Subgroups — #31820026

Order 17792.

1094 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22581682	C₂	1	1	yes	maximal, characteristic, complemented
2	#22581673	C₂	1	3	yes	maximal, complemented
3	#22581642	C₂	1	6	yes	maximal, complemented
4	Q₈×D₅₅₆	C₂	1	1	yes	maximal, characteristic, complemented
5	#22581610	C₂	1	2	yes	maximal, complemented
6	#22581608	C₂	1	3	yes	maximal, complemented
7	#22581510	C₂	1	3	yes	maximal, complemented
8	#22581485	C₂	1	6	yes	maximal, complemented
9	C₄×#14555785	C₂	1	3	yes	maximal, complemented
10	C₄×Q₈×D₁₃₉	C₂	1	2	yes	maximal, complemented
11	C₄×#41×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14555785	C₂²	1	3	yes	—
13	C₂×Q₈×D₁₃₉	C₂²	1	2	yes	—
14	#17984842	C₂²	1	3	yes	—
15	#17984842	C₂²	1	3	yes	—
16	#17984835	C₂²	1	12	yes	—
17	#17984834	C₂²	1	6	yes	—
18	#17984833	C₂²	1	6	yes	—
19	#17984829	C₂²	1	6	yes	—
20	#17984824	C₂²	1	12	yes	—
21	#17984822	C₂²	1	6	yes	—
22	#17984821	C₂²	1	6	yes	—
23	#17984821	C₂²	1	6	yes	—
24	#17984813	C₂²	1	6	yes	—
25	Q₈×Dic₁₃₉	C₂²	1	2	yes	—
26	C₄×Dic₂₇₈	C₂²	1	3	yes	—
27	C₄×Dic₂₇₈	C₂²	1	6	yes	—
28	C₄×Dic₂₇₈	C₂²	1	3	yes	—
29	#17984807	C₂²	1	6	yes	—
30	C₄×D₅₅₆	C₂²	1	3	yes	—
31	C₄×D₅₅₆	C₂²	1	1	yes	characteristic
32	C₄×C₂²⋊D₁₃₉	C₂²	1	6	yes	—
33	C₄×C₂²⋊D₁₃₉	C₂²	1	6	yes	—
34	#17984801	C₂²	1	6	yes	—
35	#17984801	C₂²	1	3	yes	—
36	#17984799	C₂²	1	6	yes	—
37	C₄⋊C₄×D₁₃₉	C₂²	1	6	yes	—
38	C₄²×D₁₃₉	C₂²	1	6	yes	—
39	C₂×#41×C₁₃₉	C₂²	1	1	yes	characteristic
40	C₄×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
41	C₄×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
42	C₄×D₄×C₁₃₉	C₂²	1	3	yes	—
43	C₄×D₄×C₁₃₉	C₂²	1	3	yes	—
44	#117×C₁₃₉	C₂²	1	3	yes	—
45	C₂×C₄²×C₁₃₉	C₂²	1	3	yes	—
46	#14555785	—	2	12	no	—
47	C₂×Dic₂₇₈	C₂³	1	6	yes	—
48	C₂×Dic₂₇₈	C₂³	1	3	yes	—
49	Q₈×D₁₃₉	—	2	8	no	—
50	C₂×D₅₅₆	C₂³	1	1	yes	characteristic
51	C₂×C₂²⋊D₁₃₉	C₂³	1	6	yes	—
52	C₂×C₄×D₁₃₉	C₂³	1	2	yes	—
53	C₂×C₄×D₁₃₉	C₂³	1	6	yes	—
54	C₂×C₄×D₁₃₉	C₂³	1	6	yes	—
55	C₄⋊Dic₁₃₉	C₂³	1	6	yes	—
56	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
57	C₄⋊Dic₁₃₉	C₂³	1	3	yes	—
58	C₄⋊Dic₁₃₉	C₂³	1	6	yes	—
59	C₄⋊Dic₁₃₉	C₂³	1	6	yes	—
60	Dic₁₃₉⋊C₄	C₂³	1	12	yes	—
61	Dic₁₃₉⋊C₄	C₂³	1	6	yes	—
62	Dic₁₃₉⋊C₄	C₂³	1	6	yes	—
63	C₂²⋊Dic₁₃₉	C₂³	1	6	yes	—
64	C₂²⋊Dic₁₃₉	C₂³	1	12	yes	—
65	(C₂×D₁₃₉)⋊C₄	C₂³	1	2	yes	—
66	(C₂×D₁₃₉)⋊C₄	C₂³	1	6	yes	—
67	C₄×Dic₁₃₉	C₂³	1	6	yes	—
68	C₄×Dic₁₃₉	C₂³	1	6	yes	—
69	C₄×Dic₁₃₉	C₂³	1	6	yes	—
70	#41×C₁₃₉	—	2	4	no	—
71	C₂×Q₈×C₁₃₉	C₂³	1	1	yes	characteristic
72	C₂×D₄×C₁₃₉	C₂³	1	3	yes	—
73	C₂²×C₄×C₁₃₉	C₂³	1	3	yes	—
74	C₂²×C₄×C₁₃₉	C₂³	1	3	yes	—
75	C₂²×C₄×C₁₃₉	C₂³	1	3	yes	—
76	C₄⋊C₄×C₁₃₉	C₂³	1	3	yes	—
77	C₄⋊C₄×C₁₃₉	C₂³	1	3	yes	—
78	C₂²⋊C₄×C₁₃₉	C₂³	1	6	yes	—
79	C₄²×C₁₃₉	C₂³	1	3	yes	—
80	C₄²×C₁₃₉	C₂³	1	3	yes	—
81	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
82	C₄²×C₁₃₉	C₂³	1	3	yes	—
83	C₂²×D₁₃₉	C₂⁴	1	2	yes	—
84	Dic₂₇₈	—	2	12	no	—
85	Dic₂₇₈	—	2	6	no	—
86	D₅₅₆	#41	1	4	yes	complemented
87	C₂²⋊D₁₃₉	—	2	12	no	—
88	C₂×Dic₁₃₉	C₂⁴	1	2	yes	—
89	C₂×Dic₁₃₉	C₂⁴	1	6	yes	—
90	C₂×Dic₁₃₉	C₂⁴	1	6	yes	—
91	C₄×D₁₃₉	—	2	12	no	—
92	C₄×D₁₃₉	—	2	12	no	—
93	C₄×D₁₃₉	—	2	4	no	—
94	C₂³×C₁₃₉	C₂⁴	1	3	yes	—
95	Q₈×C₁₃₉	#41	1	4	yes	complemented
96	D₄×C₁₃₉	—	2	6	no	—
97	C₂×C₄×C₁₃₉	—	2	6	no	—
98	C₂×C₄×C₁₃₉	—	2	6	no	—
99	C₂×C₄×C₁₃₉	—	2	6	no	—
100	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
101	C₂×C₄×C₁₃₉	C₂⁴	1	3	yes	—
102	C₂×C₄×C₁₃₉	C₂⁴	1	3	yes	—
103	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
104	C₂×C₄×C₁₃₉	C₂⁴	1	3	yes	—
105	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
106	C₂²×C₁₃₉	—	2	3	no	—
107	C₂²×C₁₃₉	—	2	3	no	—
108	C₂²×C₁₃₉	—	2	3	no	—
109	C₂²×C₁₃₉	C₂⁵	1	1	yes	characteristic
110	C₂×D₁₃₉	—	2	2	no	—
111	C₂×D₁₃₉	—	2	2	no	—
112	C₂×D₁₃₉	C₂×#41	1	4	yes	—
113	C₄×C₁₃₉	—	2	1	no	—
114	C₄×C₁₃₉	—	2	3	no	—
115	C₄×C₁₃₉	—	2	3	no	—
116	C₄×C₁₃₉	C₂×#41	1	2	yes	—
117	C₄×C₁₃₉	C₂×#41	1	6	yes	—
118	C₄×C₁₃₉	2₋¹⁺⁴	1	2	yes	—
119	Dic₁₃₉	—	2	6	no	—
120	Dic₁₃₉	—	2	6	no	—
121	Dic₁₃₉	—	2	2	no	—
122	C₂×C₁₃₉	—	4	3	no	—
123	C₂×C₁₃₉	C₂²×#41	1	1	yes	characteristic
124	C₂×C₁₃₉	C₂²×#41	1	1	yes	characteristic
125	C₂×C₁₃₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
126	D₁₃₉	—	2	4	no	—
127	C₁₃₉	#3482	1	1	yes	characteristic, complemented, Sylow
128	#3482	—	139	1	no	maximal, Sylow
129	#557	—	139	1	no	—
130	#556	—	139	3	no	—
131	#552	—	139	6	no	—
132	D₄×Q₈	—	139	1	no	—
133	#548	—	139	2	no	—
134	#547	—	139	3	no	—
135	#533	—	139	3	no	—
136	#529	—	139	6	no	—
137	C₄×#41	—	139	3	no	—
138	C₄×#41	D₁₃₉	1	1	yes	characteristic, complemented
139	C₂×C₄×Q₈	—	139	2	no	—
140	C₂×#41	—	139	3	no	—
141	C₂×#41	C₂×D₁₃₉	1	1	yes	characteristic
142	C₂²×Q₈	—	139	2	no	—
143	C₄⋊Q₈	—	139	3	no	—
144	C₄⋊Q₈	—	139	3	no	—
145	#126	—	139	12	no	—
146	#125	—	139	6	no	—
147	#125	—	139	6	no	—
148	#124	—	139	6	no	—
149	#123	—	139	12	no	—
150	C₂²⋊Q₈	—	139	6	no	—
151	C₂²⋊Q₈	—	139	6	no	—
152	C₂²⋊Q₈	—	139	6	no	—
153	#121	—	139	6	no	—
154	C₄×Q₈	—	139	2	no	—
155	C₄×Q₈	—	139	6	no	—
156	C₄×Q₈	—	139	6	no	—
157	C₄×Q₈	—	139	3	no	—
158	C₄×Q₈	—	139	3	no	—
159	C₄×Q₈	C₂×D₁₃₉	1	1	yes	characteristic
160	C₄×Q₈	C₂×D₁₃₉	1	1	yes	characteristic
161	C₄×D₄	—	139	6	no	—
162	C₄×D₄	—	139	6	no	—
163	C₄×D₄	—	139	1	no	—
164	C₄×D₄	—	139	3	no	—
165	C₄×D₄	C₂×D₁₃₉	1	3	yes	—
166	C₄×D₄	C₂×D₁₃₉	1	3	yes	—
167	#117	—	139	3	no	—
168	#117	—	139	6	no	—
169	#117	—	139	6	no	—
170	#117	C₂×D₁₃₉	1	3	yes	—
171	C₂×C₄⋊C₄	—	139	6	no	—
172	C₂×C₄²	—	139	6	no	—
173	C₂×C₄²	C₂×D₁₃₉	1	3	yes	—
174	#41	—	2	4	no	—
175	#41	—	278	12	no	—
176	C₂×Q₈	—	278	8	no	—
177	C₂×Q₈	—	139	6	no	—
178	C₂×Q₈	—	139	3	no	—
179	C₂×Q₈	C₂²×D₁₃₉	1	1	yes	characteristic
180	C₂×D₄	—	139	6	no	—
181	C₂×D₄	—	139	1	no	—
182	C₂×D₄	C₂²×D₁₃₉	1	3	yes	—
183	C₂²×C₄	—	139	6	no	—
184	C₂²×C₄	—	139	6	no	—
185	C₂²×C₄	—	139	2	no	—
186	C₂²×C₄	C₂²×D₁₃₉	1	3	yes	—
187	C₂²×C₄	C₂²×D₁₃₉	1	3	yes	—
188	C₂²×C₄	C₂²×D₁₃₉	1	3	yes	—
189	C₄⋊C₄	—	139	12	no	—
190	C₄⋊C₄	—	139	6	no	—
191	C₄⋊C₄	—	139	6	no	—
192	C₄⋊C₄	—	139	6	no	—
193	C₄⋊C₄	—	139	1	no	—
194	C₄⋊C₄	—	139	3	no	—
195	C₄⋊C₄	—	139	6	no	—
196	C₄⋊C₄	—	139	6	no	—
197	C₄⋊C₄	C₂²×D₁₃₉	1	3	yes	—
198	C₄⋊C₄	C₂²×D₁₃₉	1	3	yes	—
199	C₂²⋊C₄	—	139	6	no	—
200	C₂²⋊C₄	—	139	12	no	—
201	C₂²⋊C₄	—	139	2	no	—
202	C₂²⋊C₄	—	139	6	no	—
203	C₂²⋊C₄	C₂²×D₁₃₉	1	6	yes	—
204	C₄²	—	139	6	no	—
205	C₄²	—	139	6	no	—
206	C₄²	—	139	6	no	—
207	C₄²	C₂²×D₁₃₉	1	3	yes	—
208	C₄²	C₂²×D₁₃₉	1	3	yes	—
209	C₄²	C₂²×D₁₃₉	1	3	yes	—
210	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
211	C₂³	—	139	2	no	—
212	C₂³	C₂³×D₁₃₉	1	3	yes	—
213	Q₈	—	278	6	no	—
214	Q₈	—	278	12	no	—
215	Q₈	#14555785	1	4	yes	complemented
216	D₄	—	2	6	no	—
217	D₄	—	278	12	no	—
218	D₄	—	139	4	no	—
219	C₂×C₄	—	2	6	no	—
220	C₂×C₄	—	2	6	no	—
221	C₂×C₄	—	2	6	no	—
222	C₂×C₄	—	278	12	no	—
223	C₂×C₄	—	278	12	no	—
224	C₂×C₄	—	278	4	no	—
225	C₂×C₄	—	139	2	no	—
226	C₂×C₄	—	139	6	no	—
227	C₂×C₄	—	139	6	no	—
228	C₂×C₄	C₂³×D₁₃₉	1	3	yes	—
229	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
230	C₂×C₄	C₂³×D₁₃₉	1	3	yes	—
231	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
232	C₂×C₄	C₂³×D₁₃₉	1	3	yes	—
233	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
234	C₂²	—	2	3	no	—
235	C₂²	—	2	3	no	—
236	C₂²	—	2	3	no	—
237	C₂²	—	278	2	no	—
238	C₂²	—	278	2	no	—
239	C₂²	—	139	4	no	—
240	C₂²	C₂⁴×D₁₃₉	1	1	yes	characteristic
241	C₄	—	2	3	no	—
242	C₄	—	2	1	no	—
243	C₄	—	2	3	no	—
244	C₄	—	278	6	no	—
245	C₄	—	278	6	no	—
246	C₄	—	278	2	no	—
247	C₄	#41×D₁₃₉	1	2	yes	—
248	C₄	C₂×#14555785	1	6	yes	—
249	C₄	#17984889	1	2	yes	—
250	C₂	—	4	3	no	—
251	C₂	—	278	4	no	—
252	C₂	C₂×#41×D₁₃₉	1	1	yes	characteristic
253	C₂	C₂²×#14555785	1	1	yes	characteristic
254	C₂	C₂×#17984889	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂³×D₁₃₉
commutator, socle, cosocle	C₂²×C₁₃₉	C₂⁵
Frattini	C₂²	C₂⁴×D₁₃₉
Fitting	C₄×#41×C₁₃₉	C₂
hypercenter	C₄×#41	D₁₃₉
hypocenter	C₁₃₉	#3482
LCS	G ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. #22581682	C₂	semidirect
2. #22581673	C₂	semidirect
3. #22581642	C₂	semidirect
4. Q₈×D₅₅₆	C₂	semidirect
5. #22581610	C₂	semidirect
6. #22581608	C₂	semidirect
7. #22581510	C₂	semidirect
8. #22581485	C₂	semidirect
9. C₄×#14555785	C₂	semidirect
10. C₄×Q₈×D₁₃₉	C₂	semidirect
11. C₄×#41×C₁₃₉	C₂	semidirect
86. D₅₅₆	#41	semidirect
95. Q₈×C₁₃₉	#41	semidirect
127. C₁₃₉	#3482	semidirect
138. C₄×#41	D₁₃₉	semidirect
215. Q₈	#14555785	semidirect

Maximal subgroups

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

subgroup	index
#22581682	2
#22581673	2
#22581642	2
Q₈×D₅₅₆	2
#22581610	2
#22581608	2
#22581510	2
#22581485	2
C₄×#14555785	2
C₄×Q₈×D₁₃₉	2
C₄×#41×C₁₃₉	2
#3482	139