Subgroups — C₂×#26021827 #36584393

Order 22784.

710 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151647	C₂	1	1	yes	maximal, characteristic
2	#26021827	C₂	1	8	yes	maximal, direct product factor
3	C₂×#19151370	C₂	1	2	yes	maximal
4	C₂×#19151358	C₂	1	2	yes	maximal
5	C₂×#19151355	C₂	1	1	yes	maximal, characteristic
6	C₂×#423×C₈₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×#13083558	C₂²	1	2	yes	—
8	C₂×#15667971	C₂²	1	2	yes	—
9	#19151647	—	2	2	no	—
10	#19151647	C₂²	1	4	yes	—
11	C₂×#15667948	—	2	2	no	—
12	#19151370	C₂²	1	8	yes	—
13	#19151358	C₂²	1	8	yes	—
14	#19151355	C₂²	1	4	yes	—
15	C₂²×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
16	C₂²×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
17	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
18	#423×C₈₉	—	2	2	no	—
19	#423×C₈₉	C₂²	1	4	yes	—
20	C₂×C₄⋊C₈×C₈₉	—	2	2	no	—
21	C₂×#13083558	—	2	2	no	—
22	C₂×#13083558	—	2	2	no	—
23	C₂×#13083558	—	2	2	no	—
24	C₂×#13083558	—	2	2	no	—
25	C₂×#13083558	—	2	2	no	—
26	C₂×#13083558	C₂³	1	4	yes	—
27	C₂²×#10623494	—	2	2	no	—
28	#15667971	—	2	2	no	—
29	#15667971	—	2	2	no	—
30	#15667971	—	2	2	no	—
31	#15667971	C₂³	1	4	yes	—
32	C₂×C₄⋊Dic₈₉	—	2	2	no	—
33	C₂×C₂²⋊Dic₈₉	—	2	2	no	—
34	C₂×C₄×Dic₈₉	—	2	2	no	—
35	#15667948	—	2	8	no	—
36	C₂³×C₄×C₈₉	—	2	1	no	—
37	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
38	C₂×M₁₆×C₈₉	—	2	2	no	—
39	C₂×M₁₆×C₈₉	—	2	2	no	—
40	C₂×M₁₆×C₈₉	—	2	1	no	—
41	C₂×M₁₆×C₈₉	—	2	2	no	—
42	C₂×M₁₆×C₈₉	—	2	2	no	—
43	C₂×M₁₆×C₈₉	D₄	1	2	yes	—
44	C₂×M₁₆×C₈₉	C₂³	1	2	yes	—
45	C₂×M₁₆×C₈₉	C₂³	1	2	yes	—
46	C₂²×C₈×C₈₉	—	2	1	no	—
47	C₂²×C₈×C₈₉	D₄	1	2	yes	—
48	C₂×C₄²×C₈₉	—	2	1	no	—
49	C₂×C₄²×C₈₉	—	2	1	no	—
50	C₂×C₄²×C₈₉	—	2	1	no	—
51	C₂×C₄²×C₈₉	Q₈	1	1	yes	characteristic
52	C₂×C₄²×C₈₉	Q₈	1	2	yes	—
53	C₂×C₄²×C₈₉	Q₈	1	1	yes	characteristic
54	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
55	C₄⋊C₈×C₈₉	—	2	8	no	—
56	C₂²×Dic₈₉	—	4	2	no	—
57	#13083558	—	2	4	no	—
58	#13083558	—	2	4	no	—
59	#13083558	—	4	2	no	—
60	#13083558	—	4	2	no	—
61	C₂×#10623494	—	2	4	no	—
62	C₂×#10623494	—	4	2	no	—
63	C₂×#10623494	—	4	2	no	—
64	C₄⋊Dic₈₉	—	2	4	no	—
65	C₄⋊Dic₈₉	—	4	2	no	—
66	C₂²⋊Dic₈₉	—	2	4	no	—
67	C₂²⋊Dic₈₉	—	4	2	no	—
68	C₄×Dic₈₉	—	2	4	no	—
69	C₄×Dic₈₉	—	4	2	no	—
70	C₂⁴×C₈₉	#41	1	1	yes	characteristic
71	C₂²×C₄×C₈₉	—	2	1	no	—
72	C₂²×C₄×C₈₉	—	2	2	no	—
73	C₂²×C₄×C₈₉	—	2	1	no	—
74	C₂²×C₄×C₈₉	—	2	2	no	—
75	C₂²×C₄×C₈₉	—	4	4	no	—
76	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
79	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
80	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
81	C₂²×C₄×C₈₉	#41	1	2	yes	—
82	C₂²×C₄×C₈₉	#41	1	2	yes	—
83	C₂²×C₄×C₈₉	#41	1	2	yes	—
84	C₂²×C₄×C₈₉	#41	1	2	yes	—
85	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
86	M₁₆×C₈₉	—	2	4	no	—
87	M₁₆×C₈₉	—	2	4	no	—
88	M₁₆×C₈₉	—	2	4	no	—
89	M₁₆×C₈₉	—	4	2	no	—
90	C₂×C₈×C₈₉	—	2	2	no	—
91	C₂×C₈×C₈₉	—	2	2	no	—
92	C₂×C₈×C₈₉	—	2	2	no	—
93	C₂×C₈×C₈₉	—	4	1	no	—
94	C₂×C₈×C₈₉	—	4	1	no	—
95	C₂×C₈×C₈₉	C₂×D₄	1	4	yes	—
96	C₄²×C₈₉	—	2	1	no	—
97	C₄²×C₈₉	—	2	2	no	—
98	C₄²×C₈₉	—	2	1	no	—
99	C₄²×C₈₉	—	2	2	no	—
100	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
101	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
102	C₂³×C₈₉	—	4	2	no	—
103	C₂³×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
104	C₂³×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
105	C₂³×C₈₉	#126	1	2	yes	—
106	C₂³×C₈₉	#126	1	2	yes	—
107	C₂³×C₈₉	C₂×#41	1	1	yes	characteristic
108	C₂×C₄×C₈₉	—	2	4	no	—
109	C₂×C₄×C₈₉	—	2	2	no	—
110	C₂×C₄×C₈₉	—	2	2	no	—
111	C₂×C₄×C₈₉	—	4	4	no	—
112	C₂×C₄×C₈₉	—	4	4	no	—
113	C₂×C₄×C₈₉	—	4	2	no	—
114	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
115	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
116	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
117	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
118	C₂×C₄×C₈₉	#123	1	2	yes	—
119	C₂×C₄×C₈₉	#124	1	2	yes	—
120	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
121	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
122	C₂×C₄×C₈₉	#125	1	2	yes	—
123	C₂×C₄×C₈₉	#125	1	2	yes	—
124	C₂×C₄×C₈₉	#125	1	2	yes	—
125	C₂×C₄×C₈₉	#125	1	2	yes	—
126	C₂×C₄×C₈₉	#126	1	2	yes	—
127	C₂×C₄×C₈₉	#126	1	2	yes	—
128	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
130	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
131	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
132	C₂×Dic₈₉	—	4	4	no	—
133	C₂×Dic₈₉	—	8	4	no	—
134	C₈×C₈₉	—	2	4	no	—
135	C₈×C₈₉	—	4	2	no	—
136	#10623494	—	4	4	no	—
137	C₂²×C₈₉	—	2	4	no	—
138	C₂²×C₈₉	—	2	2	no	—
139	C₂²×C₈₉	—	4	1	no	—
140	C₂²×C₈₉	—	4	1	no	—
141	C₂²×C₈₉	—	4	2	no	—
142	C₂²×C₈₉	#400	1	1	yes	characteristic
143	C₂²×C₈₉	#400	1	2	yes	—
144	C₂²×C₈₉	#400	1	1	yes	characteristic
145	C₂²×C₈₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
146	C₂²×C₈₉	C₂×#126	1	2	yes	—
147	C₄×C₈₉	—	2	4	no	—
148	C₄×C₈₉	—	4	2	no	—
149	C₄×C₈₉	#400	1	1	yes	characteristic
150	C₄×C₈₉	#400	1	2	yes	—
151	C₄×C₈₉	#400	1	1	yes	characteristic
152	C₄×C₈₉	C₂×#123	1	1	yes	characteristic
153	C₄×C₈₉	C₂×#124	1	1	yes	characteristic
154	C₄×C₈₉	C₂×#125	1	2	yes	—
155	Dic₈₉	—	8	4	no	—
156	C₂×#2102	—	89	1	no	maximal, Sylow
157	C₂×C₈₉	—	2	2	no	—
158	C₂×C₈₉	—	2	1	no	—
159	C₂×C₈₉	—	2	2	no	—
160	C₂×C₈₉	—	2	1	no	—
161	C₂×C₈₉	#2102	1	2	yes	complemented
162	C₂×C₈₉	C₂×#400	1	1	yes	characteristic
163	C₂×#430	—	89	1	no	—
164	C₂×#423	D₈₉	1	1	yes	characteristic
165	#2102	—	89	8	no	—
166	C₂×#344	—	89	2	no	—
167	C₂×#339	—	89	2	no	—
168	C₂×H(3,ℤ/4)	—	89	1	no	—
169	C₈₉	C₂×#2102	1	1	yes	characteristic, complemented, Sylow
170	C₂²×M₁₆	—	89	2	no	—
171	C₂²×M₁₆	C₂×D₈₉	1	1	yes	characteristic
172	C₂²×M₁₆	C₂×D₈₉	1	1	yes	characteristic
173	C₂×#117	—	89	2	no	—
174	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
175	#430	—	89	4	no	—
176	#430	—	178	2	no	—
177	C₂×#108	—	178	2	no	—
178	#423	—	2	2	no	—
179	#423	C₂×D₈₉	1	4	yes	—
180	C₂×C₄⋊C₈	—	2	2	no	—
181	#344	—	89	8	no	—
182	#339	—	89	8	no	—
183	H(3,ℤ/4)	—	89	4	no	—
184	C₂³×C₄	—	2	1	no	—
185	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	—	2	2	no	—
188	C₂×M₁₆	—	2	1	no	—
189	C₂×M₁₆	—	2	2	no	—
190	C₂×M₁₆	—	2	2	no	—
191	C₂×M₁₆	—	89	4	no	—
192	C₂×M₁₆	—	178	2	no	—
193	C₂×M₁₆	—	178	2	no	—
194	C₂×M₁₆	—	178	2	no	—
195	C₂×M₁₆	—	178	2	no	—
196	C₂×M₁₆	—	178	2	no	—
197	C₂×M₁₆	C₂²⋊D₈₉	1	2	yes	—
198	C₂×M₁₆	C₂²×D₈₉	1	2	yes	—
199	C₂×M₁₆	C₂²×D₈₉	1	2	yes	—
200	C₂²×C₈	—	2	1	no	—
201	C₂²×C₈	—	178	2	no	—
202	C₂²×C₈	C₂²⋊D₈₉	1	2	yes	—
203	#117	—	89	4	no	—
204	#117	—	178	2	no	—
205	#117	—	178	2	no	—
206	#117	—	178	2	no	—
207	C₂×C₄⋊C₄	—	178	2	no	—
208	C₂×C₂²⋊C₄	—	178	2	no	—
209	C₂×C₄²	—	2	1	no	—
210	C₂×C₄²	—	2	1	no	—
211	C₂×C₄²	—	2	1	no	—
212	C₂×C₄²	—	178	2	no	—
213	C₂×C₄²	Dic₁₇₈	1	1	yes	characteristic
214	C₂×C₄²	Dic₁₇₈	1	2	yes	—
215	C₂×C₄²	Dic₁₇₈	1	1	yes	characteristic
216	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
217	#108	—	178	8	no	—
218	C₄⋊C₈	—	2	8	no	—
219	C₂⁴	#13083574	1	1	yes	characteristic
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	1	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	4	4	no	—
225	C₂²×C₄	—	356	2	no	—
226	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
228	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
229	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
230	C₂²×C₄	#13083573	1	2	yes	—
231	C₂²×C₄	#13083573	1	2	yes	—
232	C₂²×C₄	#13083574	1	1	yes	characteristic
233	C₂²×C₄	#13083575	1	2	yes	—
234	C₂²×C₄	#13083575	1	2	yes	—
235	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
236	M₁₆	—	2	4	no	—
237	M₁₆	—	2	4	no	—
238	M₁₆	—	2	4	no	—
239	M₁₆	—	4	2	no	—
240	M₁₆	—	356	2	no	—
241	M₁₆	—	356	2	no	—
242	M₁₆	—	178	4	no	—
243	M₁₆	—	178	4	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	4	1	no	—
248	C₂×C₈	—	4	1	no	—
249	C₂×C₈	—	356	2	no	—
250	C₂×C₈	—	356	2	no	—
251	C₂×C₈	—	178	4	no	—
252	C₂×C₈	C₂×C₂²⋊D₈₉	1	4	yes	—
253	C₄⋊C₄	—	356	2	no	—
254	C₄⋊C₄	—	178	4	no	—
255	C₂²⋊C₄	—	356	2	no	—
256	C₂²⋊C₄	—	178	4	no	—
257	C₄²	—	2	1	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	2	1	no	—
260	C₄²	—	2	2	no	—
261	C₄²	—	356	2	no	—
262	C₄²	—	178	4	no	—
263	C₄²	C₂×Dic₁₇₈	1	2	yes	—
264	C₄²	C₂×Dic₁₇₈	1	2	yes	—
265	C₂³	—	4	2	no	—
266	C₂³	#15667991	1	1	yes	characteristic
267	C₂³	#15667991	1	1	yes	characteristic
268	C₂³	#15668007	1	2	yes	—
269	C₂³	#15668007	1	2	yes	—
270	C₂³	C₂×#13083574	1	1	yes	characteristic
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	356	4	no	—
278	C₂×C₄	—	712	4	no	—
279	C₂×C₄	#15667991	1	1	yes	characteristic
280	C₂×C₄	#15667991	1	1	yes	characteristic
281	C₂×C₄	#15667993	1	1	yes	characteristic
282	C₂×C₄	#15667993	1	1	yes	characteristic
283	C₂×C₄	#15667993	1	2	yes	—
284	C₂×C₄	#15668001	1	2	yes	—
285	C₂×C₄	#15668001	1	1	yes	characteristic
286	C₂×C₄	#15668001	1	1	yes	characteristic
287	C₂×C₄	#15668003	1	2	yes	—
288	C₂×C₄	#15668003	1	2	yes	—
289	C₂×C₄	#15668003	1	2	yes	—
290	C₂×C₄	#15668003	1	2	yes	—
291	C₂×C₄	#15668007	1	2	yes	—
292	C₂×C₄	#15668007	1	2	yes	—
293	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
294	C₂×C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
295	C₂×C₄	C₂×#13083573	1	2	yes	—
296	C₂×C₄	C₂×#13083575	1	2	yes	—
297	C₈	—	2	4	no	—
298	C₈	—	4	2	no	—
299	C₈	—	356	4	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	2	4	no	—
302	C₂²	—	4	1	no	—
303	C₂²	—	4	1	no	—
304	C₂²	—	4	2	no	—
305	C₂²	#19151537	1	1	yes	characteristic
306	C₂²	#19151537	1	2	yes	—
307	C₂²	#19151537	1	1	yes	characteristic
308	C₂²	C₂×#15667991	1	1	yes	characteristic
309	C₂²	C₂×#15668007	1	2	yes	—
310	C₄	—	2	4	no	—
311	C₄	—	4	2	no	—
312	C₄	—	712	4	no	—
313	C₄	#19151537	1	1	yes	characteristic
314	C₄	#19151537	1	2	yes	—
315	C₄	#19151537	1	1	yes	characteristic
316	C₄	C₂×#15667993	1	1	yes	characteristic
317	C₄	C₂×#15668001	1	1	yes	characteristic
318	C₄	C₂×#15668003	1	2	yes	—
319	C₂	—	2	1	no	—
320	C₂	—	2	2	no	—
321	C₂	—	2	1	no	—
322	C₂	—	2	2	no	—
323	C₂	#26021827	1	2	yes	direct product factor
324	C₂	C₂×#19151537	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#15668001
commutator, cosocle	C₂²×C₄×C₈₉	C₂⁴
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₂×#423×C₈₉	C₂
socle	C₂²×C₈₉	#400
prime radical	C₂⁴×C₈₉	#41
hypercenter	C₂×#423	D₈₉
hypocenter	C₈₉	C₂×#2102
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
2. #26021827	C₂	direct (N × H)
161. C₂×C₈₉	#2102	semidirect
169. C₈₉	C₂×#2102	semidirect
323. C₂	#26021827	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×#19151647	2
#26021827	2
C₂×#19151370	2
C₂×#19151358	2
C₂×#19151355	2
C₂×#423×C₈₉	2
C₂×#2102	89

Subgroups — C2×#26021827 #36584393

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26021827 #36584393