Subgroups — #13273152

Order 1792.

462 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#89471	C₂	1	1	yes	maximal, characteristic
2	C₇×#1942	C₂	1	1	yes	maximal, characteristic
3	C₂×#89287	C₂	1	2	yes	maximal
4	C₂×#89287	C₂	1	2	yes	maximal
5	#11725463	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₈	—	2	1	no	—
7	C₂²×Dic₂₈	C₂²	1	2	yes	—
8	C₂×#6432	—	2	1	no	—
9	C₂×C₇×#107	C₂²	1	1	yes	characteristic
10	C₂×C₇×#107	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
12	C₇×#389	—	2	1	no	—
13	C₂²×C₇⋊C₁₆	C₂²	1	2	yes	—
14	#89471	—	2	4	no	—
15	C₇×#340	—	2	1	no	—
16	#89287	—	2	4	no	—
17	#89287	—	2	4	no	—
18	#13409	—	7	1	no	maximal, Sylow
19	C₂²×Dic₁₄	—	2	2	no	—
20	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
21	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
22	C₂×C₇×C₄⋊C₄	—	2	1	no	—
23	C₂×C₇×C₄⋊C₄	—	2	1	no	—
24	C₂×C₇×C₄⋊C₄	—	4	1	no	—
25	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
26	C₂×C₄⋊Dic₇	—	4	1	no	—
27	C₂×C₄⋊Dic₇	—	4	1	no	—
28	C₂×Dic₂₈	—	2	2	no	—
29	C₂×Dic₂₈	—	2	2	no	—
30	C₂×Dic₂₈	—	2	2	no	—
31	C₂×Dic₂₈	—	2	2	no	—
32	C₂×Dic₂₈	—	2	4	no	—
33	C₂×Dic₂₈	—	2	2	no	—
34	C₂×Dic₂₈	—	4	2	no	—
35	C₂×Dic₂₈	—	4	2	no	—
36	C₂×Dic₂₈	D₄	1	4	yes	—
37	#6432	—	2	4	no	—
38	#6432	—	4	2	no	—
39	C₇×#107	—	2	1	no	—
40	C₇×#107	—	2	2	no	—
41	C₇×#107	D₄	1	2	yes	—
42	C₄×C₇×C₈	—	2	2	no	—
43	C₇×#95	—	4	1	no	—
44	C₂×C₇⋊C₁₆	—	2	2	no	—
45	C₂×C₇⋊C₁₆	—	2	2	no	—
46	C₂×C₇⋊C₁₆	—	2	2	no	—
47	C₂×#494	—	7	1	no	—
48	C₂×#358	—	7	2	no	—
49	C₂×#358	—	7	2	no	—
50	#1942	D₇	1	1	yes	characteristic
51	#1374	—	7	1	no	—
52	C₂²×C₄×C₇	—	4	1	no	—
53	C₂²×C₄×C₇	—	4	1	no	—
54	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
55	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
56	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×Dic₇	—	4	2	no	—
58	C₂×Dic₁₄	—	2	4	no	—
59	C₂×Dic₁₄	—	4	2	no	—
60	C₂×Dic₁₄	—	4	2	no	—
61	C₂×Dic₁₄	—	4	2	no	—
62	C₂×Dic₁₄	—	4	2	no	—
63	C₂×Dic₁₄	—	4	2	no	—
64	C₂×C₇×C₈	—	2	1	no	—
65	C₂×C₇×C₈	—	2	1	no	—
66	C₂×C₇×C₈	SD₁₆	1	2	yes	—
67	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
68	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
69	C₂×C₇×C₈	#41	1	2	yes	—
70	C₂×C₇×C₈	#41	1	1	yes	characteristic
71	C₇×C₄⋊C₄	—	2	2	no	—
72	C₇×C₄⋊C₄	—	4	1	no	—
73	C₇×C₄⋊C₄	—	4	1	no	—
74	C₇×C₄⋊C₄	—	4	1	no	—
75	C₇×C₄⋊C₄	—	8	2	no	—
76	C₄²×C₇	—	2	1	no	—
77	C₄²×C₇	—	2	1	no	—
78	C₄⋊Dic₇	—	4	4	no	—
79	C₄⋊Dic₇	—	4	4	no	—
80	Dic₂₈	—	2	4	no	—
81	Dic₂₈	—	2	4	no	—
82	Dic₂₈	—	4	8	no	—
83	Dic₂₈	—	4	2	no	—
84	Dic₂₈	—	4	2	no	—
85	C₇⋊C₁₆	—	4	2	no	—
86	C₂²×Q₁₆	—	7	2	no	—
87	C₂²×Q₁₆	—	14	1	no	—
88	C₂×C₄⋊Q₈	—	14	1	no	—
89	C₂²×C₁₆	—	7	2	no	—
90	#494	—	14	4	no	—
91	C₂×#107	C₂×D₇	1	1	yes	characteristic
92	C₂×#107	C₂×D₇	1	1	yes	characteristic
93	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
94	#389	—	2	1	no	—
95	#358	—	14	4	no	—
96	#358	—	14	4	no	—
97	#340	—	2	1	no	—
98	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
99	C₂×C₄×C₇	—	2	1	no	—
100	C₂×C₄×C₇	—	2	1	no	—
101	C₂×C₄×C₇	—	2	1	no	—
102	C₂×C₄×C₇	—	2	1	no	—
103	C₂×C₄×C₇	—	4	2	no	—
104	C₂×C₄×C₇	—	8	1	no	—
105	C₂×C₄×C₇	—	8	1	no	—
106	C₂×C₄×C₇	—	8	1	no	—
107	C₂×C₄×C₇	—	8	1	no	—
108	C₂×C₄×C₇	—	8	1	no	—
109	C₂×C₄×C₇	SD₃₂	1	2	yes	—
110	C₂×C₄×C₇	Q₃₂	1	2	yes	—
111	C₂×C₄×C₇	#120	1	1	yes	characteristic
112	C₂×C₄×C₇	#121	1	1	yes	characteristic
113	C₂×C₄×C₇	#124	1	1	yes	characteristic
114	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
115	C₂×C₄×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
116	C₂×C₄×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
117	C₂×Dic₇	—	4	4	no	—
118	C₂×Dic₇	—	8	2	no	—
119	C₂×Dic₇	—	8	2	no	—
120	Dic₁₄	—	4	4	no	—
121	Dic₁₄	—	4	4	no	—
122	Dic₁₄	—	8	2	no	—
123	Dic₁₄	—	8	2	no	—
124	C₇×C₈	—	2	2	no	—
125	C₇×C₈	#121	1	2	yes	—
126	C₇×C₈	#123	1	2	yes	—
127	C₂²×Q₈	—	14	2	no	—
128	C₂×Q₁₆	—	7	4	no	—
129	C₂×Q₁₆	—	14	4	no	—
130	C₂×Q₁₆	—	14	2	no	—
131	C₂×Q₁₆	—	14	2	no	—
132	C₂×Q₁₆	—	14	2	no	—
133	C₂×Q₁₆	—	14	2	no	—
134	C₂×Q₁₆	—	14	2	no	—
135	C₂×Q₁₆	—	28	2	no	—
136	C₂×Q₁₆	—	28	2	no	—
137	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
138	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
139	C₄⋊Q₈	—	14	4	no	—
140	C₄⋊Q₈	—	28	2	no	—
141	C₂×C₄⋊C₄	—	2	1	no	—
142	C₂×C₄⋊C₄	—	2	1	no	—
143	C₂×C₄⋊C₄	—	4	1	no	—
144	C₂×C₄⋊C₄	—	28	1	no	—
145	C₂×C₄⋊C₄	—	28	1	no	—
146	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
147	C₂×C₁₆	—	14	2	no	—
148	C₂×C₁₆	—	14	2	no	—
149	C₂×C₁₆	—	14	2	no	—
150	#107	—	2	1	no	—
151	#107	—	2	2	no	—
152	#107	D₂₈	1	2	yes	—
153	C₄×C₈	—	2	2	no	—
154	#95	—	4	1	no	—
155	C₂²×C₇	#447	1	1	yes	characteristic
156	C₂²×C₇	#459	1	1	yes	characteristic
157	C₂²×C₇	#486	1	1	yes	characteristic
158	C₂²×C₇	C₂×SD₃₂	1	1	yes	characteristic
159	C₂²×C₇	C₂×Q₃₂	1	1	yes	characteristic
160	C₂²×C₇	#510	1	1	yes	characteristic
161	C₂²×C₇	#510	1	1	yes	characteristic
162	C₄×C₇	—	2	2	no	—
163	C₄×C₇	—	2	2	no	—
164	C₄×C₇	—	8	2	no	—
165	C₄×C₇	—	16	1	no	—
166	C₄×C₇	#394	1	1	yes	characteristic
167	C₄×C₇	#450	1	1	yes	characteristic
168	C₄×C₇	#460	1	1	yes	characteristic
169	C₄×C₇	#490	1	1	yes	characteristic
170	Dic₇	—	8	4	no	—
171	C₂×Q₈	—	14	4	no	—
172	C₂×Q₈	—	28	2	no	—
173	C₂×Q₈	—	28	2	no	—
174	C₂×Q₈	—	28	2	no	—
175	C₂×Q₈	—	28	2	no	—
176	C₂×Q₈	—	28	2	no	—
177	C₂²×C₄	—	4	1	no	—
178	C₂²×C₄	—	4	1	no	—
179	C₂²×C₄	—	28	2	no	—
180	C₂²×C₄	#1141	1	1	yes	characteristic
181	C₂²×C₄	#1141	1	1	yes	characteristic
182	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
183	Q₁₆	—	14	4	no	—
184	Q₁₆	—	14	4	no	—
185	Q₁₆	—	28	8	no	—
186	Q₁₆	—	28	2	no	—
187	Q₁₆	—	28	2	no	—
188	C₂×C₈	—	2	1	no	—
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	#1143	1	2	yes	—
191	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
192	C₂×C₈	#1157	1	1	yes	characteristic
193	C₂×C₈	D₄×D₇	1	2	yes	—
194	C₂×C₈	#1161	1	2	yes	—
195	C₄⋊C₄	—	2	2	no	—
196	C₄⋊C₄	—	4	1	no	—
197	C₄⋊C₄	—	4	1	no	—
198	C₄⋊C₄	—	4	1	no	—
199	C₄⋊C₄	—	8	2	no	—
200	C₄⋊C₄	—	28	4	no	—
201	C₄⋊C₄	—	28	4	no	—
202	C₄²	—	2	1	no	—
203	C₄²	—	2	1	no	—
204	C₁₆	—	28	2	no	—
205	C₂×C₇	#2041	1	1	yes	characteristic
206	C₂×C₇	#2185	1	1	yes	characteristic
207	C₂×C₇	#2188	1	1	yes	characteristic
208	C₂×C₇	#2208	1	1	yes	characteristic
209	C₂×C₇	#2209	1	1	yes	characteristic
210	C₂×C₇	#2241	1	1	yes	characteristic
211	C₂×C₇	#2245	1	1	yes	characteristic
212	C₂³	C₂×#1141	1	1	yes	characteristic
213	Q₈	—	28	4	no	—
214	Q₈	—	28	4	no	—
215	Q₈	—	56	2	no	—
216	Q₈	—	56	2	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	8	1	no	—
223	C₂×C₄	—	8	1	no	—
224	C₂×C₄	—	8	1	no	—
225	C₂×C₄	—	8	1	no	—
226	C₂×C₄	—	8	1	no	—
227	C₂×C₄	—	28	4	no	—
228	C₂×C₄	—	56	2	no	—
229	C₂×C₄	—	56	2	no	—
230	C₂×C₄	#6401	1	2	yes	—
231	C₂×C₄	#6403	1	2	yes	—
232	C₂×C₄	#6493	1	1	yes	characteristic
233	C₂×C₄	#6500	1	1	yes	characteristic
234	C₂×C₄	C₂×#1143	1	1	yes	characteristic
235	C₂×C₄	#6510	1	1	yes	characteristic
236	C₂×C₄	#6512	1	1	yes	characteristic
237	C₂×C₄	#6512	1	1	yes	characteristic
238	C₈	—	2	2	no	—
239	C₈	#6457	1	2	yes	—
240	C₈	#6458	1	2	yes	—
241	C₇	#13409	1	1	yes	characteristic, complemented, Sylow
242	C₂²	#89787	1	1	yes	characteristic
243	C₂²	#89808	1	1	yes	characteristic
244	C₂²	#89860	1	1	yes	characteristic
245	C₂²	C₂×#6401	1	1	yes	characteristic
246	C₂²	C₂×#6403	1	1	yes	characteristic
247	C₂²	#89967	1	1	yes	characteristic
248	C₂²	#89967	1	1	yes	characteristic
249	C₄	—	2	2	no	—
250	C₄	—	2	2	no	—
251	C₄	—	8	2	no	—
252	C₄	—	16	1	no	—
253	C₄	—	56	4	no	—
254	C₄	#89763	1	1	yes	characteristic
255	C₄	#89799	1	1	yes	characteristic
256	C₄	#89818	1	1	yes	characteristic
257	C₄	#89839	1	1	yes	characteristic
258	C₂	#11727381	1	1	yes	characteristic
259	C₂	#11728646	1	1	yes	characteristic
260	C₂	#11728670	1	1	yes	characteristic
261	C₂	#11728831	1	1	yes	characteristic
262	C₂	#11728841	1	1	yes	characteristic
263	C₂	#11729107	1	1	yes	characteristic
264	C₂	#11729136	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#1141
commutator, cosocle	C₂²×C₇×C₈	C₂³
Frattini	C₂²×C₈	C₂²×D₇
Fitting	C₇×#1942	C₂
socle, prime radical	C₂³×C₇	C₂×D₈
hypercenter	#1942	D₇
hypocenter	C₇	#13409
LCS	G ▷ C₂²×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1942

Complements

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

N	H ≅ G/N	type
241. C₇	#13409	semidirect

Maximal subgroups

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

subgroup	index
C₂×#89471	2
C₇×#1942	2
C₂×#89287	2
#11725463	2
#13409	7