Subgroups — #13253933

Order 1792.

430 conjugacy classes of subgroups.

The subgroup lattice has 272 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×#1831	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	C₄	1	2	yes	—
7	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
8	C₄³×C₇	C₂²	1	1	yes	characteristic
9	#89708	C₄	1	2	yes	—
10	#89707	C₂²	1	1	yes	characteristic
11	#89707	C₂²	1	1	yes	characteristic
12	#89700	—	2	1	no	—
13	#89700	—	2	1	no	—
14	#89700	—	2	1	no	—
15	#89700	—	2	1	no	—
16	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
17	C₂×C₄×C₇⋊C₈	—	2	1	no	—
18	C₇×#336	—	2	1	no	—
19	C₇×#336	—	2	1	no	—
20	#89319	—	2	1	no	—
21	#9901	—	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	4	no	—
40	C₇×C₄⋊C₈	—	2	4	no	—
41	C₇⋊#95	—	2	2	no	—
42	C₇⋊#95	—	2	2	no	—
43	C₄⋊(C₇⋊C₈)	—	2	2	no	—
44	C₄⋊(C₇⋊C₈)	Q₈	1	4	yes	—
45	C₄×C₇⋊C₈	—	4	2	no	—
46	#2859	—	7	1	no	—
47	#1831	D₇	1	1	yes	characteristic
48	C₄×C₄⋊C₈	—	7	1	no	—
49	#1296	—	7	2	no	—
50	#1296	—	7	2	no	—
51	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×Dic₇	—	2	2	no	—
59	C₂²×Dic₇	—	2	2	no	—
60	C₂×C₇×C₈	—	4	2	no	—
61	C₂×C₇×C₈	—	4	2	no	—
62	C₂×C₇×C₈	—	4	2	no	—
63	C₄²×C₇	—	2	2	no	—
64	C₄²×C₇	—	2	2	no	—
65	C₄²×C₇	—	2	2	no	—
66	C₄²×C₇	—	2	2	no	—
67	C₄²×C₇	—	2	2	no	—
68	C₄²×C₇	—	2	1	no	—
69	C₄²×C₇	—	2	1	no	—
70	C₄²×C₇	C₂×Q₈	1	2	yes	—
71	C₄²×C₇	#41	1	2	yes	—
72	C₄⋊Dic₇	—	4	2	no	—
73	C₄⋊Dic₇	—	4	2	no	—
74	C₄⋊Dic₇	—	4	2	no	—
75	C₄⋊Dic₇	—	4	1	no	—
76	C₄⋊Dic₇	—	4	1	no	—
77	C₄⋊Dic₇	—	4	2	no	—
78	C₄⋊Dic₇	—	4	2	no	—
79	Dic₇⋊C₄	—	4	4	no	—
80	Dic₇⋊C₄	—	4	4	no	—
81	C₂×C₇⋊C₈	—	2	4	no	—
82	C₂×C₇⋊C₈	—	4	2	no	—
83	C₂×C₇⋊C₈	—	4	2	no	—
84	C₂×#125	—	7	2	no	—
85	C₂×C₄⋊C₈	—	7	2	no	—
86	C₂×C₄⋊C₈	C₂×D₇	1	2	yes	—
87	C₂×C₄×C₈	—	14	1	no	—
88	#389	—	14	1	no	—
89	#389	—	14	1	no	—
90	#389	—	14	1	no	—
91	#389	—	14	1	no	—
92	#384	—	7	1	no	—
93	#384	—	7	1	no	—
94	#382	—	7	2	no	—
95	C₄³	C₂×D₇	1	1	yes	characteristic
96	#336	—	2	1	no	—
97	#336	—	2	1	no	—
98	#336	—	14	1	no	—
99	C₂³×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	—	2	1	no	—
104	C₂×C₄×C₇	—	2	1	no	—
105	C₂×C₄×C₇	—	2	1	no	—
106	C₂×C₄×C₇	—	2	1	no	—
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	#100	1	2	yes	—
113	C₂×C₄×C₇	#101	1	2	yes	—
114	C₂×C₄×C₇	C₄×Q₈	1	2	yes	—
115	C₂×C₄×C₇	C₂²⋊Q₈	1	2	yes	—
116	C₂×C₄×C₇	#124	1	1	yes	characteristic
117	C₂×C₄×C₇	C₄⋊Q₈	1	1	yes	characteristic
118	C₂×C₄×C₇	#135	1	2	yes	—
119	C₂×C₄×C₇	#135	1	2	yes	—
120	C₂×C₄×C₇	#135	1	2	yes	—
121	C₂×C₄×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
122	C₂×C₄×C₇	#137	1	1	yes	characteristic
123	C₂×Dic₇	—	4	2	no	—
124	C₂×Dic₇	—	4	2	no	—
125	C₂×Dic₇	—	4	2	no	—
126	C₂×Dic₇	—	4	2	no	—
127	C₂×Dic₇	—	4	2	no	—
128	C₂×Dic₇	—	4	2	no	—
129	C₇×C₈	—	8	2	no	—
130	C₇⋊C₈	—	4	4	no	—
131	C₂²×C₈	—	2	2	no	—
132	C₂²×C₈	—	14	2	no	—
133	#125	—	28	4	no	—
134	C₂×C₄⋊C₄	—	7	2	no	—
135	C₂×C₄⋊C₄	—	7	2	no	—
136	C₂×C₄⋊C₄	—	14	2	no	—
137	C₂×C₄⋊C₄	—	14	2	no	—
138	C₂×C₄⋊C₄	—	14	1	no	—
139	C₂×C₄⋊C₄	—	14	1	no	—
140	C₂×C₄⋊C₄	—	14	1	no	—
141	C₂×C₄⋊C₄	—	14	1	no	—
142	C₂×C₄²	C₄×D₇	1	1	yes	characteristic
143	C₂×C₄²	C₄×D₇	1	1	yes	characteristic
144	C₂×C₄²	D₂₈	1	1	yes	characteristic
145	C₂×C₄²	D₂₈	1	1	yes	characteristic
146	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
147	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
148	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
149	C₄⋊C₈	—	2	4	no	—
150	C₄⋊C₈	—	7	4	no	—
151	C₄⋊C₈	—	14	2	no	—
152	C₄×C₈	—	28	2	no	—
153	#95	—	14	2	no	—
154	#95	—	14	2	no	—
155	C₂²×C₇	#391	1	1	yes	characteristic
156	C₂²×C₇	C₂×#100	1	1	yes	characteristic
157	C₂²×C₇	C₂×#101	1	1	yes	characteristic
158	C₂²×C₇	#416	1	1	yes	characteristic
159	C₂²×C₇	#416	1	1	yes	characteristic
160	C₂²×C₇	#416	1	1	yes	characteristic
161	C₂²×C₇	#417	1	1	yes	characteristic
162	C₄×C₇	—	2	2	no	—
163	C₄×C₇	—	2	2	no	—
164	C₄×C₇	—	2	2	no	—
165	C₄×C₇	—	4	1	no	—
166	C₄×C₇	—	4	1	no	—
167	C₄×C₇	#478	1	2	yes	—
168	C₄×C₇	#479	1	2	yes	—
169	C₄×C₇	#488	1	2	yes	—
170	C₄×C₇	#499	1	2	yes	—
171	Dic₇	—	8	2	no	—
172	Dic₇	—	8	2	no	—
173	C₂²×C₄	—	14	2	no	—
174	C₂²×C₄	—	14	2	no	—
175	C₂²×C₄	(C₂×D₇)⋊C₄	1	1	yes	characteristic
176	C₂²×C₄	(C₂×D₇)⋊C₄	1	1	yes	characteristic
177	C₂²×C₄	(C₂×D₇)⋊C₄	1	1	yes	characteristic
178	C₂²×C₄	(C₂×D₇)⋊C₄	1	1	yes	characteristic
179	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
180	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
181	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
182	C₂×C₈	—	4	2	no	—
183	C₂×C₈	—	4	2	no	—
184	C₂×C₈	—	4	2	no	—
185	C₂×C₈	—	14	4	no	—
186	C₂×C₈	—	28	2	no	—
187	C₂×C₈	—	28	2	no	—
188	C₄⋊C₄	—	28	2	no	—
189	C₄⋊C₄	—	28	2	no	—
190	C₄⋊C₄	—	28	4	no	—
191	C₄⋊C₄	—	28	2	no	—
192	C₄⋊C₄	—	28	1	no	—
193	C₄⋊C₄	—	28	1	no	—
194	C₄⋊C₄	—	28	2	no	—
195	C₄⋊C₄	—	28	2	no	—
196	C₄⋊C₄	—	28	4	no	—
197	C₄²	—	2	2	no	—
198	C₄²	—	2	1	no	—
199	C₄²	—	2	1	no	—
200	C₄²	—	2	2	no	—
201	C₄²	—	2	2	no	—
202	C₄²	—	2	2	no	—
203	C₄²	—	2	2	no	—
204	C₄²	#1159	1	2	yes	—
205	C₄²	Q₈×D₇	1	2	yes	—
206	C₂×C₇	#1543	1	1	yes	characteristic
207	C₂×C₇	#1544	1	1	yes	characteristic
208	C₂×C₇	#1551	1	1	yes	characteristic
209	C₂×C₇	#1552	1	1	yes	characteristic
210	C₂×C₇	#1989	1	1	yes	characteristic
211	C₂×C₇	#1991	1	1	yes	characteristic
212	C₂×C₇	#1997	1	1	yes	characteristic
213	C₂³	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	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	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	28	2	no	—
227	C₂×C₄	—	28	2	no	—
228	C₂×C₄	—	28	2	no	—
229	C₂×C₄	—	28	2	no	—
230	C₂×C₄	—	28	2	no	—
231	C₂×C₄	—	28	2	no	—
232	C₂×C₄	#6397	1	2	yes	—
233	C₂×C₄	#6398	1	2	yes	—
234	C₂×C₄	#6450	1	2	yes	—
235	C₂×C₄	#6460	1	2	yes	—
236	C₂×C₄	#6467	1	2	yes	—
237	C₂×C₄	#6471	1	1	yes	characteristic
238	C₂×C₄	#6472	1	1	yes	characteristic
239	C₂×C₄	#6499	1	1	yes	characteristic
240	C₂×C₄	#6507	1	1	yes	characteristic
241	C₂×C₄	#6513	1	2	yes	—
242	C₂×C₄	#6513	1	2	yes	—
243	C₈	—	8	2	no	—
244	C₈	—	28	4	no	—
245	C₇	#9901	1	1	yes	characteristic, complemented, Sylow
246	C₂²	#89746	1	1	yes	characteristic
247	C₂²	#89774	1	1	yes	characteristic
248	C₂²	#89774	1	1	yes	characteristic
249	C₂²	#89885	1	1	yes	characteristic
250	C₂²	C₂×#6397	1	1	yes	characteristic
251	C₂²	#89905	1	1	yes	characteristic
252	C₂²	C₂×#6398	1	1	yes	characteristic
253	C₄	—	2	2	no	—
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	4	1	no	—
257	C₄	—	4	1	no	—
258	C₄	—	56	2	no	—
259	C₄	—	56	2	no	—
260	C₄	#89601	1	2	yes	—
261	C₄	#89619	1	2	yes	—
262	C₄	#89828	1	2	yes	—
263	C₄	#89836	1	2	yes	—
264	C₂	#11727308	1	1	yes	characteristic
265	C₂	#11728215	1	1	yes	characteristic
266	C₂	#11728278	1	1	yes	characteristic
267	C₂	#11728527	1	1	yes	characteristic
268	C₂	#11728531	1	1	yes	characteristic
269	C₂	#11728570	1	1	yes	characteristic
270	C₂	#11728574	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×#1831	C₂
socle, prime radical	C₂³×C₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	#1831	D₇
hypocenter	C₇	#9901
LCS	G ▷ C₂²×C₄×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

Complements

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

N	H ≅ G/N	type
245. C₇	#9901	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×#1831	2
C₄×C₄⋊(C₇⋊C₈)	2
#11725508	2
#9901	7