²×C₄×C₁₀₉	—	4	2	no	—
69	C₂²×C₄×C₁₀₉	—	4	1	no	—
70	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
71	C₂²×C₄×C₁₀₉	D₈	1	1	yes	characteristic
72	C₂²×C₄×C₁₀₉	SD₁₆	1	2	yes	—
73	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
74	D₈×C₁₀₉	—	2	4	no	—
75	D₈×C₁₀₉	—	4	2	no	—
76	M₁₆×C₁₀₉	—	2	4	no	—
77	M₁₆×C₁₀₉	—	2	4	no	—
78	C₂×C₈×C₁₀₉	—	2	1	no	—
79	C₂×C₈×C₁₀₉	—	2	1	no	—
80	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	2	yes	—
81	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₁₀₉	C₂×D₄	1	2	yes	—
83	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₁₀₉	#41	1	2	yes	—
85	C₄⋊C₄×C₁₀₉	—	2	2	no	—
86	C₄⋊C₄×C₁₀₉	—	2	4	no	—
87	C₄⋊C₄×C₁₀₉	—	4	2	no	—
88	C₄⋊C₄×C₁₀₉	—	4	1	no	—
89	C₂²⋊C₄×C₁₀₉	—	4	2	no	—
90	C₂²⋊C₄×C₁₀₉	—	4	2	no	—
91	C₂²⋊C₄×C₁₀₉	—	4	4	no	—
92	C₄²×C₁₀₉	—	4	2	no	—
93	C₄²×C₁₀₉	—	4	2	no	—
94	C₁₆×C₁₀₉	—	4	2	no	—
95	C₂³×C₁₀₉	—	2	1	no	—
96	C₂³×C₁₀₉	—	2	1	no	—
97	C₂³×C₁₀₉	—	4	2	no	—
98	C₂³×C₁₀₉	—	4	2	no	—
99	C₂³×C₁₀₉	—	8	2	no	—
100	C₂³×C₁₀₉	—	8	2	no	—
101	C₂³×C₁₀₉	—	8	1	no	—
102	C₂³×C₁₀₉	—	8	1	no	—
103	C₂³×C₁₀₉	#102	1	2	yes	—
104	C₂³×C₁₀₉	C₂×D₈	1	1	yes	characteristic
105	D₄×C₁₀₉	—	4	2	no	—
106	D₄×C₁₀₉	—	4	2	no	—
107	D₄×C₁₀₉	—	8	8	no	—
108	D₄×C₁₀₉	—	8	1	no	—
109	D₄×C₁₀₉	—	8	1	no	—
110	C₂×C₄×C₁₀₉	—	2	1	no	—
111	C₂×C₄×C₁₀₉	—	2	1	no	—
112	C₂×C₄×C₁₀₉	—	2	2	no	—
113	C₂×C₄×C₁₀₉	—	2	2	no	—
114	C₂×C₄×C₁₀₉	—	2	2	no	—
115	C₂×C₄×C₁₀₉	—	2	2	no	—
116	C₂×C₄×C₁₀₉	—	4	2	no	—
117	C₂×C₄×C₁₀₉	—	4	4	no	—
118	C₂×C₄×C₁₀₉	—	8	2	no	—
119	C₂×C₄×C₁₀₉	—	8	4	no	—
120	C₂×C₄×C₁₀₉	#102	1	2	yes	—
121	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₁₀₉	#120	1	1	yes	characteristic
123	C₂×C₄×C₁₀₉	#123	1	1	yes	characteristic
124	C₂×C₄×C₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
125	C₂×C₄×C₁₀₉	AGL(1,ℤ/8)	1	1	yes	characteristic
126	C₂×C₄×C₁₀₉	#137	1	1	yes	characteristic
127	C₈×C₁₀₉	—	2	2	no	—
128	C₈×C₁₀₉	C₄×D₄	1	2	yes	—
129	C₈×C₁₀₉	#121	1	2	yes	—
130	C₂²×C₁₀₉	—	2	2	no	—
131	C₂²×C₁₀₉	—	2	2	no	—
132	C₂²×C₁₀₉	—	4	2	no	—
133	C₂²×C₁₀₉	—	4	2	no	—
134	C₂²×C₁₀₉	—	8	1	no	—
135	C₂²×C₁₀₉	—	8	2	no	—
136	C₂²×C₁₀₉	—	8	2	no	—
137	C₂²×C₁₀₉	—	8	2	no	—
138	C₂²×C₁₀₉	—	16	2	no	—
139	C₂²×C₁₀₉	—	16	2	no	—
140	C₂²×C₁₀₉	C₂×#102	1	1	yes	characteristic
141	C₂²×C₁₀₉	#447	1	1	yes	characteristic
142	C₂²×C₁₀₉	#480	1	1	yes	characteristic
143	C₄×C₁₀₉	—	2	2	no	—
144	C₄×C₁₀₉	—	8	2	no	—
145	C₄×C₁₀₉	—	8	4	no	—
146	C₄×C₁₀₉	#386	1	1	yes	characteristic
147	C₄×C₁₀₉	#417	1	1	yes	characteristic
148	C₄×C₁₀₉	#448	1	1	yes	characteristic
149	C₄×C₁₀₉	#481	1	1	yes	characteristic
150	#12999	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
151	C₂×C₁₀₉	—	2	1	no	—
152	C₂×C₁₀₉	—	2	1	no	—
153	C₂×C₁₀₉	—	4	2	no	—
154	C₂×C₁₀₉	—	16	2	no	—
155	C₂×C₁₀₉	#1874	1	1	yes	characteristic
156	C₂×C₁₀₉	#2190	1	2	yes	—
157	C₂×#469	C₂×C₁₀₉	1	1	yes	characteristic
158	#2911	C₂×C₁₀₉	1	1	yes	characteristic, complemented
159	#2140	C₂×C₁₀₉	1	2	yes	complemented
160	C₂×#349	C₂×C₁₀₉	1	1	yes	characteristic, complemented
161	#1386	C₂×C₁₀₉	1	2	yes	complemented
162	C₁₀₉	#12999	1	1	yes	characteristic, direct product factor, Sylow
163	C₂²×D₈	C₂²×C₁₀₉	1	1	yes	characteristic
164	C₂²×M₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
165	C₂×#121	—	2	1	no	—
166	C₂×#117	—	2	1	no	—
167	C₂×M₃₂	C₂²×C₁₀₉	1	2	yes	—
168	#469	—	2	2	no	—
169	#469	C₄×C₁₀₉	1	4	yes	complemented
170	#428	—	2	4	no	—
171	#427	—	2	1	no	—
172	#427	C₂²×C₁₀₉	1	2	yes	—
173	C₂×#106	C₂²×C₁₀₉	1	1	yes	characteristic
174	C₂×#102	—	2	1	no	—
175	#357	—	2	4	no	—
176	#349	—	2	2	no	—
177	C₂²×D₄	—	2	1	no	—
178	C₂²×D₄	—	4	1	no	—
179	C₂³×C₄	D₄×C₁₀₉	1	1	yes	characteristic
180	C₂×D₈	—	2	2	no	—
181	C₂×D₈	—	2	1	no	—
182	C₂×D₈	—	2	2	no	—
183	C₂×D₈	C₂×C₄×C₁₀₉	1	2	yes	complemented
184	C₂×M₁₆	—	2	2	no	—
185	C₂×M₁₆	—	2	2	no	—
186	C₂×M₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
187	C₂×M₁₆	D₄×C₁₀₉	1	2	yes	—
188	C₂²×C₈	D₄×C₁₀₉	1	1	yes	characteristic
189	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
190	#121	—	2	4	no	—
191	#121	—	4	2	no	—
192	#117	—	2	2	no	—
193	#117	—	2	2	no	—
194	#117	—	4	1	no	—
195	#117	—	4	1	no	—
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₄⋊C₄	—	2	1	no	—
198	C₂×C₂²⋊C₄	—	4	1	no	—
199	C₂×C₂²⋊C₄	—	4	1	no	—
200	C₂×C₄²	—	4	1	no	—
201	M₃₂	—	2	2	no	—
202	M₃₂	—	2	2	no	—
203	C₂×C₁₆	—	2	2	no	—
204	#107	—	2	2	no	—
205	#107	D₄×C₁₀₉	1	4	yes	complemented
206	#106	—	2	2	no	—
207	#106	—	2	1	no	—
208	#106	—	2	2	no	—
209	#106	C₂×C₄×C₁₀₉	1	2	yes	complemented
210	#102	—	2	4	no	—
211	C₂⁴	—	4	1	no	—
212	C₂⁴	D₈×C₁₀₉	1	1	yes	characteristic
213	C₂×D₄	—	2	2	no	—
214	C₂×D₄	—	4	4	no	—
215	C₂×D₄	—	4	1	no	—
216	C₂×D₄	—	4	1	no	—
217	C₂×D₄	—	4	1	no	—
218	C₂×D₄	—	4	1	no	—
219	C₂×D₄	—	4	1	no	—
220	C₂×D₄	—	8	4	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	2	1	no	—
223	C₂²×C₄	—	2	1	no	—
224	C₂²×C₄	—	2	1	no	—
225	C₂²×C₄	—	4	2	no	—
226	C₂²×C₄	—	4	1	no	—
227	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	2	yes	—
228	C₂²×C₄	D₈×C₁₀₉	1	1	yes	characteristic
229	C₂²×C₄	SD₁₆×C₁₀₉	1	2	yes	—
230	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
231	D₈	—	2	4	no	—
232	D₈	—	4	2	no	—
233	M₁₆	—	2	4	no	—
234	M₁₆	—	2	4	no	—
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	—	2	1	no	—
237	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	2	yes	—
238	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
239	C₂×C₈	C₂×D₄×C₁₀₉	1	2	yes	—
240	C₂×C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
241	C₂×C₈	#41×C₁₀₉	1	2	yes	—
242	C₄⋊C₄	—	2	2	no	—
243	C₄⋊C₄	—	2	4	no	—
244	C₄⋊C₄	—	4	2	no	—
245	C₄⋊C₄	—	4	1	no	—
246	C₂²⋊C₄	—	4	2	no	—
247	C₂²⋊C₄	—	4	2	no	—
248	C₂²⋊C₄	—	4	4	no	—
249	C₄²	—	4	2	no	—
250	C₄²	—	4	2	no	—
251	C₁₆	—	4	2	no	—
252	C₂³	—	2	1	no	—
253	C₂³	—	2	1	no	—
254	C₂³	—	4	2	no	—
255	C₂³	—	4	2	no	—
256	C₂³	—	8	2	no	—
257	C₂³	—	8	2	no	—
258	C₂³	—	8	1	no	—
259	C₂³	—	8	1	no	—
260	C₂³	#102×C₁₀₉	1	2	yes	—
261	C₂³	C₂×D₈×C₁₀₉	1	1	yes	characteristic
262	D₄	—	4	2	no	—
263	D₄	—	4	2	no	—
264	D₄	—	8	8	no	—
265	D₄	—	8	1	no	—
266	D₄	—	8	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	8	2	no	—
276	C₂×C₄	—	8	4	no	—
277	C₂×C₄	#102×C₁₀₉	1	2	yes	—
278	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
279	C₂×C₄	#120×C₁₀₉	1	1	yes	characteristic
280	C₂×C₄	#123×C₁₀₉	1	1	yes	characteristic
281	C₂×C₄	C₂×SD₁₆×C₁₀₉	1	1	yes	characteristic
282	C₂×C₄	AGL(1,ℤ/8)×C₁₀₉	1	1	yes	characteristic
283	C₂×C₄	#137×C₁₀₉	1	1	yes	characteristic
284	C₈	—	2	2	no	—
285	C₈	C₄×D₄×C₁₀₉	1	2	yes	—
286	C₈	#121×C₁₀₉	1	2	yes	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	4	2	no	—
290	C₂²	—	4	2	no	—
291	C₂²	—	8	1	no	—
292	C₂²	—	8	2	no	—
293	C₂²	—	8	2	no	—
294	C₂²	—	8	2	no	—
295	C₂²	—	16	2	no	—
296	C₂²	—	16	2	no	—
297	C₂²	C₂×#102×C₁₀₉	1	1	yes	characteristic
298	C₂²	#447×C₁₀₉	1	1	yes	characteristic
299	C₂²	#480×C₁₀₉	1	1	yes	characteristic
300	C₄	—	2	2	no	—
301	C₄	—	8	2	no	—
302	C₄	—	8	4	no	—
303	C₄	#386×C₁₀₉	1	1	yes	characteristic
304	C₄	#417×C₁₀₉	1	1	yes	characteristic
305	C₄	#448×C₁₀₉	1	1	yes	characteristic
306	C₄	#481×C₁₀₉	1	1	yes	characteristic
307	C₂	—	2	1	no	—
308	C₂	—	2	1	no	—
309	C₂	—	4	2	no	—
310	C₂	—	16	2	no	—
311	C₂	C₁₀₉×#1874	1	1	yes	characteristic
312	C₂	C₁₀₉×#2190	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₀₉	#447
commutator	C₂×C₈	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₀₉
prime radical	C₂×#469×C₁₀₉	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₂²×C₄×C₁₀₉ ◁ C₂²×M₁₆×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₀₉×#2911	C₂	semidirect
3. C₁₀₉×#2140	C₂	semidirect
4. C₂×#349×C₁₀₉	C₂	semidirect
5. C₁₀₉×#1386	C₂	semidirect
12. #469×C₁₀₉	C₄	semidirect
26. C₂×D₈×C₁₀₉	C₂×C₄	semidirect
48. #107×C₁₀₉	D₄	semidirect
52. #106×C₁₀₉	C₂×C₄	semidirect
150. #12999	C₁₀₉	direct (N × H)
158. #2911	C₂×C₁₀₉	semidirect
159. #2140	C₂×C₁₀₉	semidirect
160. C₂×#349	C₂×C₁₀₉	semidirect
161. #1386	C₂×C₁₀₉	semidirect
162. C₁₀₉	#12999	direct (N × H)
169. #469	C₄×C₁₀₉	semidirect
183. C₂×D₈	C₂×C₄×C₁₀₉	semidirect
205. #107	D₄×C₁₀₉	semidirect
209. #106	C₂×C₄×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#469×C₁₀₉	2
C₁₀₉×#2911	2
C₁₀₉×#2140	2
C₂×#349×C₁₀₉	2
C₁₀₉×#1386	2
#12999	109

Subgroups — C109×#12999 #42130993

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#12999 #42130993