Subgroups — #13272167

Order 1792.

510 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×D₁₆	C₂	1	1	yes	maximal, characteristic
2	#11729792	C₂	1	2	yes	maximal, complemented
3	C₂×#89357	C₂	1	2	yes	maximal, complemented
4	C₂×#89299	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₁₆×Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₇×D₈	C₂²	1	2	yes	—
7	C₂×C₇×D₁₆	—	2	2	no	—
8	C₂×C₇×D₁₆	—	2	2	no	—
9	C₂×C₇×D₁₆	C₄	1	4	yes	complemented
10	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
11	#89993	—	2	2	no	—
12	C₂×#6406	—	2	2	no	—
13	C₂×#6392	C₂²	1	2	yes	—
14	C₂×C₈×Dic₇	C₂²	1	1	yes	characteristic
15	C₂²×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
16	#89357	—	2	4	no	—
17	#89299	—	2	2	no	—
18	C₁₆×Dic₇	—	2	2	no	—
19	#13235	—	7	1	no	maximal, Sylow
20	C₂²×C₇×D₄	—	2	2	no	—
21	C₂×C₇×D₈	—	2	2	no	—
22	C₂×C₇×D₈	—	2	2	no	—
23	C₂×C₇×D₈	—	2	2	no	—
24	C₂×C₇×D₈	—	2	2	no	—
25	C₂×C₇×D₈	—	2	2	no	—
26	C₂×C₇×D₈	C₂×C₄	1	4	yes	—
27	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
28	C₂×C₂²⋊Dic₇	—	4	4	no	—
29	C₂×C₄⋊Dic₇	—	2	2	no	—
30	C₂×C₄×Dic₇	D₄	1	1	yes	characteristic
31	C₂²×C₇⋊C₈	D₄	1	1	yes	characteristic
32	C₇×D₁₆	—	2	8	no	—
33	C₂×C₇×C₁₆	C₂×C₄	1	2	yes	—
34	C₂×C₇×C₁₆	D₄	1	2	yes	—
35	C₂×C₇×C₁₆	D₄	1	2	yes	—
36	#6406	—	4	4	no	—
37	#6392	—	2	2	no	—
38	#6392	—	2	2	no	—
39	C₈×Dic₇	—	2	1	no	—
40	C₈×Dic₇	—	2	1	no	—
41	C₂×C₇⋊C₁₆	—	2	1	no	—
42	C₂×C₇⋊C₁₆	—	2	1	no	—
43	C₂×C₇⋊C₁₆	D₄	1	2	yes	—
44	C₂²×D₁₆	D₇	1	1	yes	characteristic
45	C₂×#366	—	7	1	no	—
46	C₂×#357	—	7	2	no	—
47	C₂×C₄×C₁₆	—	7	1	no	—
48	#1970	—	7	2	no	—
49	C₂⁴×C₇	—	4	2	no	—
50	C₂×C₇×D₄	—	2	4	no	—
51	C₂×C₇×D₄	—	4	2	no	—
52	C₂×C₇×D₄	—	4	2	no	—
53	C₂×C₇×D₄	—	4	2	no	—
54	C₂×C₇×D₄	—	4	2	no	—
55	C₂×C₇×D₄	—	4	2	no	—
56	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×Dic₇	—	4	2	no	—
58	C₂²×Dic₇	D₈	1	2	yes	—
59	C₇×D₈	—	2	4	no	—
60	C₇×D₈	—	2	4	no	—
61	C₇×D₈	—	4	2	no	—
62	C₇×D₈	—	4	2	no	—
63	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
65	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
66	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
67	C₂×C₇×C₈	#41	1	1	yes	characteristic
68	C₂×C₇×C₈	#41	1	1	yes	characteristic
69	C₂²⋊Dic₇	—	8	8	no	—
70	C₄⋊Dic₇	—	4	2	no	—
71	C₄⋊Dic₇	—	4	2	no	—
72	C₄×Dic₇	—	2	1	no	—
73	C₄×Dic₇	—	2	1	no	—
74	C₂×C₇⋊C₈	—	2	1	no	—
75	C₂×C₇⋊C₈	—	2	1	no	—
76	C₂×C₇⋊C₈	D₈	1	2	yes	—
77	C₇×C₁₆	C₂²⋊C₄	1	4	yes	complemented
78	C₇⋊C₁₆	—	2	2	no	—
79	C₂²×D₈	C₂×D₇	1	2	yes	—
80	C₂×D₁₆	—	2	2	no	—
81	C₂×D₁₆	—	2	2	no	—
82	C₂×D₁₆	Dic₇	1	4	yes	complemented
83	C₂²×C₁₆	—	7	1	no	—
84	C₂²×C₁₆	C₂×D₇	1	1	yes	characteristic
85	C₂×#107	—	7	2	no	—
86	C₂×#102	—	14	2	no	—
87	C₂×C₄×C₈	—	7	1	no	—
88	#390	—	14	2	no	—
89	#366	—	14	2	no	—
90	#357	—	14	4	no	—
91	C₄×C₁₆	—	14	2	no	—
92	C₂³×C₇	—	4	4	no	—
93	C₂³×C₇	—	8	2	no	—
94	C₂³×C₇	—	8	2	no	—
95	C₂³×C₇	—	8	4	no	—
96	C₂³×C₇	—	8	4	no	—
97	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
98	C₇×D₄	—	4	4	no	—
99	C₇×D₄	—	4	4	no	—
100	C₇×D₄	—	8	2	no	—
101	C₇×D₄	—	8	2	no	—
102	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₇	#121	1	1	yes	characteristic
104	C₂×C₄×C₇	#127	1	1	yes	characteristic
105	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
106	C₂×C₄×C₇	#135	1	1	yes	characteristic
107	C₂×C₄×C₇	#135	1	1	yes	characteristic
108	C₂×Dic₇	—	2	2	no	—
109	C₂×Dic₇	—	2	2	no	—
110	C₂×Dic₇	—	8	2	no	—
111	C₂×Dic₇	—	8	4	no	—
112	C₂×Dic₇	D₁₆	1	4	yes	complemented
113	C₇×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₇×C₈	C₄×D₄	1	1	yes	characteristic
115	C₇×C₈	#121	1	1	yes	characteristic
116	C₇×C₈	#124	1	1	yes	characteristic
117	C₇⋊C₈	—	2	2	no	—
118	C₂²×D₄	—	2	2	no	—
119	C₂×D₈	—	2	2	no	—
120	C₂×D₈	—	2	2	no	—
121	C₂×D₈	—	2	2	no	—
122	C₂×D₈	—	2	2	no	—
123	C₂×D₈	—	2	2	no	—
124	C₂×D₈	C₂×Dic₇	1	4	yes	—
125	C₂²×C₈	—	7	1	no	—
126	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
127	C₂×C₄⋊C₄	—	14	2	no	—
128	C₂×C₂²⋊C₄	—	28	4	no	—
129	C₂×C₄²	—	7	1	no	—
130	D₁₆	—	2	8	no	—
131	C₂×C₁₆	—	7	2	no	—
132	C₂×C₁₆	—	14	1	no	—
133	C₂×C₁₆	—	14	1	no	—
134	C₂×C₁₆	C₂×Dic₇	1	2	yes	—
135	C₂×C₁₆	C₂²⋊D₇	1	2	yes	—
136	C₂×C₁₆	C₂²⋊D₇	1	2	yes	—
137	#107	—	14	2	no	—
138	#107	—	14	2	no	—
139	#102	—	28	4	no	—
140	C₄×C₈	—	14	1	no	—
141	C₄×C₈	—	14	1	no	—
142	C₂²×C₇	—	8	8	no	—
143	C₂²×C₇	—	8	4	no	—
144	C₂²×C₇	—	16	4	no	—
145	C₂²×C₇	—	16	2	no	—
146	C₂²×C₇	—	16	2	no	—
147	C₂²×C₇	C₄×D₈	1	1	yes	characteristic
148	C₂²×C₇	#466	1	1	yes	characteristic
149	C₂²×C₇	#493	1	1	yes	characteristic
150	C₂²×C₇	C₂×D₁₆	1	2	yes	—
151	C₂²×C₇	#508	1	2	yes	—
152	C₄×C₇	#390	1	1	yes	characteristic
153	C₄×C₇	C₄×D₈	1	1	yes	characteristic
154	C₄×C₇	#466	1	1	yes	characteristic
155	C₄×C₇	#495	1	1	yes	characteristic
156	Dic₇	—	2	4	no	—
157	Dic₇	—	16	2	no	—
158	C₂⁴	—	4	2	no	—
159	C₂×D₄	—	2	4	no	—
160	C₂×D₄	—	4	2	no	—
161	C₂×D₄	—	4	2	no	—
162	C₂×D₄	—	4	2	no	—
163	C₂×D₄	—	4	2	no	—
164	C₂×D₄	—	4	2	no	—
165	C₂²×C₄	—	7	2	no	—
166	C₂²×C₄	—	28	2	no	—
167	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
168	D₈	—	2	4	no	—
169	D₈	—	2	4	no	—
170	D₈	—	4	2	no	—
171	D₈	—	4	2	no	—
172	C₂×C₈	—	7	2	no	—
173	C₂×C₈	—	14	1	no	—
174	C₂×C₈	—	14	1	no	—
175	C₂×C₈	D₄×D₇	1	1	yes	characteristic
176	C₂×C₈	#1159	1	1	yes	characteristic
177	C₂×C₈	#1159	1	1	yes	characteristic
178	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
179	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
180	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
181	C₄⋊C₄	—	28	2	no	—
182	C₄⋊C₄	—	28	2	no	—
183	C₂²⋊C₄	—	56	8	no	—
184	C₄²	—	14	1	no	—
185	C₄²	—	14	1	no	—
186	C₁₆	—	14	2	no	—
187	C₁₆	C₂²⋊Dic₇	1	4	yes	complemented
188	C₂×C₇	—	16	4	no	—
189	C₂×C₇	#1970	1	1	yes	characteristic
190	C₂×C₇	C₄×D₁₆	1	2	yes	—
191	C₂×C₇	#2215	1	2	yes	—
192	C₂×C₇	#2246	1	1	yes	characteristic
193	C₂×C₇	#2249	1	1	yes	characteristic
194	C₂³	—	4	4	no	—
195	C₂³	—	8	2	no	—
196	C₂³	—	8	2	no	—
197	C₂³	—	8	4	no	—
198	C₂³	—	8	4	no	—
199	C₂³	D₇×D₈	1	1	yes	characteristic
200	D₄	—	4	4	no	—
201	D₄	—	4	4	no	—
202	D₄	—	8	2	no	—
203	D₄	—	8	2	no	—
204	C₂×C₄	—	7	4	no	—
205	C₂×C₄	—	14	2	no	—
206	C₂×C₄	—	14	2	no	—
207	C₂×C₄	—	56	2	no	—
208	C₂×C₄	—	56	4	no	—
209	C₂×C₄	D₇×D₈	1	1	yes	characteristic
210	C₂×C₄	#6475	1	1	yes	characteristic
211	C₂×C₄	#6475	1	1	yes	characteristic
212	C₂×C₄	D₄×Dic₇	1	1	yes	characteristic
213	C₂×C₄	#6501	1	1	yes	characteristic
214	C₂×C₄	#6503	1	1	yes	characteristic
215	C₈	—	14	2	no	—
216	C₈	D₄×Dic₇	1	1	yes	characteristic
217	C₈	#6499	1	1	yes	characteristic
218	C₈	#6501	1	1	yes	characteristic
219	C₈	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
220	C₇	#13235	1	1	yes	characteristic, complemented, Sylow
221	C₂²	—	8	8	no	—
222	C₂²	—	8	4	no	—
223	C₂²	—	16	2	no	—
224	C₂²	—	16	2	no	—
225	C₂²	—	16	4	no	—
226	C₂²	D₇×D₁₆	1	2	yes	—
227	C₂²	#89684	1	2	yes	—
228	C₂²	D₈×Dic₇	1	1	yes	characteristic
229	C₂²	#89923	1	1	yes	characteristic
230	C₂²	#89929	1	1	yes	characteristic
231	C₄	—	14	4	no	—
232	C₄	—	112	2	no	—
233	C₄	D₈×Dic₇	1	1	yes	characteristic
234	C₄	#89927	1	1	yes	characteristic
235	C₄	#89929	1	1	yes	characteristic
236	C₄	#89993	1	1	yes	characteristic
237	C₂	—	16	4	no	—
238	C₂	Dic₇×D₁₆	1	2	yes	—
239	C₂	#11729218	1	1	yes	characteristic
240	C₂	#11729219	1	1	yes	characteristic
241	C₂	#11729223	1	2	yes	—
242	C₂	#11729792	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₇×D₈
commutator	C₂×C₇×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇
Fitting, prime radical	C₂²×C₇×D₁₆	C₂
socle	C₂³×C₇	C₂×D₈
cosocle	C₂²×C₇×C₈	C₂³
hypercenter	C₂²×D₁₆	D₇
hypocenter	C₇	#13235
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×D₁₆

Complements

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

N	H ≅ G/N	type
2. #11729792	C₂	semidirect
3. C₂×#89357	C₂	semidirect
4. C₂×#89299	C₂	semidirect
5. C₂×C₁₆×Dic₇	C₂	semidirect
9. C₂×C₇×D₁₆	C₄	semidirect
77. C₇×C₁₆	C₂²⋊C₄	semidirect
82. C₂×D₁₆	Dic₇	semidirect
112. C₂×Dic₇	D₁₆	semidirect
187. C₁₆	C₂²⋊Dic₇	semidirect
220. C₇	#13235	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₇×D₁₆	2
#11729792	2
C₂×#89357	2
C₂×#89299	2
C₂×C₁₆×Dic₇	2
#13235	7