Subgroups — #13258512

Order 1792.

366 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#1845	C₂	1	1	yes	maximal, characteristic
2	#11727542	C₂	1	1	yes	maximal, characteristic
3	#11727342	C₂	1	2	yes	maximal
4	#11725460	C₂	1	1	yes	maximal, characteristic
5	#11725440	C₂	1	2	yes	maximal
6	C₂×C₄×Dic₁₄	C₂²	1	1	yes	characteristic
7	C₂×C₇×#107	—	2	1	no	—
8	C₂×C₇×#107	—	2	1	no	—
9	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
10	C₇×#384	C₂²	1	2	yes	—
11	C₂×#6384	—	2	2	no	—
12	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
14	#89345	—	2	1	no	—
15	#89345	—	2	1	no	—
16	#89324	—	2	2	no	—
17	#10618	—	7	1	no	maximal, Sylow
18	C₂²×Dic₁₄	D₄	1	1	yes	characteristic
19	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
20	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
21	C₂×C₇×C₄⋊C₄	—	2	2	no	—
22	C₂×C₇×C₄⋊C₄	—	2	2	no	—
23	C₂×C₇×C₄⋊C₄	—	2	2	no	—
24	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
25	C₂×C₄⋊Dic₇	D₄	1	1	yes	characteristic
26	C₂×Dic₇⋊C₄	—	2	1	no	—
27	C₂×C₄×Dic₇	—	2	1	no	—
28	C₂²×C₇⋊C₈	—	2	1	no	—
29	C₂²×C₇⋊C₈	—	2	2	no	—
30	C₄×Dic₁₄	—	2	2	no	—
31	C₄×Dic₁₄	—	2	2	no	—
32	C₇×#107	—	4	2	no	—
33	C₇×#107	—	4	2	no	—
34	C₄×C₇×C₈	—	2	2	no	—
35	#6384	—	4	4	no	—
36	C₄⋊(C₇⋊C₈)	—	2	2	no	—
37	C₄⋊(C₇⋊C₈)	—	2	2	no	—
38	C₄⋊(C₇⋊C₈)	—	2	2	no	—
39	#1984	—	7	1	no	—
40	#1845	D₇	1	1	yes	characteristic
41	#1803	—	7	2	no	—
42	#1296	—	7	2	no	—
43	#1277	—	7	1	no	—
44	C₂²×C₄×C₇	—	4	2	no	—
45	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₇	#41	1	1	yes	characteristic
48	C₂²×Dic₇	—	2	1	no	—
49	C₂²×Dic₇	—	2	1	no	—
50	C₂×Dic₁₄	—	2	2	no	—
51	C₂×Dic₁₄	—	2	2	no	—
52	C₂×Dic₁₄	—	2	1	no	—
53	C₂×Dic₁₄	Q₁₆	1	2	yes	—
54	C₂×C₇×C₈	—	2	1	no	—
55	C₂×C₇×C₈	—	2	1	no	—
56	C₂×C₇×C₈	—	2	1	no	—
57	C₂×C₇×C₈	—	2	1	no	—
58	C₂×C₇×C₈	SD₁₆	1	2	yes	—
59	C₂×C₇×C₈	Q₁₆	1	2	yes	—
60	C₇×C₄⋊C₄	—	4	2	no	—
61	C₇×C₄⋊C₄	—	4	2	no	—
62	C₇×C₄⋊C₄	—	4	2	no	—
63	C₇×C₄⋊C₄	—	4	2	no	—
64	C₇×C₄⋊C₄	—	8	2	no	—
65	C₄²×C₇	#41	1	2	yes	—
66	C₄²×C₇	#41	1	2	yes	—
67	C₄⋊Dic₇	—	2	2	no	—
68	Dic₇⋊C₄	—	4	2	no	—
69	C₄×Dic₇	—	4	1	no	—
70	C₄×Dic₇	—	4	1	no	—
71	C₂×C₇⋊C₈	—	4	2	no	—
72	C₂×C₇⋊C₈	—	4	1	no	—
73	C₂×C₇⋊C₈	—	4	2	no	—
74	C₂×C₇⋊C₈	—	4	1	no	—
75	C₂×C₇⋊C₈	—	4	2	no	—
76	C₂×C₇⋊C₈	—	4	1	no	—
77	C₂×C₄×Q₈	—	7	1	no	—
78	C₂×#107	—	2	1	no	—
79	C₂×#107	—	2	1	no	—
80	C₂×C₄⋊C₈	—	7	2	no	—
81	C₂×C₄⋊C₈	—	7	1	no	—
82	C₂×#103	—	14	2	no	—
83	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
84	#384	C₂×D₇	1	2	yes	—
85	#340	—	14	2	no	—
86	#336	—	14	1	no	—
87	#336	—	14	1	no	—
88	C₂³×C₇	#121	1	1	yes	characteristic
89	C₂×C₄×C₇	—	2	1	no	—
90	C₂×C₄×C₇	—	2	1	no	—
91	C₂×C₄×C₇	—	8	2	no	—
92	C₂×C₄×C₇	—	8	2	no	—
93	C₂×C₄×C₇	—	8	2	no	—
94	C₂×C₄×C₇	#123	1	2	yes	—
95	C₂×C₄×C₇	#123	1	1	yes	characteristic
96	C₂×C₄×C₇	#124	1	1	yes	characteristic
97	C₂×C₄×C₇	#126	1	2	yes	—
98	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
99	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
100	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
101	C₂×C₄×C₇	#135	1	1	yes	characteristic
102	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
103	C₂×C₄×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
104	C₂×C₄×C₇	#137	1	1	yes	characteristic
105	C₂×Dic₇	—	2	2	no	—
106	C₂×Dic₇	—	4	1	no	—
107	C₂×Dic₇	—	4	1	no	—
108	C₂×Dic₇	—	4	1	no	—
109	C₂×Dic₇	—	4	1	no	—
110	C₂×Dic₇	—	4	1	no	—
111	Dic₁₄	—	2	4	no	—
112	Dic₁₄	—	4	2	no	—
113	C₇×C₈	—	2	2	no	—
114	C₇×C₈	—	2	2	no	—
115	C₇⋊C₈	—	8	2	no	—
116	C₇⋊C₈	—	8	1	no	—
117	C₂²×Q₈	—	7	1	no	—
118	C₂²×C₈	—	14	2	no	—
119	C₂²×C₈	—	14	1	no	—
120	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
121	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
122	C₄×Q₈	—	14	2	no	—
123	C₄×Q₈	—	14	2	no	—
124	C₂×C₄⋊C₄	—	2	2	no	—
125	C₂×C₄⋊C₄	—	2	2	no	—
126	C₂×C₄⋊C₄	—	2	2	no	—
127	C₂×C₄⋊C₄	—	7	1	no	—
128	C₂×C₄⋊C₄	—	14	1	no	—
129	C₂×C₄²	—	14	1	no	—
130	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
131	#107	—	4	2	no	—
132	#107	—	4	2	no	—
133	C₄⋊C₈	—	14	2	no	—
134	C₄⋊C₈	—	14	2	no	—
135	C₄⋊C₈	—	14	2	no	—
136	#103	—	28	4	no	—
137	C₄×C₈	—	2	2	no	—
138	C₂²×C₇	#397	1	1	yes	characteristic
139	C₂²×C₇	#460	1	1	yes	characteristic
140	C₂²×C₇	#462	1	1	yes	characteristic
141	C₂²×C₇	#467	1	1	yes	characteristic
142	C₂²×C₇	#469	1	1	yes	characteristic
143	C₂²×C₇	#472	1	1	yes	characteristic
144	C₂²×C₇	#473	1	1	yes	characteristic
145	C₄×C₇	—	2	2	no	—
146	C₄×C₇	—	16	2	no	—
147	C₄×C₇	#482	1	2	yes	—
148	C₄×C₇	#483	1	1	yes	characteristic
149	C₄×C₇	#484	1	2	yes	—
150	C₄×C₇	#484	1	1	yes	characteristic
151	C₄×C₇	#487	1	1	yes	characteristic
152	C₄×C₇	#489	1	1	yes	characteristic
153	Dic₇	—	4	2	no	—
154	Dic₇	—	8	1	no	—
155	C₂×Q₈	—	7	2	no	—
156	C₂×Q₈	—	14	1	no	—
157	C₂×Q₈	—	14	2	no	—
158	C₂×Q₈	—	14	2	no	—
159	C₂²×C₄	—	4	2	no	—
160	C₂²×C₄	—	14	1	no	—
161	C₂²×C₄	—	14	1	no	—
162	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
163	C₂²×C₄	#1159	1	1	yes	characteristic
164	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	2	1	no	—
168	C₂×C₈	—	2	1	no	—
169	C₂×C₈	—	28	2	no	—
170	C₂×C₈	—	28	2	no	—
171	C₂×C₈	—	28	2	no	—
172	C₂×C₈	—	28	1	no	—
173	C₂×C₈	—	28	1	no	—
174	C₂×C₈	—	28	1	no	—
175	C₂×C₈	#1142	1	2	yes	—
176	C₂×C₈	#1144	1	2	yes	—
177	C₄⋊C₄	—	4	2	no	—
178	C₄⋊C₄	—	4	2	no	—
179	C₄⋊C₄	—	4	2	no	—
180	C₄⋊C₄	—	4	2	no	—
181	C₄⋊C₄	—	8	2	no	—
182	C₄⋊C₄	—	14	2	no	—
183	C₄⋊C₄	—	28	2	no	—
184	C₄²	—	28	1	no	—
185	C₄²	—	28	1	no	—
186	C₄²	#1157	1	2	yes	—
187	C₄²	#1161	1	2	yes	—
188	C₂×C₇	#1670	1	1	yes	characteristic
189	C₂×C₇	#1676	1	1	yes	characteristic
190	C₂×C₇	#1688	1	1	yes	characteristic
191	C₂×C₇	#1694	1	1	yes	characteristic
192	C₂×C₇	#2072	1	1	yes	characteristic
193	C₂×C₇	#2076	1	1	yes	characteristic
194	C₂×C₇	#2078	1	1	yes	characteristic
195	C₂³	#6501	1	1	yes	characteristic
196	Q₈	—	14	4	no	—
197	Q₈	—	28	2	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	8	2	no	—
201	C₂×C₄	—	8	2	no	—
202	C₂×C₄	—	8	2	no	—
203	C₂×C₄	—	14	2	no	—
204	C₂×C₄	—	28	1	no	—
205	C₂×C₄	—	28	1	no	—
206	C₂×C₄	—	28	1	no	—
207	C₂×C₄	—	28	1	no	—
208	C₂×C₄	—	28	1	no	—
209	C₂×C₄	#6457	1	2	yes	—
210	C₂×C₄	#6461	1	2	yes	—
211	C₂×C₄	#6474	1	2	yes	—
212	C₂×C₄	D₇×Q₁₆	1	1	yes	characteristic
213	C₂×C₄	#6482	1	1	yes	characteristic
214	C₂×C₄	#6492	1	1	yes	characteristic
215	C₂×C₄	#6495	1	1	yes	characteristic
216	C₂×C₄	C₂×#1142	1	1	yes	characteristic
217	C₂×C₄	#6505	1	1	yes	characteristic
218	C₂×C₄	C₂×#1144	1	1	yes	characteristic
219	C₂×C₄	#6510	1	1	yes	characteristic
220	C₈	—	2	2	no	—
221	C₈	—	2	2	no	—
222	C₈	—	56	1	no	—
223	C₈	—	56	2	no	—
224	C₇	#10618	1	1	yes	characteristic, complemented, Sylow
225	C₂²	#89765	1	1	yes	characteristic
226	C₂²	#89847	1	1	yes	characteristic
227	C₂²	#89862	1	1	yes	characteristic
228	C₂²	#89931	1	1	yes	characteristic
229	C₂²	#89932	1	1	yes	characteristic
230	C₂²	#89960	1	1	yes	characteristic
231	C₂²	#89964	1	1	yes	characteristic
232	C₄	—	2	2	no	—
233	C₄	—	16	2	no	—
234	C₄	—	28	2	no	—
235	C₄	—	56	1	no	—
236	C₄	#89656	1	2	yes	—
237	C₄	#89657	1	2	yes	—
238	C₄	#89742	1	1	yes	characteristic
239	C₄	#89743	1	1	yes	characteristic
240	C₄	#89861	1	1	yes	characteristic
241	C₄	#89865	1	1	yes	characteristic
242	C₂	#11727343	1	1	yes	characteristic
243	C₂	#11728391	1	1	yes	characteristic
244	C₂	#11728401	1	1	yes	characteristic
245	C₂	#11729041	1	1	yes	characteristic
246	C₂	#11729062	1	1	yes	characteristic
247	C₂	#11729132	1	1	yes	characteristic
248	C₂	#11729133	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#6501
commutator, cosocle	C₂×C₄²×C₇	C₂³
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₇×#1845	C₂
socle, prime radical	C₂³×C₇	#121
hypercenter	#1845	D₇
hypocenter	C₇	#10618
LCS	G ▷ C₂×C₄²×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

Complements

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

N	H ≅ G/N	type
224. C₇	#10618	semidirect

Maximal subgroups

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

subgroup	index
C₇×#1845	2
#11727542	2
#11727342	2
#11725460	2
#11725440	2
#10618	7