Subgroups — #13509156

Order 1792.

578 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3160	C₂	1	1	yes	maximal, characteristic
2	C₂×#89471	C₂	1	1	yes	maximal, characteristic
3	#11729136	C₂	1	4	yes	maximal, complemented
4	#11729107	C₂	1	4	yes	maximal, complemented
5	#11728408	C₂	1	4	yes	maximal
6	C₄×#6386	C₂	1	1	yes	maximal, characteristic
7	C₇×#533	—	2	1	no	—
8	C₂²×Dic₂₈	—	2	1	no	—
9	C₂²×Dic₂₈	C₂²	1	2	yes	—
10	C₂×#6432	—	2	1	no	—
11	C₇×#500	C₂²	1	2	yes	—
12	C₇×#499	—	2	1	no	—
13	C₇×#499	C₂²	1	2	yes	—
14	C₇×#498	—	2	1	no	—
15	C₇×#427	—	2	1	no	—
16	C₇×#427	C₂²	1	2	yes	—
17	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
18	C₂×#6386	C₂²	1	2	yes	—
19	#89471	—	2	1	no	—
20	#89471	—	2	1	no	—
21	#89471	C₂²	1	2	yes	—
22	#89471	C₂²	1	2	yes	—
23	#89287	C₂²	1	16	yes	—
24	#89256	C₂²	1	2	yes	—
25	C₄×C₇⋊C₁₆	C₂²	1	2	yes	—
26	#33705	—	7	1	no	maximal, Sylow
27	C₂²×Dic₁₄	—	2	2	no	—
28	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
29	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
30	C₇×C₄⋊Q₈	—	2	2	no	—
31	C₇×#125	—	2	2	no	—
32	C₇×C₂²⋊Q₈	—	4	2	no	—
33	C₄×C₇×Q₈	—	4	2	no	—
34	C₇×#117	—	2	2	no	—
35	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
36	C₂×C₄⋊Dic₇	—	4	2	no	—
37	C₂×Dic₂₈	—	2	8	no	—
38	C₂×Dic₂₈	—	2	2	no	—
39	C₂×Dic₂₈	—	2	2	no	—
40	C₂×Dic₂₈	—	2	2	no	—
41	C₂×Dic₂₈	—	4	4	no	—
42	C₂×Dic₂₈	C₂³	1	4	yes	—
43	#6432	—	2	2	no	—
44	#6432	—	2	2	no	—
45	#6432	—	4	2	no	—
46	C₇×#107	—	2	2	no	—
47	C₇×#107	C₂³	1	4	yes	—
48	C₇×#106	—	2	2	no	—
49	C₇×#106	—	2	2	no	—
50	C₄×C₇×C₈	D₄	1	1	yes	characteristic
51	C₄×C₇×C₈	D₄	1	1	yes	characteristic
52	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
53	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
54	#6386	—	2	4	no	—
55	C₂×C₇⋊C₁₆	C₂³	1	4	yes	—
56	#3160	D₇	1	1	yes	characteristic
57	C₂×#494	—	7	1	no	—
58	#2245	—	7	4	no	—
59	#2241	—	7	4	no	—
60	#2141	—	7	4	no	—
61	C₄×M₃₂	—	7	1	no	—
62	C₂×C₇×Q₈	—	4	2	no	—
63	C₂²×C₄×C₇	D₈	1	2	yes	—
64	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×Dic₇	—	4	2	no	—
66	C₂×Dic₁₄	—	2	4	no	—
67	C₂×Dic₁₄	—	4	4	no	—
68	C₂×Dic₁₄	—	4	4	no	—
69	C₂×Dic₁₄	—	4	2	no	—
70	C₂×C₇×C₈	SD₁₆	1	4	yes	—
71	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
72	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
73	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
74	C₂×C₇×C₈	#41	1	4	yes	—
75	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
76	C₇×C₄⋊C₄	—	2	4	no	—
77	C₇×C₄⋊C₄	—	4	4	no	—
78	C₇×C₄⋊C₄	—	4	2	no	—
79	C₇×C₂²⋊C₄	—	4	2	no	—
80	C₄²×C₇	—	4	2	no	—
81	C₄²×C₇	D₈	1	2	yes	—
82	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
84	C₄⋊Dic₇	—	4	4	no	—
85	C₄⋊Dic₇	—	4	4	no	—
86	Dic₂₈	—	2	8	no	—
87	Dic₂₈	—	4	4	no	—
88	Dic₂₈	—	4	4	no	—
89	Dic₂₈	—	4	4	no	—
90	C₇⋊C₁₆	—	2	4	no	—
91	C₂²×Q₁₆	—	7	2	no	—
92	C₂²×Q₁₆	—	14	1	no	—
93	#533	—	2	1	no	—
94	C₂×C₄⋊Q₈	—	14	1	no	—
95	C₂×M₃₂	—	7	2	no	—
96	#500	C₂×D₇	1	2	yes	—
97	#499	—	2	1	no	—
98	#499	C₂×D₇	1	2	yes	—
99	#498	—	2	1	no	—
100	#494	—	7	2	no	—
101	#494	—	7	2	no	—
102	#494	—	14	1	no	—
103	#494	—	14	1	no	—
104	#427	—	2	1	no	—
105	#427	C₂×D₇	1	2	yes	—
106	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
107	#358	—	7	16	no	—
108	#346	—	7	2	no	—
109	C₄×C₁₆	—	7	2	no	—
110	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
111	C₇×Q₈	—	8	4	no	—
112	C₂×C₄×C₇	—	2	4	no	—
113	C₂×C₄×C₇	—	4	4	no	—
114	C₂×C₄×C₇	#124	1	2	yes	—
115	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
116	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
117	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
118	C₂×C₄×C₇	C₂×SD₁₆	1	2	yes	—
119	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
120	C₂×Dic₇	—	4	4	no	—
121	C₂×Dic₇	—	8	4	no	—
122	Dic₁₄	—	4	4	no	—
123	Dic₁₄	—	4	4	no	—
124	Dic₁₄	—	8	4	no	—
125	C₇×C₈	#124	1	2	yes	—
126	C₇×C₈	C₂×SD₁₆	1	2	yes	—
127	C₇×C₈	C₂×SD₁₆	1	2	yes	—
128	C₇×C₈	C₂×#41	1	2	yes	—
129	C₂²×Q₈	—	14	2	no	—
130	C₂×Q₁₆	—	7	4	no	—
131	C₂×Q₁₆	—	14	8	no	—
132	C₂×Q₁₆	—	14	2	no	—
133	C₂×Q₁₆	—	14	2	no	—
134	C₂×Q₁₆	—	14	2	no	—
135	C₂×Q₁₆	—	28	4	no	—
136	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
137	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
138	C₄⋊Q₈	—	2	2	no	—
139	C₄⋊Q₈	—	14	2	no	—
140	C₄⋊Q₈	—	14	2	no	—
141	C₄⋊Q₈	—	28	2	no	—
142	#125	—	2	2	no	—
143	C₂²⋊Q₈	—	4	2	no	—
144	C₄×Q₈	—	4	2	no	—
145	#117	—	2	2	no	—
146	C₂×C₄⋊C₄	—	28	2	no	—
147	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
148	M₃₂	—	14	4	no	—
149	C₂×C₁₆	—	7	4	no	—
150	#107	—	2	2	no	—
151	#107	C₂²×D₇	1	4	yes	—
152	#106	—	2	2	no	—
153	#106	—	2	2	no	—
154	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
155	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
156	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
157	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
158	C₂²×C₇	—	2	2	no	—
159	C₂²×C₇	#486	1	2	yes	—
160	C₂²×C₇	C₂²×D₈	1	1	yes	characteristic
161	C₄×C₇	—	2	2	no	—
162	C₄×C₇	—	8	4	no	—
163	C₄×C₇	#486	1	2	yes	—
164	C₄×C₇	#510	1	4	yes	—
165	C₄×C₇	C₂×#124	1	1	yes	characteristic
166	C₄×C₇	C₂²×SD₁₆	1	1	yes	characteristic
167	Dic₇	—	8	4	no	—
168	C₂×Q₈	—	4	2	no	—
169	C₂×Q₈	—	14	4	no	—
170	C₂×Q₈	—	28	2	no	—
171	C₂×Q₈	—	28	4	no	—
172	C₂×Q₈	—	28	4	no	—
173	C₂²×C₄	—	28	2	no	—
174	C₂²×C₄	#1141	1	2	yes	—
175	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
176	Q₁₆	—	14	8	no	—
177	Q₁₆	—	28	4	no	—
178	Q₁₆	—	28	4	no	—
179	Q₁₆	—	28	4	no	—
180	C₂×C₈	#1143	1	4	yes	—
181	C₂×C₈	#1161	1	4	yes	—
182	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
183	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
184	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
185	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
186	C₄⋊C₄	—	2	4	no	—
187	C₄⋊C₄	—	4	4	no	—
188	C₄⋊C₄	—	4	2	no	—
189	C₄⋊C₄	—	28	4	no	—
190	C₄⋊C₄	—	28	4	no	—
191	C₂²⋊C₄	—	4	2	no	—
192	C₄²	—	4	2	no	—
193	C₄²	#1141	1	2	yes	—
194	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
195	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
196	C₁₆	—	14	4	no	—
197	C₂×C₇	—	2	2	no	—
198	C₂×C₇	C₂×#486	1	1	yes	characteristic
199	C₂×C₇	C₂×#510	1	2	yes	—
200	C₂³	C₂×#1141	1	1	yes	characteristic
201	Q₈	—	8	4	no	—
202	Q₈	—	28	4	no	—
203	Q₈	—	28	4	no	—
204	Q₈	—	56	4	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	4	4	no	—
207	C₂×C₄	—	28	4	no	—
208	C₂×C₄	—	56	4	no	—
209	C₂×C₄	C₂×#1141	1	2	yes	—
210	C₂×C₄	C₂×#1141	1	1	yes	characteristic
211	C₂×C₄	C₂×#1141	1	2	yes	—
212	C₂×C₄	C₂×#1143	1	2	yes	—
213	C₂×C₄	#6510	1	2	yes	—
214	C₂×C₄	C₂²×C₂²⋊D₇	1	1	yes	characteristic
215	C₈	C₂×#1143	1	2	yes	—
216	C₈	C₂×#1143	1	2	yes	—
217	C₈	#6510	1	2	yes	—
218	C₈	C₂×#1161	1	2	yes	—
219	C₇	#33705	1	1	yes	characteristic, complemented, Sylow
220	C₂²	—	2	2	no	—
221	C₂²	#89860	1	2	yes	—
222	C₂²	C₂²×#1141	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	8	4	no	—
225	C₄	—	56	4	no	—
226	C₄	#89860	1	2	yes	—
227	C₄	#89967	1	4	yes	—
228	C₄	C₂²×#1143	1	1	yes	characteristic
229	C₄	C₂×#6510	1	1	yes	characteristic
230	C₂	—	2	2	no	—
231	C₂	C₂×#89860	1	1	yes	characteristic
232	C₂	C₂×#89967	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#1141
commutator, cosocle	C₂×C₇×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₇
Fitting	C₇×#3160	C₂
socle	C₂²×C₇	C₂²×D₈
prime radical	C₂³×C₇	C₂×D₈
hypercenter	#3160	D₇
hypocenter	C₇	#33705
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
3. #11729136	C₂	semidirect
4. #11729107	C₂	semidirect
219. C₇	#33705	semidirect

Maximal subgroups

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

subgroup	index
C₇×#3160	2
C₂×#89471	2
#11729136	2
#11729107	2
#11728408	2
C₄×#6386	2
#33705	7