Subgroups — #13324474

Order 1792.

560 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11736761	C₂	1	2	yes	maximal
2	#11735920	C₂	1	1	yes	maximal, characteristic
3	C₇×#1568	C₂	1	1	yes	maximal, characteristic
4	#11728557	C₂	1	2	yes	maximal
5	#11727478	C₂	1	1	yes	maximal, characteristic
6	#11726318	C₂	1	2	yes	maximal, complemented
7	#11726311	C₂	1	2	yes	maximal, complemented
8	#11726278	C₂	1	2	yes	maximal, complemented
9	#11726271	C₂	1	2	yes	maximal, complemented
10	#90161	C₂²	1	2	yes	—
11	C₇×#432	C₂²	1	2	yes	—
12	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
13	#89888	C₄	1	2	yes	—
14	#89888	C₄	1	2	yes	—
15	#89697	C₂²	1	1	yes	characteristic
16	#89697	C₂²	1	1	yes	characteristic
17	#89603	C₂²	1	4	yes	—
18	#89602	C₂²	1	4	yes	—
19	#89492	—	2	4	no	—
20	#89473	—	2	4	no	—
21	#89471	C₄	1	1	yes	characteristic
22	C₄×Dic₂₈	C₂²	1	2	yes	—
23	C₄×Dic₂₈	C₂²	1	2	yes	—
24	#89454	C₄	1	1	yes	characteristic
25	#89453	C₄	1	2	yes	—
26	C₇×#335	C₂²	1	2	yes	—
27	C₇×#334	C₂²	1	2	yes	—
28	#89279	C₂²	1	4	yes	—
29	#89279	C₂²	1	4	yes	—
30	#89250	C₂²	1	1	yes	characteristic
31	#89250	C₂²	1	1	yes	characteristic
32	C₈×C₇⋊C₈	C₂²	1	1	yes	characteristic
33	C₈×C₇⋊C₈	C₂²	1	1	yes	characteristic
34	#17972	—	7	1	no	maximal, Sylow
35	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
38	#6489	—	2	2	no	—
39	#6487	—	2	2	no	—
40	#6487	C₂×C₄	1	4	yes	—
41	C₂×Dic₂₈	C₂×C₄	1	2	yes	—
42	C₂×Dic₂₈	C₂×C₄	1	2	yes	—
43	#6433	—	2	2	no	—
44	#6432	C₂×C₄	1	2	yes	—
45	#6432	C₂×C₄	1	2	yes	—
46	C₄×Dic₁₄	—	2	2	no	—
47	C₄×Dic₁₄	C₂³	1	4	yes	—
48	C₇×C₄⋊C₈	C₂³	1	4	yes	—
49	C₇×C₂²⋊C₈	—	2	2	no	—
50	C₇×#97	—	2	2	no	—
51	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
52	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
53	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
54	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
55	C₂²⋊(C₇⋊C₈)	—	2	1	no	—
56	C₂²⋊(C₇⋊C₈)	D₄	1	2	yes	—
57	#6392	C₂×C₄	1	2	yes	—
58	#6392	C₂×C₄	1	2	yes	—
59	#6390	C₂×C₄	1	4	yes	—
60	#6390	C₂×C₄	1	4	yes	—
61	#6389	—	2	4	no	—
62	Dic₇⋊C₈	—	2	4	no	—
63	Dic₇⋊C₈	—	2	4	no	—
64	C₈×Dic₇	—	2	4	no	—
65	C₄⋊(C₇⋊C₈)	—	2	1	no	—
66	C₄⋊(C₇⋊C₈)	D₄	1	2	yes	—
67	C₄×C₇⋊C₈	C₂³	1	1	yes	characteristic
68	C₄×C₇⋊C₈	C₂³	1	1	yes	characteristic
69	C₄×C₇⋊C₈	C₂³	1	1	yes	characteristic
70	C₄×C₇⋊C₈	C₂³	1	1	yes	characteristic
71	#3170	—	7	1	no	—
72	#2985	—	7	2	no	—
73	#1593	—	7	2	no	—
74	#1591	—	7	2	no	—
75	#1579	—	7	2	no	—
76	C₈×Q₁₆	—	7	2	no	—
77	#1568	D₇	1	1	yes	characteristic
78	#1497	—	7	2	no	—
79	#1450	—	7	1	no	—
80	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₇	#41	1	1	yes	characteristic
83	C₂×Dic₁₄	—	2	2	no	—
84	C₂×Dic₁₄	C₂²×C₄	1	4	yes	—
85	C₂×C₇×C₈	—	2	2	no	—
86	C₂×C₇×C₈	—	2	2	no	—
87	C₂×C₇×C₈	—	2	4	no	—
88	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₇	#41	1	1	yes	characteristic
95	C₄²×C₇	C₂⁴	1	1	yes	characteristic
96	C₂²⋊Dic₇	—	2	4	no	—
97	C₄⋊Dic₇	—	2	2	no	—
98	C₄⋊Dic₇	—	2	2	no	—
99	C₄⋊Dic₇	—	2	2	no	—
100	C₄⋊Dic₇	C₂²×C₄	1	4	yes	—
101	Dic₇⋊C₄	—	2	4	no	—
102	Dic₇⋊C₄	—	2	4	no	—
103	C₄×Dic₇	—	2	4	no	—
104	C₂×C₇⋊C₈	—	2	1	no	—
105	C₂×C₇⋊C₈	—	2	1	no	—
106	C₂×C₇⋊C₈	C₂×D₄	1	2	yes	—
107	C₂×C₇⋊C₈	C₂×D₄	1	2	yes	—
108	Dic₂₈	—	2	4	no	—
109	Dic₂₈	—	2	4	no	—
110	#533	—	7	2	no	—
111	#500	—	7	1	no	—
112	#494	—	7	1	no	—
113	#487	—	7	2	no	—
114	#467	—	7	2	no	—
115	#467	—	7	2	no	—
116	#446	—	7	4	no	—
117	#446	—	14	4	no	—
118	C₈×Q₈	—	7	4	no	—
119	C₄×Q₁₆	—	7	2	no	—
120	C₄×Q₁₆	—	7	2	no	—
121	#433	—	14	4	no	—
122	#432	C₂×D₇	1	2	yes	—
123	#431	—	7	1	no	—
124	#431	—	7	1	no	—
125	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
126	#335	C₂×D₇	1	2	yes	—
127	#334	C₂×D₇	1	2	yes	—
128	Q₈⋊C₈	—	7	4	no	—
129	Q₈⋊C₈	—	7	4	no	—
130	#322	—	7	1	no	—
131	#322	—	7	1	no	—
132	C₈²	—	7	1	no	—
133	C₈²	—	7	1	no	—
134	C₂³×C₇	C₄×D₄	1	1	yes	characteristic
135	C₂×C₄×C₇	—	2	1	no	—
136	C₂×C₄×C₇	—	2	1	no	—
137	C₂×C₄×C₇	—	2	2	no	—
138	C₂×C₄×C₇	—	4	1	no	—
139	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
140	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
143	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
145	C₂×Dic₇	—	2	4	no	—
146	C₂×Dic₇	—	2	4	no	—
147	Dic₁₄	—	2	4	no	—
148	Dic₁₄	—	2	4	no	—
149	Dic₁₄	—	8	2	no	—
150	C₇×C₈	—	2	1	no	—
151	C₇×C₈	—	2	1	no	—
152	C₇×C₈	—	4	4	no	—
153	C₇×C₈	#131	1	2	yes	—
154	C₇×C₈	#131	1	2	yes	—
155	C₇⋊C₈	—	2	2	no	—
156	C₇⋊C₈	—	2	2	no	—
157	C₇⋊C₈	—	4	1	no	—
158	C₇⋊C₈	—	4	1	no	—
159	C₂×Q₁₆	—	7	2	no	—
160	C₂×Q₁₆	—	7	2	no	—
161	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
162	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
163	C₄⋊Q₈	—	7	2	no	—
164	C₄⋊Q₈	—	7	2	no	—
165	#125	—	14	2	no	—
166	C₂²⋊Q₈	—	7	4	no	—
167	C₂²⋊Q₈	—	14	2	no	—
168	C₄×Q₈	—	7	4	no	—
169	C₄×Q₈	—	14	2	no	—
170	#117	—	14	2	no	—
171	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
172	#107	—	7	2	no	—
173	#107	—	7	2	no	—
174	C₄⋊C₈	—	7	2	no	—
175	C₄⋊C₈	—	14	4	no	—
176	C₄⋊C₈	—	14	4	no	—
177	C₄⋊C₈	—	14	1	no	—
178	C₄⋊C₈	C₂²×D₇	1	4	yes	—
179	#103	—	7	4	no	—
180	#103	—	7	4	no	—
181	C₂²⋊C₈	—	2	2	no	—
182	C₂²⋊C₈	—	7	2	no	—
183	C₂²⋊C₈	—	14	1	no	—
184	#97	—	2	2	no	—
185	#97	—	14	4	no	—
186	C₄×C₈	—	7	1	no	—
187	C₄×C₈	—	7	1	no	—
188	C₄×C₈	—	7	1	no	—
189	C₄×C₈	—	7	1	no	—
190	C₄×C₈	—	14	4	no	—
191	C₄×C₈	C₄×D₇	1	1	yes	characteristic
192	C₄×C₈	C₄×D₇	1	1	yes	characteristic
193	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
194	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
195	C₂²×C₇	—	2	1	no	—
196	C₂²×C₇	—	2	1	no	—
197	C₂²×C₇	—	2	1	no	—
198	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
199	C₄×C₇	—	2	1	no	—
200	C₄×C₇	—	2	1	no	—
201	C₄×C₇	—	2	1	no	—
202	C₄×C₇	—	2	1	no	—
203	C₄×C₇	C₂×#131	1	1	yes	characteristic
204	C₄×C₇	C₂×#131	1	1	yes	characteristic
205	C₄×C₇	#578	1	2	yes	—
206	Dic₇	—	4	4	no	—
207	Dic₇	—	4	4	no	—
208	C₂×Q₈	—	7	4	no	—
209	C₂×Q₈	—	14	2	no	—
210	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
211	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
212	C₂²×C₄	#1161	1	1	yes	characteristic
213	Q₁₆	—	14	4	no	—
214	Q₁₆	—	14	4	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	7	2	no	—
219	C₂×C₈	—	7	2	no	—
220	C₂×C₈	—	14	1	no	—
221	C₂×C₈	—	14	1	no	—
222	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
223	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
224	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
225	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
226	C₄⋊C₄	—	7	4	no	—
227	C₄⋊C₄	—	14	4	no	—
228	C₄⋊C₄	—	14	2	no	—
229	C₄⋊C₄	—	14	2	no	—
230	C₄⋊C₄	—	14	2	no	—
231	C₄⋊C₄	—	14	4	no	—
232	C₂²⋊C₄	—	14	4	no	—
233	C₄²	—	14	4	no	—
234	C₄²	C₂×C₄×D₇	1	1	yes	characteristic
235	C₄²	D₄×D₇	1	1	yes	characteristic
236	C₄²	#1161	1	1	yes	characteristic
237	C₄²	C₂³×D₇	1	1	yes	characteristic
238	C₂×C₇	—	4	1	no	—
239	C₂×C₇	#2931	1	1	yes	characteristic
240	C₂×C₇	#2950	1	1	yes	characteristic
241	C₂×C₇	#2957	1	1	yes	characteristic
242	C₂³	#6456	1	1	yes	characteristic
243	Q₈	—	14	4	no	—
244	Q₈	—	14	4	no	—
245	Q₈	—	56	2	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	14	4	no	—
251	C₂×C₄	—	14	4	no	—
252	C₂×C₄	#6456	1	1	yes	characteristic
253	C₂×C₄	#6456	1	1	yes	characteristic
254	C₂×C₄	#6456	1	1	yes	characteristic
255	C₂×C₄	C₂²×C₄×D₇	1	1	yes	characteristic
256	C₂×C₄	C₂×D₄×D₇	1	1	yes	characteristic
257	C₂×C₄	C₂×#1161	1	1	yes	characteristic
258	C₈	—	2	1	no	—
259	C₈	—	2	1	no	—
260	C₈	—	4	4	no	—
261	C₈	—	14	2	no	—
262	C₈	—	14	2	no	—
263	C₈	—	28	1	no	—
264	C₈	—	28	1	no	—
265	C₈	#6470	1	2	yes	—
266	C₈	#6470	1	2	yes	—
267	C₇	#17972	1	1	yes	characteristic, complemented, Sylow
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	C₂×#6456	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	28	4	no	—
277	C₄	—	28	4	no	—
278	C₄	C₂×#6470	1	1	yes	characteristic
279	C₄	C₂×#6470	1	1	yes	characteristic
280	C₄	#90462	1	2	yes	—
281	C₂	—	4	1	no	—
282	C₂	#11736486	1	1	yes	characteristic
283	C₂	#11736614	1	1	yes	characteristic
284	C₂	#11736694	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₇
commutator	C₂×C₄×C₇	C₂³×C₄
Frattini	C₄²	C₂³×D₇
Fitting	C₇×#1568	C₂
socle	C₂²×C₇	C₂×C₄×D₄
prime radical	C₂³×C₇	C₄×D₄
cosocle	C₄²×C₇	C₂⁴
hypercenter	#1568	D₇
hypocenter	C₇	#17972
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1568

Complements

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

N	H ≅ G/N	type
6. #11726318	C₂	semidirect
7. #11726311	C₂	semidirect
8. #11726278	C₂	semidirect
9. #11726271	C₂	semidirect
267. C₇	#17972	semidirect

Maximal subgroups

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

subgroup	index
#11736761	2
#11735920	2
C₇×#1568	2
#11728557	2
#11727478	2
#11726318	2
#11726311	2
#11726278	2
#11726271	2
#17972	7