Subgroups — #13272171

Order 1792.

438 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₅₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₇×C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×#89298	C₂	1	2	yes	maximal
4	C₂×#89299	C₂	1	1	yes	maximal, characteristic
5	#11727529	C₂	1	2	yes	maximal
6	C₂²×Dic₂₈	C₂²	1	2	yes	—
7	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
8	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
10	C₂×#6392	C₂²	1	2	yes	—
11	C₂×#6390	—	2	2	no	—
12	#89701	—	2	2	no	—
13	C₂×Dic₅₆	—	2	2	no	—
14	C₂×Dic₅₆	—	2	2	no	—
15	C₂×Dic₅₆	C₄	1	4	yes	complemented
16	C₄×C₇×C₁₆	—	2	2	no	—
17	#89299	—	2	2	no	—
18	#89298	—	2	4	no	—
19	#13236	—	7	1	no	maximal, Sylow
20	C₂²×Dic₁₄	—	2	2	no	—
21	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
22	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
23	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
24	C₂×C₄⋊Dic₇	—	2	2	no	—
25	C₂×Dic₂₈	—	2	2	no	—
26	C₂×Dic₂₈	—	2	2	no	—
27	C₂×Dic₂₈	—	2	2	no	—
28	C₂×Dic₂₈	—	2	2	no	—
29	C₂×Dic₂₈	—	2	2	no	—
30	C₂×Dic₂₈	C₂×C₄	1	4	yes	—
31	C₂×C₇×C₁₆	—	2	1	no	—
32	C₂×C₇×C₁₆	—	2	1	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	C₂×C₇×C₁₆	D₄	1	2	yes	—
37	C₄×C₇×C₈	—	2	1	no	—
38	C₄×C₇×C₈	—	2	1	no	—
39	C₇⋊#95	—	4	4	no	—
40	#6392	—	2	2	no	—
41	#6392	—	2	2	no	—
42	#6390	—	4	4	no	—
43	Dic₅₆	—	2	8	no	—
44	C₂²×Q₃₂	—	7	1	no	—
45	C₂×#366	—	7	1	no	—
46	C₂×#358	—	7	2	no	—
47	C₂×C₄×C₁₆	D₇	1	1	yes	characteristic
48	#1969	—	7	2	no	—
49	C₂²×C₄×C₇	D₈	1	2	yes	—
50	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
51	C₂²×Dic₇	—	4	2	no	—
52	C₂²×Dic₇	—	4	2	no	—
53	C₂×Dic₁₄	—	2	4	no	—
54	C₂×Dic₁₄	—	4	2	no	—
55	C₂×Dic₁₄	—	4	2	no	—
56	C₂×Dic₁₄	—	4	2	no	—
57	C₂×Dic₁₄	—	4	2	no	—
58	C₂×Dic₁₄	—	4	2	no	—
59	C₂×C₇×C₈	—	2	1	no	—
60	C₂×C₇×C₈	—	2	1	no	—
61	C₂×C₇×C₈	D₈	1	2	yes	—
62	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₇×C₈	C₂×D₄	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	#41	1	1	yes	characteristic
67	C₂×C₇×C₈	#41	1	1	yes	characteristic
68	C₄²×C₇	—	2	1	no	—
69	C₄²×C₇	—	2	1	no	—
70	C₄⋊Dic₇	—	4	2	no	—
71	C₄⋊Dic₇	—	4	2	no	—
72	Dic₂₈	—	2	4	no	—
73	Dic₂₈	—	2	4	no	—
74	Dic₂₈	—	4	2	no	—
75	Dic₂₈	—	4	2	no	—
76	C₇×C₁₆	—	2	2	no	—
77	C₇×C₁₆	C₂²⋊C₄	1	4	yes	—
78	C₂²×Q₁₆	—	7	2	no	—
79	C₂×Q₃₂	—	7	4	no	—
80	C₂×Q₃₂	—	14	2	no	—
81	C₂×Q₃₂	—	14	2	no	—
82	C₂²×C₁₆	C₂×D₇	1	1	yes	characteristic
83	C₂²×C₁₆	C₂×D₇	1	1	yes	characteristic
84	C₂×#107	—	7	2	no	—
85	C₂×#103	—	14	2	no	—
86	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
87	#391	—	14	2	no	—
88	#366	—	14	2	no	—
89	#358	—	14	4	no	—
90	C₄×C₁₆	—	2	2	no	—
91	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
92	C₂×C₄×C₇	—	2	2	no	—
93	C₂×C₄×C₇	—	2	2	no	—
94	C₂×C₄×C₇	Q₃₂	1	4	yes	complemented
95	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄×C₇	#121	1	1	yes	characteristic
97	C₂×C₄×C₇	#127	1	1	yes	characteristic
98	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
99	C₂×C₄×C₇	#135	1	1	yes	characteristic
100	C₂×C₄×C₇	#135	1	1	yes	characteristic
101	C₂×Dic₇	—	4	4	no	—
102	C₂×Dic₇	—	8	2	no	—
103	C₂×Dic₇	—	8	2	no	—
104	C₂×Dic₇	—	8	2	no	—
105	C₂×Dic₇	—	8	4	no	—
106	Dic₁₄	—	4	4	no	—
107	Dic₁₄	—	4	4	no	—
108	Dic₁₄	—	8	2	no	—
109	Dic₁₄	—	8	2	no	—
110	C₇×C₈	—	2	2	no	—
111	C₇×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₇×C₈	C₄×D₄	1	1	yes	characteristic
113	C₇×C₈	#121	1	1	yes	characteristic
114	C₇×C₈	#124	1	1	yes	characteristic
115	C₂²×Q₈	—	14	2	no	—
116	C₂×Q₁₆	—	7	4	no	—
117	C₂×Q₁₆	—	14	2	no	—
118	C₂×Q₁₆	—	14	2	no	—
119	C₂×Q₁₆	—	14	2	no	—
120	C₂×Q₁₆	—	14	2	no	—
121	C₂×Q₁₆	—	14	2	no	—
122	C₂²×C₈	D₂₈	1	1	yes	characteristic
123	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
124	C₂×C₄⋊C₄	—	14	2	no	—
125	C₂×C₄²	D₂₈	1	1	yes	characteristic
126	Q₃₂	—	14	8	no	—
127	C₂×C₁₆	—	2	1	no	—
128	C₂×C₁₆	—	2	1	no	—
129	C₂×C₁₆	C₄×D₇	1	2	yes	—
130	C₂×C₁₆	D₂₈	1	2	yes	—
131	C₂×C₁₆	D₂₈	1	2	yes	—
132	C₂×C₁₆	C₂²⋊D₇	1	2	yes	—
133	#107	—	14	2	no	—
134	#107	—	14	2	no	—
135	#103	—	28	4	no	—
136	C₄×C₈	—	2	1	no	—
137	C₄×C₈	—	2	1	no	—
138	#95	—	28	4	no	—
139	C₂²×C₇	C₄×D₈	1	1	yes	characteristic
140	C₂²×C₇	#466	1	1	yes	characteristic
141	C₂²×C₇	#493	1	1	yes	characteristic
142	C₂²×C₇	C₂×Q₃₂	1	2	yes	—
143	C₂²×C₇	#508	1	2	yes	—
144	C₄×C₇	—	2	4	no	—
145	C₄×C₇	#390	1	1	yes	characteristic
146	C₄×C₇	C₄×D₈	1	1	yes	characteristic
147	C₄×C₇	#466	1	1	yes	characteristic
148	C₄×C₇	#495	1	1	yes	characteristic
149	Dic₇	—	8	4	no	—
150	Dic₇	—	16	2	no	—
151	C₂×Q₈	—	14	4	no	—
152	C₂×Q₈	—	28	2	no	—
153	C₂×Q₈	—	28	2	no	—
154	C₂×Q₈	—	28	2	no	—
155	C₂×Q₈	—	28	2	no	—
156	C₂×Q₈	—	28	2	no	—
157	C₂²×C₄	—	28	2	no	—
158	C₂²×C₄	—	28	2	no	—
159	C₂²×C₄	D₅₆	1	2	yes	—
160	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
161	Q₁₆	—	14	4	no	—
162	Q₁₆	—	14	4	no	—
163	Q₁₆	—	28	2	no	—
164	Q₁₆	—	28	2	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	D₅₆	1	2	yes	—
168	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
169	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
170	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
171	C₂×C₈	#1157	1	1	yes	characteristic
172	C₂×C₈	#1157	1	1	yes	characteristic
173	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
174	C₄⋊C₄	—	28	2	no	—
175	C₄⋊C₄	—	28	2	no	—
176	C₄²	—	2	1	no	—
177	C₄²	—	2	1	no	—
178	C₁₆	—	2	2	no	—
179	C₁₆	(C₂×D₇)⋊C₄	1	4	yes	—
180	C₂×C₇	#1970	1	1	yes	characteristic
181	C₂×C₇	C₄×Q₃₂	1	2	yes	—
182	C₂×C₇	#2216	1	2	yes	—
183	C₂×C₇	#2247	1	1	yes	characteristic
184	C₂×C₇	#2249	1	1	yes	characteristic
185	C₂³	C₂×D₅₆	1	1	yes	characteristic
186	Q₈	—	28	4	no	—
187	Q₈	—	28	4	no	—
188	Q₈	—	56	2	no	—
189	Q₈	—	56	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	28	4	no	—
193	C₂×C₄	—	56	2	no	—
194	C₂×C₄	—	56	2	no	—
195	C₂×C₄	—	56	4	no	—
196	C₂×C₄	—	56	2	no	—
197	C₂×C₄	Dic₅₆	1	4	yes	complemented
198	C₂×C₄	C₄×D₂₈	1	1	yes	characteristic
199	C₂×C₄	#6437	1	1	yes	characteristic
200	C₂×C₄	C₂×D₅₆	1	1	yes	characteristic
201	C₂×C₄	#6467	1	1	yes	characteristic
202	C₂×C₄	#6467	1	1	yes	characteristic
203	C₂×C₄	#6493	1	1	yes	characteristic
204	C₈	—	2	2	no	—
205	C₈	C₄×D₂₈	1	1	yes	characteristic
206	C₈	#6438	1	1	yes	characteristic
207	C₈	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
208	C₈	#6493	1	1	yes	characteristic
209	C₇	#13236	1	1	yes	characteristic, complemented, Sylow
210	C₂²	C₄×D₅₆	1	1	yes	characteristic
211	C₂²	#89467	1	1	yes	characteristic
212	C₂²	#89676	1	2	yes	—
213	C₂²	C₂×Dic₅₆	1	2	yes	—
214	C₂²	#89887	1	1	yes	characteristic
215	C₄	—	2	4	no	—
216	C₄	—	56	4	no	—
217	C₄	—	112	2	no	—
218	C₄	C₄×D₅₆	1	1	yes	characteristic
219	C₄	#89468	1	1	yes	characteristic
220	C₄	#89711	1	1	yes	characteristic
221	C₄	#89887	1	1	yes	characteristic
222	C₂	#11726816	1	1	yes	characteristic
223	C₂	C₄×Dic₅₆	1	2	yes	—
224	C₂	#11726820	1	1	yes	characteristic
225	C₂	#11727530	1	1	yes	characteristic
226	C₂	#11729170	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₅₆
commutator	C₂×C₇×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇
Fitting	C₂×C₄×C₇×C₁₆	C₂
socle, prime radical	C₂³×C₇	C₂×D₈
cosocle	C₂²×C₇×C₈	C₂³
hypercenter	C₂×C₄×C₁₆	D₇
hypocenter	C₇	#13236
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
15. C₂×Dic₅₆	C₄	semidirect
94. C₂×C₄×C₇	Q₃₂	semidirect
197. C₂×C₄	Dic₅₆	semidirect
209. C₇	#13236	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Dic₅₆	2
C₂×C₄×C₇×C₁₆	2
C₂×#89298	2
C₂×#89299	2
#11727529	2
#13236	7