Subgroups — #13252839

Order 1792.

570 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11730756	C₂	1	1	yes	maximal, characteristic
2	C₄×C₇×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#89277	C₂	1	4	yes	maximal
4	#11727148	C₂	1	1	yes	maximal, characteristic
5	C₂×#6432	—	2	1	no	—
6	C₂×#6432	—	2	1	no	—
7	C₂×#6432	C₂²	1	2	yes	—
8	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
9	C₂×C₄×C₇×C₈	—	2	1	no	—
10	C₄³×C₇	C₂²	1	1	yes	characteristic
11	#89707	—	2	2	no	—
12	#89707	C₄	1	4	yes	—
13	C₂×C₄⋊(C₇⋊C₈)	—	2	2	no	—
14	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
15	C₇×#336	—	2	1	no	—
16	#89277	—	2	8	no	—
17	#9694	—	7	1	no	maximal, Sylow
18	C₂²×Dic₁₄	—	2	2	no	—
19	C₂²×C₇×C₈	—	2	2	no	—
20	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₇	D₄	1	2	yes	—
22	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
23	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
24	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
25	C₂×C₄⋊Dic₇	—	2	4	no	—
26	C₂×C₄⋊Dic₇	—	2	4	no	—
27	C₂×C₄⋊Dic₇	—	2	2	no	—
28	C₂×C₄⋊Dic₇	C₂×C₄	1	4	yes	—
29	C₂²×C₇⋊C₈	—	2	2	no	—
30	#6432	—	2	4	no	—
31	#6432	—	2	4	no	—
32	#6432	—	4	4	no	—
33	#6432	—	4	2	no	—
34	#6432	—	4	2	no	—
35	C₇×C₄⋊C₈	—	2	2	no	—
36	C₇×C₄⋊C₈	D₄	1	4	yes	—
37	C₄×C₇×C₈	—	4	2	no	—
38	C₄⋊(C₇⋊C₈)	—	2	4	no	—
39	C₄⋊(C₇⋊C₈)	—	4	4	no	—
40	#2868	—	7	1	no	—
41	#1842	—	7	1	no	—
42	C₄×C₄⋊C₈	D₇	1	1	yes	characteristic
43	C₂×#332	—	7	4	no	—
44	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
49	C₂²×Dic₇	—	2	4	no	—
50	C₂×Dic₁₄	—	4	4	no	—
51	C₂×Dic₁₄	—	4	2	no	—
52	C₂×Dic₁₄	—	4	2	no	—
53	C₂×Dic₁₄	—	4	4	no	—
54	C₂×C₇×C₈	—	2	4	no	—
55	C₂×C₇×C₈	—	4	2	no	—
56	C₂×C₇×C₈	—	4	2	no	—
57	C₄²×C₇	—	2	4	no	—
58	C₄²×C₇	—	2	2	no	—
59	C₄²×C₇	D₈	1	2	yes	—
60	C₄²×C₇	SD₁₆	1	4	yes	—
61	C₄²×C₇	SD₁₆	1	2	yes	—
62	C₄²×C₇	Q₁₆	1	4	yes	—
63	C₄²×C₇	C₂×D₄	1	2	yes	—
64	C₄²×C₇	#41	1	2	yes	—
65	C₄⋊Dic₇	—	2	8	no	—
66	C₄⋊Dic₇	—	4	8	no	—
67	C₄⋊Dic₇	—	4	8	no	—
68	C₄⋊Dic₇	—	4	2	no	—
69	C₄⋊Dic₇	—	4	2	no	—
70	C₂×C₇⋊C₈	—	4	2	no	—
71	C₂×C₇⋊C₈	—	4	4	no	—
72	C₂×C₄⋊Q₈	—	7	2	no	—
73	C₂×C₄⋊Q₈	—	14	1	no	—
74	C₂×C₄⋊Q₈	—	14	1	no	—
75	C₂×C₄⋊C₈	—	7	2	no	—
76	C₂×C₄⋊C₈	—	14	2	no	—
77	C₂×C₄⋊C₈	C₂×D₇	1	2	yes	—
78	C₂×C₄×C₈	—	2	1	no	—
79	#384	—	7	4	no	—
80	#384	—	14	2	no	—
81	C₄³	C₂×D₇	1	1	yes	characteristic
82	#336	—	2	1	no	—
83	#332	—	14	8	no	—
84	C₂³×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₇	—	2	2	no	—
86	C₂×C₄×C₇	—	2	2	no	—
87	C₂×C₄×C₇	—	2	2	no	—
88	C₂×C₄×C₇	#101	1	4	yes	—
89	C₂×C₄×C₇	#102	1	4	yes	—
90	C₂×C₄×C₇	#103	1	4	yes	—
91	C₂×C₄×C₇	#103	1	4	yes	—
92	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
93	C₂×C₄×C₇	#121	1	2	yes	—
94	C₂×C₄×C₇	#124	1	1	yes	characteristic
95	C₂×C₄×C₇	#127	1	1	yes	characteristic
96	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
97	C₂×C₄×C₇	C₂×SD₁₆	1	2	yes	—
98	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
99	C₂×C₄×C₇	C₂×Q₁₆	1	2	yes	—
100	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
101	C₂×Dic₇	—	4	4	no	—
102	C₂×Dic₇	—	4	4	no	—
103	C₂×Dic₇	—	4	4	no	—
104	Dic₁₄	—	8	4	no	—
105	Dic₁₄	—	8	4	no	—
106	C₇×C₈	—	4	4	no	—
107	C₇⋊C₈	—	8	2	no	—
108	C₂²×Q₈	—	14	2	no	—
109	C₂²×C₈	—	2	2	no	—
110	C₂²×C₈	—	14	2	no	—
111	C₄⋊Q₈	—	14	4	no	—
112	C₄⋊Q₈	—	14	4	no	—
113	C₄⋊Q₈	—	28	4	no	—
114	C₄⋊Q₈	—	28	2	no	—
115	C₄⋊Q₈	—	28	2	no	—
116	C₂×C₄⋊C₄	—	7	4	no	—
117	C₂×C₄⋊C₄	—	14	4	no	—
118	C₂×C₄⋊C₄	—	14	4	no	—
119	C₂×C₄⋊C₄	—	14	2	no	—
120	C₂×C₄²	C₄×D₇	1	2	yes	—
121	C₂×C₄²	D₂₈	1	2	yes	—
122	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
123	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
124	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
125	C₄⋊C₈	—	2	2	no	—
126	C₄⋊C₈	—	14	4	no	—
127	C₄⋊C₈	—	28	4	no	—
128	C₄⋊C₈	D₂₈	1	4	yes	—
129	C₄×C₈	—	4	2	no	—
130	C₂²×C₇	#390	1	1	yes	characteristic
131	C₂²×C₇	C₂×#101	1	2	yes	—
132	C₂²×C₇	C₂×#102	1	1	yes	characteristic
133	C₂²×C₇	C₂×#103	1	2	yes	—
134	C₂²×C₇	#418	1	1	yes	characteristic
135	C₄×C₇	—	2	4	no	—
136	C₄×C₇	—	2	2	no	—
137	C₄×C₇	#332	1	8	yes	complemented
138	C₄×C₇	#459	1	2	yes	—
139	C₄×C₇	#460	1	2	yes	—
140	C₄×C₇	#487	1	2	yes	—
141	C₄×C₇	#492	1	1	yes	characteristic
142	C₄×C₇	#494	1	1	yes	characteristic
143	Dic₇	—	8	4	no	—
144	C₂×Q₈	—	28	4	no	—
145	C₂×Q₈	—	28	2	no	—
146	C₂×Q₈	—	28	2	no	—
147	C₂×Q₈	—	28	4	no	—
148	C₂²×C₄	—	14	4	no	—
149	C₂²×C₄	(C₂×D₇)⋊C₄	1	2	yes	—
150	C₂²×C₄	(C₂×D₇)⋊C₄	1	2	yes	—
151	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
152	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
153	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
154	C₂×C₈	—	2	4	no	—
155	C₂×C₈	—	4	2	no	—
156	C₂×C₈	—	4	2	no	—
157	C₂×C₈	—	28	2	no	—
158	C₂×C₈	—	28	4	no	—
159	C₄⋊C₄	—	14	8	no	—
160	C₄⋊C₄	—	28	8	no	—
161	C₄⋊C₄	—	28	2	no	—
162	C₄⋊C₄	—	28	2	no	—
163	C₄⋊C₄	—	28	8	no	—
164	C₄²	—	2	2	no	—
165	C₄²	—	2	4	no	—
166	C₄²	#1132	1	4	yes	—
167	C₄²	Dic₂₈	1	4	yes	—
168	C₄²	#1141	1	2	yes	—
169	C₄²	#1143	1	2	yes	—
170	C₄²	C₂×D₂₈	1	2	yes	—
171	C₄²	#1157	1	2	yes	—
172	C₂×C₇	C₂×#332	1	2	yes	—
173	C₂×C₇	#1549	1	2	yes	—
174	C₂×C₇	#1956	1	1	yes	characteristic
175	C₂×C₇	#1966	1	1	yes	characteristic
176	C₂×C₇	#1982	1	1	yes	characteristic
177	C₂³	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
178	Q₈	—	56	4	no	—
179	Q₈	—	56	4	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	28	4	no	—
184	C₂×C₄	—	28	4	no	—
185	C₂×C₄	—	28	4	no	—
186	C₂×C₄	#6383	1	4	yes	—
187	C₂×C₄	#6390	1	4	yes	—
188	C₂×C₄	#6390	1	4	yes	—
189	C₂×C₄	#6398	1	4	yes	—
190	C₂×C₄	C₄×D₂₈	1	2	yes	—
191	C₂×C₄	#6437	1	1	yes	characteristic
192	C₂×C₄	#6438	1	1	yes	characteristic
193	C₂×C₄	C₂×#1132	1	2	yes	—
194	C₂×C₄	C₂×Dic₂₈	1	2	yes	—
195	C₂×C₄	#6493	1	2	yes	—
196	C₂×C₄	C₂×#1141	1	1	yes	characteristic
197	C₂×C₄	C₂×#1143	1	1	yes	characteristic
198	C₂×C₄	#6512	1	2	yes	—
199	C₈	—	4	4	no	—
200	C₈	—	56	2	no	—
201	C₇	#9694	1	1	yes	characteristic, complemented, Sylow
202	C₂²	#89711	1	1	yes	characteristic
203	C₂²	C₂×#6383	1	1	yes	characteristic
204	C₂²	#89773	1	1	yes	characteristic
205	C₂²	C₂×#6390	1	2	yes	—
206	C₂²	C₂×#6398	1	2	yes	—
207	C₄	—	2	2	no	—
208	C₄	—	2	4	no	—
209	C₄	—	56	4	no	—
210	C₄	#89277	1	8	yes	complemented
211	C₄	#89453	1	2	yes	—
212	C₄	#89465	1	1	yes	characteristic
213	C₄	#89471	1	1	yes	characteristic
214	C₄	#89787	1	2	yes	—
215	C₄	#89799	1	2	yes	—
216	C₂	#11727241	1	1	yes	characteristic
217	C₂	#11727525	1	1	yes	characteristic
218	C₂	#11727631	1	1	yes	characteristic
219	C₂	C₂×#89277	1	2	yes	—
220	C₂	#11728534	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×(C₂×D₇)⋊C₄
commutator	C₂²×C₄×C₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₄×C₇×C₄⋊C₈	C₂
socle, prime radical	C₂³×C₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	C₄×C₄⋊C₈	D₇
hypocenter	C₇	#9694
LCS	G ▷ C₂²×C₄×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
137. C₄×C₇	#332	semidirect
201. C₇	#9694	semidirect
210. C₄	#89277	semidirect

Maximal subgroups

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

subgroup	index
#11730756	2
C₄×C₇×C₄⋊C₈	2
C₂×#89277	2
#11727148	2
#9694	7