Subgroups — C₂×C₄×#15667958 #36707701

Order 22784.

678 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	C₂²×#15667958	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₃₅₆	C₂	1	1	yes	maximal, characteristic
3	C₄×#15667958	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19151423	C₂	1	1	yes	maximal, characteristic
5	C₂×#19151402	C₂	1	1	yes	maximal, characteristic
6	C₂×#19151401	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄×#13083549	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Q₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₃₅₆	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₁₇₈	—	2	1	no	—
11	C₂×#15667958	C₄	1	8	yes	direct product factor
12	C₂×#15667958	C₂²	1	4	yes	—
13	C₂²×#13083549	C₂²	1	1	yes	characteristic
14	C₂²×#13083549	C₂²	1	1	yes	characteristic
15	C₄×Dic₃₅₆	—	2	2	no	—
16	C₄×Dic₃₅₆	C₂²	1	4	yes	—
17	C₂×#15667946	C₂²	1	1	yes	characteristic
18	C₂×#15667937	—	2	1	no	—
19	#19151423	C₂²	1	4	yes	—
20	#19151402	C₂²	1	4	yes	—
21	#19151401	C₂²	1	4	yes	—
22	C₄×#13083549	C₂²	1	4	yes	—
23	C₂²×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
24	C₂×C₄×Q₈×C₈₉	—	2	1	no	—
25	C₄×Q₁₆×C₈₉	—	2	2	no	—
26	C₄×Q₁₆×C₈₉	C₂²	1	4	yes	—
27	C₂×#107×C₈₉	C₂²	1	1	yes	characteristic
28	C₂×#103×C₈₉	—	2	1	no	—
29	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
30	C₂²×Dic₁₇₈	—	2	1	no	—
31	C₂×Dic₃₅₆	—	2	1	no	—
32	C₂×Dic₃₅₆	—	2	2	no	—
33	C₂×Dic₃₅₆	C₂×C₄	1	4	yes	—
34	C₂×Dic₃₅₆	C₂³	1	2	yes	—
35	C₄×Dic₁₇₈	—	2	4	no	—
36	C₄×Dic₁₇₈	—	4	2	no	—
37	C₂×C₄⋊Dic₈₉	—	2	1	no	—
38	C₂×#13083553	—	4	1	no	—
39	C₂×C₄×Dic₈₉	—	4	1	no	—
40	#15667958	C₂×C₄	1	16	yes	direct product factor
41	C₂×#13083549	—	2	2	no	—
42	C₂×#13083549	C₂×C₄	1	4	yes	—
43	C₂×#13083549	C₂³	1	2	yes	—
44	C₂×#13083549	C₂³	1	2	yes	—
45	#15667946	—	2	1	no	—
46	#15667946	C₂³	1	2	yes	—
47	#15667937	—	2	4	no	—
48	C₂²×Q₈×C₈₉	—	2	1	no	—
49	C₂×Q₁₆×C₈₉	—	2	1	no	—
50	C₂×Q₁₆×C₈₉	—	2	2	no	—
51	C₂×Q₁₆×C₈₉	C₂×C₄	1	4	yes	—
52	C₂×Q₁₆×C₈₉	C₂³	1	2	yes	—
53	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
54	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
55	C₄×Q₈×C₈₉	—	2	4	no	—
56	C₄×Q₈×C₈₉	—	4	2	no	—
57	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
58	C₂×C₄⋊C₄×C₈₉	—	4	1	no	—
59	C₂×C₄²×C₈₉	—	4	1	no	—
60	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
61	#107×C₈₉	—	2	1	no	—
62	#107×C₈₉	C₂³	1	2	yes	—
63	#103×C₈₉	—	2	4	no	—
64	C₄×C₈×C₈₉	D₄	1	2	yes	—
65	C₄×C₈×C₈₉	C₂³	1	2	yes	—
66	C₂×Dic₁₇₈	—	2	4	no	—
67	C₂×Dic₁₇₈	—	2	2	no	—
68	C₂×Dic₁₇₈	—	4	1	no	—
69	C₂×Dic₁₇₈	—	4	2	no	—
70	C₂²×Dic₈₉	—	4	1	no	—
71	C₂²×Dic₈₉	—	4	1	no	—
72	Dic₃₅₆	—	2	4	no	—
73	Dic₃₅₆	—	2	2	no	—
74	Dic₃₅₆	C₂²×C₄	1	4	yes	—
75	C₄⋊Dic₈₉	—	2	2	no	—
76	C₄⋊Dic₈₉	—	4	1	no	—
77	#13083553	—	4	4	no	—
78	C₄×Dic₈₉	—	4	4	no	—
79	#13083549	—	2	2	no	—
80	#13083549	C₂²×C₄	1	4	yes	—
81	C₂×Q₈×C₈₉	—	2	4	no	—
82	C₂×Q₈×C₈₉	—	2	2	no	—
83	C₂×Q₈×C₈₉	—	4	1	no	—
84	C₂×Q₈×C₈₉	—	4	2	no	—
85	C₂²×C₄×C₈₉	—	4	1	no	—
86	C₂²×C₄×C₈₉	—	4	1	no	—
87	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
88	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
89	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
90	Q₁₆×C₈₉	—	2	4	no	—
91	Q₁₆×C₈₉	—	2	2	no	—
92	Q₁₆×C₈₉	C₂²×C₄	1	4	yes	—
93	C₂×C₈×C₈₉	—	2	2	no	—
94	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
95	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
96	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₈₉	#41	1	2	yes	—
98	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
99	C₄⋊C₄×C₈₉	—	2	2	no	—
100	C₄⋊C₄×C₈₉	—	4	1	no	—
101	C₄⋊C₄×C₈₉	—	4	4	no	—
102	C₄²×C₈₉	—	4	4	no	—
103	C₄²×C₈₉	D₈	1	2	yes	—
104	C₄²×C₈₉	C₂×D₄	1	2	yes	—
105	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
106	Q₈×C₈₉	—	2	4	no	—
107	Q₈×C₈₉	—	4	2	no	—
108	Q₈×C₈₉	—	4	4	no	—
109	C₂×C₄×C₈₉	—	2	2	no	—
110	C₂×C₄×C₈₉	—	4	2	no	—
111	C₂×C₄×C₈₉	—	4	4	no	—
112	C₂×C₄×C₈₉	—	4	2	no	—
113	C₂×C₄×C₈₉	—	8	2	no	—
114	C₂×C₄×C₈₉	Q₃₂	1	4	yes	complemented
115	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
117	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
118	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
119	C₂×C₄×C₈₉	#135	1	2	yes	—
120	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
121	C₂×Dic₈₉	—	4	2	no	—
122	C₂×Dic₈₉	—	4	4	no	—
123	C₂×Dic₈₉	—	4	2	no	—
124	C₂×Dic₈₉	—	8	2	no	—
125	Dic₁₇₈	—	2	4	no	—
126	Dic₁₇₈	—	4	2	no	—
127	Dic₁₇₈	—	4	4	no	—
128	C₈×C₈₉	—	2	2	no	—
129	C₈×C₈₉	C₄×D₄	1	2	yes	—
130	C₈×C₈₉	C₂³×C₄	1	1	yes	characteristic
131	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
132	C₂²×C₈₉	C₄×D₈	1	2	yes	—
133	C₂²×C₈₉	C₂×Q₃₂	1	2	yes	—
134	C₂²×C₈₉	#508	1	2	yes	—
135	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
136	C₄×C₈₉	—	2	2	no	—
137	C₄×C₈₉	—	4	4	no	—
138	C₄×C₈₉	—	8	2	no	—
139	C₄×C₈₉	C₄×D₈	1	2	yes	—
140	C₄×C₈₉	C₂×Q₃₂	1	4	yes	complemented
141	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
142	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
143	Dic₈₉	—	4	4	no	—
144	Dic₈₉	—	8	2	no	—
145	C₂×C₄×Q₃₂	—	89	1	no	maximal, Sylow
146	C₂×C₈₉	C₄×Q₃₂	1	4	yes	complemented
147	C₂×C₈₉	C₂×C₄×D₈	1	1	yes	characteristic
148	C₂×C₈₉	C₂²×Q₃₂	1	1	yes	characteristic
149	C₂×C₈₉	C₂×#508	1	1	yes	characteristic
150	C₂²×Q₃₂	—	89	1	no	—
151	C₂×C₄×Q₁₆	—	89	1	no	—
152	C₂×C₄×Q₁₆	D₈₉	1	1	yes	characteristic
153	C₄×Q₃₂	—	89	8	no	—
154	C₂×#366	—	89	1	no	—
155	C₂×#358	—	89	1	no	—
156	C₂×#358	—	89	1	no	—
157	C₂×C₄×C₁₆	—	89	1	no	—
158	C₈₉	C₂×C₄×Q₃₂	1	1	yes	characteristic, complemented, Sylow
159	C₂²×Q₁₆	—	89	1	no	—
160	C₂²×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
161	C₂×C₄×Q₈	—	2	1	no	—
162	C₂×C₄×Q₈	—	178	1	no	—
163	C₂×Q₃₂	—	89	8	no	—
164	C₂×Q₃₂	—	89	4	no	—
165	C₂²×C₁₆	—	89	1	no	—
166	C₂²×C₁₆	—	89	1	no	—
167	C₄×Q₁₆	—	2	2	no	—
168	C₄×Q₁₆	—	89	4	no	—
169	C₄×Q₁₆	—	178	2	no	—
170	C₄×Q₁₆	C₂×D₈₉	1	4	yes	—
171	C₂×#107	—	89	1	no	—
172	C₂×#107	C₂×D₈₉	1	1	yes	characteristic
173	C₂×#103	—	2	1	no	—
174	C₂×#103	—	178	1	no	—
175	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
176	#366	—	89	4	no	—
177	#358	—	89	4	no	—
178	#358	—	89	4	no	—
179	C₄×C₁₆	—	89	4	no	—
180	C₂²×Q₈	—	2	1	no	—
181	C₂²×Q₈	—	178	1	no	—
182	C₂×Q₁₆	—	2	1	no	—
183	C₂×Q₁₆	—	2	2	no	—
184	C₂×Q₁₆	—	89	4	no	—
185	C₂×Q₁₆	—	89	2	no	—
186	C₂×Q₁₆	—	178	1	no	—
187	C₂×Q₁₆	—	178	2	no	—
188	C₂×Q₁₆	C₄×D₈₉	1	4	yes	—
189	C₂×Q₁₆	C₂²×D₈₉	1	2	yes	—
190	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
191	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
192	C₄×Q₈	—	2	4	no	—
193	C₄×Q₈	—	4	2	no	—
194	C₄×Q₈	—	356	2	no	—
195	C₄×Q₈	—	178	4	no	—
196	C₂×C₄⋊C₄	—	2	1	no	—
197	C₂×C₄⋊C₄	—	4	1	no	—
198	C₂×C₄⋊C₄	—	356	1	no	—
199	C₂×C₄⋊C₄	—	178	1	no	—
200	C₂×C₄²	—	4	1	no	—
201	C₂×C₄²	—	356	1	no	—
202	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
203	Q₃₂	—	89	16	no	—
204	C₂×C₁₆	—	89	4	no	—
205	C₂×C₁₆	—	89	2	no	—
206	C₂×C₁₆	—	89	2	no	—
207	C₂×C₁₆	—	178	2	no	—
208	#107	—	2	1	no	—
209	#107	—	89	2	no	—
210	#107	—	178	1	no	—
211	#107	C₂²×D₈₉	1	2	yes	—
212	#103	—	2	4	no	—
213	#103	—	178	4	no	—
214	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
215	C₄×C₈	C₂²×D₈₉	1	2	yes	—
216	C₂×Q₈	—	2	4	no	—
217	C₂×Q₈	—	2	2	no	—
218	C₂×Q₈	—	4	1	no	—
219	C₂×Q₈	—	4	2	no	—
220	C₂×Q₈	—	356	1	no	—
221	C₂×Q₈	—	356	2	no	—
222	C₂×Q₈	—	178	4	no	—
223	C₂×Q₈	—	178	2	no	—
224	C₂²×C₄	—	4	1	no	—
225	C₂²×C₄	—	4	1	no	—
226	C₂²×C₄	—	356	1	no	—
227	C₂²×C₄	—	356	1	no	—
228	C₂²×C₄	#13083559	1	1	yes	characteristic
229	C₂²×C₄	#13083559	1	1	yes	characteristic
230	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
231	Q₁₆	—	2	4	no	—
232	Q₁₆	—	2	2	no	—
233	Q₁₆	—	89	4	no	—
234	Q₁₆	—	178	4	no	—
235	Q₁₆	—	178	2	no	—
236	Q₁₆	C₂×C₄×D₈₉	1	4	yes	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
239	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
240	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
241	C₂×C₈	#13083574	1	2	yes	—
242	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	4	1	no	—
245	C₄⋊C₄	—	4	4	no	—
246	C₄⋊C₄	—	356	4	no	—
247	C₄⋊C₄	—	356	1	no	—
248	C₄⋊C₄	—	178	2	no	—
249	C₄²	—	4	4	no	—
250	C₄²	—	356	4	no	—
251	C₄²	#13083559	1	2	yes	—
252	C₄²	C₂×C₂²⋊D₈₉	1	2	yes	—
253	C₁₆	—	89	4	no	—
254	C₁₆	—	178	2	no	—
255	C₂³	C₂×#13083559	1	1	yes	characteristic
256	Q₈	—	2	4	no	—
257	Q₈	—	4	4	no	—
258	Q₈	—	4	2	no	—
259	Q₈	—	356	4	no	—
260	Q₈	—	356	2	no	—
261	Q₈	—	178	4	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	8	2	no	—
267	C₂×C₄	—	356	4	no	—
268	C₂×C₄	—	356	2	no	—
269	C₂×C₄	—	356	2	no	—
270	C₂×C₄	—	712	2	no	—
271	C₂×C₄	#15667958	1	4	yes	direct product factor
272	C₂×C₄	C₄×C₂²⋊D₈₉	1	2	yes	—
273	C₂×C₄	C₂×#13083559	1	2	yes	—
274	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
275	C₂×C₄	C₂×#13083559	1	2	yes	—
276	C₂×C₄	#15668032	1	2	yes	—
277	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
278	C₈	—	2	2	no	—
279	C₈	C₄×C₂²⋊D₈₉	1	2	yes	—
280	C₈	C₂²×C₄×D₈₉	1	1	yes	characteristic
281	C₈	C₂×#13083574	1	1	yes	characteristic
282	C₂²	C₄×#13083559	1	2	yes	—
283	C₂²	C₂×#15667958	1	2	yes	—
284	C₂²	#19152028	1	2	yes	—
285	C₂²	C₂²×#13083559	1	1	yes	characteristic
286	C₄	—	2	2	no	—
287	C₄	—	4	4	no	—
288	C₄	—	8	2	no	—
289	C₄	—	356	4	no	—
290	C₄	—	712	2	no	—
291	C₄	C₄×#13083559	1	2	yes	—
292	C₄	C₂×#15667958	1	4	yes	direct product factor
293	C₄	C₂×C₄×C₂²⋊D₈₉	1	1	yes	characteristic
294	C₄	C₂×#15668032	1	1	yes	characteristic
295	C₂	C₄×#15667958	1	4	yes	direct product factor
296	C₂	C₂×C₄×#13083559	1	1	yes	characteristic
297	C₂	C₂²×#15667958	1	1	yes	characteristic
298	C₂	C₂×#19152028	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#13083559
commutator	C₈×C₈₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂×C₄×Q₁₆×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×D₈
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	C₂×C₄×Q₁₆	D₈₉
hypocenter	C₈₉	C₂×C₄×Q₃₂
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
3. C₄×#15667958	C₂	direct (N × H)
11. C₂×#15667958	C₄	direct (N × H)
40. #15667958	C₂×C₄	direct (N × H)
114. C₂×C₄×C₈₉	Q₃₂	semidirect
140. C₄×C₈₉	C₂×Q₃₂	semidirect
146. C₂×C₈₉	C₄×Q₃₂	semidirect
158. C₈₉	C₂×C₄×Q₃₂	semidirect
271. C₂×C₄	#15667958	direct (N × H)
292. C₄	C₂×#15667958	direct (N × H)
295. C₂	C₄×#15667958	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#15667958	2
C₂×C₄×Dic₃₅₆	2
C₄×#15667958	2
C₂×#19151423	2
C₂×#19151402	2
C₂×#19151401	2
C₂×C₄×#13083549	2
C₂×C₄×Q₁₆×C₈₉	2
C₂×C₄×Q₃₂	89

Subgroups — C2×C4×#15667958 #36707701

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#15667958 #36707701