Subgroups — C₂×#20295898 #28378710

Order 13952.

488 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×#16812778	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×#89137	C₂	1	1	yes	maximal, characteristic
3	#20295898	C₂	1	4	yes	maximal, direct product factor
4	C₂×M₁₆×D₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂³×#89137	—	2	1	no	—
6	#16812778	—	2	1	no	—
7	#16812778	—	2	1	no	—
8	#16812778	C₄	1	2	yes	complemented
9	#16812778	C₂²	1	2	yes	—
10	C₂×C₄⋊#89137	C₄	1	2	yes	complemented
11	C₂×C₂²⋊#89137	—	2	1	no	—
12	C₂×C₄×#89137	—	2	1	no	—
13	C₂×C₄×#89137	—	2	1	no	—
14	C₂×C₄×#89137	—	2	1	no	—
15	C₂×C₄×#89137	—	2	1	no	—
16	C₂×C₄×#89137	C₄	1	2	yes	complemented
17	C₂×C₄×#89137	C₄	1	2	yes	complemented
18	C₂×C₄×#89137	C₂²	1	2	yes	—
19	C₂²×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
20	C₂×#13182013	C₄	1	1	yes	characteristic, complemented
21	C₂×#13182012	—	2	1	no	—
22	M₁₆×D₁₀₉	—	2	1	no	—
23	M₁₆×D₁₀₉	—	2	1	no	—
24	M₁₆×D₁₀₉	C₄	1	2	yes	complemented
25	M₁₆×D₁₀₉	C₂²	1	2	yes	—
26	C₂×C₈×D₁₀₉	—	2	1	no	—
27	C₂×M₁₆×C₁₀₉	C₄	1	1	yes	characteristic, complemented
28	C₂²×#89137	—	2	8	no	—
29	C₂²×#89137	—	2	2	no	—
30	C₂²×#89137	—	2	2	no	—
31	C₂²×#89137	—	2	2	no	—
32	C₂²×#89137	—	2	2	no	—
33	C₄⋊#89137	—	2	2	no	—
34	C₄⋊#89137	C₂×C₄	1	4	yes	complemented
35	C₂²⋊#89137	—	2	2	no	—
36	C₂²⋊#89137	—	2	2	no	—
37	C₄×#89137	—	2	2	no	—
38	C₄×#89137	—	2	2	no	—
39	C₄×#89137	—	2	2	no	—
40	C₄×#89137	—	2	2	no	—
41	C₄×#89137	—	2	2	no	—
42	C₄×#89137	C₂×C₄	1	4	yes	complemented
43	C₂³×D₁₀₉	D₄	1	1	yes	characteristic
44	C₂²×Dic₁₀₉	Q₈	1	1	yes	characteristic
45	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
51	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
52	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
53	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
54	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
55	C₂×C₄×D₁₀₉	Q₈	1	1	yes	characteristic
56	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
57	#13182013	—	2	1	no	—
58	#13182013	C₂×C₄	1	2	yes	complemented
59	#13182012	—	2	2	no	—
60	#13182012	—	2	2	no	—
61	C₂×#11724784	—	2	1	no	—
62	C₈×D₁₀₉	—	2	2	no	—
63	C₈×D₁₀₉	—	2	2	no	—
64	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
65	M₁₆×C₁₀₉	—	2	1	no	—
66	M₁₆×C₁₀₉	C₂×C₄	1	2	yes	complemented
67	C₂×C₈×C₁₀₉	—	2	1	no	—
68	C₂³×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
69	C₂²×D₁₀₉	—	2	4	no	—
70	C₂²×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
75	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
76	C₂×#89137	—	2	8	no	—
77	C₂×#89137	—	2	8	no	—
78	C₂×#89137	—	2	8	no	—
79	C₂×#89137	—	2	4	no	—
80	C₂×#89137	—	2	4	no	—
81	C₂×#89137	—	4	4	no	—
82	C₂×C₄×C₁₀₉	C₄²	1	1	yes	characteristic
83	C₂×C₄×C₁₀₉	C₄²	1	1	yes	characteristic
84	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
88	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
89	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
90	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
91	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
92	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
93	C₂×Dic₁₀₉	C₂×Q₈	1	1	yes	characteristic
94	C₄×D₁₀₉	C₄²	1	1	yes	characteristic
95	C₄×D₁₀₉	C₄²	1	1	yes	characteristic
96	C₄×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
97	C₄×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
98	C₄×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
99	C₄×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
100	C₄×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
101	C₄×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
102	C₄×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
103	C₄×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
104	C₄×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
105	C₄×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
106	C₄×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
107	C₄×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
108	C₄×D₁₀₉	C₂×D₄	1	1	yes	characteristic
109	C₄×D₁₀₉	C₂×D₄	1	1	yes	characteristic
110	C₈×C₁₀₉	—	2	2	no	—
111	#11724784	—	2	2	no	—
112	C₂²×C₁₀₉	—	2	2	no	—
113	C₂²×C₁₀₉	#95	1	1	yes	characteristic
114	C₂²×C₁₀₉	#95	1	1	yes	characteristic
115	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×D₁₀₉	—	2	2	no	—
117	C₂×D₁₀₉	—	2	2	no	—
118	C₂×D₁₀₉	—	2	2	no	—
119	C₂×D₁₀₉	—	2	2	no	—
120	C₂×D₁₀₉	—	2	2	no	—
121	C₂×D₁₀₉	#95	1	1	yes	characteristic
122	C₂×D₁₀₉	#95	1	1	yes	characteristic
123	C₂×D₁₀₉	#104	1	4	yes	—
124	C₂×D₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×D₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	#89137	—	2	8	no	—
127	#89137	—	4	4	no	—
128	C₄×C₁₀₉	#95	1	1	yes	characteristic
129	C₄×C₁₀₉	#95	1	1	yes	characteristic
130	C₄×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
131	C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	Dic₁₀₉	#95	1	1	yes	characteristic
133	Dic₁₀₉	#95	1	1	yes	characteristic
134	Dic₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
135	Dic₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₂×C₁₀₉	—	2	1	no	—
137	C₂×C₁₀₉	—	2	1	no	—
138	C₂×C₁₀₉	#339	1	2	yes	complemented
139	C₂×C₁₀₉	C₂×#95	1	1	yes	characteristic
140	D₁₀₉	—	2	1	no	—
141	D₁₀₉	—	2	1	no	—
142	D₁₀₉	#339	1	2	yes	complemented
143	D₁₀₉	C₂×#104	1	2	yes	—
144	C₂×#339	—	109	1	no	maximal, Sylow
145	C₁₀₉	C₂×#339	1	1	yes	characteristic, complemented, Sylow
146	C₂²×M₁₆	—	109	1	no	—
147	C₂×#117	—	109	1	no	—
148	C₂²×C₄²	—	109	1	no	—
149	#339	—	109	4	no	—
150	C₂³×C₄	—	109	1	no	—
151	C₂³×C₄	—	218	1	no	—
152	C₂×M₁₆	—	109	1	no	—
153	C₂×M₁₆	—	109	2	no	—
154	C₂×M₁₆	—	109	2	no	—
155	C₂×M₁₆	—	218	1	no	—
156	C₂×M₁₆	—	218	1	no	—
157	C₂×M₁₆	—	218	1	no	—
158	C₂×M₁₆	#89137	1	1	yes	characteristic, complemented
159	C₂²×C₈	—	218	1	no	—
160	#117	—	109	2	no	—
161	#117	—	109	2	no	—
162	#117	—	218	1	no	—
163	#117	—	218	1	no	—
164	C₂×C₄⋊C₄	—	109	2	no	—
165	C₂×C₂²⋊C₄	—	218	1	no	—
166	C₂×C₄²	—	109	2	no	—
167	C₂×C₄²	—	109	2	no	—
168	C₂×C₄²	—	109	2	no	—
169	C₂×C₄²	—	218	1	no	—
170	C₂×C₄²	—	218	1	no	—
171	C₂×C₄²	—	218	1	no	—
172	C₂×C₄²	—	218	1	no	—
173	C₂⁴	—	109	1	no	—
174	C₂²×C₄	—	109	1	no	—
175	C₂²×C₄	—	109	1	no	—
176	C₂²×C₄	—	109	1	no	—
177	C₂²×C₄	—	109	1	no	—
178	C₂²×C₄	—	109	1	no	—
179	C₂²×C₄	—	109	1	no	—
180	C₂²×C₄	—	109	1	no	—
181	C₂²×C₄	—	109	1	no	—
182	C₂²×C₄	—	109	1	no	—
183	C₂²×C₄	—	109	1	no	—
184	C₂²×C₄	—	109	1	no	—
185	C₂²×C₄	—	109	1	no	—
186	C₂²×C₄	—	109	1	no	—
187	C₂²×C₄	—	218	2	no	—
188	C₂²×C₄	—	218	2	no	—
189	C₂²×C₄	—	218	2	no	—
190	C₂²×C₄	—	218	2	no	—
191	C₂²×C₄	—	218	8	no	—
192	C₂²×C₄	C₂×#89137	1	1	yes	characteristic
193	M₁₆	—	2	1	no	—
194	M₁₆	—	109	2	no	—
195	M₁₆	—	218	2	no	—
196	M₁₆	—	218	2	no	—
197	M₁₆	—	218	1	no	—
198	M₁₆	C₂×#89137	1	2	yes	complemented
199	C₂×C₈	—	2	1	no	—
200	C₂×C₈	—	218	2	no	—
201	C₂×C₈	—	218	2	no	—
202	C₂×C₈	—	218	1	no	—
203	C₄⋊C₄	—	109	4	no	—
204	C₄⋊C₄	—	218	2	no	—
205	C₂²⋊C₄	—	218	2	no	—
206	C₂²⋊C₄	—	218	2	no	—
207	C₄²	—	109	4	no	—
208	C₄²	—	218	2	no	—
209	C₄²	—	218	2	no	—
210	C₄²	—	218	2	no	—
211	C₄²	—	218	2	no	—
212	C₄²	—	218	2	no	—
213	C₂³	—	109	1	no	—
214	C₂³	—	109	1	no	—
215	C₂³	—	109	1	no	—
216	C₂³	—	109	1	no	—
217	C₂³	—	109	1	no	—
218	C₂³	—	109	1	no	—
219	C₂³	—	218	4	no	—
220	C₂³	C₄⋊#89137	1	1	yes	characteristic
221	C₂×C₄	—	109	1	no	—
222	C₂×C₄	—	109	1	no	—
223	C₂×C₄	—	109	1	no	—
224	C₂×C₄	—	109	1	no	—
225	C₂×C₄	—	109	1	no	—
226	C₂×C₄	—	109	1	no	—
227	C₂×C₄	—	109	1	no	—
228	C₂×C₄	—	109	1	no	—
229	C₂×C₄	—	109	1	no	—
230	C₂×C₄	—	109	1	no	—
231	C₂×C₄	—	109	1	no	—
232	C₂×C₄	—	109	1	no	—
233	C₂×C₄	—	109	1	no	—
234	C₂×C₄	—	109	1	no	—
235	C₂×C₄	—	109	1	no	—
236	C₂×C₄	—	109	1	no	—
237	C₂×C₄	—	109	1	no	—
238	C₂×C₄	—	109	1	no	—
239	C₂×C₄	—	109	1	no	—
240	C₂×C₄	—	109	1	no	—
241	C₂×C₄	—	109	1	no	—
242	C₂×C₄	—	109	1	no	—
243	C₂×C₄	—	436	4	no	—
244	C₂×C₄	—	218	4	no	—
245	C₂×C₄	—	218	4	no	—
246	C₂×C₄	—	218	8	no	—
247	C₂×C₄	—	218	8	no	—
248	C₂×C₄	—	218	8	no	—
249	C₂×C₄	C₄×#89137	1	1	yes	characteristic
250	C₂×C₄	C₄×#89137	1	1	yes	characteristic
251	C₂×C₄	C₂²⋊#89137	1	1	yes	characteristic
252	C₂×C₄	C₂²⋊#89137	1	1	yes	characteristic
253	C₂×C₄	C₄⋊#89137	1	1	yes	characteristic
254	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
255	C₈	—	2	2	no	—
256	C₈	—	218	2	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	109	1	no	—
259	C₂²	—	109	1	no	—
260	C₂²	—	109	4	no	—
261	C₂²	—	109	1	no	—
262	C₂²	—	109	1	no	—
263	C₂²	—	218	2	no	—
264	C₂²	—	218	2	no	—
265	C₂²	—	218	2	no	—
266	C₂²	—	218	2	no	—
267	C₂²	—	218	2	no	—
268	C₂²	(C₂×#89137)⋊C₄	1	1	yes	characteristic
269	C₂²	(C₂×#89137)⋊C₄	1	1	yes	characteristic
270	C₂²	C₂×C₄⋊#89137	1	1	yes	characteristic
271	C₄	—	109	1	no	—
272	C₄	—	109	1	no	—
273	C₄	—	109	1	no	—
274	C₄	—	109	1	no	—
275	C₄	—	436	4	no	—
276	C₄	—	218	8	no	—
277	C₄	(C₂×#89137)⋊C₄	1	1	yes	characteristic
278	C₄	(C₂×#89137)⋊C₄	1	1	yes	characteristic
279	C₄	C₂×C₄×#89137	1	1	yes	characteristic
280	C₄	C₂×C₂²⋊#89137	1	1	yes	characteristic
281	C₂	—	2	1	no	—
282	C₂	—	2	1	no	—
283	C₂	—	109	2	no	—
284	C₂	—	109	2	no	—
285	C₂	—	218	1	no	—
286	C₂	—	218	1	no	—
287	C₂	#20295898	1	2	yes	direct product factor
288	C₂	C₂×(C₂×#89137)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊#89137
commutator	C₄×C₁₀₉	C₂×C₄²
Frattini	C₂×C₄	C₂²×#89137
Fitting	C₂×M₁₆×C₁₀₉	C₄
socle	C₂²×C₁₀₉	#95
prime radical	C₂³×D₁₀₉	D₄
cosocle	C₂×C₄×D₁₀₉	C₂³
hypercenter	C₂×M₁₆	#89137
hypocenter	C₁₀₉	C₂×#339
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
3. #20295898	C₂	direct (N × H)
8. #16812778	C₄	semidirect
10. C₂×C₄⋊#89137	C₄	semidirect
16. C₂×C₄×#89137	C₄	semidirect
17. C₂×C₄×#89137	C₄	semidirect
20. C₂×#13182013	C₄	semidirect
24. M₁₆×D₁₀₉	C₄	semidirect
27. C₂×M₁₆×C₁₀₉	C₄	semidirect
34. C₄⋊#89137	C₂×C₄	semidirect
42. C₄×#89137	C₂×C₄	semidirect
58. #13182013	C₂×C₄	semidirect
66. M₁₆×C₁₀₉	C₂×C₄	semidirect
138. C₂×C₁₀₉	#339	semidirect
142. D₁₀₉	#339	semidirect
145. C₁₀₉	C₂×#339	semidirect
158. C₂×M₁₆	#89137	semidirect
198. M₁₆	C₂×#89137	semidirect
287. C₂	#20295898	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×#16812778	2
C₂²×C₄×#89137	2
#20295898	2
C₂×M₁₆×D₁₀₉	2
C₂×#339	109

Subgroups — C2×#20295898 #28378710

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20295898 #28378710