Subgroups — C₂×#20295585 #28376332

Order 13952.

1984 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₂²⋊D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295585	C₂	1	16	yes	maximal, direct product factor
3	C₂×#16812682	C₂	1	2	yes	maximal, complemented
4	C₂×#16812680	C₂	1	2	yes	maximal, complemented
5	C₂×#16812671	C₂	1	4	yes	maximal, complemented
6	C₂×#16812668	C₂	1	2	yes	maximal, complemented
7	C₂×#16812667	C₂	1	2	yes	maximal, complemented
8	C₂²×C₂²⋊Dic₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×D₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂⁴×D₁₀₉	C₂²	1	1	yes	characteristic, complemented
11	C₂²×C₂²⋊D₁₀₉	—	2	8	no	—
12	C₂²×C₂²⋊D₁₀₉	C₂²	1	2	yes	—
13	C₂²×C₂²⋊D₁₀₉	C₂²	1	2	yes	complemented
14	C₂²×C₂²⋊D₁₀₉	C₂²	1	4	yes	complemented
15	C₂²×C₂²⋊D₁₀₉	C₂²	1	8	yes	complemented
16	C₂³×Dic₁₀₉	C₂²	1	1	yes	characteristic, complemented
17	C₂³×Dic₁₀₉	C₂²	1	2	yes	complemented
18	#16812682	C₂²	1	16	yes	complemented
19	#16812680	C₂²	1	16	yes	complemented
20	#16812671	C₂²	1	32	yes	complemented
21	#16812668	C₂²	1	16	yes	complemented
22	#16812667	C₂²	1	16	yes	complemented
23	C₂×#13182008	C₂²	1	4	yes	complemented
24	C₂×C₂²⋊Dic₁₀₉	C₂²	1	2	yes	complemented
25	C₂×C₂²⋊Dic₁₀₉	C₂²	1	2	yes	complemented
26	C₂×C₂²⋊Dic₁₀₉	C₂²	1	4	yes	complemented
27	C₂×C₂²⋊Dic₁₀₉	C₂²	1	8	yes	complemented
28	C₂×#13182006	C₂²	1	4	yes	complemented
29	C₂⁵×C₁₀₉	C₂²	1	1	yes	characteristic
30	C₂⁵×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
31	C₂²×D₄×C₁₀₉	—	2	8	no	—
32	C₂²×D₄×C₁₀₉	C₂²	1	2	yes	—
33	C₂²×D₄×C₁₀₉	C₂²	1	2	yes	complemented
34	C₂²×D₄×C₁₀₉	C₂²	1	8	yes	complemented
35	C₂³×C₄×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
36	C₂³×D₁₀₉	—	2	4	no	—
37	C₂³×D₁₀₉	—	4	4	no	—
38	C₂³×D₁₀₉	C₂³	1	2	yes	—
39	C₂³×D₁₀₉	C₂³	1	4	yes	complemented
40	C₂×C₂²⋊D₁₀₉	—	2	8	no	—
41	C₂×C₂²⋊D₁₀₉	—	2	8	no	—
42	C₂×C₂²⋊D₁₀₉	—	2	32	no	—
43	C₂×C₂²⋊D₁₀₉	—	4	8	no	—
44	C₂×C₂²⋊D₁₀₉	C₂³	1	8	yes	—
45	C₂×C₂²⋊D₁₀₉	C₂³	1	8	yes	complemented
46	C₂×C₂²⋊D₁₀₉	C₂³	1	16	yes	complemented
47	C₂²×Dic₁₀₉	—	2	4	no	—
48	C₂²×Dic₁₀₉	—	4	4	no	—
49	C₂²×Dic₁₀₉	C₂³	1	4	yes	—
50	C₂²×Dic₁₀₉	C₂³	1	2	yes	—
51	C₂²×Dic₁₀₉	C₂³	1	4	yes	complemented
52	C₂²×Dic₁₀₉	C₂³	1	8	yes	complemented
53	#13182008	C₂³	1	16	yes	complemented
54	C₂²⋊Dic₁₀₉	C₂³	1	8	yes	complemented
55	C₂²⋊Dic₁₀₉	C₂³	1	8	yes	complemented
56	C₂²⋊Dic₁₀₉	C₂³	1	16	yes	complemented
57	#13182006	C₂³	1	16	yes	complemented
58	C₂⁴×C₁₀₉	—	2	4	no	—
59	C₂⁴×C₁₀₉	—	4	4	no	—
60	C₂⁴×C₁₀₉	—	4	4	no	—
61	C₂⁴×C₁₀₉	D₄	1	8	yes	complemented
62	C₂⁴×C₁₀₉	C₂³	1	2	yes	—
63	C₂⁴×C₁₀₉	C₂³	1	1	yes	characteristic
64	C₂⁴×C₁₀₉	C₂³	1	1	yes	characteristic
65	C₂⁴×C₁₀₉	C₂³	1	1	yes	characteristic
66	C₂⁴×C₁₀₉	C₂³	1	4	yes	—
67	C₂⁴×C₁₀₉	C₂³	1	4	yes	complemented
68	C₂×D₄×C₁₀₉	—	2	8	no	—
69	C₂×D₄×C₁₀₉	—	2	8	no	—
70	C₂×D₄×C₁₀₉	—	2	32	no	—
71	C₂×D₄×C₁₀₉	C₂³	1	8	yes	—
72	C₂×D₄×C₁₀₉	C₂³	1	8	yes	complemented
73	C₂²×C₄×C₁₀₉	—	2	4	no	—
74	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
75	C₂²×C₄×C₁₀₉	C₂³	1	4	yes	complemented
76	C₂³×C₁₀₉	—	2	4	no	—
77	C₂³×C₁₀₉	—	2	4	no	—
78	C₂³×C₁₀₉	—	2	4	no	—
79	C₂³×C₁₀₉	—	2	4	no	—
80	C₂³×C₁₀₉	—	2	8	no	—
81	C₂³×C₁₀₉	—	2	8	no	—
82	C₂³×C₁₀₉	—	4	16	no	—
83	C₂³×C₁₀₉	—	4	8	no	—
84	C₂³×C₁₀₉	—	4	8	no	—
85	C₂³×C₁₀₉	—	4	16	no	—
86	C₂³×C₁₀₉	C₂×D₄	1	4	yes	—
87	C₂³×C₁₀₉	C₂×D₄	1	4	yes	—
88	C₂³×C₁₀₉	C₂×D₄	1	4	yes	—
89	C₂³×C₁₀₉	C₂×D₄	1	16	yes	complemented
90	C₂³×C₁₀₉	C₂⁴	1	4	yes	—
91	C₂³×C₁₀₉	C₂⁴	1	2	yes	—
92	C₂³×C₁₀₉	C₂⁴	1	2	yes	—
93	C₂³×C₁₀₉	C₂⁴	1	2	yes	—
94	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
95	C₂²×D₁₀₉	—	2	4	no	—
96	C₂²×D₁₀₉	—	2	8	no	—
97	C₂²×D₁₀₉	—	2	8	no	—
98	C₂²×D₁₀₉	—	4	8	no	—
99	C₂²×D₁₀₉	—	4	16	no	—
100	C₂²×D₁₀₉	C₂⁴	1	4	yes	—
101	D₄×C₁₀₉	—	2	16	no	—
102	D₄×C₁₀₉	—	2	16	no	—
103	C₂×C₄×C₁₀₉	—	2	4	no	—
104	C₂×C₄×C₁₀₉	—	2	8	no	—
105	C₂×C₄×C₁₀₉	C₂⁴	1	4	yes	—
106	C₂²⋊D₁₀₉	—	2	16	no	—
107	C₂²⋊D₁₀₉	—	2	16	no	—
108	C₂²⋊D₁₀₉	—	4	16	no	—
109	C₂×Dic₁₀₉	—	2	4	no	—
110	C₂×Dic₁₀₉	—	2	8	no	—
111	C₂×Dic₁₀₉	—	2	8	no	—
112	C₂×Dic₁₀₉	—	4	8	no	—
113	C₂×Dic₁₀₉	C₂⁴	1	8	yes	—
114	C₂×Dic₁₀₉	C₂⁴	1	4	yes	—
115	C₂²×C₁₀₉	—	2	8	no	—
116	C₂²×C₁₀₉	—	2	8	no	—
117	C₂²×C₁₀₉	—	2	8	no	—
118	C₂²×C₁₀₉	—	2	4	no	—
119	C₂²×C₁₀₉	—	2	4	no	—
120	C₂²×C₁₀₉	—	2	4	no	—
121	C₂²×C₁₀₉	—	2	4	no	—
122	C₂²×C₁₀₉	—	2	8	no	—
123	C₂²×C₁₀₉	—	4	16	no	—
124	C₂²×C₁₀₉	—	4	16	no	—
125	C₂²×C₁₀₉	—	4	8	no	—
126	C₂²×C₁₀₉	C₂²×D₄	1	4	yes	—
127	C₂²×C₁₀₉	C₂²×D₄	1	4	yes	—
128	C₂²×C₁₀₉	C₂²×D₄	1	4	yes	—
129	C₂²×C₁₀₉	C₂²×D₄	1	2	yes	—
130	C₂²×C₁₀₉	H(5,𝔽₂)	1	4	yes	—
131	C₂²×C₁₀₉	C₂⁵	1	1	yes	characteristic
132	C₂×D₁₀₉	—	2	4	no	—
133	C₂×D₁₀₉	—	2	8	no	—
134	C₂×D₁₀₉	—	4	16	no	—
135	C₂×D₁₀₉	—	4	8	no	—
136	C₄×C₁₀₉	—	2	4	no	—
137	Dic₁₀₉	—	2	4	no	—
138	Dic₁₀₉	—	2	8	no	—
139	C₂×C₁₀₉	—	2	4	no	—
140	C₂×C₁₀₉	—	2	4	no	—
141	C₂×C₁₀₉	—	2	4	no	—
142	C₂×C₁₀₉	—	4	4	no	—
143	C₂×C₁₀₉	#534	1	4	yes	complemented
144	C₂×C₁₀₉	C₂³×D₄	1	1	yes	characteristic
145	C₂×C₁₀₉	C₂×H(5,𝔽₂)	1	2	yes	—
146	D₁₀₉	—	4	4	no	—
147	C₂×#534	—	109	1	no	maximal, Sylow
148	C₁₀₉	C₂×#534	1	1	yes	characteristic, complemented, Sylow
149	C₂³×D₄	—	109	1	no	—
150	C₂³×D₄	D₁₀₉	1	1	yes	characteristic, complemented
151	#534	—	109	16	no	—
152	C₂×#123	—	109	2	no	—
153	C₂×#123	—	109	2	no	—
154	C₂×#121	—	109	4	no	—
155	C₂×#120	—	109	2	no	—
156	C₂×#120	—	109	2	no	—
157	C₂²×C₂²⋊C₄	—	109	1	no	—
158	C₂⁵	—	109	1	no	—
159	C₂⁵	C₂×D₁₀₉	1	1	yes	characteristic
160	C₂⁵	C₂×D₁₀₉	1	1	yes	characteristic, complemented
161	C₂²×D₄	—	2	8	no	—
162	C₂²×D₄	—	109	4	no	—
163	C₂²×D₄	—	109	2	no	—
164	C₂²×D₄	—	109	2	no	—
165	C₂²×D₄	—	109	8	no	—
166	C₂²×D₄	—	218	8	no	—
167	C₂²×D₄	C₂×D₁₀₉	1	2	yes	—
168	C₂²×D₄	C₂×D₁₀₉	1	2	yes	complemented
169	C₂²×D₄	C₂×D₁₀₉	1	8	yes	complemented
170	C₂³×C₄	—	109	2	no	—
171	C₂³×C₄	—	109	1	no	—
172	C₂³×C₄	C₂×D₁₀₉	1	1	yes	characteristic, complemented
173	#123	—	109	16	no	—
174	#123	—	109	16	no	—
175	#121	—	109	32	no	—
176	#120	—	109	16	no	—
177	#120	—	109	16	no	—
178	C₂×C₄⋊C₄	—	109	4	no	—
179	C₂×C₂²⋊C₄	—	109	2	no	—
180	C₂×C₂²⋊C₄	—	109	2	no	—
181	C₂×C₂²⋊C₄	—	109	8	no	—
182	C₂×C₂²⋊C₄	—	109	4	no	—
183	C₂×C₂²⋊C₄	—	109	4	no	—
184	C₂⁴	—	2	4	no	—
185	C₂⁴	—	4	4	no	—
186	C₂⁴	—	4	4	no	—
187	C₂⁴	—	109	4	no	—
188	C₂⁴	—	109	2	no	—
189	C₂⁴	—	436	4	no	—
190	C₂⁴	—	218	4	no	—
191	C₂⁴	C₂²⋊D₁₀₉	1	8	yes	complemented
192	C₂⁴	C₂²×D₁₀₉	1	2	yes	—
193	C₂⁴	C₂²×D₁₀₉	1	1	yes	characteristic
194	C₂⁴	C₂²×D₁₀₉	1	1	yes	characteristic
195	C₂⁴	C₂²×D₁₀₉	1	1	yes	characteristic
196	C₂⁴	C₂²×D₁₀₉	1	4	yes	—
197	C₂⁴	C₂²×D₁₀₉	1	4	yes	complemented
198	C₂×D₄	—	2	8	no	—
199	C₂×D₄	—	2	8	no	—
200	C₂×D₄	—	2	32	no	—
201	C₂×D₄	—	109	16	no	—
202	C₂×D₄	—	109	8	no	—
203	C₂×D₄	—	109	8	no	—
204	C₂×D₄	—	436	8	no	—
205	C₂×D₄	—	218	8	no	—
206	C₂×D₄	—	218	8	no	—
207	C₂×D₄	—	218	32	no	—
208	C₂×D₄	C₂²×D₁₀₉	1	8	yes	—
209	C₂×D₄	C₂²×D₁₀₉	1	8	yes	complemented
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	109	8	no	—
212	C₂²×C₄	—	109	4	no	—
213	C₂²×C₄	—	109	4	no	—
214	C₂²×C₄	—	109	2	no	—
215	C₂²×C₄	—	436	4	no	—
216	C₂²×C₄	—	218	4	no	—
217	C₂²×C₄	C₂²×D₁₀₉	1	2	yes	—
218	C₂²×C₄	C₂²×D₁₀₉	1	4	yes	complemented
219	C₄⋊C₄	—	109	16	no	—
220	C₂²⋊C₄	—	109	8	no	—
221	C₂²⋊C₄	—	109	8	no	—
222	C₂²⋊C₄	—	109	16	no	—
223	C₂²⋊C₄	—	109	16	no	—
224	C₂³	—	2	4	no	—
225	C₂³	—	2	4	no	—
226	C₂³	—	2	4	no	—
227	C₂³	—	2	4	no	—
228	C₂³	—	2	8	no	—
229	C₂³	—	2	8	no	—
230	C₂³	—	4	16	no	—
231	C₂³	—	4	8	no	—
232	C₂³	—	4	8	no	—
233	C₂³	—	4	16	no	—
234	C₂³	—	109	4	no	—
235	C₂³	—	436	16	no	—
236	C₂³	—	436	8	no	—
237	C₂³	—	218	8	no	—
238	C₂³	—	218	4	no	—
239	C₂³	—	218	8	no	—
240	C₂³	C₂×C₂²⋊D₁₀₉	1	4	yes	—
241	C₂³	C₂×C₂²⋊D₁₀₉	1	4	yes	—
242	C₂³	C₂×C₂²⋊D₁₀₉	1	4	yes	—
243	C₂³	C₂×C₂²⋊D₁₀₉	1	16	yes	complemented
244	C₂³	C₂³×D₁₀₉	1	4	yes	—
245	C₂³	C₂³×D₁₀₉	1	2	yes	—
246	C₂³	C₂³×D₁₀₉	1	2	yes	—
247	C₂³	C₂³×D₁₀₉	1	2	yes	—
248	C₂³	C₂³×D₁₀₉	1	1	yes	characteristic
249	D₄	—	2	16	no	—
250	D₄	—	2	16	no	—
251	D₄	—	436	16	no	—
252	D₄	—	218	16	no	—
253	D₄	—	218	16	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	8	no	—
256	C₂×C₄	—	109	8	no	—
257	C₂×C₄	—	109	4	no	—
258	C₂×C₄	—	436	8	no	—
259	C₂×C₄	—	218	8	no	—
260	C₂×C₄	—	218	4	no	—
261	C₂×C₄	—	218	8	no	—
262	C₂×C₄	C₂³×D₁₀₉	1	4	yes	—
263	C₂²	—	2	8	no	—
264	C₂²	—	2	8	no	—
265	C₂²	—	2	8	no	—
266	C₂²	—	2	4	no	—
267	C₂²	—	2	4	no	—
268	C₂²	—	2	4	no	—
269	C₂²	—	2	4	no	—
270	C₂²	—	2	8	no	—
271	C₂²	—	4	16	no	—
272	C₂²	—	4	16	no	—
273	C₂²	—	4	8	no	—
274	C₂²	—	436	16	no	—
275	C₂²	—	436	8	no	—
276	C₂²	—	218	4	no	—
277	C₂²	—	218	8	no	—
278	C₂²	C₂²×C₂²⋊D₁₀₉	1	4	yes	—
279	C₂²	C₂²×C₂²⋊D₁₀₉	1	4	yes	—
280	C₂²	C₂²×C₂²⋊D₁₀₉	1	4	yes	—
281	C₂²	C₂²×C₂²⋊D₁₀₉	1	2	yes	—
282	C₂²	#16812744	1	4	yes	—
283	C₂²	C₂⁴×D₁₀₉	1	1	yes	characteristic
284	C₄	—	2	4	no	—
285	C₄	—	218	4	no	—
286	C₄	—	218	8	no	—
287	C₂	—	2	4	no	—
288	C₂	—	2	4	no	—
289	C₂	—	2	4	no	—
290	C₂	—	4	4	no	—
291	C₂	—	436	4	no	—
292	C₂	#20295585	1	4	yes	direct product factor
293	C₂	C₂³×C₂²⋊D₁₀₉	1	1	yes	characteristic
294	C₂	C₂×#16812744	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂³×D₁₀₉
commutator, cosocle	C₂²×C₁₀₉	C₂⁵
Frattini	C₂²	C₂⁴×D₁₀₉
Fitting	C₂³×D₄×C₁₀₉	C₂
socle	C₂³×C₁₀₉	C₂⁴
hypercenter	C₂³×D₄	D₁₀₉
hypocenter	C₁₀₉	C₂×#534
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂³×D₄

Complements

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

N	H ≅ G/N	type
1. C₂³×C₂²⋊D₁₀₉	C₂	semidirect
2. #20295585	C₂	direct (N × H)
3. C₂×#16812682	C₂	semidirect
4. C₂×#16812680	C₂	semidirect
5. C₂×#16812671	C₂	semidirect
6. C₂×#16812668	C₂	semidirect
7. C₂×#16812667	C₂	semidirect
8. C₂²×C₂²⋊Dic₁₀₉	C₂	semidirect
9. C₂³×D₄×C₁₀₉	C₂	semidirect
10. C₂⁴×D₁₀₉	C₂²	semidirect
13. C₂²×C₂²⋊D₁₀₉	C₂²	semidirect
14. C₂²×C₂²⋊D₁₀₉	C₂²	semidirect
15. C₂²×C₂²⋊D₁₀₉	C₂²	semidirect
16. C₂³×Dic₁₀₉	C₂²	semidirect
17. C₂³×Dic₁₀₉	C₂²	semidirect
18. #16812682	C₂²	semidirect
19. #16812680	C₂²	semidirect
20. #16812671	C₂²	semidirect
21. #16812668	C₂²	semidirect
22. #16812667	C₂²	semidirect
23. C₂×#13182008	C₂²	semidirect
24. C₂×C₂²⋊Dic₁₀₉	C₂²	semidirect
25. C₂×C₂²⋊Dic₁₀₉	C₂²	semidirect
26. C₂×C₂²⋊Dic₁₀₉	C₂²	semidirect
27. C₂×C₂²⋊Dic₁₀₉	C₂²	semidirect
28. C₂×#13182006	C₂²	semidirect
30. C₂⁵×C₁₀₉	C₂²	semidirect
33. C₂²×D₄×C₁₀₉	C₂²	semidirect
34. C₂²×D₄×C₁₀₉	C₂²	semidirect
35. C₂³×C₄×C₁₀₉	C₂²	semidirect
39. C₂³×D₁₀₉	C₂³	semidirect
45. C₂×C₂²⋊D₁₀₉	C₂³	semidirect
46. C₂×C₂²⋊D₁₀₉	C₂³	semidirect
51. C₂²×Dic₁₀₉	C₂³	semidirect
52. C₂²×Dic₁₀₉	C₂³	semidirect
53. #13182008	C₂³	semidirect
54. C₂²⋊Dic₁₀₉	C₂³	semidirect
55. C₂²⋊Dic₁₀₉	C₂³	semidirect
56. C₂²⋊Dic₁₀₉	C₂³	semidirect
57. #13182006	C₂³	semidirect
61. C₂⁴×C₁₀₉	D₄	semidirect
67. C₂⁴×C₁₀₉	C₂³	semidirect
72. C₂×D₄×C₁₀₉	C₂³	semidirect
75. C₂²×C₄×C₁₀₉	C₂³	semidirect
89. C₂³×C₁₀₉	C₂×D₄	semidirect
143. C₂×C₁₀₉	#534	semidirect
148. C₁₀₉	C₂×#534	semidirect
150. C₂³×D₄	D₁₀₉	semidirect
160. C₂⁵	C₂×D₁₀₉	semidirect
168. C₂²×D₄	C₂×D₁₀₉	semidirect
169. C₂²×D₄	C₂×D₁₀₉	semidirect
172. C₂³×C₄	C₂×D₁₀₉	semidirect
191. C₂⁴	C₂²⋊D₁₀₉	semidirect
197. C₂⁴	C₂²×D₁₀₉	semidirect
209. C₂×D₄	C₂²×D₁₀₉	semidirect
218. C₂²×C₄	C₂²×D₁₀₉	semidirect
243. C₂³	C₂×C₂²⋊D₁₀₉	semidirect
292. C₂	#20295585	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₂²⋊D₁₀₉	2
#20295585	2
C₂×#16812682	2
C₂×#16812680	2
C₂×#16812671	2
C₂×#16812668	2
C₂×#16812667	2
C₂²×C₂²⋊Dic₁₀₉	2
C₂³×D₄×C₁₀₉	2
C₂×#534	109

Subgroups — C2×#20295585 #28376332

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20295585 #28376332