Subgroups — #13253928

Order 1792.

450 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11730752	C₂	1	1	yes	maximal, characteristic
2	C₇×#1842	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊(C₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#11725508	C₂	1	2	yes	maximal
5	#11725508	C₂	1	2	yes	maximal
6	C₂×#6433	—	2	1	no	—
7	C₂×C₇×C₄⋊C₈	—	2	1	no	—
8	C₂×C₇×C₄⋊C₈	—	2	1	no	—
9	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
10	C₄³×C₇	C₂²	1	1	yes	characteristic
11	#89708	—	2	1	no	—
12	#89707	C₂²	1	1	yes	characteristic
13	#89707	C₂²	1	1	yes	characteristic
14	#89700	—	2	1	no	—
15	#89700	—	2	1	no	—
16	#89700	C₄	1	2	yes	—
17	#89700	C₄	1	2	yes	—
18	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
19	C₂×C₄×C₇⋊C₈	—	2	1	no	—
20	#89319	—	2	1	no	—
21	#9900	—	7	1	no	maximal, Sylow
22	C₂²×C₇×C₈	—	2	2	no	—
23	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
30	C₂×C₄⋊Dic₇	—	2	1	no	—
31	C₂×C₄⋊Dic₇	—	2	1	no	—
32	C₂×C₄⋊Dic₇	—	2	1	no	—
33	C₂×C₄⋊Dic₇	—	2	1	no	—
34	C₂×C₄⋊Dic₇	C₂×C₄	1	2	yes	—
35	C₂×C₄⋊Dic₇	C₂×C₄	1	2	yes	—
36	C₂×Dic₇⋊C₄	—	2	2	no	—
37	C₂×Dic₇⋊C₄	—	2	2	no	—
38	C₂²×C₇⋊C₈	—	2	2	no	—
39	#6433	—	4	2	no	—
40	#6433	—	4	2	no	—
41	C₇×C₄⋊C₈	—	2	4	no	—
42	C₇×C₄⋊C₈	—	4	2	no	—
43	C₇×C₄⋊C₈	—	4	2	no	—
44	C₇⋊#95	—	2	2	no	—
45	C₇⋊#95	—	2	2	no	—
46	C₄⋊(C₇⋊C₈)	—	2	2	no	—
47	C₄⋊(C₇⋊C₈)	Q₈	1	4	yes	—
48	C₄×C₇⋊C₈	—	4	2	no	—
49	#2859	—	7	1	no	—
50	#1842	D₇	1	1	yes	characteristic
51	C₄×C₄⋊C₈	—	7	1	no	—
52	#1296	—	7	2	no	—
53	#1296	—	7	2	no	—
54	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
61	C₂²×Dic₇	—	2	2	no	—
62	C₂²×Dic₇	—	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	C₄²×C₇	—	2	2	no	—
67	C₄²×C₇	—	2	1	no	—
68	C₄²×C₇	—	2	1	no	—
69	C₄²×C₇	—	4	1	no	—
70	C₄²×C₇	—	4	1	no	—
71	C₄²×C₇	D₈	1	2	yes	—
72	C₄²×C₇	SD₁₆	1	2	yes	—
73	C₄²×C₇	SD₁₆	1	2	yes	—
74	C₄²×C₇	Q₁₆	1	2	yes	—
75	C₄²×C₇	C₂×Q₈	1	2	yes	—
76	C₄²×C₇	#41	1	2	yes	—
77	C₄⋊Dic₇	—	4	1	no	—
78	C₄⋊Dic₇	—	4	1	no	—
79	C₄⋊Dic₇	—	4	1	no	—
80	C₄⋊Dic₇	—	4	1	no	—
81	C₄⋊Dic₇	—	4	2	no	—
82	C₄⋊Dic₇	—	4	1	no	—
83	C₄⋊Dic₇	—	4	1	no	—
84	C₄⋊Dic₇	—	4	2	no	—
85	C₄⋊Dic₇	—	4	2	no	—
86	Dic₇⋊C₄	—	4	4	no	—
87	Dic₇⋊C₄	—	4	4	no	—
88	C₂×C₇⋊C₈	—	2	4	no	—
89	C₂×C₇⋊C₈	—	4	2	no	—
90	C₂×C₇⋊C₈	—	4	2	no	—
91	C₂×#125	—	14	1	no	—
92	C₂×C₄⋊C₈	—	2	1	no	—
93	C₂×C₄⋊C₈	—	2	1	no	—
94	C₂×C₄⋊C₈	—	7	2	no	—
95	C₂×C₄⋊C₈	C₂×D₇	1	2	yes	—
96	C₂×C₄×C₈	—	14	1	no	—
97	#389	—	7	2	no	—
98	#389	—	7	2	no	—
99	#389	—	14	1	no	—
100	#389	—	14	1	no	—
101	#384	—	7	1	no	—
102	#384	—	7	1	no	—
103	#382	—	14	1	no	—
104	C₄³	C₂×D₇	1	1	yes	characteristic
105	#336	—	14	1	no	—
106	C₂³×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₇	—	2	1	no	—
108	C₂×C₄×C₇	—	2	1	no	—
109	C₂×C₄×C₇	—	2	1	no	—
110	C₂×C₄×C₇	—	2	1	no	—
111	C₂×C₄×C₇	—	2	1	no	—
112	C₂×C₄×C₇	—	2	1	no	—
113	C₂×C₄×C₇	—	2	1	no	—
114	C₂×C₄×C₇	—	2	1	no	—
115	C₂×C₄×C₇	—	2	1	no	—
116	C₂×C₄×C₇	—	2	1	no	—
117	C₂×C₄×C₇	#102	1	2	yes	—
118	C₂×C₄×C₇	#102	1	2	yes	—
119	C₂×C₄×C₇	#103	1	2	yes	—
120	C₂×C₄×C₇	#103	1	2	yes	—
121	C₂×C₄×C₇	C₄×Q₈	1	2	yes	—
122	C₂×C₄×C₇	C₂²⋊Q₈	1	2	yes	—
123	C₂×C₄×C₇	#124	1	1	yes	characteristic
124	C₂×C₄×C₇	C₄⋊Q₈	1	1	yes	characteristic
125	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
127	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
128	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
129	C₂×C₄×C₇	#135	1	2	yes	—
130	C₂×C₄×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
131	C₂×C₄×C₇	#137	1	1	yes	characteristic
132	C₂×Dic₇	—	4	2	no	—
133	C₂×Dic₇	—	4	2	no	—
134	C₂×Dic₇	—	4	2	no	—
135	C₂×Dic₇	—	4	2	no	—
136	C₂×Dic₇	—	4	2	no	—
137	C₂×Dic₇	—	4	2	no	—
138	C₇×C₈	—	8	2	no	—
139	C₇⋊C₈	—	4	4	no	—
140	C₂²×C₈	—	2	2	no	—
141	C₂²×C₈	—	14	2	no	—
142	#125	—	28	2	no	—
143	#125	—	28	2	no	—
144	C₂×C₄⋊C₄	—	7	2	no	—
145	C₂×C₄⋊C₄	—	7	2	no	—
146	C₂×C₄⋊C₄	—	14	2	no	—
147	C₂×C₄⋊C₄	—	14	2	no	—
148	C₂×C₄⋊C₄	—	14	1	no	—
149	C₂×C₄⋊C₄	—	14	1	no	—
150	C₂×C₄⋊C₄	—	14	1	no	—
151	C₂×C₄⋊C₄	—	14	1	no	—
152	C₂×C₄²	C₄×D₇	1	1	yes	characteristic
153	C₂×C₄²	C₄×D₇	1	1	yes	characteristic
154	C₂×C₄²	D₂₈	1	1	yes	characteristic
155	C₂×C₄²	D₂₈	1	1	yes	characteristic
156	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
157	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
158	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
159	C₄⋊C₈	—	2	4	no	—
160	C₄⋊C₈	—	4	2	no	—
161	C₄⋊C₈	—	4	2	no	—
162	C₄⋊C₈	—	7	4	no	—
163	C₄⋊C₈	—	14	2	no	—
164	C₄×C₈	—	28	2	no	—
165	#95	—	14	2	no	—
166	#95	—	14	2	no	—
167	C₂²×C₇	#391	1	1	yes	characteristic
168	C₂²×C₇	#413	1	1	yes	characteristic
169	C₂²×C₇	#413	1	1	yes	characteristic
170	C₂²×C₇	C₂×#102	1	1	yes	characteristic
171	C₂²×C₇	C₂×#103	1	1	yes	characteristic
172	C₂²×C₇	#416	1	1	yes	characteristic
173	C₂²×C₇	#417	1	1	yes	characteristic
174	C₄×C₇	—	2	2	no	—
175	C₄×C₇	—	2	2	no	—
176	C₄×C₇	—	2	2	no	—
177	C₄×C₇	—	2	2	no	—
178	C₄×C₇	—	2	2	no	—
179	C₄×C₇	#478	1	2	yes	—
180	C₄×C₇	#479	1	2	yes	—
181	C₄×C₇	#486	1	1	yes	characteristic
182	C₄×C₇	#487	1	1	yes	characteristic
183	C₄×C₇	#498	1	1	yes	characteristic
184	C₄×C₇	#500	1	1	yes	characteristic
185	Dic₇	—	8	2	no	—
186	Dic₇	—	8	2	no	—
187	C₂²×C₄	—	14	2	no	—
188	C₂²×C₄	—	14	2	no	—
189	C₂²×C₄	(C₂×D₇)⋊C₄	1	1	yes	characteristic
190	C₂²×C₄	(C₂×D₇)⋊C₄	1	1	yes	characteristic
191	C₂²×C₄	(C₂×D₇)⋊C₄	1	1	yes	characteristic
192	C₂²×C₄	(C₂×D₇)⋊C₄	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
194	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
195	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
196	C₂×C₈	—	4	2	no	—
197	C₂×C₈	—	4	2	no	—
198	C₂×C₈	—	4	2	no	—
199	C₂×C₈	—	14	4	no	—
200	C₂×C₈	—	28	2	no	—
201	C₂×C₈	—	28	2	no	—
202	C₄⋊C₄	—	28	2	no	—
203	C₄⋊C₄	—	28	2	no	—
204	C₄⋊C₄	—	28	2	no	—
205	C₄⋊C₄	—	28	1	no	—
206	C₄⋊C₄	—	28	1	no	—
207	C₄⋊C₄	—	28	1	no	—
208	C₄⋊C₄	—	28	1	no	—
209	C₄⋊C₄	—	28	1	no	—
210	C₄⋊C₄	—	28	1	no	—
211	C₄⋊C₄	—	28	4	no	—
212	C₄⋊C₄	—	28	4	no	—
213	C₄²	—	2	2	no	—
214	C₄²	—	2	1	no	—
215	C₄²	—	2	1	no	—
216	C₄²	—	4	1	no	—
217	C₄²	—	4	1	no	—
218	C₄²	#1141	1	2	yes	—
219	C₄²	#1142	1	2	yes	—
220	C₄²	#1143	1	2	yes	—
221	C₄²	#1144	1	2	yes	—
222	C₄²	#1159	1	2	yes	—
223	C₄²	Q₈×D₇	1	2	yes	—
224	C₂×C₇	#1540	1	1	yes	characteristic
225	C₂×C₇	#1542	1	1	yes	characteristic
226	C₂×C₇	#1547	1	1	yes	characteristic
227	C₂×C₇	#1550	1	1	yes	characteristic
228	C₂×C₇	#1989	1	1	yes	characteristic
229	C₂×C₇	#1990	1	1	yes	characteristic
230	C₂×C₇	#1998	1	1	yes	characteristic
231	C₂³	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	28	2	no	—
243	C₂×C₄	—	28	2	no	—
244	C₂×C₄	—	28	2	no	—
245	C₂×C₄	—	28	2	no	—
246	C₂×C₄	—	28	2	no	—
247	C₂×C₄	—	28	2	no	—
248	C₂×C₄	#6383	1	2	yes	—
249	C₂×C₄	#6383	1	2	yes	—
250	C₂×C₄	#6384	1	2	yes	—
251	C₂×C₄	#6384	1	2	yes	—
252	C₂×C₄	#6450	1	2	yes	—
253	C₂×C₄	#6460	1	2	yes	—
254	C₂×C₄	#6467	1	2	yes	—
255	C₂×C₄	#6471	1	1	yes	characteristic
256	C₂×C₄	#6472	1	1	yes	characteristic
257	C₂×C₄	C₂×#1141	1	1	yes	characteristic
258	C₂×C₄	C₂×#1142	1	1	yes	characteristic
259	C₂×C₄	#6499	1	1	yes	characteristic
260	C₂×C₄	C₂×#1143	1	1	yes	characteristic
261	C₂×C₄	C₂×#1144	1	1	yes	characteristic
262	C₂×C₄	#6507	1	1	yes	characteristic
263	C₈	—	8	2	no	—
264	C₈	—	28	4	no	—
265	C₇	#9900	1	1	yes	characteristic, complemented, Sylow
266	C₂²	C₂×#6383	1	1	yes	characteristic
267	C₂²	C₂×#6384	1	1	yes	characteristic
268	C₂²	#89746	1	1	yes	characteristic
269	C₂²	#89885	1	1	yes	characteristic
270	C₂²	#89904	1	1	yes	characteristic
271	C₂²	#89904	1	1	yes	characteristic
272	C₂²	#89905	1	1	yes	characteristic
273	C₄	—	2	2	no	—
274	C₄	—	2	2	no	—
275	C₄	—	2	2	no	—
276	C₄	—	2	2	no	—
277	C₄	—	2	2	no	—
278	C₄	—	56	2	no	—
279	C₄	—	56	2	no	—
280	C₄	#89601	1	2	yes	—
281	C₄	#89619	1	2	yes	—
282	C₄	#89842	1	1	yes	characteristic
283	C₄	#89850	1	1	yes	characteristic
284	C₄	#89851	1	1	yes	characteristic
285	C₄	#89853	1	1	yes	characteristic
286	C₂	#11727336	1	1	yes	characteristic
287	C₂	#11728185	1	1	yes	characteristic
288	C₂	#11728191	1	1	yes	characteristic
289	C₂	#11728215	1	1	yes	characteristic
290	C₂	#11728257	1	1	yes	characteristic
291	C₂	#11728263	1	1	yes	characteristic
292	C₂	#11728279	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₇)⋊C₄
commutator	C₂²×C₄×C₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₇×#1842	C₂
socle, prime radical	C₂³×C₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	#1842	D₇
hypocenter	C₇	#9900
LCS	G ▷ C₂²×C₄×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1842

Complements

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

N	H ≅ G/N	type
265. C₇	#9900	semidirect

Maximal subgroups

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

subgroup	index
#11730752	2
C₇×#1842	2
C₄×C₄⋊(C₇⋊C₈)	2
#11725508	2
#9900	7