Subgroups — #13243259

Order 1792.

722 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#431	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊(C₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#1136	C₂	1	1	yes	maximal, characteristic
4	#11725509	C₂	1	4	yes	maximal, complemented
5	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
6	C₂²×#1136	C₄	1	2	yes	—
7	C₂³×C₇⋊C₈	—	2	1	no	—
8	C₇×#431	—	2	2	no	—
9	C₇×#431	C₄	1	4	yes	—
10	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
11	C₂×C₇×C₂²⋊C₈	—	2	1	no	—
12	C₂×C₄×C₇×C₈	—	2	1	no	—
13	C₂×C₄⋊(C₇⋊C₈)	—	2	2	no	—
14	C₂×C₄⋊(C₇⋊C₈)	—	2	1	no	—
15	C₂×C₄⋊(C₇⋊C₈)	C₄	1	4	yes	—
16	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
17	C₄×#1136	—	2	2	no	—
18	C₄×#1136	—	2	2	no	—
19	C₂×#6377	C₄	1	2	yes	—
20	C₂×C₄×C₇⋊C₈	C₂²	1	2	yes	—
21	#7873	—	7	1	no	maximal, Sylow
22	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₇	Q₈	1	1	yes	characteristic
25	C₂²×C₇×C₈	—	2	2	no	—
26	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₇	D₄	1	2	yes	—
30	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₇	D₄	1	2	yes	—
32	C₂×C₄²×C₇	Q₈	1	1	yes	characteristic
33	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
34	C₂×#1136	—	2	4	no	—
35	C₂×#1136	—	2	4	no	—
36	C₂×#1136	C₈	1	8	yes	complemented
37	C₂²×C₇⋊C₈	—	2	4	no	—
38	C₂²×C₇⋊C₈	—	2	2	no	—
39	C₂²×C₇⋊C₈	—	4	4	no	—
40	C₂²×C₇⋊C₈	C₂×C₄	1	4	yes	—
41	C₇×C₄⋊C₈	—	2	2	no	—
42	C₇×C₄⋊C₈	C₂×C₄	1	4	yes	—
43	C₇×C₂²⋊C₈	—	2	4	no	—
44	C₄×C₇×C₈	—	2	4	no	—
45	C₄⋊(C₇⋊C₈)	—	2	4	no	—
46	C₄⋊(C₇⋊C₈)	—	2	2	no	—
47	C₄⋊(C₇⋊C₈)	C₂×C₄	1	4	yes	—
48	#6377	—	2	4	no	—
49	C₄×C₇⋊C₈	—	2	2	no	—
50	C₄×C₇⋊C₈	—	2	2	no	—
51	C₂×#431	D₇	1	1	yes	characteristic
52	C₂²×C₄⋊C₈	—	7	1	no	—
53	C₂×C₄×M₁₆	—	7	1	no	—
54	#1277	—	7	4	no	—
55	C₂⁴×C₇	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₇	—	2	2	no	—
57	C₂²×C₄×C₇	—	2	2	no	—
58	C₂²×C₄×C₇	—	4	2	no	—
59	C₂²×C₄×C₇	C₄²	1	2	yes	—
60	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₇	C₂×C₈	1	4	yes	—
69	C₂²×C₄×C₇	M₁₆	1	4	yes	—
70	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
73	C₂×C₇×C₈	—	2	4	no	—
74	C₂×C₇×C₈	—	4	2	no	—
75	C₂×C₇×C₈	—	4	2	no	—
76	C₄²×C₇	C₄²	1	2	yes	—
77	C₄²×C₇	C₂²⋊C₄	1	2	yes	—
78	C₄²×C₇	C₂²⋊C₄	1	2	yes	—
79	C₄²×C₇	C₂²⋊C₄	1	2	yes	—
80	C₄²×C₇	C₄⋊C₄	1	2	yes	—
81	C₄²×C₇	C₄⋊C₄	1	2	yes	—
82	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
86	#1136	—	2	16	no	—
87	C₂×C₇⋊C₈	—	2	4	no	—
88	C₂×C₇⋊C₈	—	2	4	no	—
89	C₂×C₇⋊C₈	—	2	4	no	—
90	C₂×C₇⋊C₈	—	4	8	no	—
91	C₂×C₇⋊C₈	—	4	4	no	—
92	C₂×C₇⋊C₈	C₂×C₈	1	8	yes	complemented
93	C₂²×M₁₆	—	7	2	no	—
94	C₂³×C₈	—	14	1	no	—
95	C₂²×C₄²	C₂×D₇	1	1	yes	characteristic
96	#431	—	2	2	no	—
97	#431	Dic₇	1	4	yes	—
98	C₂×C₄⋊C₈	—	7	4	no	—
99	C₂×C₄⋊C₈	—	7	2	no	—
100	C₂×C₄⋊C₈	—	14	2	no	—
101	C₂×C₄⋊C₈	—	14	1	no	—
102	C₂×C₄⋊C₈	C₂×D₇	1	2	yes	—
103	C₂×C₂²⋊C₈	—	2	1	no	—
104	C₄×M₁₆	—	14	2	no	—
105	C₄×M₁₆	—	14	2	no	—
106	C₂×#97	—	7	2	no	—
107	C₂×C₄×C₈	—	2	1	no	—
108	C₂×C₄×C₈	—	7	2	no	—
109	C₂³×C₇	—	2	2	no	—
110	C₂³×C₇	—	2	2	no	—
111	C₂³×C₇	#95	1	2	yes	—
112	C₂³×C₇	C₄⋊C₈	1	4	yes	—
113	C₂³×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₇	—	2	2	no	—
115	C₂×C₄×C₇	—	2	2	no	—
116	C₂×C₄×C₇	—	2	2	no	—
117	C₂×C₄×C₇	—	2	4	no	—
118	C₂×C₄×C₇	—	2	2	no	—
119	C₂×C₄×C₇	—	2	2	no	—
120	C₂×C₄×C₇	—	4	2	no	—
121	C₂×C₄×C₇	—	4	2	no	—
122	C₂×C₄×C₇	#95	1	2	yes	—
123	C₂×C₄×C₇	#95	1	2	yes	—
124	C₂×C₄×C₇	#95	1	2	yes	—
125	C₂×C₄×C₇	C₄×C₈	1	4	yes	—
126	C₂×C₄×C₇	#97	1	4	yes	—
127	C₂×C₄×C₇	C₂²⋊C₈	1	8	yes	—
128	C₂×C₄×C₇	C₄⋊C₈	1	4	yes	—
129	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
130	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
131	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₇	C₂²×C₈	1	2	yes	—
136	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
137	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
138	C₂×C₄×C₇	#137	1	2	yes	—
139	C₇×C₈	—	4	4	no	—
140	C₇⋊C₈	—	2	8	no	—
141	C₇⋊C₈	—	4	4	no	—
142	C₂³×C₄	Dic₁₄	1	1	yes	characteristic
143	C₂³×C₄	C₄×D₇	1	1	yes	characteristic
144	C₂³×C₄	C₂²⋊D₇	1	1	yes	characteristic
145	C₂×M₁₆	—	7	8	no	—
146	C₂×M₁₆	—	14	4	no	—
147	C₂×M₁₆	—	14	4	no	—
148	C₂²×C₈	—	2	2	no	—
149	C₂²×C₈	—	7	4	no	—
150	C₂²×C₈	—	14	4	no	—
151	C₂²×C₈	—	14	2	no	—
152	C₂²×C₈	—	28	4	no	—
153	C₂×C₄²	Dic₁₄	1	1	yes	characteristic
154	C₂×C₄²	C₄×D₇	1	1	yes	characteristic
155	C₂×C₄²	C₄×D₇	1	2	yes	—
156	C₂×C₄²	D₂₈	1	2	yes	—
157	C₂×C₄²	C₂×Dic₇	1	2	yes	—
158	C₂×C₄²	C₂²⋊D₇	1	2	yes	—
159	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
160	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
161	C₄⋊C₈	—	2	2	no	—
162	C₄⋊C₈	—	7	4	no	—
163	C₄⋊C₈	—	14	4	no	—
164	C₄⋊C₈	—	14	2	no	—
165	C₄⋊C₈	C₂×Dic₇	1	4	yes	—
166	C₂²⋊C₈	—	2	4	no	—
167	#97	—	14	4	no	—
168	C₄×C₈	—	2	4	no	—
169	C₄×C₈	—	14	2	no	—
170	C₄×C₈	—	14	2	no	—
171	C₂²×C₇	—	2	8	no	—
172	C₂²×C₇	—	2	2	no	—
173	C₂²×C₇	—	2	2	no	—
174	C₂²×C₇	#336	1	4	yes	—
175	C₂²×C₇	C₂×#95	1	1	yes	characteristic
176	C₂²×C₇	C₂×C₄⋊C₈	1	2	yes	—
177	C₂²×C₇	#428	1	2	yes	—
178	C₂²×C₇	#430	1	2	yes	—
179	C₄×C₇	—	2	2	no	—
180	C₄×C₇	—	2	2	no	—
181	C₄×C₇	—	2	1	no	—
182	C₄×C₇	—	2	1	no	—
183	C₄×C₇	#336	1	4	yes	—
184	C₄×C₇	C₂×C₄×C₈	1	1	yes	characteristic
185	C₄×C₇	C₂×#97	1	1	yes	characteristic
186	C₄×C₇	C₂×C₂²⋊C₈	1	2	yes	—
187	C₄×C₇	#417	1	2	yes	—
188	C₄×C₇	#418	1	1	yes	characteristic
189	C₄×C₇	#419	1	1	yes	characteristic
190	C₂⁴	Dic₇⋊C₄	1	1	yes	characteristic
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	4	2	no	—
194	C₂²×C₄	C₈×D₇	1	4	yes	—
195	C₂²×C₄	#1131	1	4	yes	—
196	C₂²×C₄	C₄×Dic₇	1	2	yes	—
197	C₂²×C₄	Dic₇⋊C₄	1	1	yes	characteristic
198	C₂²×C₄	Dic₇⋊C₄	1	1	yes	characteristic
199	C₂²×C₄	Dic₇⋊C₄	1	2	yes	—
200	C₂²×C₄	Dic₇⋊C₄	1	1	yes	characteristic
201	C₂²×C₄	C₄⋊Dic₇	1	2	yes	—
202	C₂²×C₄	(C₂×D₇)⋊C₄	1	2	yes	—
203	C₂²×C₄	(C₂×D₇)⋊C₄	1	2	yes	—
204	C₂²×C₄	C₂²⋊Dic₇	1	2	yes	—
205	C₂²×C₄	C₂×Dic₁₄	1	1	yes	characteristic
206	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
207	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
208	M₁₆	—	14	16	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	4	2	no	—
211	C₂×C₈	—	4	2	no	—
212	C₂×C₈	—	7	8	no	—
213	C₂×C₈	—	14	4	no	—
214	C₂×C₈	—	14	4	no	—
215	C₂×C₈	—	14	4	no	—
216	C₂×C₈	—	28	8	no	—
217	C₂×C₈	—	28	4	no	—
218	C₄²	C₄×Dic₇	1	2	yes	—
219	C₄²	Dic₇⋊C₄	1	2	yes	—
220	C₄²	C₄⋊Dic₇	1	2	yes	—
221	C₄²	(C₂×D₇)⋊C₄	1	2	yes	—
222	C₄²	(C₂×D₇)⋊C₄	1	2	yes	—
223	C₄²	C₂²⋊Dic₇	1	2	yes	—
224	C₄²	C₂×C₄×D₇	1	1	yes	characteristic
225	C₄²	C₂×D₂₈	1	1	yes	characteristic
226	C₄²	C₂²×Dic₇	1	1	yes	characteristic
227	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
228	C₂×C₇	—	2	4	no	—
229	C₂×C₇	#1565	1	4	yes	—
230	C₂×C₇	C₂×#336	1	1	yes	characteristic
231	C₂×C₇	#1736	1	1	yes	characteristic
232	C₂×C₇	#1738	1	1	yes	characteristic
233	C₂³	—	2	2	no	—
234	C₂³	—	2	2	no	—
235	C₂³	Dic₇⋊C₈	1	4	yes	—
236	C₂³	C₇⋊#95	1	2	yes	—
237	C₂³	C₂×Dic₇⋊C₄	1	1	yes	characteristic
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	C₈×Dic₇	1	4	yes	—
247	C₂×C₄	Dic₇⋊C₈	1	4	yes	—
248	C₂×C₄	#6389	1	4	yes	—
249	C₂×C₄	(C₂×D₇)⋊C₈	1	8	yes	—
250	C₂×C₄	C₇⋊#95	1	2	yes	—
251	C₂×C₄	C₇⋊#95	1	2	yes	—
252	C₂×C₄	C₇⋊#95	1	2	yes	—
253	C₂×C₄	C₂×C₈×D₇	1	2	yes	—
254	C₂×C₄	C₂×#1131	1	2	yes	—
255	C₂×C₄	C₂×C₄×Dic₇	1	1	yes	characteristic
256	C₂×C₄	C₂×Dic₇⋊C₄	1	1	yes	characteristic
257	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
258	C₂×C₄	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
259	C₂×C₄	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
260	C₂×C₄	#6512	1	2	yes	—
261	C₂×C₄	#6514	1	2	yes	—
262	C₂×C₄	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
263	C₈	—	4	4	no	—
264	C₈	—	14	8	no	—
265	C₈	—	28	4	no	—
266	C₇	#7873	1	1	yes	characteristic, complemented, Sylow
267	C₂²	—	2	2	no	—
268	C₂²	—	2	8	no	—
269	C₂²	—	2	2	no	—
270	C₂²	#89345	1	4	yes	—
271	C₂²	#89767	1	2	yes	—
272	C₂²	C₂×Dic₇⋊C₈	1	2	yes	—
273	C₂²	#89896	1	2	yes	—
274	C₂²	C₂×C₇⋊#95	1	1	yes	characteristic
275	C₄	—	2	2	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	2	2	no	—
279	C₄	#89345	1	4	yes	—
280	C₄	#89773	1	1	yes	characteristic
281	C₄	#89776	1	1	yes	characteristic
282	C₄	C₂×C₈×Dic₇	1	1	yes	characteristic
283	C₄	C₂×#6389	1	1	yes	characteristic
284	C₄	C₂×(C₂×D₇)⋊C₈	1	2	yes	—
285	C₄	#90004	1	2	yes	—
286	C₂	—	2	4	no	—
287	C₂	#11728500	1	4	yes	—
288	C₂	#11729335	1	1	yes	characteristic
289	C₂	C₂×#89345	1	1	yes	characteristic
290	C₂	#11729748	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₇
commutator	C₄×C₇	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₂×C₇×#431	C₂
socle	C₂³×C₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₇	C₄⋊C₄
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	C₂×#431	D₇
hypocenter	C₇	#7873
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
4. #11725509	C₂	semidirect
36. C₂×#1136	C₈	semidirect
92. C₂×C₇⋊C₈	C₂×C₈	semidirect
266. C₇	#7873	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×#431	2
C₂²×C₄⋊(C₇⋊C₈)	2
C₂×C₄×#1136	2
#11725509	2
#7873	7