Subgroups — #13273409

Order 1792.

382 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#89454	C₂	1	1	yes	maximal, characteristic
2	C₂×#89299	C₂	1	2	yes	maximal
3	#11727524	C₂	1	1	yes	maximal, characteristic
4	C₇×#1374	C₂	1	1	yes	maximal, characteristic
5	#11725612	C₂	1	2	yes	maximal
6	C₂×#6432	—	2	1	no	—
7	C₂²×C₇×C₁₆	C₂²	1	2	yes	—
8	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
9	C₂×#6392	—	2	1	no	—
10	C₂×#6392	C₂²	1	1	yes	characteristic
11	C₂×#6392	C₂²	1	2	yes	—
12	C₂×#6392	C₂²	1	1	yes	characteristic
13	#89700	—	2	1	no	—
14	#89454	—	2	4	no	—
15	#89346	—	2	1	no	—
16	#89299	—	2	4	no	—
17	#13451	—	7	1	no	maximal, Sylow
18	C₂²×Dic₁₄	—	4	1	no	—
19	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
20	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
21	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
22	C₂×C₄⋊Dic₇	—	2	1	no	—
23	C₂×C₄⋊Dic₇	—	2	2	no	—
24	C₂×C₄⋊Dic₇	—	2	1	no	—
25	C₂×C₄⋊Dic₇	—	4	1	no	—
26	C₂×Dic₇⋊C₄	—	4	1	no	—
27	#6432	—	2	4	no	—
28	#6432	—	4	2	no	—
29	C₂×C₇×C₁₆	—	2	2	no	—
30	C₂×C₇×C₁₆	—	2	2	no	—
31	C₂×C₇×C₁₆	Q₈	1	4	yes	complemented
32	C₄×C₇×C₈	—	2	2	no	—
33	C₇⋊#95	—	4	1	no	—
34	#6392	—	2	4	no	—
35	#6392	—	2	2	no	—
36	#6392	—	2	1	no	—
37	#6392	—	2	2	no	—
38	#6392	D₄	1	4	yes	—
39	#6392	Q₈	1	2	yes	complemented
40	C₂×#500	—	7	1	no	—
41	C₂×#366	—	7	2	no	—
42	#1942	—	7	1	no	—
43	#1380	—	7	2	no	—
44	#1374	D₇	1	1	yes	characteristic
45	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
46	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
47	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
48	C₂²×Dic₇	—	4	1	no	—
49	C₂²×Dic₇	—	4	2	no	—
50	C₂²×Dic₇	—	4	1	no	—
51	C₂×Dic₁₄	—	4	4	no	—
52	C₂×Dic₁₄	—	8	2	no	—
53	C₂×Dic₁₄	—	8	2	no	—
54	C₂×C₇×C₈	—	2	1	no	—
55	C₂×C₇×C₈	—	2	1	no	—
56	C₂×C₇×C₈	Q₁₆	1	2	yes	—
57	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
58	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
59	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₇×C₈	#41	1	1	yes	characteristic
61	C₄²×C₇	—	2	1	no	—
62	C₄²×C₇	—	2	1	no	—
63	C₄⋊Dic₇	—	2	4	no	—
64	C₄⋊Dic₇	—	2	2	no	—
65	C₄⋊Dic₇	—	4	4	no	—
66	C₄⋊Dic₇	—	4	2	no	—
67	C₄⋊Dic₇	—	4	1	no	—
68	C₄⋊Dic₇	—	4	1	no	—
69	C₄⋊Dic₇	—	4	1	no	—
70	Dic₇⋊C₄	—	8	2	no	—
71	C₇×C₁₆	—	2	4	no	—
72	C₂×C₄⋊Q₈	—	14	1	no	—
73	C₂²×C₁₆	C₂×D₇	1	2	yes	—
74	#500	—	14	4	no	—
75	C₂×#107	—	7	2	no	—
76	C₂×#107	—	7	1	no	—
77	C₂×#107	—	7	1	no	—
78	C₂×#107	—	14	1	no	—
79	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
80	#389	—	14	1	no	—
81	#366	—	14	4	no	—
82	#340	—	14	1	no	—
83	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
84	C₂×C₄×C₇	—	2	1	no	—
85	C₂×C₄×C₇	—	2	1	no	—
86	C₂×C₄×C₇	—	2	1	no	—
87	C₂×C₄×C₇	—	2	1	no	—
88	C₂×C₄×C₇	D₁₆	1	2	yes	—
89	C₂×C₄×C₇	Q₃₂	1	2	yes	—
90	C₂×C₄×C₇	#120	1	1	yes	characteristic
91	C₂×C₄×C₇	C₂²⋊Q₈	1	1	yes	characteristic
92	C₂×C₄×C₇	C₄⋊Q₈	1	1	yes	characteristic
93	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
94	C₂×C₄×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
95	C₂×C₄×C₇	#137	1	1	yes	characteristic
96	C₂×Dic₇	—	4	4	no	—
97	C₂×Dic₇	—	4	2	no	—
98	C₂×Dic₇	—	8	1	no	—
99	C₂×Dic₇	—	8	2	no	—
100	C₂×Dic₇	—	8	2	no	—
101	C₂×Dic₇	—	8	1	no	—
102	C₂×Dic₇	—	8	1	no	—
103	C₂×Dic₇	—	8	1	no	—
104	C₂×Dic₇	—	8	1	no	—
105	Dic₁₄	—	8	8	no	—
106	C₇×C₈	—	2	2	no	—
107	C₇×C₈	C₂²⋊Q₈	1	2	yes	—
108	C₇×C₈	C₄⋊Q₈	1	2	yes	—
109	C₂²×Q₈	—	28	1	no	—
110	C₂²×C₈	D₂₈	1	1	yes	characteristic
111	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
112	C₄⋊Q₈	—	14	4	no	—
113	C₄⋊Q₈	—	28	2	no	—
114	C₂×C₄⋊C₄	—	14	2	no	—
115	C₂×C₄⋊C₄	—	14	1	no	—
116	C₂×C₄⋊C₄	—	14	1	no	—
117	C₂×C₄⋊C₄	—	28	1	no	—
118	C₂×C₄⋊C₄	—	28	1	no	—
119	C₂×C₄²	D₂₈	1	1	yes	characteristic
120	C₂×C₁₆	—	2	2	no	—
121	C₂×C₁₆	—	2	2	no	—
122	C₂×C₁₆	Dic₁₄	1	4	yes	complemented
123	#107	—	7	4	no	—
124	#107	—	7	2	no	—
125	#107	—	14	4	no	—
126	#107	—	14	2	no	—
127	#107	—	14	1	no	—
128	#107	—	14	2	no	—
129	C₄×C₈	—	2	2	no	—
130	#95	—	28	1	no	—
131	C₂²×C₇	#447	1	1	yes	characteristic
132	C₂²×C₇	#474	1	1	yes	characteristic
133	C₂²×C₇	#500	1	1	yes	characteristic
134	C₂²×C₇	C₂×D₁₆	1	1	yes	characteristic
135	C₂²×C₇	C₂×Q₃₂	1	1	yes	characteristic
136	C₂²×C₇	#509	1	1	yes	characteristic
137	C₂²×C₇	#510	1	1	yes	characteristic
138	C₄×C₇	—	2	2	no	—
139	C₄×C₇	—	2	2	no	—
140	C₄×C₇	#395	1	1	yes	characteristic
141	C₄×C₇	#451	1	1	yes	characteristic
142	C₄×C₇	#477	1	1	yes	characteristic
143	C₄×C₇	#501	1	1	yes	characteristic
144	Dic₇	—	8	4	no	—
145	Dic₇	—	8	2	no	—
146	Dic₇	—	16	1	no	—
147	C₂×Q₈	—	28	4	no	—
148	C₂×Q₈	—	56	2	no	—
149	C₂×Q₈	—	56	2	no	—
150	C₂²×C₄	—	28	1	no	—
151	C₂²×C₄	—	28	2	no	—
152	C₂²×C₄	—	28	1	no	—
153	C₂²×C₄	D₅₆	1	1	yes	characteristic
154	C₂²×C₄	D₅₆	1	1	yes	characteristic
155	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	2	1	no	—
158	C₂×C₈	Dic₂₈	1	2	yes	—
159	C₂×C₈	C₂×Dic₁₄	1	2	yes	—
160	C₂×C₈	D₄×D₇	1	2	yes	—
161	C₂×C₈	#1159	1	1	yes	characteristic
162	C₂×C₈	Q₈×D₇	1	1	yes	characteristic
163	C₄⋊C₄	—	14	4	no	—
164	C₄⋊C₄	—	14	2	no	—
165	C₄⋊C₄	—	28	4	no	—
166	C₄⋊C₄	—	28	2	no	—
167	C₄⋊C₄	—	28	1	no	—
168	C₄⋊C₄	—	28	1	no	—
169	C₄⋊C₄	—	28	1	no	—
170	C₄⋊C₄	—	56	2	no	—
171	C₄²	—	2	1	no	—
172	C₄²	—	2	1	no	—
173	C₁₆	—	2	4	no	—
174	C₂×C₇	#2057	1	1	yes	characteristic
175	C₂×C₇	#2184	1	1	yes	characteristic
176	C₂×C₇	#2188	1	1	yes	characteristic
177	C₂×C₇	#2224	1	1	yes	characteristic
178	C₂×C₇	#2227	1	1	yes	characteristic
179	C₂×C₇	#2252	1	1	yes	characteristic
180	C₂×C₇	#2255	1	1	yes	characteristic
181	C₂³	C₂×D₅₆	1	1	yes	characteristic
182	Q₈	—	56	8	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	28	2	no	—
188	C₂×C₄	—	28	4	no	—
189	C₂×C₄	—	56	1	no	—
190	C₂×C₄	—	56	2	no	—
191	C₂×C₄	—	56	2	no	—
192	C₂×C₄	—	56	1	no	—
193	C₂×C₄	—	56	1	no	—
194	C₂×C₄	—	56	1	no	—
195	C₂×C₄	—	56	1	no	—
196	C₂×C₄	D₁₁₂	1	2	yes	—
197	C₂×C₄	Dic₅₆	1	2	yes	—
198	C₂×C₄	#6432	1	1	yes	characteristic
199	C₂×C₄	#6445	1	1	yes	characteristic
200	C₂×C₄	#6460	1	1	yes	characteristic
201	C₂×C₄	C₂×Dic₂₈	1	1	yes	characteristic
202	C₂×C₄	#6471	1	1	yes	characteristic
203	C₂×C₄	#6472	1	1	yes	characteristic
204	C₈	—	2	2	no	—
205	C₈	#6441	1	2	yes	—
206	C₈	#6451	1	2	yes	—
207	C₇	#13451	1	1	yes	characteristic, complemented, Sylow
208	C₂²	#89454	1	1	yes	characteristic
209	C₂²	#89504	1	1	yes	characteristic
210	C₂²	#89618	1	1	yes	characteristic
211	C₂²	C₂×D₁₁₂	1	1	yes	characteristic
212	C₂²	C₂×Dic₅₆	1	1	yes	characteristic
213	C₂²	#89680	1	1	yes	characteristic
214	C₂²	#89681	1	1	yes	characteristic
215	C₄	—	2	2	no	—
216	C₄	—	2	2	no	—
217	C₄	—	56	4	no	—
218	C₄	—	56	2	no	—
219	C₄	—	112	1	no	—
220	C₄	#89428	1	1	yes	characteristic
221	C₄	#89474	1	1	yes	characteristic
222	C₄	#89511	1	1	yes	characteristic
223	C₄	#89622	1	1	yes	characteristic
224	C₂	#11726386	1	1	yes	characteristic
225	C₂	#11726798	1	1	yes	characteristic
226	C₂	#11726823	1	1	yes	characteristic
227	C₂	#11726862	1	1	yes	characteristic
228	C₂	#11726867	1	1	yes	characteristic
229	C₂	#11727043	1	1	yes	characteristic
230	C₂	#11727050	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₅₆
commutator, cosocle	C₂²×C₇×C₈	C₂³
Frattini	C₂²×C₈	C₂²×D₇
Fitting	C₇×#1374	C₂
socle, prime radical	C₂³×C₇	C₂×D₈
hypercenter	#1374	D₇
hypocenter	C₇	#13451
LCS	G ▷ C₂²×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1374

Complements

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

N	H ≅ G/N	type
31. C₂×C₇×C₁₆	Q₈	semidirect
39. #6392	Q₈	semidirect
122. C₂×C₁₆	Dic₁₄	semidirect
207. C₇	#13451	semidirect

Maximal subgroups

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

subgroup	index
C₂×#89454	2
C₂×#89299	2
#11727524	2
C₇×#1374	2
#11725612	2
#13451	7