Subgroups — #31819605

Order 17792.

1540 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×#14555785	C₂	1	1	yes	maximal, characteristic
2	#22581434	C₂	1	4	yes	maximal, complemented
3	#22581433	C₂	1	4	yes	maximal, complemented
4	#22581430	C₂	1	2	yes	maximal, complemented
5	#22581427	C₂	1	2	yes	maximal, complemented
6	#22581426	C₂	1	4	yes	maximal, complemented
7	C₄×#14555785	C₂	1	4	yes	maximal, complemented
8	#22581421	C₂	1	4	yes	maximal, complemented
9	C₂×#17984801	C₂	1	1	yes	maximal, characteristic, complemented
10	#117×D₁₃₉	C₂	1	4	yes	maximal, complemented
11	C₂×#117×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14555785	C₄	1	16	yes	complemented
13	C₂×#14555785	C₂²	1	2	yes	—
14	C₂×#14555785	C₂²	1	2	yes	—
15	C₂×#14555785	C₂²	1	2	yes	—
16	C₂×#14555785	C₂²	1	2	yes	—
17	C₂²×Dic₂₇₈	C₂²	1	1	yes	characteristic
18	C₂²×D₅₅₆	C₂²	1	1	yes	characteristic
19	C₂²×C₂²⋊D₁₃₉	C₂²	1	2	yes	—
20	C₂²×C₄×D₁₃₉	C₂²	1	2	yes	—
21	C₄×Dic₂₇₈	C₂²	1	8	yes	complemented
22	#17984807	C₂²	1	8	yes	complemented
23	C₄×D₅₅₆	C₂²	1	8	yes	complemented
24	#17984804	C₂²	1	8	yes	complemented
25	C₄×C₂²⋊D₁₃₉	C₂²	1	16	yes	complemented
26	(C₂²⋊D₁₃₉)⋊C₄	C₂²	1	16	yes	complemented
27	#17984801	C₂²	1	2	yes	—
28	#17984801	C₂²	1	2	yes	—
29	#17984801	C₂²	1	2	yes	complemented
30	#17984801	C₂²	1	2	yes	complemented
31	#17984800	C₂²	1	8	yes	complemented
32	#17984799	C₂²	1	8	yes	complemented
33	#17984798	C₂²	1	8	yes	—
34	C₂×C₄⋊Dic₁₃₉	C₂²	1	2	yes	complemented
35	C₄⋊C₄×D₁₃₉	C₂²	1	8	yes	complemented
36	C₂×C₂²⋊Dic₁₃₉	C₂²	1	2	yes	—
37	C₂²⋊C₄×D₁₃₉	C₂²	1	8	yes	—
38	C₂×C₄×Dic₁₃₉	C₂²	1	2	yes	complemented
39	C₄²×D₁₃₉	C₂²	1	8	yes	complemented
40	C₂³×C₄×C₁₃₉	C₂²	1	1	yes	characteristic
41	#117×C₁₃₉	C₂²	1	2	yes	—
42	#117×C₁₃₉	C₂²	1	2	yes	—
43	#117×C₁₃₉	C₂²	1	2	yes	complemented
44	#117×C₁₃₉	C₂²	1	2	yes	complemented
45	C₂×C₄⋊C₄×C₁₃₉	C₂²	1	2	yes	complemented
46	C₂×C₂²⋊C₄×C₁₃₉	C₂²	1	2	yes	—
47	C₂×C₄²×C₁₃₉	C₂²	1	2	yes	complemented
48	C₂³×D₁₃₉	C₂³	1	2	yes	—
49	#14555785	C₂×C₄	1	16	yes	complemented
50	#14555785	C₂×C₄	1	16	yes	complemented
51	#14555785	C₂×C₄	1	16	yes	complemented
52	#14555785	C₂×C₄	1	16	yes	complemented
53	C₂×Dic₂₇₈	C₂×C₄	1	8	yes	complemented
54	C₂×Dic₂₇₈	C₂³	1	2	yes	—
55	C₂×Dic₂₇₈	C₂³	1	2	yes	—
56	C₂×D₅₅₆	C₂×C₄	1	8	yes	complemented
57	C₂×D₅₅₆	C₂³	1	2	yes	—
58	C₂×D₅₅₆	C₂³	1	2	yes	—
59	C₂×C₂²⋊D₁₃₉	C₂×C₄	1	16	yes	complemented
60	C₂×C₂²⋊D₁₃₉	C₂³	1	4	yes	—
61	C₂×C₂²⋊D₁₃₉	C₂³	1	4	yes	—
62	C₂²×Dic₁₃₉	C₂³	1	4	yes	—
63	C₂²×Dic₁₃₉	C₂³	1	2	yes	—
64	C₂×C₄×D₁₃₉	C₂×C₄	1	16	yes	complemented
65	C₂×C₄×D₁₃₉	C₂³	1	16	yes	—
66	C₂×C₄×D₁₃₉	C₂³	1	4	yes	—
67	C₂×C₄×D₁₃₉	C₂³	1	4	yes	—
68	C₄⋊Dic₁₃₉	C₂³	1	4	yes	—
69	C₄⋊Dic₁₃₉	C₂³	1	4	yes	—
70	Dic₁₃₉⋊C₄	C₂³	1	16	yes	—
71	C₂²⋊Dic₁₃₉	C₂³	1	4	yes	—
72	C₂²⋊Dic₁₃₉	C₂³	1	4	yes	—
73	(C₂×D₁₃₉)⋊C₄	C₂³	1	16	yes	—
74	C₄×Dic₁₃₉	C₂³	1	4	yes	—
75	C₄×Dic₁₃₉	C₂³	1	16	yes	—
76	C₄×Dic₁₃₉	C₂³	1	4	yes	—
77	C₂⁴×C₁₃₉	C₂³	1	1	yes	characteristic
78	C₂²×C₄×C₁₃₉	C₂×C₄	1	8	yes	complemented
79	C₂²×C₄×C₁₃₉	C₂³	1	4	yes	—
80	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
81	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
82	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
83	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
84	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
85	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
86	C₄⋊C₄×C₁₃₉	C₂³	1	4	yes	—
87	C₄⋊C₄×C₁₃₉	C₂³	1	4	yes	—
88	C₂²⋊C₄×C₁₃₉	C₂³	1	4	yes	—
89	C₂²⋊C₄×C₁₃₉	C₂³	1	4	yes	—
90	C₄²×C₁₃₉	C₂³	1	4	yes	—
91	C₄²×C₁₃₉	C₂³	1	4	yes	—
92	C₂²×D₁₃₉	—	4	4	no	—
93	C₂²×D₁₃₉	C₂²×C₄	1	8	yes	—
94	C₂²×D₁₃₉	C₂⁴	1	4	yes	—
95	Dic₂₇₈	C₂²×C₄	1	8	yes	—
96	Dic₂₇₈	C₂²×C₄	1	8	yes	—
97	D₅₅₆	C₂²×C₄	1	8	yes	—
98	D₅₅₆	C₂²×C₄	1	8	yes	—
99	C₂²⋊D₁₃₉	C₂²×C₄	1	16	yes	—
100	C₂²⋊D₁₃₉	C₂²×C₄	1	16	yes	—
101	C₂×Dic₁₃₉	—	4	4	no	—
102	C₂×Dic₁₃₉	C₂²×C₄	1	8	yes	—
103	C₂×Dic₁₃₉	C₂⁴	1	8	yes	—
104	C₂×Dic₁₃₉	C₂⁴	1	4	yes	—
105	C₄×D₁₃₉	—	4	16	no	—
106	C₄×D₁₃₉	C₂²×C₄	1	16	yes	—
107	C₄×D₁₃₉	C₂²×C₄	1	16	yes	—
108	C₂³×C₁₃₉	—	4	2	no	—
109	C₂³×C₁₃₉	C₂²×C₄	1	4	yes	—
110	C₂³×C₁₃₉	C₂⁴	1	1	yes	characteristic
111	C₂³×C₁₃₉	C₂⁴	1	1	yes	characteristic
112	C₂³×C₁₃₉	C₂⁴	1	1	yes	characteristic
113	C₂×C₄×C₁₃₉	—	4	4	no	—
114	C₂×C₄×C₁₃₉	C₂²×C₄	1	4	yes	—
115	C₂×C₄×C₁₃₉	C₂²×C₄	1	4	yes	—
116	C₂×C₄×C₁₃₉	C₂²×C₄	1	4	yes	—
117	C₂×C₄×C₁₃₉	C₂²×C₄	1	4	yes	—
118	C₂×C₄×C₁₃₉	C₂²×C₄	1	4	yes	—
119	C₂×C₄×C₁₃₉	C₂²×C₄	1	4	yes	—
120	C₂×C₄×C₁₃₉	C₂⁴	1	8	yes	—
121	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
122	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
123	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
124	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
125	C₂²×C₁₃₉	—	2	2	no	—
126	C₂²×C₁₃₉	—	2	2	no	—
127	C₂²×C₁₃₉	—	2	2	no	—
128	C₂²×C₁₃₉	—	4	4	no	—
129	C₂²×C₁₃₉	C₂³×C₄	1	2	yes	—
130	C₂²×C₁₃₉	C₂³×C₄	1	2	yes	—
131	C₂²×C₁₃₉	C₂³×C₄	1	2	yes	—
132	C₂²×C₁₃₉	C₂⁵	1	1	yes	characteristic
133	C₂×D₁₃₉	—	2	8	no	—
134	C₂×D₁₃₉	—	4	8	no	—
135	C₂×D₁₃₉	C₂³×C₄	1	8	yes	—
136	C₄×C₁₃₉	—	2	8	no	—
137	C₄×C₁₃₉	C₂³×C₄	1	2	yes	—
138	C₄×C₁₃₉	C₂³×C₄	1	2	yes	—
139	C₄×C₁₃₉	C₂³×C₄	1	2	yes	—
140	C₄×C₁₃₉	C₂³×C₄	1	2	yes	—
141	Dic₁₃₉	—	2	8	no	—
142	Dic₁₃₉	C₂³×C₄	1	8	yes	—
143	C₂×C₁₃₉	—	2	2	no	—
144	C₂×C₁₃₉	—	2	2	no	—
145	C₂×C₁₃₉	—	2	2	no	—
146	C₂×C₁₃₉	C₂⁴×C₄	1	1	yes	characteristic
147	C₂×C₁₃₉	#585	1	2	yes	—
148	D₁₃₉	—	2	8	no	—
149	C₁₃₉	#3428	1	1	yes	characteristic, complemented, Sylow
150	#3428	—	139	1	no	maximal, Sylow
151	C₂²×#41	—	139	1	no	—
152	#520	—	139	4	no	—
153	#520	—	139	4	no	—
154	#519	—	139	2	no	—
155	#518	—	139	2	no	—
156	#518	—	139	4	no	—
157	C₄×#41	—	139	4	no	—
158	C₄×#41	—	139	4	no	—
159	C₂×#117	—	139	1	no	—
160	C₂×#117	—	139	4	no	—
161	C₂×#117	D₁₃₉	1	1	yes	characteristic, complemented
162	C₂×#41	—	139	16	no	—
163	C₂×#41	—	139	2	no	—
164	C₂×#41	—	139	2	no	—
165	C₂×#41	—	139	2	no	—
166	C₂×#41	—	139	2	no	—
167	C₂²×Q₈	—	139	1	no	—
168	C₂²×D₄	—	139	2	no	—
169	C₂²×D₄	—	139	1	no	—
170	C₂³×C₄	—	139	2	no	—
171	C₂³×C₄	C₂×D₁₃₉	1	1	yes	characteristic
172	C₄×Q₈	—	139	8	no	—
173	C₄×Q₈	—	139	8	no	—
174	C₄×D₄	—	139	16	no	—
175	C₄×D₄	—	139	8	no	—
176	C₄×D₄	—	139	16	no	—
177	C₄×D₄	—	139	8	no	—
178	#117	—	139	2	no	—
179	#117	—	139	2	no	—
180	#117	—	139	2	no	—
181	#117	—	139	2	no	—
182	#117	—	139	8	no	—
183	#117	—	139	8	no	—
184	#117	—	139	8	no	—
185	#117	C₂×D₁₃₉	1	2	yes	—
186	#117	C₂×D₁₃₉	1	2	yes	—
187	#117	C₂×D₁₃₉	1	2	yes	complemented
188	#117	C₂×D₁₃₉	1	2	yes	complemented
189	C₂×C₄⋊C₄	—	139	2	no	—
190	C₂×C₄⋊C₄	—	139	8	no	—
191	C₂×C₄⋊C₄	C₂×D₁₃₉	1	2	yes	complemented
192	C₂×C₂²⋊C₄	—	139	2	no	—
193	C₂×C₂²⋊C₄	—	139	8	no	—
194	C₂×C₂²⋊C₄	C₂×D₁₃₉	1	2	yes	—
195	C₂×C₄²	—	139	2	no	—
196	C₂×C₄²	—	139	8	no	—
197	C₂×C₄²	C₂×D₁₃₉	1	2	yes	complemented
198	C₂⁴	—	139	2	no	—
199	C₂⁴	C₂²×D₁₃₉	1	1	yes	characteristic
200	#41	—	139	16	no	—
201	#41	—	139	16	no	—
202	#41	—	139	16	no	—
203	#41	—	139	16	no	—
204	C₂×Q₈	—	139	8	no	—
205	C₂×Q₈	—	139	2	no	—
206	C₂×Q₈	—	139	2	no	—
207	C₂×D₄	—	139	16	no	—
208	C₂×D₄	—	139	4	no	—
209	C₂×D₄	—	139	4	no	—
210	C₂×D₄	—	139	8	no	—
211	C₂×D₄	—	139	2	no	—
212	C₂×D₄	—	139	2	no	—
213	C₂²×C₄	—	139	4	no	—
214	C₂²×C₄	—	139	16	no	—
215	C₂²×C₄	—	139	2	no	—
216	C₂²×C₄	—	139	16	no	—
217	C₂²×C₄	—	139	4	no	—
218	C₂²×C₄	—	139	4	no	—
219	C₂²×C₄	C₄×D₁₃₉	1	8	yes	complemented
220	C₂²×C₄	C₂²×D₁₃₉	1	4	yes	—
221	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
222	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
223	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
224	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
225	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
226	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
227	C₄⋊C₄	—	139	4	no	—
228	C₄⋊C₄	—	139	16	no	—
229	C₄⋊C₄	—	139	4	no	—
230	C₄⋊C₄	C₂²×D₁₃₉	1	4	yes	—
231	C₄⋊C₄	C₂²×D₁₃₉	1	4	yes	—
232	C₂²⋊C₄	—	139	4	no	—
233	C₂²⋊C₄	—	139	4	no	—
234	C₂²⋊C₄	—	139	16	no	—
235	C₂²⋊C₄	C₂²×D₁₃₉	1	4	yes	—
236	C₂²⋊C₄	C₂²×D₁₃₉	1	4	yes	—
237	C₄²	—	139	4	no	—
238	C₄²	—	139	16	no	—
239	C₄²	—	139	4	no	—
240	C₄²	C₂²×D₁₃₉	1	4	yes	—
241	C₄²	C₂²×D₁₃₉	1	4	yes	—
242	C₂³	—	4	2	no	—
243	C₂³	—	556	4	no	—
244	C₂³	—	139	4	no	—
245	C₂³	—	139	8	no	—
246	C₂³	C₂×C₄×D₁₃₉	1	4	yes	—
247	C₂³	C₂³×D₁₃₉	1	1	yes	characteristic
248	C₂³	C₂³×D₁₃₉	1	1	yes	characteristic
249	C₂³	C₂³×D₁₃₉	1	1	yes	characteristic
250	Q₈	—	139	8	no	—
251	Q₈	—	139	8	no	—
252	D₄	—	139	16	no	—
253	D₄	—	139	16	no	—
254	D₄	—	139	8	no	—
255	D₄	—	139	8	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	556	16	no	—
258	C₂×C₄	—	556	4	no	—
259	C₂×C₄	—	139	8	no	—
260	C₂×C₄	—	139	8	no	—
261	C₂×C₄	—	139	4	no	—
262	C₂×C₄	—	139	16	no	—
263	C₂×C₄	—	139	16	no	—
264	C₂×C₄	C₂×C₄×D₁₃₉	1	4	yes	—
265	C₂×C₄	C₂×C₄×D₁₃₉	1	4	yes	—
266	C₂×C₄	C₂×C₄×D₁₃₉	1	4	yes	—
267	C₂×C₄	C₂×C₄×D₁₃₉	1	4	yes	—
268	C₂×C₄	C₂×C₄×D₁₃₉	1	4	yes	—
269	C₂×C₄	C₂×C₄×D₁₃₉	1	4	yes	—
270	C₂×C₄	C₂³×D₁₃₉	1	8	yes	—
271	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
272	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
273	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
274	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
275	C₂²	—	2	2	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	4	4	no	—
279	C₂²	—	278	8	no	—
280	C₂²	—	556	8	no	—
281	C₂²	—	139	8	no	—
282	C₂²	C₂²×C₄×D₁₃₉	1	2	yes	—
283	C₂²	C₂²×C₄×D₁₃₉	1	2	yes	—
284	C₂²	C₂²×C₄×D₁₃₉	1	2	yes	—
285	C₂²	C₂⁴×D₁₃₉	1	1	yes	characteristic
286	C₄	—	2	8	no	—
287	C₄	—	278	8	no	—
288	C₄	—	139	8	no	—
289	C₄	C₂²×C₄×D₁₃₉	1	2	yes	—
290	C₄	C₂²×C₄×D₁₃₉	1	2	yes	—
291	C₄	C₂²×C₄×D₁₃₉	1	2	yes	—
292	C₄	C₂²×C₄×D₁₃₉	1	2	yes	—
293	C₂	—	2	2	no	—
294	C₂	—	2	2	no	—
295	C₂	—	2	2	no	—
296	C₂	—	278	8	no	—
297	C₂	C₂³×C₄×D₁₃₉	1	1	yes	characteristic
298	C₂	#22581789	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂³×D₁₃₉
commutator	C₂×C₁₃₉	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₁₃₉
Fitting	C₂×#117×C₁₃₉	C₂
socle, cosocle	C₂²×C₁₃₉	C₂⁵
prime radical	C₂²×#14555785	C₂
hypercenter	C₂×#117	D₁₃₉
hypocenter	C₁₃₉	#3428
LCS	G ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
2. #22581434	C₂	semidirect
3. #22581433	C₂	semidirect
4. #22581430	C₂	semidirect
5. #22581427	C₂	semidirect
6. #22581426	C₂	semidirect
7. C₄×#14555785	C₂	semidirect
8. #22581421	C₂	semidirect
9. C₂×#17984801	C₂	semidirect
10. #117×D₁₃₉	C₂	semidirect
11. C₂×#117×C₁₃₉	C₂	semidirect
12. C₂×#14555785	C₄	semidirect
21. C₄×Dic₂₇₈	C₂²	semidirect
22. #17984807	C₂²	semidirect
23. C₄×D₅₅₆	C₂²	semidirect
24. #17984804	C₂²	semidirect
25. C₄×C₂²⋊D₁₃₉	C₂²	semidirect
26. (C₂²⋊D₁₃₉)⋊C₄	C₂²	semidirect
29. #17984801	C₂²	semidirect
30. #17984801	C₂²	semidirect
31. #17984800	C₂²	semidirect
32. #17984799	C₂²	semidirect
34. C₂×C₄⋊Dic₁₃₉	C₂²	semidirect
35. C₄⋊C₄×D₁₃₉	C₂²	semidirect
38. C₂×C₄×Dic₁₃₉	C₂²	semidirect
39. C₄²×D₁₃₉	C₂²	semidirect
43. #117×C₁₃₉	C₂²	semidirect
44. #117×C₁₃₉	C₂²	semidirect
45. C₂×C₄⋊C₄×C₁₃₉	C₂²	semidirect
47. C₂×C₄²×C₁₃₉	C₂²	semidirect
49. #14555785	C₂×C₄	semidirect
50. #14555785	C₂×C₄	semidirect
51. #14555785	C₂×C₄	semidirect
52. #14555785	C₂×C₄	semidirect
53. C₂×Dic₂₇₈	C₂×C₄	semidirect
56. C₂×D₅₅₆	C₂×C₄	semidirect
59. C₂×C₂²⋊D₁₃₉	C₂×C₄	semidirect
64. C₂×C₄×D₁₃₉	C₂×C₄	semidirect
78. C₂²×C₄×C₁₃₉	C₂×C₄	semidirect
149. C₁₃₉	#3428	semidirect
161. C₂×#117	D₁₃₉	semidirect
187. #117	C₂×D₁₃₉	semidirect
188. #117	C₂×D₁₃₉	semidirect
191. C₂×C₄⋊C₄	C₂×D₁₃₉	semidirect
197. C₂×C₄²	C₂×D₁₃₉	semidirect
219. C₂²×C₄	C₄×D₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14555785	2
#22581434	2
#22581433	2
#22581430	2
#22581427	2
#22581426	2
C₄×#14555785	2
#22581421	2
C₂×#17984801	2
#117×D₁₃₉	2
C₂×#117×C₁₃₉	2
#3428	139