Subgroups — #13246763

Order 1792.

342 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#446	C₂	1	1	yes	maximal, characteristic
2	#11729379	C₂	1	2	yes	maximal
3	#11727486	C₂	1	1	yes	maximal, characteristic
4	#11727486	C₂	1	1	yes	maximal, characteristic
5	#11725514	C₂	1	2	yes	maximal
6	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
7	C₇×#446	—	2	1	no	—
8	C₇×#446	—	2	1	no	—
9	C₇×#446	—	2	1	no	—
10	C₇×#446	—	2	1	no	—
11	C₂×C₇×C₄⋊C₈	C₄	1	2	yes	—
12	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₇×#97	C₄	1	2	yes	—
14	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
15	#89698	C₂²	1	2	yes	—
16	#89698	C₂²	1	2	yes	—
17	#89323	—	2	4	no	—
18	#89253	—	2	1	no	—
19	#89253	—	2	1	no	—
20	#89252	—	2	1	no	—
21	#89252	—	2	1	no	—
22	#8625	—	7	1	no	maximal, Sylow
23	C₂²×C₇×Q₈	Q₈	1	1	yes	characteristic
24	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
26	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
27	C₄×C₇×Q₈	—	2	1	no	—
28	C₄×C₇×Q₈	—	2	1	no	—
29	C₄×C₇×Q₈	—	2	1	no	—
30	C₄×C₇×Q₈	—	2	1	no	—
31	C₂×C₇×C₄⋊C₄	D₄	1	2	yes	—
32	C₂×C₇×C₄⋊C₄	Q₈	1	1	yes	characteristic
33	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
35	C₇×C₄⋊C₈	—	2	2	no	—
36	C₇×C₄⋊C₈	—	2	2	no	—
37	C₇×C₄⋊C₈	—	2	1	no	—
38	C₇×C₄⋊C₈	—	2	1	no	—
39	C₇×#97	—	2	2	no	—
40	C₇×#97	—	2	2	no	—
41	C₄×C₇×C₈	D₄	1	1	yes	characteristic
42	C₄×C₇×C₈	D₄	1	1	yes	characteristic
43	C₄×C₇×C₈	D₄	1	1	yes	characteristic
44	C₄×C₇×C₈	D₄	1	1	yes	characteristic
45	C₂²⋊(C₇⋊C₈)	—	2	2	no	—
46	C₂²⋊(C₇⋊C₈)	—	2	2	no	—
47	C₄⋊(C₇⋊C₈)	—	2	2	no	—
48	C₄⋊(C₇⋊C₈)	—	2	2	no	—
49	#6377	—	2	2	no	—
50	#6377	—	2	2	no	—
51	C₂×#446	D₇	1	1	yes	characteristic
52	#1568	—	7	1	no	—
53	#1568	—	7	1	no	—
54	#1467	—	7	2	no	—
55	#1290	—	7	2	no	—
56	C₂×C₇×Q₈	—	2	1	no	—
57	C₂×C₇×Q₈	—	2	1	no	—
58	C₂×C₇×Q₈	—	4	2	no	—
59	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
60	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
61	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₇×C₈	—	2	2	no	—
65	C₂×C₇×C₈	—	2	2	no	—
66	C₂×C₇×C₈	—	4	1	no	—
67	C₂×C₇×C₈	—	4	2	no	—
68	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
72	C₇×C₄⋊C₄	—	2	2	no	—
73	C₇×C₄⋊C₄	—	2	2	no	—
74	C₇×C₄⋊C₄	—	2	1	no	—
75	C₇×C₄⋊C₄	—	2	1	no	—
76	C₄²×C₇	—	2	2	no	—
77	C₄²×C₇	—	2	2	no	—
78	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₇	#41	1	1	yes	characteristic
81	C₄²×C₇	#41	1	1	yes	characteristic
82	C₂×C₇⋊C₈	—	4	2	no	—
83	C₂×C₇⋊C₈	—	4	2	no	—
84	C₂×C₄×Q₈	C₂×D₇	1	1	yes	characteristic
85	#446	—	2	1	no	—
86	#446	—	2	1	no	—
87	#446	—	2	1	no	—
88	#446	—	2	1	no	—
89	#432	—	7	2	no	—
90	#432	—	7	2	no	—
91	C₂×C₄⋊C₈	Dic₇	1	2	yes	—
92	C₂×C₄⋊C₈	C₂×D₇	1	1	yes	characteristic
93	C₂×#97	Dic₇	1	2	yes	—
94	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
95	#335	—	14	1	no	—
96	#335	—	14	1	no	—
97	#334	—	14	1	no	—
98	#334	—	14	1	no	—
99	#324	—	14	4	no	—
100	C₂³×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₇×Q₈	—	4	2	no	—
102	C₇×Q₈	—	4	2	no	—
103	C₂×C₄×C₇	—	2	2	no	—
104	C₂×C₄×C₇	—	2	1	no	—
105	C₂×C₄×C₇	—	2	1	no	—
106	C₂×C₄×C₇	—	2	2	no	—
107	C₂×C₄×C₇	—	2	2	no	—
108	C₂×C₄×C₇	—	2	4	no	—
109	C₂×C₄×C₇	—	4	1	no	—
110	C₂×C₄×C₇	—	4	2	no	—
111	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₇	#117	1	1	yes	characteristic
113	C₂×C₄×C₇	#121	1	1	yes	characteristic
114	C₂×C₄×C₇	#121	1	1	yes	characteristic
115	C₂×C₄×C₇	C₂²⋊Q₈	1	1	yes	characteristic
116	C₂×C₄×C₇	C₂²⋊Q₈	1	1	yes	characteristic
117	C₇×C₈	—	2	1	no	—
118	C₇×C₈	—	2	1	no	—
119	C₇×C₈	—	2	1	no	—
120	C₇×C₈	—	4	2	no	—
121	C₇×C₈	#101	1	2	yes	—
122	C₇⋊C₈	—	8	2	no	—
123	C₇⋊C₈	—	8	2	no	—
124	C₂²×Q₈	Dic₁₄	1	1	yes	characteristic
125	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
126	C₂²×C₈	C₂×Dic₇	1	2	yes	—
127	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
128	C₄×Q₈	—	2	1	no	—
129	C₄×Q₈	—	2	1	no	—
130	C₄×Q₈	—	2	1	no	—
131	C₄×Q₈	—	2	1	no	—
132	C₂×C₄⋊C₄	Dic₁₄	1	1	yes	characteristic
133	C₂×C₄⋊C₄	D₂₈	1	2	yes	—
134	C₂×C₄²	C₂×Dic₇	1	2	yes	—
135	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
136	C₄⋊C₈	—	2	2	no	—
137	C₄⋊C₈	—	2	2	no	—
138	C₄⋊C₈	—	2	1	no	—
139	C₄⋊C₈	—	2	1	no	—
140	C₄⋊C₈	—	14	2	no	—
141	C₄⋊C₈	—	14	2	no	—
142	C₂²⋊C₈	—	14	2	no	—
143	C₂²⋊C₈	—	14	2	no	—
144	#97	—	2	2	no	—
145	#97	—	2	2	no	—
146	#97	—	14	2	no	—
147	#97	—	14	2	no	—
148	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
149	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
150	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
151	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
152	C₂²×C₇	—	2	1	no	—
153	C₂²×C₇	—	2	1	no	—
154	C₂²×C₇	#380	1	1	yes	characteristic
155	C₂²×C₇	#430	1	2	yes	—
156	C₄×C₇	—	2	1	no	—
157	C₄×C₇	—	2	1	no	—
158	C₄×C₇	—	2	1	no	—
159	C₄×C₇	—	2	1	no	—
160	C₄×C₇	—	2	1	no	—
161	C₄×C₇	—	4	2	no	—
162	C₄×C₇	—	4	2	no	—
163	C₄×C₇	C₂×#101	1	1	yes	characteristic
164	C₄×C₇	#413	1	1	yes	characteristic
165	C₂×Q₈	—	2	1	no	—
166	C₂×Q₈	—	2	1	no	—
167	C₂×Q₈	—	4	2	no	—
168	C₂²×C₄	C₄⋊Dic₇	1	2	yes	—
169	C₂²×C₄	C₄⋊Dic₇	1	2	yes	—
170	C₂²×C₄	C₂×Dic₁₄	1	1	yes	characteristic
171	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
172	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	—	4	1	no	—
176	C₂×C₈	—	4	2	no	—
177	C₂×C₈	—	28	2	no	—
178	C₂×C₈	—	28	2	no	—
179	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
180	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
181	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
182	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
183	C₄⋊C₄	—	2	1	no	—
184	C₄⋊C₄	—	2	1	no	—
185	C₄⋊C₄	—	2	2	no	—
186	C₄⋊C₄	—	2	2	no	—
187	C₄²	—	2	2	no	—
188	C₄²	—	2	2	no	—
189	C₄²	#1157	1	1	yes	characteristic
190	C₄²	#1157	1	1	yes	characteristic
191	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
192	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
193	C₂×C₇	—	2	2	no	—
194	C₂×C₇	#1801	1	1	yes	characteristic
195	C₂×C₇	#1811	1	1	yes	characteristic
196	C₂×C₇	#1812	1	1	yes	characteristic
197	C₂³	C₂×C₄⋊Dic₇	1	1	yes	characteristic
198	Q₈	—	4	2	no	—
199	Q₈	—	4	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	4	1	no	—
207	C₂×C₄	—	4	2	no	—
208	C₂×C₄	#6487	1	1	yes	characteristic
209	C₂×C₄	#6487	1	1	yes	characteristic
210	C₂×C₄	#6489	1	1	yes	characteristic
211	C₂×C₄	#6493	1	1	yes	characteristic
212	C₂×C₄	#6493	1	1	yes	characteristic
213	C₂×C₄	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
214	C₈	—	2	1	no	—
215	C₈	—	2	1	no	—
216	C₈	—	2	1	no	—
217	C₈	—	4	2	no	—
218	C₈	—	56	2	no	—
219	C₈	—	56	2	no	—
220	C₈	#6410	1	2	yes	—
221	C₇	#8625	1	1	yes	characteristic, complemented, Sylow
222	C₂²	—	2	1	no	—
223	C₂²	—	2	1	no	—
224	C₂²	#89897	1	2	yes	—
225	C₂²	#89984	1	1	yes	characteristic
226	C₄	—	2	1	no	—
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	4	2	no	—
232	C₄	—	4	2	no	—
233	C₄	C₂×#6410	1	1	yes	characteristic
234	C₄	#90009	1	1	yes	characteristic
235	C₂	—	2	2	no	—
236	C₂	#11729391	1	1	yes	characteristic
237	C₂	#11729754	1	1	yes	characteristic
238	C₂	#11729766	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#89984
commutator	C₂²×C₄×C₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₂×C₇×#446	C₂
socle	C₂²×C₇	#380
prime radical	C₂³×C₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	C₂×#446	D₇
hypocenter	C₇	#8625
LCS	G ▷ C₂²×C₄×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#446

Complements

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

N	H ≅ G/N	type
221. C₇	#8625	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×#446	2
#11729379	2
#11727486	2
#11725514	2
#8625	7