Subgroups — C₂×C₁₀₉×#2643 #42152559

Order 27904.

2546 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₉×#2643	C₂	1	16	yes	maximal, direct product factor
2	C₂×#401×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#400×C₁₀₉	C₂	1	3	yes	maximal, complemented
4	C₂×#397×C₁₀₉	C₂	1	3	yes	maximal, complemented
5	C₂×#396×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#386×C₁₀₉	C₂	1	3	yes	maximal
7	C₂×#385×C₁₀₉	C₂	1	3	yes	maximal, complemented
8	C₂²×#95×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
10	C₂⁴×C₄×C₁₀₉	C₂²	1	3	yes	—
11	C₂²×C₄⋊C₄×C₁₀₉	C₂²	1	3	yes	—
12	C₂²×C₄⋊C₄×C₁₀₉	C₂²	1	3	yes	—
13	C₂²×C₂²⋊C₄×C₁₀₉	C₂²	1	3	yes	—
14	C₂²×C₂²⋊C₄×C₁₀₉	C₂²	1	3	yes	—
15	C₂²×C₂²⋊C₄×C₁₀₉	C₂²	1	3	yes	—
16	#401×C₁₀₉	C₂²	1	8	yes	complemented
17	#400×C₁₀₉	C₂²	1	24	yes	complemented
18	#397×C₁₀₉	C₂²	1	24	yes	complemented
19	#396×C₁₀₉	C₂²	1	8	yes	complemented
20	#386×C₁₀₉	C₂²	1	24	yes	—
21	#385×C₁₀₉	C₂²	1	24	yes	complemented
22	C₂×#95×C₁₀₉	C₂²	1	1	yes	characteristic
23	C₂×#95×C₁₀₉	C₂²	1	3	yes	—
24	C₂×#95×C₁₀₉	C₂²	1	3	yes	—
25	C₂×#95×C₁₀₉	C₂²	1	3	yes	—
26	C₂×#95×C₁₀₉	C₂²	1	3	yes	—
27	C₂×#95×C₁₀₉	C₂²	1	3	yes	—
28	C₂×#95×C₁₀₉	C₂²	1	8	yes	complemented
29	C₂⁵×C₁₀₉	C₂³	1	1	yes	characteristic
30	C₂⁵×C₁₀₉	C₂³	1	1	yes	characteristic
31	C₂²×D₄×C₁₀₉	—	2	12	no	—
32	C₂²×D₄×C₁₀₉	C₂³	1	4	yes	—
33	C₂³×C₄×C₁₀₉	—	2	12	no	—
34	C₂³×C₄×C₁₀₉	—	4	6	no	—
35	C₂³×C₄×C₁₀₉	—	4	6	no	—
36	C₂³×C₄×C₁₀₉	C₂³	1	3	yes	—
37	C₂³×C₄×C₁₀₉	C₂³	1	3	yes	—
38	C₂³×C₄×C₁₀₉	C₂³	1	3	yes	—
39	C₂³×C₄×C₁₀₉	C₂³	1	3	yes	—
40	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
41	C₂³×C₄×C₁₀₉	C₂³	1	12	yes	—
42	C₂×C₄⋊C₄×C₁₀₉	—	4	6	no	—
43	C₂×C₄⋊C₄×C₁₀₉	—	4	6	no	—
44	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	12	yes	—
45	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	12	yes	—
46	C₂×C₂²⋊C₄×C₁₀₉	—	2	12	no	—
47	C₂×C₂²⋊C₄×C₁₀₉	—	4	6	no	—
48	C₂×C₂²⋊C₄×C₁₀₉	—	4	6	no	—
49	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	12	yes	—
50	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	12	yes	—
51	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	12	yes	—
52	#95×C₁₀₉	C₂³	1	4	yes	—
53	#95×C₁₀₉	C₂³	1	12	yes	—
54	#95×C₁₀₉	C₂³	1	12	yes	—
55	#95×C₁₀₉	C₂³	1	12	yes	—
56	#95×C₁₀₉	C₂³	1	12	yes	—
57	#95×C₁₀₉	C₂³	1	12	yes	—
58	C₂⁴×C₁₀₉	—	2	6	no	—
59	C₂⁴×C₁₀₉	—	2	2	no	—
60	C₂⁴×C₁₀₉	—	2	6	no	—
61	C₂⁴×C₁₀₉	—	2	6	no	—
62	C₂⁴×C₁₀₉	—	2	2	no	—
63	C₂⁴×C₁₀₉	#41	1	12	yes	—
64	C₂⁴×C₁₀₉	C₂⁴	1	2	yes	—
65	C₂⁴×C₁₀₉	C₂⁴	1	2	yes	—
66	C₂⁴×C₁₀₉	C₂⁴	1	1	yes	characteristic
67	C₂×D₄×C₁₀₉	—	2	24	no	—
68	C₂×D₄×C₁₀₉	—	4	16	no	—
69	C₂²×C₄×C₁₀₉	—	2	6	no	—
70	C₂²×C₄×C₁₀₉	—	2	6	no	—
71	C₂²×C₄×C₁₀₉	—	2	24	no	—
72	C₂²×C₄×C₁₀₉	—	2	6	no	—
73	C₂²×C₄×C₁₀₉	—	2	6	no	—
74	C₂²×C₄×C₁₀₉	—	2	6	no	—
75	C₂²×C₄×C₁₀₉	—	2	6	no	—
76	C₂²×C₄×C₁₀₉	—	2	6	no	—
77	C₂²×C₄×C₁₀₉	—	2	6	no	—
78	C₂²×C₄×C₁₀₉	—	2	6	no	—
79	C₂²×C₄×C₁₀₉	—	2	6	no	—
80	C₂²×C₄×C₁₀₉	—	2	6	no	—
81	C₂²×C₄×C₁₀₉	—	2	6	no	—
82	C₂²×C₄×C₁₀₉	—	2	6	no	—
83	C₂²×C₄×C₁₀₉	—	4	12	no	—
84	C₂²×C₄×C₁₀₉	—	4	12	no	—
85	C₂²×C₄×C₁₀₉	—	4	48	no	—
86	C₂²×C₄×C₁₀₉	C₂⁴	1	2	yes	—
87	C₂²×C₄×C₁₀₉	C₂⁴	1	6	yes	—
88	C₂²×C₄×C₁₀₉	C₂⁴	1	6	yes	—
89	C₂²×C₄×C₁₀₉	C₂⁴	1	6	yes	—
90	C₂²×C₄×C₁₀₉	C₂⁴	1	6	yes	—
91	C₄⋊C₄×C₁₀₉	—	4	12	no	—
92	C₄⋊C₄×C₁₀₉	—	4	12	no	—
93	C₂²⋊C₄×C₁₀₉	—	2	24	no	—
94	C₂²⋊C₄×C₁₀₉	—	4	12	no	—
95	C₂²⋊C₄×C₁₀₉	—	4	12	no	—
96	C₂³×C₁₀₉	—	2	24	no	—
97	C₂³×C₁₀₉	—	2	24	no	—
98	C₂³×C₁₀₉	—	2	6	no	—
99	C₂³×C₁₀₉	—	2	2	no	—
100	C₂³×C₁₀₉	—	2	6	no	—
101	C₂³×C₁₀₉	—	2	6	no	—
102	C₂³×C₁₀₉	—	2	2	no	—
103	C₂³×C₁₀₉	—	4	4	no	—
104	C₂³×C₁₀₉	—	4	12	no	—
105	C₂³×C₁₀₉	—	4	12	no	—
106	C₂³×C₁₀₉	#126	1	16	yes	—
107	C₂³×C₁₀₉	C₂×#41	1	12	yes	—
108	C₂³×C₁₀₉	C₂×#41	1	6	yes	—
109	C₂³×C₁₀₉	H(5,𝔽₂)	1	6	yes	—
110	C₂³×C₁₀₉	2₋¹⁺⁴	1	2	yes	—
111	C₂³×C₁₀₉	C₂⁵	1	1	yes	characteristic
112	D₄×C₁₀₉	—	4	16	no	—
113	C₂×C₄×C₁₀₉	—	2	6	no	—
114	C₂×C₄×C₁₀₉	—	2	6	no	—
115	C₂×C₄×C₁₀₉	—	2	6	no	—
116	C₂×C₄×C₁₀₉	—	2	6	no	—
117	C₂×C₄×C₁₀₉	—	2	6	no	—
118	C₂×C₄×C₁₀₉	—	2	6	no	—
119	C₂×C₄×C₁₀₉	—	2	6	no	—
120	C₂×C₄×C₁₀₉	—	2	6	no	—
121	C₂×C₄×C₁₀₉	—	2	6	no	—
122	C₂×C₄×C₁₀₉	—	2	6	no	—
123	C₂×C₄×C₁₀₉	—	2	6	no	—
124	C₂×C₄×C₁₀₉	—	2	6	no	—
125	C₂×C₄×C₁₀₉	—	2	6	no	—
126	C₂×C₄×C₁₀₉	—	4	48	no	—
127	C₂×C₄×C₁₀₉	—	4	4	no	—
128	C₂×C₄×C₁₀₉	—	4	12	no	—
129	C₂×C₄×C₁₀₉	—	4	12	no	—
130	C₂×C₄×C₁₀₉	—	4	12	no	—
131	C₂×C₄×C₁₀₉	—	4	12	no	—
132	C₂²×C₁₀₉	—	2	32	no	—
133	C₂²×C₁₀₉	—	2	12	no	—
134	C₂²×C₁₀₉	—	2	12	no	—
135	C₂²×C₁₀₉	—	4	2	no	—
136	C₂²×C₁₀₉	—	4	6	no	—
137	C₂²×C₁₀₉	—	4	6	no	—
138	C₂²×C₁₀₉	—	8	8	no	—
139	C₂²×C₁₀₉	C₂×#126	1	8	yes	—
140	C₂²×C₁₀₉	C₂×#126	1	4	yes	—
141	C₂²×C₁₀₉	#538	1	12	yes	—
142	C₂²×C₁₀₉	#539	1	12	yes	—
143	C₂²×C₁₀₉	C₂²×#41	1	3	yes	—
144	C₂²×C₁₀₉	C₂×H(5,𝔽₂)	1	3	yes	—
145	C₂²×C₁₀₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
146	C₄×C₁₀₉	—	4	2	no	—
147	C₄×C₁₀₉	—	4	6	no	—
148	C₄×C₁₀₉	—	4	6	no	—
149	C₄×C₁₀₉	—	4	6	no	—
150	C₄×C₁₀₉	—	4	6	no	—
151	C₂×#2643	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
152	C₂×C₁₀₉	—	2	8	no	—
153	C₂×C₁₀₉	—	8	2	no	—
154	C₂×C₁₀₉	#2643	1	8	yes	direct product factor
155	C₂×C₁₀₉	C₂²×#126	1	1	yes	characteristic
156	C₂×C₁₀₉	C₂×#538	1	3	yes	—
157	C₂×C₁₀₉	C₂×#539	1	3	yes	—
158	#2643	C₂×C₁₀₉	1	16	yes	direct product factor
159	C₂×#401	C₂×C₁₀₉	1	1	yes	characteristic, complemented
160	C₂×#400	C₂×C₁₀₉	1	3	yes	complemented
161	C₂×#397	C₂×C₁₀₉	1	3	yes	complemented
162	C₂×#396	C₂×C₁₀₉	1	1	yes	characteristic, complemented
163	C₂×#386	C₂×C₁₀₉	1	3	yes	—
164	C₂×#385	C₂×C₁₀₉	1	3	yes	complemented
165	C₂²×#95	C₂×C₁₀₉	1	1	yes	characteristic, complemented
166	C₁₀₉	C₂×#2643	1	1	yes	characteristic, direct product factor, Sylow
167	C₂³×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
168	C₂⁴×C₄	C₂²×C₁₀₉	1	3	yes	—
169	C₂²×C₄⋊C₄	C₂²×C₁₀₉	1	3	yes	—
170	C₂²×C₄⋊C₄	C₂²×C₁₀₉	1	3	yes	—
171	C₂²×C₂²⋊C₄	C₂²×C₁₀₉	1	3	yes	—
172	C₂²×C₂²⋊C₄	C₂²×C₁₀₉	1	3	yes	—
173	C₂²×C₂²⋊C₄	C₂²×C₁₀₉	1	3	yes	—
174	#401	C₂²×C₁₀₉	1	8	yes	complemented
175	#400	C₂²×C₁₀₉	1	24	yes	complemented
176	#397	C₂²×C₁₀₉	1	24	yes	complemented
177	#396	C₂²×C₁₀₉	1	8	yes	complemented
178	#386	C₂²×C₁₀₉	1	24	yes	—
179	#385	C₂²×C₁₀₉	1	24	yes	complemented
180	C₂×#95	C₂²×C₁₀₉	1	1	yes	characteristic
181	C₂×#95	C₂²×C₁₀₉	1	3	yes	—
182	C₂×#95	C₂²×C₁₀₉	1	3	yes	—
183	C₂×#95	C₂²×C₁₀₉	1	3	yes	—
184	C₂×#95	C₂²×C₁₀₉	1	3	yes	—
185	C₂×#95	C₂²×C₁₀₉	1	3	yes	—
186	C₂×#95	C₂²×C₁₀₉	1	8	yes	complemented
187	C₂⁵	C₂³×C₁₀₉	1	1	yes	characteristic
188	C₂⁵	C₂³×C₁₀₉	1	1	yes	characteristic
189	C₂²×D₄	—	2	12	no	—
190	C₂²×D₄	C₂³×C₁₀₉	1	4	yes	—
191	C₂³×C₄	—	2	12	no	—
192	C₂³×C₄	—	4	6	no	—
193	C₂³×C₄	—	4	6	no	—
194	C₂³×C₄	C₂³×C₁₀₉	1	3	yes	—
195	C₂³×C₄	C₂³×C₁₀₉	1	3	yes	—
196	C₂³×C₄	C₂³×C₁₀₉	1	3	yes	—
197	C₂³×C₄	C₂³×C₁₀₉	1	3	yes	—
198	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
199	C₂³×C₄	C₂³×C₁₀₉	1	12	yes	—
200	C₂×C₄⋊C₄	—	4	6	no	—
201	C₂×C₄⋊C₄	—	4	6	no	—
202	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	12	yes	—
203	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	12	yes	—
204	C₂×C₂²⋊C₄	—	2	12	no	—
205	C₂×C₂²⋊C₄	—	4	6	no	—
206	C₂×C₂²⋊C₄	—	4	6	no	—
207	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	12	yes	—
208	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	12	yes	—
209	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	12	yes	—
210	#95	C₂³×C₁₀₉	1	4	yes	—
211	#95	C₂³×C₁₀₉	1	12	yes	—
212	#95	C₂³×C₁₀₉	1	12	yes	—
213	#95	C₂³×C₁₀₉	1	12	yes	—
214	#95	C₂³×C₁₀₉	1	12	yes	—
215	#95	C₂³×C₁₀₉	1	12	yes	—
216	C₂⁴	—	2	6	no	—
217	C₂⁴	—	2	2	no	—
218	C₂⁴	—	2	6	no	—
219	C₂⁴	—	2	6	no	—
220	C₂⁴	—	2	2	no	—
221	C₂⁴	#41×C₁₀₉	1	12	yes	—
222	C₂⁴	C₂⁴×C₁₀₉	1	2	yes	—
223	C₂⁴	C₂⁴×C₁₀₉	1	2	yes	—
224	C₂⁴	C₂⁴×C₁₀₉	1	1	yes	characteristic
225	C₂×D₄	—	2	24	no	—
226	C₂×D₄	—	4	16	no	—
227	C₂²×C₄	—	2	6	no	—
228	C₂²×C₄	—	2	6	no	—
229	C₂²×C₄	—	2	24	no	—
230	C₂²×C₄	—	2	6	no	—
231	C₂²×C₄	—	2	6	no	—
232	C₂²×C₄	—	2	6	no	—
233	C₂²×C₄	—	2	6	no	—
234	C₂²×C₄	—	2	6	no	—
235	C₂²×C₄	—	2	6	no	—
236	C₂²×C₄	—	2	6	no	—
237	C₂²×C₄	—	2	6	no	—
238	C₂²×C₄	—	2	6	no	—
239	C₂²×C₄	—	2	6	no	—
240	C₂²×C₄	—	2	6	no	—
241	C₂²×C₄	—	4	12	no	—
242	C₂²×C₄	—	4	12	no	—
243	C₂²×C₄	—	4	48	no	—
244	C₂²×C₄	C₂⁴×C₁₀₉	1	2	yes	—
245	C₂²×C₄	C₂⁴×C₁₀₉	1	6	yes	—
246	C₂²×C₄	C₂⁴×C₁₀₉	1	6	yes	—
247	C₂²×C₄	C₂⁴×C₁₀₉	1	6	yes	—
248	C₂²×C₄	C₂⁴×C₁₀₉	1	6	yes	—
249	C₄⋊C₄	—	4	12	no	—
250	C₄⋊C₄	—	4	12	no	—
251	C₂²⋊C₄	—	2	24	no	—
252	C₂²⋊C₄	—	4	12	no	—
253	C₂²⋊C₄	—	4	12	no	—
254	C₂³	—	2	24	no	—
255	C₂³	—	2	24	no	—
256	C₂³	—	2	6	no	—
257	C₂³	—	2	2	no	—
258	C₂³	—	2	6	no	—
259	C₂³	—	2	6	no	—
260	C₂³	—	2	2	no	—
261	C₂³	—	4	4	no	—
262	C₂³	—	4	12	no	—
263	C₂³	—	4	12	no	—
264	C₂³	#126×C₁₀₉	1	16	yes	—
265	C₂³	C₂×#41×C₁₀₉	1	12	yes	—
266	C₂³	C₂×#41×C₁₀₉	1	6	yes	—
267	C₂³	H(5,𝔽₂)×C₁₀₉	1	6	yes	—
268	C₂³	2₋¹⁺⁴×C₁₀₉	1	2	yes	—
269	C₂³	C₂⁵×C₁₀₉	1	1	yes	characteristic
270	D₄	—	4	16	no	—
271	C₂×C₄	—	2	6	no	—
272	C₂×C₄	—	2	6	no	—
273	C₂×C₄	—	2	6	no	—
274	C₂×C₄	—	2	6	no	—
275	C₂×C₄	—	2	6	no	—
276	C₂×C₄	—	2	6	no	—
277	C₂×C₄	—	2	6	no	—
278	C₂×C₄	—	2	6	no	—
279	C₂×C₄	—	2	6	no	—
280	C₂×C₄	—	2	6	no	—
281	C₂×C₄	—	2	6	no	—
282	C₂×C₄	—	2	6	no	—
283	C₂×C₄	—	2	6	no	—
284	C₂×C₄	—	4	48	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	4	12	no	—
287	C₂×C₄	—	4	12	no	—
288	C₂×C₄	—	4	12	no	—
289	C₂×C₄	—	4	12	no	—
290	C₂²	—	2	12	no	—
291	C₂²	—	2	12	no	—
292	C₂²	—	2	32	no	—
293	C₂²	—	4	6	no	—
294	C₂²	—	4	6	no	—
295	C₂²	—	4	2	no	—
296	C₂²	—	8	8	no	—
297	C₂²	C₂×#126×C₁₀₉	1	8	yes	—
298	C₂²	C₂×#126×C₁₀₉	1	4	yes	—
299	C₂²	#538×C₁₀₉	1	12	yes	—
300	C₂²	#539×C₁₀₉	1	12	yes	—
301	C₂²	C₂²×#41×C₁₀₉	1	3	yes	—
302	C₂²	C₂×H(5,𝔽₂)×C₁₀₉	1	3	yes	—
303	C₂²	C₂×2₋¹⁺⁴×C₁₀₉	1	1	yes	characteristic
304	C₄	—	4	6	no	—
305	C₄	—	4	6	no	—
306	C₄	—	4	6	no	—
307	C₄	—	4	6	no	—
308	C₄	—	4	2	no	—
309	C₂	—	2	8	no	—
310	C₂	—	8	2	no	—
311	C₂	C₁₀₉×#2643	1	8	yes	direct product factor
312	C₂	C₂²×#126×C₁₀₉	1	1	yes	characteristic
313	C₂	C₂×#538×C₁₀₉	1	3	yes	—
314	C₂	C₂×#539×C₁₀₉	1	3	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₁₀₉	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₁₀₉
prime radical	C₂³×D₄×C₁₀₉	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₀₉×#2643	C₂	direct (N × H)
2. C₂×#401×C₁₀₉	C₂	semidirect
3. C₂×#400×C₁₀₉	C₂	semidirect
4. C₂×#397×C₁₀₉	C₂	semidirect
5. C₂×#396×C₁₀₉	C₂	semidirect
7. C₂×#385×C₁₀₉	C₂	semidirect
8. C₂²×#95×C₁₀₉	C₂	semidirect
16. #401×C₁₀₉	C₂²	semidirect
17. #400×C₁₀₉	C₂²	semidirect
18. #397×C₁₀₉	C₂²	semidirect
19. #396×C₁₀₉	C₂²	semidirect
21. #385×C₁₀₉	C₂²	semidirect
28. C₂×#95×C₁₀₉	C₂²	semidirect
151. C₂×#2643	C₁₀₉	direct (N × H)
154. C₂×C₁₀₉	#2643	direct (N × H)
158. #2643	C₂×C₁₀₉	direct (N × H)
159. C₂×#401	C₂×C₁₀₉	semidirect
160. C₂×#400	C₂×C₁₀₉	semidirect
161. C₂×#397	C₂×C₁₀₉	semidirect
162. C₂×#396	C₂×C₁₀₉	semidirect
164. C₂×#385	C₂×C₁₀₉	semidirect
165. C₂²×#95	C₂×C₁₀₉	semidirect
166. C₁₀₉	C₂×#2643	direct (N × H)
174. #401	C₂²×C₁₀₉	semidirect
175. #400	C₂²×C₁₀₉	semidirect
176. #397	C₂²×C₁₀₉	semidirect
177. #396	C₂²×C₁₀₉	semidirect
179. #385	C₂²×C₁₀₉	semidirect
186. C₂×#95	C₂²×C₁₀₉	semidirect
311. C₂	C₁₀₉×#2643	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×#2643	2
C₂×#401×C₁₀₉	2
C₂×#400×C₁₀₉	2
C₂×#397×C₁₀₉	2
C₂×#396×C₁₀₉	2
C₂×#386×C₁₀₉	2
C₂×#385×C₁₀₉	2
C₂²×#95×C₁₀₉	2
C₂×#2643	109

Subgroups — C2×C109×#2643 #42152559

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₀₉×#2643 #42152559