Subgroups — C₂²⋊Q₈×Dic₂₆₉ #46813421

Order 34432.

656 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×Dic₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₄×Dic₂₆₉	C₂	1	1	yes	maximal, characteristic
3	#31770512	C₂	1	1	yes	maximal, characteristic, complemented
4	#31770504	C₂	1	1	yes	maximal, characteristic, complemented
5	#31770491	C₂	1	2	yes	maximal, complemented
6	#31770480	C₂	1	2	yes	maximal
7	#31770465	C₂	1	1	yes	maximal, characteristic
8	C₄⋊C₄×Dic₂₆₉	C₂	1	2	yes	maximal, complemented
9	C₄⋊C₄×Dic₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²⋊C₄×Dic₂₆₉	C₂	1	2	yes	maximal
11	C₂×C₂²⋊Q₈×C₂₆₉	C₂	1	1	yes	maximal, characteristic
12	C₂³×Dic₂₆₉	C₂²	1	1	yes	characteristic
13	C₂³×Dic₂₆₉	C₂²	1	1	yes	characteristic
14	Q₈×Dic₂₆₉	—	2	4	no	—
15	C₂×C₄⋊Dic₂₆₉	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₂₆₉	C₂²	1	2	yes	—
17	C₂×C₄⋊Dic₂₆₉	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Dic₂₆₉	C₂²	1	2	yes	—
19	C₂×C₂²⋊Dic₂₆₉	C₂²	1	2	yes	—
20	C₂×C₄×Dic₂₆₉	—	2	4	no	—
21	C₂×C₄×Dic₂₆₉	C₂²	1	2	yes	—
22	C₂×C₄×Dic₂₆₉	C₂²	1	2	yes	—
23	C₂×C₄×Dic₂₆₉	C₂²	1	1	yes	characteristic
24	C₂×C₄×Dic₂₆₉	C₂²	1	1	yes	characteristic
25	C₂×C₄×Dic₂₆₉	C₂²	1	1	yes	characteristic
26	C₂×C₄×Dic₂₆₉	C₂²	1	1	yes	characteristic
27	#22575360	C₂²	1	2	yes	—
28	#22575360	C₂²	1	2	yes	—
29	#22575360	C₂²	1	2	yes	—
30	#22575360	C₂²	1	2	yes	—
31	#22575360	C₂²	1	2	yes	—
32	C₂²×Q₈×C₂₆₉	C₂²	1	1	yes	characteristic
33	C₂³×C₄×C₂₆₉	C₂²	1	1	yes	characteristic
34	C₂²⋊Q₈×C₂₆₉	C₄	1	8	yes	complemented
35	C₂×C₄⋊C₄×C₂₆₉	C₂²	1	2	yes	—
36	C₂×C₄⋊C₄×C₂₆₉	C₂²	1	1	yes	characteristic
37	C₂×C₂²⋊C₄×C₂₆₉	C₂²	1	2	yes	—
38	C₂²×Dic₂₆₉	—	2	2	no	—
39	C₂²×Dic₂₆₉	—	2	2	no	—
40	C₂²×Dic₂₆₉	—	2	2	no	—
41	C₂²×Dic₂₆₉	—	2	2	no	—
42	C₂²×Dic₂₆₉	—	2	2	no	—
43	C₂²×Dic₂₆₉	Q₈	1	4	yes	complemented
44	C₂²×Dic₂₆₉	C₂³	1	2	yes	—
45	C₂²×Dic₂₆₉	C₂³	1	2	yes	—
46	C₂²×Dic₂₆₉	C₂³	1	1	yes	characteristic
47	C₂²×Dic₂₆₉	C₂³	1	1	yes	characteristic
48	C₂²×Dic₂₆₉	C₂³	1	1	yes	characteristic
49	C₂²×Dic₂₆₉	C₂³	1	1	yes	characteristic
50	C₄⋊Dic₂₆₉	—	2	4	no	—
51	C₄⋊Dic₂₆₉	—	2	4	no	—
52	C₄⋊Dic₂₆₉	—	2	2	no	—
53	C₄⋊Dic₂₆₉	—	2	2	no	—
54	C₂²⋊Dic₂₆₉	—	2	4	no	—
55	C₄×Dic₂₆₉	—	2	2	no	—
56	C₄×Dic₂₆₉	—	2	2	no	—
57	C₄×Dic₂₆₉	—	2	4	no	—
58	C₄×Dic₂₆₉	—	2	4	no	—
59	C₄×Dic₂₆₉	—	2	2	no	—
60	C₄×Dic₂₆₉	D₄	1	4	yes	—
61	C₂⁴×C₂₆₉	C₂³	1	1	yes	characteristic
62	C₂×Q₈×C₂₆₉	—	2	2	no	—
63	C₂×Q₈×C₂₆₉	—	2	2	no	—
64	C₂×Q₈×C₂₆₉	C₂×C₄	1	4	yes	complemented
65	C₂²×C₄×C₂₆₉	—	2	2	no	—
66	C₂²×C₄×C₂₆₉	—	2	2	no	—
67	C₂²×C₄×C₂₆₉	C₂×C₄	1	4	yes	—
68	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
69	C₂²×C₄×C₂₆₉	C₂³	1	2	yes	—
70	C₂²×C₄×C₂₆₉	C₂³	1	2	yes	—
71	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
72	C₄⋊C₄×C₂₆₉	C₂×C₄	1	4	yes	complemented
73	C₄⋊C₄×C₂₆₉	C₂×C₄	1	8	yes	complemented
74	C₂²⋊C₄×C₂₆₉	C₂×C₄	1	8	yes	—
75	C₂×Dic₂₆₉	—	2	8	no	—
76	C₂×Dic₂₆₉	—	2	1	no	—
77	C₂×Dic₂₆₉	—	2	1	no	—
78	C₂×Dic₂₆₉	—	2	1	no	—
79	C₂×Dic₂₆₉	—	2	1	no	—
80	C₂×Dic₂₆₉	—	2	1	no	—
81	C₂×Dic₂₆₉	—	2	1	no	—
82	C₂×Dic₂₆₉	—	2	8	no	—
83	C₂×Dic₂₆₉	—	2	2	no	—
84	C₂×Dic₂₆₉	—	2	2	no	—
85	C₂×Dic₂₆₉	—	2	2	no	—
86	C₂×Dic₂₆₉	—	2	2	no	—
87	C₂×Dic₂₆₉	—	2	2	no	—
88	C₂×Dic₂₆₉	—	2	2	no	—
89	C₂×Dic₂₆₉	C₂×D₄	1	2	yes	—
90	C₂×Dic₂₆₉	C₂×D₄	1	2	yes	—
91	C₂×Dic₂₆₉	C₂×Q₈	1	2	yes	—
92	C₂×Dic₂₆₉	C₂×Q₈	1	2	yes	—
93	C₂×Dic₂₆₉	#41	1	2	yes	—
94	C₂×Dic₂₆₉	#41	1	2	yes	—
95	C₂³×C₂₆₉	—	2	1	no	—
96	C₂³×C₂₆₉	—	2	1	no	—
97	C₂³×C₂₆₉	—	2	1	no	—
98	C₂³×C₂₆₉	—	2	1	no	—
99	C₂³×C₂₆₉	C₂²×C₄	1	2	yes	—
100	C₂³×C₂₆₉	C₂×Q₈	1	2	yes	—
101	C₂³×C₂₆₉	#41	1	2	yes	—
102	C₂³×C₂₆₉	C₂⁴	1	1	yes	characteristic
103	Q₈×C₂₆₉	—	2	8	no	—
104	C₂×C₄×C₂₆₉	—	2	1	no	—
105	C₂×C₄×C₂₆₉	—	2	2	no	—
106	C₂×C₄×C₂₆₉	—	2	1	no	—
107	C₂×C₄×C₂₆₉	—	2	2	no	—
108	C₂×C₄×C₂₆₉	—	2	2	no	—
109	C₂×C₄×C₂₆₉	—	2	2	no	—
110	C₂×C₄×C₂₆₉	—	2	8	no	—
111	C₂×C₄×C₂₆₉	C₂²×C₄	1	4	yes	—
112	C₂×C₄×C₂₆₉	C₂²×C₄	1	4	yes	—
113	C₂×C₄×C₂₆₉	C₂²×C₄	1	2	yes	—
114	C₂×C₄×C₂₆₉	C₂²×C₄	1	2	yes	—
115	C₂×C₄×C₂₆₉	C₂×D₄	1	2	yes	—
116	C₂×C₄×C₂₆₉	#41	1	2	yes	—
117	C₂²×C₂₆₉	—	2	2	no	—
118	C₂²×C₂₆₉	—	2	2	no	—
119	C₂²×C₂₆₉	—	2	2	no	—
120	C₂²×C₂₆₉	—	2	2	no	—
121	C₂²×C₂₆₉	—	2	2	no	—
122	C₂²×C₂₆₉	—	2	2	no	—
123	C₂²×C₂₆₉	C₄×Q₈	1	4	yes	complemented
124	C₂²×C₂₆₉	C₂³×C₄	1	1	yes	characteristic
125	C₂²×C₂₆₉	C₂²×D₄	1	1	yes	characteristic
126	C₂²×C₂₆₉	C₂²×Q₈	1	1	yes	characteristic
127	C₂²×C₂₆₉	C₂×#41	1	1	yes	characteristic
128	C₂²×C₂₆₉	C₂×#41	1	1	yes	characteristic
129	C₂²×C₂₆₉	C₂×#41	1	1	yes	characteristic
130	C₂²×C₂₆₉	C₂×#41	1	1	yes	characteristic
131	C₄×C₂₆₉	—	2	2	no	—
132	C₄×C₂₆₉	—	2	4	no	—
133	C₄×C₂₆₉	—	2	4	no	—
134	C₄×C₂₆₉	C₄×D₄	1	4	yes	—
135	Dic₂₆₉	—	2	2	no	—
136	Dic₂₆₉	—	2	2	no	—
137	Dic₂₆₉	—	2	2	no	—
138	Dic₂₆₉	—	4	2	no	—
139	Dic₂₆₉	—	4	2	no	—
140	Dic₂₆₉	C₂²⋊Q₈	1	4	yes	direct product factor
141	C₂×C₂₆₉	—	2	4	no	—
142	C₂×C₂₆₉	C₂×C₄×D₄	1	1	yes	characteristic
143	C₂×C₂₆₉	C₂×C₄×Q₈	1	1	yes	characteristic
144	C₂×C₂₆₉	C₄×#41	1	1	yes	characteristic
145	C₂×C₂₆₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
146	C₂×C₂₆₉	#525	1	1	yes	characteristic
147	C₂×C₂₆₉	#529	1	1	yes	characteristic
148	C₂×C₂₆₉	#533	1	1	yes	characteristic
149	C₂₆₉	C₄×C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
150	C₄×C₂²⋊Q₈	—	269	1	no	maximal, Sylow
151	C₂×C₂²⋊Q₈	D₂₆₉	1	1	yes	characteristic
152	C₂×C₄×Q₈	—	269	1	no	—
153	C₂²×C₄²	—	269	1	no	—
154	#391	—	269	1	no	—
155	#389	—	269	1	no	—
156	#387	—	269	2	no	—
157	#385	—	269	2	no	—
158	#380	—	269	1	no	—
159	C₄×C₄⋊C₄	—	269	2	no	—
160	C₄×C₄⋊C₄	—	269	1	no	—
161	C₄×C₂²⋊C₄	—	269	2	no	—
162	C₂²×Q₈	C₂×D₂₆₉	1	1	yes	characteristic
163	C₂³×C₄	—	269	1	no	—
164	C₂³×C₄	—	269	1	no	—
165	C₂³×C₄	C₂×D₂₆₉	1	1	yes	characteristic
166	C₂²⋊Q₈	Dic₂₆₉	1	8	yes	direct product factor
167	C₄×Q₈	—	538	4	no	—
168	C₂×C₄⋊C₄	—	269	1	no	—
169	C₂×C₄⋊C₄	—	269	2	no	—
170	C₂×C₄⋊C₄	—	269	1	no	—
171	C₂×C₄⋊C₄	—	269	2	no	—
172	C₂×C₄⋊C₄	C₂×D₂₆₉	1	2	yes	—
173	C₂×C₄⋊C₄	C₂×D₂₆₉	1	1	yes	characteristic
174	C₂×C₂²⋊C₄	—	269	2	no	—
175	C₂×C₂²⋊C₄	C₂×D₂₆₉	1	2	yes	—
176	C₂×C₄²	—	269	1	no	—
177	C₂×C₄²	—	269	2	no	—
178	C₂×C₄²	—	269	2	no	—
179	C₂×C₄²	—	269	1	no	—
180	C₂×C₄²	—	269	1	no	—
181	C₂×C₄²	—	269	1	no	—
182	C₂×C₄²	—	538	4	no	—
183	#95	—	269	2	no	—
184	#95	—	269	2	no	—
185	#95	—	269	2	no	—
186	#95	—	269	2	no	—
187	#95	—	269	2	no	—
188	C₂⁴	C₂²×D₂₆₉	1	1	yes	characteristic
189	C₂×Q₈	—	2	2	no	—
190	C₂×Q₈	—	2	2	no	—
191	C₂×Q₈	C₂×Dic₂₆₉	1	4	yes	complemented
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	269	1	no	—
195	C₂²×C₄	—	269	4	no	—
196	C₂²×C₄	—	269	1	no	—
197	C₂²×C₄	—	269	1	no	—
198	C₂²×C₄	—	269	1	no	—
199	C₂²×C₄	—	269	2	no	—
200	C₂²×C₄	—	269	2	no	—
201	C₂²×C₄	—	538	2	no	—
202	C₂²×C₄	—	538	2	no	—
203	C₂²×C₄	—	538	2	no	—
204	C₂²×C₄	—	538	2	no	—
205	C₂²×C₄	—	538	2	no	—
206	C₂²×C₄	C₂×Dic₂₆₉	1	4	yes	—
207	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
208	C₂²×C₄	C₂²×D₂₆₉	1	2	yes	—
209	C₂²×C₄	C₂²×D₂₆₉	1	2	yes	—
210	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
211	C₄⋊C₄	—	538	4	no	—
212	C₄⋊C₄	—	538	4	no	—
213	C₄⋊C₄	—	538	2	no	—
214	C₄⋊C₄	—	538	2	no	—
215	C₄⋊C₄	C₂×Dic₂₆₉	1	4	yes	complemented
216	C₄⋊C₄	C₂×Dic₂₆₉	1	8	yes	complemented
217	C₂²⋊C₄	—	538	4	no	—
218	C₂²⋊C₄	C₂×Dic₂₆₉	1	8	yes	—
219	C₄²	—	269	4	no	—
220	C₄²	—	538	2	no	—
221	C₄²	—	538	2	no	—
222	C₄²	—	538	4	no	—
223	C₄²	—	538	4	no	—
224	C₄²	—	538	2	no	—
225	C₂³	—	2	1	no	—
226	C₂³	—	2	1	no	—
227	C₂³	—	2	1	no	—
228	C₂³	—	2	1	no	—
229	C₂³	C₂²×Dic₂₆₉	1	2	yes	—
230	C₂³	Q₈×D₂₆₉	1	2	yes	—
231	C₂³	#16864403	1	2	yes	—
232	C₂³	C₂³×D₂₆₉	1	1	yes	characteristic
233	Q₈	—	2	8	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	8	no	—
241	C₂×C₄	—	269	2	no	—
242	C₂×C₄	—	269	2	no	—
243	C₂×C₄	—	269	2	no	—
244	C₂×C₄	—	269	2	no	—
245	C₂×C₄	—	269	2	no	—
246	C₂×C₄	—	269	2	no	—
247	C₂×C₄	—	538	8	no	—
248	C₂×C₄	—	538	8	no	—
249	C₂×C₄	—	538	1	no	—
250	C₂×C₄	—	538	1	no	—
251	C₂×C₄	—	538	1	no	—
252	C₂×C₄	—	538	1	no	—
253	C₂×C₄	—	538	1	no	—
254	C₂×C₄	—	538	1	no	—
255	C₂×C₄	—	538	2	no	—
256	C₂×C₄	—	538	2	no	—
257	C₂×C₄	—	538	2	no	—
258	C₂×C₄	—	538	2	no	—
259	C₂×C₄	—	538	2	no	—
260	C₂×C₄	—	538	2	no	—
261	C₂×C₄	C₂²×Dic₂₆₉	1	4	yes	—
262	C₂×C₄	C₂²×Dic₂₆₉	1	4	yes	—
263	C₂×C₄	C₂²×Dic₂₆₉	1	2	yes	—
264	C₂×C₄	C₂²×Dic₂₆₉	1	2	yes	—
265	C₂×C₄	D₄×D₂₆₉	1	2	yes	—
266	C₂×C₄	#16864401	1	2	yes	—
267	C₂²	—	2	2	no	—
268	C₂²	—	2	2	no	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	2	no	—
273	C₂²	Q₈×Dic₂₆₉	1	4	yes	complemented
274	C₂²	C₂³×Dic₂₆₉	1	1	yes	characteristic
275	C₂²	C₂×D₄×D₂₆₉	1	1	yes	characteristic
276	C₂²	C₂×Q₈×D₂₆₉	1	1	yes	characteristic
277	C₂²	#41×D₂₆₉	1	1	yes	characteristic
278	C₂²	#41×D₂₆₉	1	1	yes	characteristic
279	C₂²	C₂×#16864401	1	1	yes	characteristic
280	C₂²	C₂×#16864403	1	1	yes	characteristic
281	C₄	—	2	2	no	—
282	C₄	—	2	4	no	—
283	C₄	—	2	4	no	—
284	C₄	—	269	4	no	—
285	C₄	—	538	2	no	—
286	C₄	—	538	2	no	—
287	C₄	—	538	2	no	—
288	C₄	—	1076	2	no	—
289	C₄	—	1076	2	no	—
290	C₄	D₄×Dic₂₆₉	1	4	yes	—
291	C₂	—	2	4	no	—
292	C₂	C₂×D₄×Dic₂₆₉	1	1	yes	characteristic
293	C₂	C₂×Q₈×Dic₂₆₉	1	1	yes	characteristic
294	C₂	#41×Dic₂₆₉	1	1	yes	characteristic
295	C₂	C₂²⋊Q₈×D₂₆₉	1	1	yes	characteristic
296	C₂	#31771118	1	1	yes	characteristic
297	C₂	#31771141	1	1	yes	characteristic
298	C₂	#31771157	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₆₉
commutator	C₂²×C₂₆₉	C₂³×C₄
Fitting	C₂×C₂²⋊Q₈×C₂₆₉	C₂
socle, cosocle	C₂³×C₂₆₉	C₂⁴
prime radical	C₂⁴×C₂₆₉	C₂³
hypercenter	C₂×C₂²⋊Q₈	D₂₆₉
hypocenter	C₂₆₉	C₄×C₂²⋊Q₈
LCS	G ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×Dic₂₆₉	C₂	semidirect
3. #31770512	C₂	semidirect
4. #31770504	C₂	semidirect
5. #31770491	C₂	semidirect
8. C₄⋊C₄×Dic₂₆₉	C₂	semidirect
9. C₄⋊C₄×Dic₂₆₉	C₂	semidirect
34. C₂²⋊Q₈×C₂₆₉	C₄	semidirect
43. C₂²×Dic₂₆₉	Q₈	semidirect
64. C₂×Q₈×C₂₆₉	C₂×C₄	semidirect
72. C₄⋊C₄×C₂₆₉	C₂×C₄	semidirect
73. C₄⋊C₄×C₂₆₉	C₂×C₄	semidirect
123. C₂²×C₂₆₉	C₄×Q₈	semidirect
140. Dic₂₆₉	C₂²⋊Q₈	direct (N × H)
149. C₂₆₉	C₄×C₂²⋊Q₈	semidirect
166. C₂²⋊Q₈	Dic₂₆₉	direct (N × H)
191. C₂×Q₈	C₂×Dic₂₆₉	semidirect
215. C₄⋊C₄	C₂×Dic₂₆₉	semidirect
216. C₄⋊C₄	C₂×Dic₂₆₉	semidirect
273. C₂²	Q₈×Dic₂₆₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×Dic₂₆₉	2
C₂²×C₄×Dic₂₆₉	2
#31770512	2
#31770504	2
#31770491	2
#31770480	2
#31770465	2
C₄⋊C₄×Dic₂₆₉	2
C₂²⋊C₄×Dic₂₆₉	2
C₂×C₂²⋊Q₈×C₂₆₉	2
C₄×C₂²⋊Q₈	269

Subgroups — C22⋊Q8×Dic269 #46813421

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²⋊Q₈×Dic₂₆₉ #46813421