Subgroups — #13246916

Order 1792.

418 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#89451	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#334	C₂	1	1	yes	maximal, characteristic
3	#11728239	C₂	1	2	yes	maximal
4	#11727516	C₂	1	1	yes	maximal, characteristic
5	#11725508	C₂	1	2	yes	maximal
6	C₂×C₄×Dic₁₄	C₂²	1	1	yes	characteristic
7	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
8	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
9	C₂×#6390	—	2	2	no	—
10	C₂×#6389	—	2	1	no	—
11	C₂×Dic₇⋊C₈	—	2	1	no	—
12	#89707	C₂²	1	2	yes	—
13	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
14	#89451	—	2	2	no	—
15	#89451	C₄	1	4	yes	complemented
16	C₇×#334	—	2	2	no	—
17	#89346	—	2	1	no	—
18	#89346	—	2	1	no	—
19	#89346	—	2	2	no	—
20	#8657	—	7	1	no	maximal, Sylow
21	C₂²×Dic₁₄	D₄	1	1	yes	characteristic
22	C₂²×C₇×C₈	—	2	2	no	—
23	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
24	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
26	C₂×C₄⋊Dic₇	—	2	2	no	—
27	C₂×C₄⋊Dic₇	—	2	2	no	—
28	C₂×C₄⋊Dic₇	—	2	2	no	—
29	C₂×C₄⋊Dic₇	D₄	1	1	yes	characteristic
30	C₂×Dic₇⋊C₄	—	2	1	no	—
31	C₂×C₄×Dic₇	—	2	1	no	—
32	C₂²×C₇⋊C₈	—	2	1	no	—
33	C₄×Dic₁₄	—	2	1	no	—
34	C₄×Dic₁₄	—	2	1	no	—
35	C₄×Dic₁₄	C₂×C₄	1	2	yes	—
36	C₄×Dic₁₄	Q₈	1	2	yes	—
37	C₇×C₄⋊C₈	—	2	2	no	—
38	C₇×C₄⋊C₈	—	2	2	no	—
39	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
40	C₄×C₇×C₈	D₄	1	2	yes	—
41	#6390	—	4	4	no	—
42	#6389	—	2	4	no	—
43	Dic₇⋊C₈	—	2	4	no	—
44	C₄⋊(C₇⋊C₈)	—	2	1	no	—
45	C₄⋊(C₇⋊C₈)	C₂×C₄	1	2	yes	—
46	C₂×#446	—	7	1	no	—
47	#1834	—	7	1	no	—
48	#1803	—	7	2	no	—
49	C₂×#334	D₇	1	1	yes	characteristic
50	#1296	—	7	2	no	—
51	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₇	#41	1	1	yes	characteristic
54	C₂²×Dic₇	—	2	1	no	—
55	C₂²×Dic₇	—	2	1	no	—
56	C₂²×Dic₇	—	4	2	no	—
57	C₂×Dic₁₄	—	2	2	no	—
58	C₂×Dic₁₄	—	2	2	no	—
59	C₂×Dic₁₄	—	2	1	no	—
60	C₂×Dic₁₄	C₄⋊C₄	1	2	yes	—
61	C₂×C₇×C₈	—	2	2	no	—
62	C₂×C₇×C₈	—	2	2	no	—
63	C₂×C₇×C₈	—	4	2	no	—
64	C₂×C₇×C₈	—	4	2	no	—
65	C₂×C₇×C₈	—	4	2	no	—
66	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
67	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
68	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
69	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
71	C₄²×C₇	#41	1	1	yes	characteristic
72	C₄⋊Dic₇	—	2	1	no	—
73	C₄⋊Dic₇	—	4	2	no	—
74	C₄⋊Dic₇	—	4	2	no	—
75	C₄⋊Dic₇	—	4	2	no	—
76	C₄⋊Dic₇	—	4	2	no	—
77	C₄⋊Dic₇	—	8	2	no	—
78	C₄⋊Dic₇	C₄⋊C₄	1	2	yes	—
79	Dic₇⋊C₄	—	2	2	no	—
80	Dic₇⋊C₄	—	2	2	no	—
81	C₄×Dic₇	—	2	2	no	—
82	C₄×Dic₇	—	2	2	no	—
83	C₂×C₇⋊C₈	—	2	2	no	—
84	C₂×C₇⋊C₈	—	4	2	no	—
85	C₂×C₄×Q₈	—	7	1	no	—
86	#446	—	7	4	no	—
87	#446	—	14	2	no	—
88	C₂×C₄⋊C₈	—	7	1	no	—
89	C₂×C₄⋊C₈	—	14	1	no	—
90	C₂×C₄⋊C₈	C₂×D₇	1	2	yes	—
91	C₂×#103	—	14	2	no	—
92	C₂×#97	—	14	1	no	—
93	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
94	#384	—	7	2	no	—
95	#340	—	14	2	no	—
96	#340	—	14	1	no	—
97	#340	—	14	1	no	—
98	#334	—	2	2	no	—
99	C₂³×C₇	#121	1	1	yes	characteristic
100	C₂×C₄×C₇	—	2	2	no	—
101	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₇	#117	1	1	yes	characteristic
104	C₂×C₄×C₇	#121	1	1	yes	characteristic
105	C₂×C₄×C₇	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂×C₄×C₇	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂×C₄×C₇	#135	1	2	yes	—
108	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
109	C₂×C₄×C₇	#137	1	2	yes	—
110	C₂×C₄×C₇	#137	1	2	yes	—
111	C₂×Dic₇	—	2	2	no	—
112	C₂×Dic₇	—	2	2	no	—
113	C₂×Dic₇	—	4	1	no	—
114	C₂×Dic₇	—	4	1	no	—
115	C₂×Dic₇	—	4	2	no	—
116	C₂×Dic₇	—	8	2	no	—
117	C₂×Dic₇	—	8	2	no	—
118	C₂×Dic₇	—	8	2	no	—
119	Dic₁₄	—	2	4	no	—
120	Dic₁₄	—	4	2	no	—
121	C₇×C₈	—	2	2	no	—
122	C₇×C₈	—	2	2	no	—
123	C₇×C₈	—	8	2	no	—
124	C₇⋊C₈	—	4	2	no	—
125	C₂²×Q₈	—	7	1	no	—
126	C₂²×C₈	—	2	2	no	—
127	C₂²×C₈	—	14	1	no	—
128	C₂²×C₈	D₂₈	1	1	yes	characteristic
129	C₂²×C₈	D₂₈	1	1	yes	characteristic
130	C₄×Q₈	—	7	2	no	—
131	C₄×Q₈	—	7	2	no	—
132	C₄×Q₈	—	14	1	no	—
133	C₄×Q₈	—	14	1	no	—
134	C₂×C₄⋊C₄	—	7	1	no	—
135	C₂×C₄⋊C₄	—	14	1	no	—
136	C₂×C₄⋊C₄	—	14	2	no	—
137	C₂×C₄⋊C₄	—	14	2	no	—
138	C₂×C₄⋊C₄	—	14	2	no	—
139	C₂×C₄²	—	14	1	no	—
140	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
141	C₄⋊C₈	—	2	2	no	—
142	C₄⋊C₈	—	2	2	no	—
143	C₄⋊C₈	—	7	2	no	—
144	C₄⋊C₈	—	14	4	no	—
145	C₄⋊C₈	—	14	1	no	—
146	#103	—	28	4	no	—
147	#97	—	14	4	no	—
148	C₄×C₈	C₄×D₇	1	2	yes	—
149	C₄×C₈	C₂²⋊D₇	1	2	yes	—
150	C₂²×C₇	#380	1	1	yes	characteristic
151	C₂²×C₇	#464	1	2	yes	—
152	C₂²×C₇	#469	1	1	yes	characteristic
153	C₂²×C₇	#470	1	1	yes	characteristic
154	C₂²×C₇	#471	1	2	yes	—
155	C₄×C₇	—	2	1	no	—
156	C₄×C₇	—	2	1	no	—
157	C₄×C₇	#416	1	1	yes	characteristic
158	C₄×C₇	#419	1	1	yes	characteristic
159	C₄×C₇	#428	1	2	yes	—
160	C₄×C₇	#468	1	2	yes	—
161	C₄×C₇	#479	1	2	yes	—
162	Dic₇	—	4	2	no	—
163	Dic₇	—	4	2	no	—
164	Dic₇	—	16	2	no	—
165	C₂×Q₈	—	7	2	no	—
166	C₂×Q₈	—	14	1	no	—
167	C₂×Q₈	—	14	2	no	—
168	C₂×Q₈	—	14	2	no	—
169	C₂²×C₄	—	14	1	no	—
170	C₂²×C₄	—	14	1	no	—
171	C₂²×C₄	—	28	2	no	—
172	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
173	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
174	C₂²×C₄	#1161	1	1	yes	characteristic
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	—	4	2	no	—
178	C₂×C₈	—	4	2	no	—
179	C₂×C₈	—	4	2	no	—
180	C₂×C₈	—	14	2	no	—
181	C₂×C₈	—	28	2	no	—
182	C₂×C₈	(C₂×D₇)⋊C₄	1	2	yes	—
183	C₂×C₈	(C₂×D₇)⋊C₄	1	2	yes	—
184	C₄⋊C₄	—	7	2	no	—
185	C₄⋊C₄	—	14	2	no	—
186	C₄⋊C₄	—	14	1	no	—
187	C₄⋊C₄	—	14	2	no	—
188	C₄⋊C₄	—	28	2	no	—
189	C₄⋊C₄	—	28	2	no	—
190	C₄⋊C₄	—	28	2	no	—
191	C₄⋊C₄	—	28	2	no	—
192	C₄⋊C₄	—	56	2	no	—
193	C₄²	—	14	2	no	—
194	C₄²	—	14	2	no	—
195	C₄²	C₂×C₄×D₇	1	1	yes	characteristic
196	C₄²	#1159	1	1	yes	characteristic
197	C₄²	Q₈×D₇	1	1	yes	characteristic
198	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
199	C₂×C₇	#1702	1	2	yes	—
200	C₂×C₇	#1704	1	2	yes	—
201	C₂×C₇	#1805	1	1	yes	characteristic
202	C₂×C₇	#1809	1	1	yes	characteristic
203	C₂×C₇	#1815	1	1	yes	characteristic
204	C₂³	#6458	1	1	yes	characteristic
205	Q₈	—	14	4	no	—
206	Q₈	—	28	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	14	2	no	—
209	C₂×C₄	—	14	2	no	—
210	C₂×C₄	—	28	1	no	—
211	C₂×C₄	—	28	1	no	—
212	C₂×C₄	—	28	2	no	—
213	C₂×C₄	—	56	2	no	—
214	C₂×C₄	—	56	2	no	—
215	C₂×C₄	—	56	2	no	—
216	C₂×C₄	D₇×C₄⋊C₄	1	1	yes	characteristic
217	C₂×C₄	#6455	1	1	yes	characteristic
218	C₂×C₄	#6460	1	1	yes	characteristic
219	C₂×C₄	#6467	1	2	yes	—
220	C₂×C₄	#6472	1	2	yes	—
221	C₂×C₄	#6477	1	2	yes	—
222	C₂×C₄	#6478	1	2	yes	—
223	C₂×C₄	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
224	C₂×C₄	#6501	1	1	yes	characteristic
225	C₂×C₄	#6509	1	1	yes	characteristic
226	C₈	—	2	2	no	—
227	C₈	—	2	2	no	—
228	C₈	—	8	2	no	—
229	C₈	—	28	2	no	—
230	C₇	#8657	1	1	yes	characteristic, complemented, Sylow
231	C₂²	#89617	1	2	yes	—
232	C₂²	#89637	1	1	yes	characteristic
233	C₂²	#89640	1	1	yes	characteristic
234	C₂²	#89670	1	2	yes	—
235	C₂²	#89759	1	1	yes	characteristic
236	C₄	—	2	1	no	—
237	C₄	—	2	1	no	—
238	C₄	—	28	2	no	—
239	C₄	—	28	2	no	—
240	C₄	—	112	2	no	—
241	C₄	#89601	1	2	yes	—
242	C₄	#89633	1	2	yes	—
243	C₄	#89885	1	1	yes	characteristic
244	C₄	#89892	1	1	yes	characteristic
245	C₄	#89946	1	2	yes	—
246	C₂	#11726308	1	2	yes	—
247	C₂	#11726309	1	2	yes	—
248	C₂	#11728240	1	1	yes	characteristic
249	C₂	#11728332	1	1	yes	characteristic
250	C₂	#11728474	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#6458
commutator	C₄²×C₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₂×C₇×#334	C₂
socle, prime radical	C₂³×C₇	#121
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	C₂×#334	D₇
hypocenter	C₇	#8657
LCS	G ▷ C₄²×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
15. #89451	C₄	semidirect
230. C₇	#8657	semidirect

Maximal subgroups

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

subgroup	index
C₂×#89451	2
C₂×C₇×#334	2
#11728239	2
#11727516	2
#11725508	2
#8657	7