Subgroups — #13324355

Order 1792.

598 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3170	C₂	1	1	yes	maximal, characteristic
2	#11740773	C₂	1	2	yes	maximal
3	#11729580	C₂	1	2	yes	maximal
4	#11728660	C₂	1	4	yes	maximal, complemented
5	#11728656	C₂	1	4	yes	maximal, complemented
6	#11727484	C₂	1	1	yes	maximal, characteristic
7	C₂×#89250	C₂	1	1	yes	maximal, characteristic
8	C₇×#533	C₂²	1	2	yes	—
9	C₇×#500	C₄	1	1	yes	characteristic
10	C₇×#494	C₄	1	1	yes	characteristic
11	C₇×#487	C₄	1	2	yes	—
12	C₇×#467	C₄	1	4	yes	—
13	C₄×C₇×Q₁₆	C₂²	1	4	yes	—
14	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
15	#89796	—	2	2	no	—
16	#89796	C₂²	1	4	yes	—
17	Q₈×C₇⋊C₈	—	2	2	no	—
18	Q₈×C₇⋊C₈	C₂²	1	4	yes	—
19	#89771	—	2	2	no	—
20	#89769	—	2	2	no	—
21	#89696	C₂²	1	2	yes	—
22	C₂×C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
24	Q₈⋊(C₇⋊C₈)	C₂²	1	8	yes	—
25	#89252	C₂²	1	4	yes	—
26	#89250	C₂²	1	2	yes	—
27	#89250	C₂²	1	2	yes	—
28	#17961	—	7	1	no	maximal, Sylow
29	C₂×C₇×Q₁₆	C₂×C₄	1	4	yes	—
30	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
31	C₇×C₄⋊Q₈	C₂×C₄	1	2	yes	—
32	C₇×C₄⋊Q₈	C₂×C₄	1	2	yes	—
33	C₇×#125	—	2	2	no	—
34	C₇×C₂²⋊Q₈	—	2	2	no	—
35	C₇×C₂²⋊Q₈	C₂×C₄	1	4	yes	—
36	C₄×C₇×Q₈	—	2	2	no	—
37	C₄×C₇×Q₈	C₂³	1	4	yes	—
38	C₇×#117	—	2	2	no	—
39	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
40	C₂×#1136	—	2	2	no	—
41	C₂²×C₇⋊C₈	—	2	1	no	—
42	C₂²×C₇⋊C₈	D₄	1	2	yes	—
43	C₇×#107	C₂×C₄	1	4	yes	—
44	C₇×#103	C₂×C₄	1	8	yes	—
45	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
46	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
47	C₄×C₇×C₈	C₂³	1	2	yes	—
48	C₄⋊(C₇⋊C₈)	—	2	2	no	—
49	C₄⋊(C₇⋊C₈)	—	2	4	no	—
50	C₄⋊(C₇⋊C₈)	—	2	4	no	—
51	C₄⋊(C₇⋊C₈)	C₂³	1	4	yes	—
52	#6377	—	2	4	no	—
53	C₄×C₇⋊C₈	—	2	1	no	—
54	C₄×C₇⋊C₈	—	2	4	no	—
55	C₄×C₇⋊C₈	D₄	1	2	yes	—
56	C₄×C₇⋊C₈	C₂³	1	2	yes	—
57	C₄×C₇⋊C₈	C₂³	1	1	yes	characteristic
58	C₄×C₇⋊C₈	C₂³	1	1	yes	characteristic
59	#3170	D₇	1	1	yes	characteristic
60	#2960	—	7	2	no	—
61	#1593	—	7	4	no	—
62	#1579	—	7	4	no	—
63	#1566	—	7	1	no	—
64	#1493	—	7	2	no	—
65	C₂×#322	—	7	1	no	—
66	C₂×C₇×Q₈	—	2	2	no	—
67	C₂×C₇×Q₈	C₂²×C₄	1	4	yes	—
68	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₇	#41	1	1	yes	characteristic
71	C₇×Q₁₆	—	2	8	no	—
72	C₂×C₇×C₈	—	2	4	no	—
73	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
74	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
75	C₇×C₄⋊C₄	—	2	2	no	—
76	C₇×C₄⋊C₄	—	2	2	no	—
77	C₇×C₄⋊C₄	—	2	2	no	—
78	C₇×C₄⋊C₄	—	2	4	no	—
79	C₇×C₄⋊C₄	—	2	4	no	—
80	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
81	C₇×C₂²⋊C₄	—	2	4	no	—
82	C₄²×C₇	—	2	4	no	—
83	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₇	#41	1	1	yes	characteristic
86	C₄²×C₇	C₂⁴	1	1	yes	characteristic
87	#1136	—	4	4	no	—
88	C₂×C₇⋊C₈	—	2	4	no	—
89	C₂×C₇⋊C₈	—	2	4	no	—
90	C₂×C₇⋊C₈	—	2	2	no	—
91	C₂×C₇⋊C₈	—	4	2	no	—
92	C₂×C₇⋊C₈	C₂×D₄	1	2	yes	—
93	C₂×C₇⋊C₈	C₂×D₄	1	2	yes	—
94	#533	C₂×D₇	1	2	yes	—
95	#500	Dic₇	1	1	yes	characteristic
96	#494	Dic₇	1	1	yes	characteristic
97	#487	Dic₇	1	2	yes	—
98	#467	Dic₇	1	4	yes	—
99	#446	—	7	4	no	—
100	#446	—	14	2	no	—
101	C₈×Q₈	—	7	4	no	—
102	C₈×Q₈	—	14	2	no	—
103	C₄×Q₁₆	C₂×D₇	1	4	yes	—
104	#424	—	14	2	no	—
105	#423	—	7	2	no	—
106	#405	—	14	2	no	—
107	C₂×C₄×C₈	—	7	1	no	—
108	C₂×C₄×C₈	—	7	1	no	—
109	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
110	#334	—	7	4	no	—
111	Q₈⋊C₈	—	7	8	no	—
112	#322	—	7	2	no	—
113	#322	—	7	2	no	—
114	C₂³×C₇	C₄×D₄	1	1	yes	characteristic
115	C₇×Q₈	—	2	8	no	—
116	C₇×Q₈	—	4	4	no	—
117	C₂×C₄×C₇	—	2	2	no	—
118	C₂×C₄×C₇	—	2	2	no	—
119	C₂×C₄×C₇	—	2	4	no	—
120	C₂×C₄×C₇	—	2	4	no	—
121	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₇	#137	1	4	yes	—
125	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
128	C₇×C₈	—	2	2	no	—
129	C₇×C₈	#131	1	4	yes	—
130	C₇⋊C₈	—	2	2	no	—
131	C₇⋊C₈	—	2	2	no	—
132	C₇⋊C₈	—	4	2	no	—
133	C₇⋊C₈	—	4	4	no	—
134	C₂×Q₁₆	C₂×Dic₇	1	4	yes	—
135	C₂×M₁₆	—	14	2	no	—
136	C₂²×C₈	—	7	2	no	—
137	C₂²×C₈	—	14	1	no	—
138	C₂²×C₈	C₂×Dic₇	1	2	yes	—
139	C₄⋊Q₈	C₂×Dic₇	1	2	yes	—
140	C₄⋊Q₈	C₂×Dic₇	1	2	yes	—
141	#125	—	2	2	no	—
142	C₂²⋊Q₈	—	2	2	no	—
143	C₂²⋊Q₈	C₂×Dic₇	1	4	yes	—
144	C₄×Q₈	—	2	2	no	—
145	C₄×Q₈	C₂²×D₇	1	4	yes	—
146	#117	—	2	2	no	—
147	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
148	#107	C₂×Dic₇	1	4	yes	—
149	C₄⋊C₈	—	7	4	no	—
150	C₄⋊C₈	—	14	4	no	—
151	C₄⋊C₈	—	14	4	no	—
152	C₄⋊C₈	—	14	2	no	—
153	#103	C₂×Dic₇	1	8	yes	—
154	#97	—	14	4	no	—
155	C₄×C₈	—	7	2	no	—
156	C₄×C₈	—	7	1	no	—
157	C₄×C₈	—	7	1	no	—
158	C₄×C₈	—	7	2	no	—
159	C₄×C₈	—	14	4	no	—
160	C₄×C₈	—	14	1	no	—
161	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
162	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
163	C₄×C₈	C₂²×D₇	1	2	yes	—
164	C₂²×C₇	—	2	1	no	—
165	C₂²×C₇	—	2	1	no	—
166	C₂²×C₇	#441	1	2	yes	—
167	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
168	C₄×C₇	—	2	1	no	—
169	C₄×C₇	—	2	1	no	—
170	C₄×C₇	—	4	4	no	—
171	C₄×C₇	—	4	4	no	—
172	C₄×C₇	#441	1	2	yes	—
173	C₄×C₇	#444	1	2	yes	—
174	C₄×C₇	C₂×#131	1	2	yes	—
175	C₄×C₇	C₂×#137	1	2	yes	—
176	C₂×Q₈	—	2	2	no	—
177	C₂×Q₈	C₂²×Dic₇	1	4	yes	—
178	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
179	C₂²×C₄	#1159	1	1	yes	characteristic
180	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
181	Q₁₆	—	2	8	no	—
182	M₁₆	—	28	4	no	—
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	—	7	2	no	—
185	C₂×C₈	—	7	2	no	—
186	C₂×C₈	—	14	4	no	—
187	C₂×C₈	—	14	4	no	—
188	C₂×C₈	—	14	2	no	—
189	C₂×C₈	—	28	2	no	—
190	C₂×C₈	C₂²×Dic₇	1	2	yes	—
191	C₂×C₈	C₂²×Dic₇	1	2	yes	—
192	C₄⋊C₄	—	2	2	no	—
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	—	2	4	no	—
196	C₄⋊C₄	—	2	4	no	—
197	C₄⋊C₄	C₂²×Dic₇	1	4	yes	—
198	C₂²⋊C₄	—	2	4	no	—
199	C₄²	—	2	4	no	—
200	C₄²	D₄×D₇	1	1	yes	characteristic
201	C₄²	#1159	1	1	yes	characteristic
202	C₄²	C₂²×Dic₇	1	1	yes	characteristic
203	C₄²	C₂³×D₇	1	1	yes	characteristic
204	C₂×C₇	—	2	2	no	—
205	C₂×C₇	#2931	1	1	yes	characteristic
206	C₂×C₇	C₂×#441	1	1	yes	characteristic
207	C₂×C₇	C₂×#444	1	1	yes	characteristic
208	C₂³	D₄×Dic₇	1	1	yes	characteristic
209	Q₈	—	2	8	no	—
210	Q₈	—	4	4	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	#6481	1	4	yes	—
216	C₂×C₄	D₄×Dic₇	1	1	yes	characteristic
217	C₂×C₄	D₄×Dic₇	1	1	yes	characteristic
218	C₂×C₄	D₄×Dic₇	1	1	yes	characteristic
219	C₂×C₄	C₂×D₄×D₇	1	1	yes	characteristic
220	C₂×C₄	C₂×#1159	1	1	yes	characteristic
221	C₂×C₄	C₂³×Dic₇	1	1	yes	characteristic
222	C₈	—	2	2	no	—
223	C₈	—	14	2	no	—
224	C₈	—	14	2	no	—
225	C₈	—	28	2	no	—
226	C₈	—	28	4	no	—
227	C₈	#6511	1	4	yes	—
228	C₇	#17961	1	1	yes	characteristic, complemented, Sylow
229	C₂²	—	2	1	no	—
230	C₂²	—	2	1	no	—
231	C₂²	#89956	1	2	yes	—
232	C₂²	C₂×D₄×Dic₇	1	1	yes	characteristic
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	4	4	no	—
236	C₄	—	4	4	no	—
237	C₄	#89956	1	2	yes	—
238	C₄	#89969	1	2	yes	—
239	C₄	C₂×#6481	1	2	yes	—
240	C₄	C₂×#6511	1	2	yes	—
241	C₂	—	2	2	no	—
242	C₂	#11740720	1	1	yes	characteristic
243	C₂	C₂×#89956	1	1	yes	characteristic
244	C₂	C₂×#89969	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×#3170	C₂
socle	C₂²×C₇	C₂×C₄×D₄
prime radical	C₂³×C₇	C₄×D₄
cosocle	C₄²×C₇	C₂⁴
hypercenter	#3170	D₇
hypocenter	C₇	#17961
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3170

Complements

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

N	H ≅ G/N	type
4. #11728660	C₂	semidirect
5. #11728656	C₂	semidirect
228. C₇	#17961	semidirect

Maximal subgroups

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

subgroup	index
C₇×#3170	2
#11740773	2
#11729580	2
#11728660	2
#11728656	2
#11727484	2
C₂×#89250	2
#17961	7