Subgroups — #31818147

Order 17792.

482 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17984864	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×Dic₅₅₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#22581612	C₂	1	2	yes	maximal, complemented
4	#22581337	C₂	1	1	yes	maximal, characteristic, complemented
5	#22581194	C₂	1	1	yes	maximal, characteristic, complemented
6	#22581189	C₂	1	1	yes	maximal, characteristic, complemented
7	#22581176	C₂	1	2	yes	maximal, complemented
8	#22581117	C₂	1	2	yes	maximal, complemented
9	#22581094	C₂	1	2	yes	maximal, complemented
10	C₄×Dic₅₅₆	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₄×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14555785	C₂²	1	2	yes	—
13	C₂²×Dic₂₇₈	C₂²	1	2	yes	—
14	#17984864	—	2	4	no	—
15	C₂×Dic₅₅₆	—	2	4	no	—
16	C₂×Dic₅₅₆	C₂²	1	1	yes	characteristic
17	C₂×Dic₅₅₆	C₂²	1	2	yes	—
18	C₂×Dic₅₅₆	C₂²	1	1	yes	characteristic
19	C₂×#14555774	C₂²	1	2	yes	—
20	C₂×#14555774	C₂²	1	2	yes	—
21	C₂×D₁₁₁₂	C₂²	1	1	yes	characteristic
22	#17984830	C₂²	1	2	yes	—
23	#17984828	—	2	2	no	—
24	#17984821	C₂²	1	2	yes	—
25	#17984820	—	2	2	no	—
26	#17984816	C₂²	1	2	yes	—
27	#17984814	—	2	2	no	—
28	C₄×Dic₂₇₈	C₂²	1	2	yes	—
29	(C₂²⋊D₁₃₉)⋊C₄	—	2	2	no	—
30	#17984776	C₂²	1	1	yes	characteristic
31	#17984767	C₂²	1	2	yes	—
32	#17984767	C₂²	1	2	yes	—
33	#17984764	C₂²	1	2	yes	—
34	C₂²×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
35	C₂²×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
36	C₄×D₄×C₁₃₉	C₂²	1	1	yes	characteristic
37	C₄⋊C₈×C₁₃₉	C₂²	1	1	yes	characteristic
38	C₂²⋊C₈×C₁₃₉	C₂²	1	1	yes	characteristic
39	C₂²⋊C₈×C₁₃₉	C₂²	1	1	yes	characteristic
40	C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
41	#14555785	—	2	4	no	—
42	#14555785	—	4	2	no	—
43	C₂×Dic₂₇₈	—	2	4	no	—
44	C₂×Dic₂₇₈	—	2	2	no	—
45	C₂×Dic₂₇₈	—	4	2	no	—
46	C₂×Dic₂₇₈	C₂³	1	2	yes	—
47	C₂×Dic₂₇₈	C₂³	1	2	yes	—
48	C₂×D₅₅₆	C₂³	1	2	yes	—
49	C₂×C₂²⋊D₁₃₉	—	2	2	no	—
50	C₂×C₂²⋊D₁₃₉	—	2	2	no	—
51	C₂²×Dic₁₃₉	—	2	2	no	—
52	C₂×C₄×D₁₃₉	—	2	2	no	—
53	Dic₅₅₆	—	2	4	no	—
54	Dic₅₅₆	—	2	2	no	—
55	Dic₅₅₆	D₄	1	4	yes	complemented
56	#14555774	—	2	4	no	—
57	#14555774	—	4	2	no	—
58	D₁₁₁₂	—	2	2	no	—
59	C₄⋊Dic₁₃₉	C₂³	1	2	yes	—
60	Dic₁₃₉⋊C₄	—	2	2	no	—
61	Dic₁₃₉⋊C₄	—	2	2	no	—
62	C₂²⋊Dic₁₃₉	—	2	2	no	—
63	(C₂×D₁₃₉)⋊C₄	—	2	2	no	—
64	(C₂×D₁₃₉)⋊C₄	—	2	2	no	—
65	(C₂×D₁₃₉)⋊C₄	—	2	2	no	—
66	C₄×Dic₁₃₉	—	2	2	no	—
67	C₂×D₄×C₁₃₉	D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
69	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
70	C₂×C₈×C₁₃₉	—	2	2	no	—
71	C₂×C₈×C₁₃₉	—	2	2	no	—
72	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
73	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
74	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
75	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₁₃₉	D₄	1	1	yes	characteristic
77	C₂²⋊C₄×C₁₃₉	D₄	1	1	yes	characteristic
78	C₂²⋊C₄×C₁₃₉	D₄	1	1	yes	characteristic
79	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
80	C₂²×D₁₃₉	—	2	2	no	—
81	Dic₂₇₈	—	2	2	no	—
82	Dic₂₇₈	—	2	2	no	—
83	Dic₂₇₈	—	2	2	no	—
84	Dic₂₇₈	—	4	4	no	—
85	Dic₂₇₈	C₂×D₄	1	4	yes	—
86	D₅₅₆	—	2	2	no	—
87	D₅₅₆	—	2	2	no	—
88	C₂²⋊D₁₃₉	—	4	4	no	—
89	C₂²⋊D₁₃₉	—	4	4	no	—
90	C₂×Dic₁₃₉	—	2	2	no	—
91	C₂×Dic₁₃₉	—	2	2	no	—
92	C₂×Dic₁₃₉	—	2	2	no	—
93	C₂×Dic₁₃₉	—	4	4	no	—
94	C₄×D₁₃₉	—	4	4	no	—
95	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
96	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
97	D₄×C₁₃₉	—	2	2	no	—
98	C₂×C₄×C₁₃₉	—	2	1	no	—
99	C₂×C₄×C₁₃₉	—	2	1	no	—
100	C₂×C₄×C₁₃₉	—	2	1	no	—
101	C₂×C₄×C₁₃₉	—	2	1	no	—
102	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
107	C₈×C₁₃₉	—	2	1	no	—
108	C₈×C₁₃₉	—	2	1	no	—
109	C₈×C₁₃₉	—	2	1	no	—
110	C₈×C₁₃₉	C₂×D₄	1	2	yes	—
111	C₂²×C₁₃₉	—	2	1	no	—
112	C₂²×C₁₃₉	—	2	1	no	—
113	C₂²×C₁₃₉	—	2	1	no	—
114	C₂²×C₁₃₉	—	2	1	no	—
115	C₂²×C₁₃₉	—	2	1	no	—
116	C₂²×C₁₃₉	#135	1	2	yes	complemented
117	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
118	C₂×D₁₃₉	—	4	2	no	—
119	C₂×D₁₃₉	—	4	2	no	—
120	C₂×D₁₃₉	—	4	2	no	—
121	C₄×C₁₃₉	—	2	1	no	—
122	C₄×C₁₃₉	—	2	1	no	—
123	C₄×C₁₃₉	—	2	1	no	—
124	C₄×C₁₃₉	—	2	1	no	—
125	C₄×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
126	C₄×C₁₃₉	H(5,𝔽₂)	1	1	yes	characteristic
127	Dic₁₃₉	—	2	4	no	—
128	Dic₁₃₉	—	4	2	no	—
129	Dic₁₃₉	—	4	2	no	—
130	C₂×C₁₃₉	—	2	2	no	—
131	C₂×C₁₃₉	—	4	1	no	—
132	C₂×C₁₃₉	D₄²	1	1	yes	characteristic
133	C₂×C₁₃₉	C₂×#135	1	1	yes	characteristic
134	C₂×C₁₃₉	#578	1	1	yes	characteristic
135	D₁₃₉	—	8	2	no	—
136	C₁₃₉	#3285	1	1	yes	characteristic, complemented, Sylow
137	#3285	—	139	1	no	maximal, Sylow
138	C₂×#135	—	139	1	no	—
139	C₂²×Q₁₆	—	139	1	no	—
140	#548	—	139	2	no	—
141	#495	—	139	1	no	—
142	#467	—	139	1	no	—
143	#466	—	139	1	no	—
144	#464	—	139	2	no	—
145	#452	—	139	2	no	—
146	#448	—	139	2	no	—
147	C₄×Q₁₆	—	139	1	no	—
148	C₈×D₄	D₁₃₉	1	1	yes	characteristic, complemented
149	C₂×#41	—	139	2	no	—
150	C₂²×Q₈	—	139	2	no	—
151	#135	—	278	4	no	—
152	C₂×Q₁₆	—	278	4	no	—
153	C₂×Q₁₆	—	139	2	no	—
154	C₂×Q₁₆	—	139	1	no	—
155	C₂×Q₁₆	—	139	1	no	—
156	C₂×SD₁₆	—	139	2	no	—
157	C₂×SD₁₆	—	139	2	no	—
158	C₂×D₈	—	139	1	no	—
159	C₂²×C₈	C₂×D₁₃₉	1	1	yes	characteristic
160	C₂²×C₈	C₂×D₁₃₉	1	1	yes	characteristic
161	#124	—	278	2	no	—
162	#124	—	139	2	no	—
163	C₂²⋊Q₈	—	278	2	no	—
164	C₂²⋊Q₈	—	139	2	no	—
165	#121	—	278	2	no	—
166	#121	—	139	2	no	—
167	C₄×Q₈	—	139	2	no	—
168	C₄×D₄	—	278	2	no	—
169	C₄×D₄	C₂×D₁₃₉	1	1	yes	characteristic
170	#107	—	139	1	no	—
171	C₄⋊C₈	C₂×D₁₃₉	1	1	yes	characteristic
172	#103	—	139	2	no	—
173	#103	—	139	2	no	—
174	#102	—	139	2	no	—
175	C₂²⋊C₈	C₂×D₁₃₉	1	1	yes	characteristic
176	C₂²⋊C₈	C₂×D₁₃₉	1	1	yes	characteristic
177	C₄×C₈	C₂×D₁₃₉	1	1	yes	characteristic
178	#41	—	278	4	no	—
179	#41	—	556	2	no	—
180	C₂×Q₈	—	278	4	no	—
181	C₂×Q₈	—	278	2	no	—
182	C₂×Q₈	—	556	2	no	—
183	C₂×Q₈	—	139	2	no	—
184	C₂×Q₈	—	139	2	no	—
185	C₂×D₄	—	278	2	no	—
186	C₂×D₄	—	278	2	no	—
187	C₂×D₄	—	139	2	no	—
188	C₂×D₄	D₅₅₆	1	1	yes	characteristic
189	C₂²×C₄	—	278	2	no	—
190	C₂²×C₄	—	278	2	no	—
191	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
192	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
193	Q₁₆	—	278	4	no	—
194	Q₁₆	—	278	2	no	—
195	Q₁₆	—	139	4	no	—
196	SD₁₆	—	278	4	no	—
197	SD₁₆	—	556	2	no	—
198	D₈	—	278	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
202	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
203	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
204	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
205	C₄⋊C₄	—	278	2	no	—
206	C₄⋊C₄	—	278	2	no	—
207	C₄⋊C₄	—	139	2	no	—
208	C₄⋊C₄	D₅₅₆	1	1	yes	characteristic
209	C₂²⋊C₄	—	278	2	no	—
210	C₂²⋊C₄	—	278	2	no	—
211	C₂²⋊C₄	—	278	2	no	—
212	C₂²⋊C₄	—	278	2	no	—
213	C₂²⋊C₄	D₅₅₆	1	1	yes	characteristic
214	C₂²⋊C₄	D₅₅₆	1	1	yes	characteristic
215	C₄²	—	278	2	no	—
216	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
217	C₂³	—	278	2	no	—
218	C₂³	C₂×D₅₅₆	1	1	yes	characteristic
219	C₂³	C₂×D₅₅₆	1	1	yes	characteristic
220	Q₈	—	278	2	no	—
221	Q₈	—	278	2	no	—
222	Q₈	—	278	2	no	—
223	Q₈	—	556	4	no	—
224	Q₈	—	139	4	no	—
225	D₄	—	2	2	no	—
226	D₄	—	278	2	no	—
227	D₄	—	278	2	no	—
228	D₄	—	556	4	no	—
229	D₄	—	556	4	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	278	2	no	—
235	C₂×C₄	—	278	2	no	—
236	C₂×C₄	—	278	2	no	—
237	C₂×C₄	—	556	4	no	—
238	C₂×C₄	—	556	4	no	—
239	C₂×C₄	C₂×D₅₅₆	1	1	yes	characteristic
240	C₂×C₄	C₂×D₅₅₆	1	1	yes	characteristic
241	C₂×C₄	C₂×D₅₅₆	1	1	yes	characteristic
242	C₂×C₄	C₂×D₅₅₆	1	1	yes	characteristic
243	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
244	C₈	—	2	1	no	—
245	C₈	—	2	1	no	—
246	C₈	—	2	1	no	—
247	C₈	D₄×D₁₃₉	1	2	yes	—
248	C₂²	—	2	1	no	—
249	C₂²	—	2	1	no	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	556	2	no	—
254	C₂²	—	556	2	no	—
255	C₂²	—	556	2	no	—
256	C₂²	#17984864	1	2	yes	complemented
257	C₂²	C₂²×D₅₅₆	1	1	yes	characteristic
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	278	4	no	—
263	C₄	—	556	2	no	—
264	C₄	—	556	2	no	—
265	C₄	C₂×D₄×D₁₃₉	1	1	yes	characteristic
266	C₄	#17984888	1	1	yes	characteristic
267	C₂	—	2	2	no	—
268	C₂	—	4	1	no	—
269	C₂	—	1112	2	no	—
270	C₂	D₄×D₅₅₆	1	1	yes	characteristic
271	C₂	C₂×#17984864	1	1	yes	characteristic
272	C₂	#22581770	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₅₅₆
commutator, cosocle	C₂×C₄×C₁₃₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	C₈×D₄×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
hypercenter	C₈×D₄	D₁₃₉
hypocenter	C₁₃₉	#3285
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#17984864	C₂	semidirect
2. C₂²×Dic₅₅₆	C₂	semidirect
3. #22581612	C₂	semidirect
4. #22581337	C₂	semidirect
5. #22581194	C₂	semidirect
6. #22581189	C₂	semidirect
7. #22581176	C₂	semidirect
8. #22581117	C₂	semidirect
9. #22581094	C₂	semidirect
10. C₄×Dic₅₅₆	C₂	semidirect
11. C₈×D₄×C₁₃₉	C₂	semidirect
55. Dic₅₅₆	D₄	semidirect
116. C₂²×C₁₃₉	#135	semidirect
136. C₁₃₉	#3285	semidirect
148. C₈×D₄	D₁₃₉	semidirect
256. C₂²	#17984864	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17984864	2
C₂²×Dic₅₅₆	2
#22581612	2
#22581337	2
#22581194	2
#22581189	2
#22581176	2
#22581117	2
#22581094	2
C₄×Dic₅₅₆	2
C₈×D₄×C₁₃₉	2
#3285	139