Subgroups — #13246915

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×C₇×#446	C₂	1	1	yes	maximal, characteristic
2	#11729380	C₂	1	2	yes	maximal
3	#11727516	C₂	1	1	yes	maximal, characteristic
4	C₂×#89252	C₂	1	1	yes	maximal, characteristic
5	#11725508	C₂	1	2	yes	maximal
6	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
7	C₇×#446	—	2	2	no	—
8	C₇×#446	C₄	1	4	yes	complemented
9	C₂×C₇×C₄⋊C₈	—	2	1	no	—
10	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₇×#97	—	2	1	no	—
12	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
13	C₂×#6409	—	2	2	no	—
14	#89707	C₂²	1	2	yes	—
15	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
16	#89346	—	2	1	no	—
17	#89346	—	2	1	no	—
18	#89324	—	2	2	no	—
19	#89252	—	2	2	no	—
20	#8657	—	7	1	no	maximal, Sylow
21	C₂²×C₇×Q₈	D₄	1	1	yes	characteristic
22	C₂²×C₇×C₈	—	2	1	no	—
23	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
24	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
25	C₄×C₇×Q₈	—	2	1	no	—
26	C₄×C₇×Q₈	—	2	1	no	—
27	C₄×C₇×Q₈	C₂×C₄	1	2	yes	—
28	C₄×C₇×Q₈	Q₈	1	2	yes	—
29	C₂×C₇×C₄⋊C₄	—	2	1	no	—
30	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₇	—	2	1	no	—
32	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
33	C₂×C₄⋊Dic₇	—	2	2	no	—
34	C₂×C₄⋊Dic₇	—	2	2	no	—
35	C₂×C₄⋊Dic₇	—	2	2	no	—
36	C₂²×C₇⋊C₈	—	2	2	no	—
37	C₇×C₄⋊C₈	—	2	4	no	—
38	C₇×C₄⋊C₈	—	2	1	no	—
39	C₇×C₄⋊C₈	C₂×C₄	1	2	yes	—
40	C₇×#97	—	2	4	no	—
41	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
42	C₄×C₇×C₈	D₄	1	2	yes	—
43	#6409	—	4	4	no	—
44	C₄⋊(C₇⋊C₈)	—	2	2	no	—
45	C₄⋊(C₇⋊C₈)	—	2	2	no	—
46	C₂×#446	D₇	1	1	yes	characteristic
47	#1834	—	7	1	no	—
48	#1803	—	7	2	no	—
49	C₂×#334	—	7	1	no	—
50	#1296	—	7	2	no	—
51	C₂×C₇×Q₈	—	2	1	no	—
52	C₂×C₇×Q₈	—	2	2	no	—
53	C₂×C₇×Q₈	—	2	2	no	—
54	C₂×C₇×Q₈	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₇	—	2	1	no	—
56	C₂²×C₄×C₇	—	2	1	no	—
57	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₇	#41	1	1	yes	characteristic
60	C₂²×Dic₇	—	4	2	no	—
61	C₂×C₇×C₈	—	2	2	no	—
62	C₂×C₇×C₈	—	2	2	no	—
63	C₂×C₇×C₈	—	2	2	no	—
64	C₂×C₇×C₈	—	4	2	no	—
65	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
66	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
67	C₇×C₄⋊C₄	—	2	2	no	—
68	C₇×C₄⋊C₄	—	2	1	no	—
69	C₇×C₄⋊C₄	—	2	2	no	—
70	C₇×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
71	C₄²×C₇	—	2	2	no	—
72	C₄²×C₇	—	2	2	no	—
73	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
74	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
76	C₄²×C₇	#41	1	1	yes	characteristic
77	C₄⋊Dic₇	—	4	2	no	—
78	C₄⋊Dic₇	—	4	2	no	—
79	C₄⋊Dic₇	—	4	2	no	—
80	C₄⋊Dic₇	—	4	2	no	—
81	C₄⋊Dic₇	—	8	2	no	—
82	C₂×C₇⋊C₈	—	4	2	no	—
83	C₂×C₇⋊C₈	—	4	2	no	—
84	C₂×C₇⋊C₈	—	4	2	no	—
85	C₂×C₄×Q₈	C₂×D₇	1	1	yes	characteristic
86	#446	—	2	2	no	—
87	#446	Dic₇	1	4	yes	complemented
88	C₂×C₄⋊C₈	—	2	1	no	—
89	C₂×C₄⋊C₈	—	7	2	no	—
90	C₂×C₄⋊C₈	C₂×D₇	1	1	yes	characteristic
91	C₂×#103	—	14	2	no	—
92	C₂×#97	—	2	1	no	—
93	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
94	#384	—	7	2	no	—
95	#340	—	14	1	no	—
96	#340	—	14	1	no	—
97	#340	—	14	2	no	—
98	#334	—	14	2	no	—
99	C₂³×C₇	#121	1	1	yes	characteristic
100	C₇×Q₈	—	2	4	no	—
101	C₇×Q₈	—	4	2	no	—
102	C₂×C₄×C₇	—	2	2	no	—
103	C₂×C₄×C₇	—	2	2	no	—
104	C₂×C₄×C₇	—	2	2	no	—
105	C₂×C₄×C₇	—	4	1	no	—
106	C₂×C₄×C₇	—	4	1	no	—
107	C₂×C₄×C₇	—	4	2	no	—
108	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₇	#117	1	1	yes	characteristic
111	C₂×C₄×C₇	#121	1	1	yes	characteristic
112	C₂×C₄×C₇	C₂²⋊Q₈	1	1	yes	characteristic
113	C₂×C₄×C₇	C₂²⋊Q₈	1	1	yes	characteristic
114	C₂×C₄×C₇	#135	1	2	yes	—
115	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
116	C₂×C₄×C₇	#137	1	2	yes	—
117	C₂×C₄×C₇	#137	1	2	yes	—
118	C₂×Dic₇	—	8	2	no	—
119	C₂×Dic₇	—	8	2	no	—
120	C₂×Dic₇	—	8	2	no	—
121	C₇×C₈	—	2	2	no	—
122	C₇×C₈	—	2	2	no	—
123	C₇×C₈	—	4	2	no	—
124	C₇⋊C₈	—	8	2	no	—
125	C₂²×Q₈	D₂₈	1	1	yes	characteristic
126	C₂²×C₈	—	2	1	no	—
127	C₂²×C₈	—	14	2	no	—
128	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
129	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
130	C₄×Q₈	—	2	1	no	—
131	C₄×Q₈	—	2	1	no	—
132	C₄×Q₈	Dic₁₄	1	2	yes	—
133	C₄×Q₈	C₂×Dic₇	1	2	yes	—
134	C₂×C₄⋊C₄	—	2	1	no	—
135	C₂×C₄⋊C₄	—	14	2	no	—
136	C₂×C₄⋊C₄	—	14	2	no	—
137	C₂×C₄⋊C₄	—	14	2	no	—
138	C₂×C₄⋊C₄	D₂₈	1	1	yes	characteristic
139	C₂×C₄²	—	2	1	no	—
140	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
141	C₄⋊C₈	—	2	4	no	—
142	C₄⋊C₈	—	2	1	no	—
143	C₄⋊C₈	—	14	2	no	—
144	C₄⋊C₈	—	14	2	no	—
145	C₄⋊C₈	C₂×Dic₇	1	2	yes	—
146	#103	—	28	4	no	—
147	#97	—	2	4	no	—
148	C₄×C₈	C₂×Dic₇	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	—	4	2	no	—
158	C₄×C₇	—	4	2	no	—
159	C₄×C₇	#416	1	1	yes	characteristic
160	C₄×C₇	#419	1	1	yes	characteristic
161	C₄×C₇	#428	1	2	yes	—
162	C₄×C₇	#468	1	2	yes	—
163	C₄×C₇	#479	1	2	yes	—
164	Dic₇	—	16	2	no	—
165	C₂×Q₈	—	2	1	no	—
166	C₂×Q₈	—	2	2	no	—
167	C₂×Q₈	—	2	2	no	—
168	C₂×Q₈	C₄⋊Dic₇	1	2	yes	—
169	C₂²×C₄	—	2	1	no	—
170	C₂²×C₄	—	2	1	no	—
171	C₂²×C₄	—	28	2	no	—
172	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
173	C₂²×C₄	#1157	1	1	yes	characteristic
174	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	—	4	2	no	—
179	C₂×C₈	—	28	2	no	—
180	C₂×C₈	—	28	2	no	—
181	C₂×C₈	—	28	2	no	—
182	C₂×C₈	C₂²⋊Dic₇	1	2	yes	—
183	C₂×C₈	C₂²⋊Dic₇	1	2	yes	—
184	C₄⋊C₄	—	2	2	no	—
185	C₄⋊C₄	—	2	2	no	—
186	C₄⋊C₄	—	2	1	no	—
187	C₄⋊C₄	—	28	2	no	—
188	C₄⋊C₄	—	28	2	no	—
189	C₄⋊C₄	—	28	2	no	—
190	C₄⋊C₄	—	28	2	no	—
191	C₄⋊C₄	—	56	2	no	—
192	C₄⋊C₄	C₄⋊Dic₇	1	2	yes	—
193	C₄²	—	2	2	no	—
194	C₄²	—	2	2	no	—
195	C₄²	C₂×Dic₁₄	1	1	yes	characteristic
196	C₄²	#1157	1	1	yes	characteristic
197	C₄²	C₂²×Dic₇	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	#6493	1	1	yes	characteristic
205	Q₈	—	2	4	no	—
206	Q₈	—	4	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	4	1	no	—
211	C₂×C₄	—	4	1	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	56	2	no	—
214	C₂×C₄	—	56	2	no	—
215	C₂×C₄	—	56	2	no	—
216	C₂×C₄	#6471	1	2	yes	—
217	C₂×C₄	#6472	1	2	yes	—
218	C₂×C₄	#6487	1	1	yes	characteristic
219	C₂×C₄	#6487	1	1	yes	characteristic
220	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
221	C₂×C₄	#6489	1	1	yes	characteristic
222	C₂×C₄	#6493	1	1	yes	characteristic
223	C₂×C₄	#6505	1	2	yes	—
224	C₂×C₄	#6513	1	2	yes	—
225	C₂×C₄	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
226	C₈	—	2	2	no	—
227	C₈	—	2	2	no	—
228	C₈	—	4	2	no	—
229	C₈	—	56	2	no	—
230	C₇	#8657	1	1	yes	characteristic, complemented, Sylow
231	C₂²	#89800	1	2	yes	—
232	C₂²	#89907	1	1	yes	characteristic
233	C₂²	#89909	1	1	yes	characteristic
234	C₂²	#89913	1	2	yes	—
235	C₂²	#89984	1	1	yes	characteristic
236	C₄	—	2	1	no	—
237	C₄	—	2	1	no	—
238	C₄	—	4	2	no	—
239	C₄	—	4	2	no	—
240	C₄	—	112	2	no	—
241	C₄	#89794	1	2	yes	—
242	C₄	#89893	1	2	yes	—
243	C₄	#89906	1	2	yes	—
244	C₄	#89997	1	1	yes	characteristic
245	C₄	#90005	1	1	yes	characteristic
246	C₂	#11728715	1	2	yes	—
247	C₂	#11728721	1	2	yes	—
248	C₂	#11729381	1	1	yes	characteristic
249	C₂	#11729742	1	1	yes	characteristic
250	C₂	#11729769	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#6493
commutator	C₄²×C₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₂×C₇×#446	C₂
socle, prime radical	C₂³×C₇	#121
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	C₂×#446	D₇
hypocenter	C₇	#8657
LCS	G ▷ C₄²×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#446

Complements

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

N	H ≅ G/N	type
8. C₇×#446	C₄	semidirect
87. #446	Dic₇	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×C₇×#446	2
#11729380	2
#11727516	2
C₂×#89252	2
#11725508	2
#8657	7