Subgroups — #13252689

Order 1792.

482 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₇×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	#11729388	C₂	1	2	yes	maximal
3	C₂×Q₈⋊(C₇⋊C₈)	C₂	1	2	yes	maximal
4	C₂×#89253	C₂	1	1	yes	maximal, characteristic
5	C₂×#89249	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇×Q₁₆	C₄	1	1	yes	characteristic
7	C₂×C₄×C₇×Q₈	C₂²	1	2	yes	—
8	C₄×C₇×Q₁₆	—	2	2	no	—
9	C₄×C₇×Q₁₆	—	2	2	no	—
10	C₂×C₇×#107	C₄	1	1	yes	characteristic
11	C₂×C₇×#103	C₄	1	2	yes	—
12	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
14	C₂×C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
16	Q₈⋊(C₇⋊C₈)	—	2	4	no	—
17	#89319	—	2	4	no	—
18	#89253	—	2	2	no	—
19	#89249	—	2	2	no	—
20	#9662	—	7	1	no	maximal, Sylow
21	C₂²×C₇×Q₈	C₂×C₄	1	2	yes	—
22	C₂×C₇×Q₁₆	—	2	4	no	—
23	C₂×C₇×Q₁₆	—	2	2	no	—
24	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
26	C₄×C₇×Q₈	—	2	2	no	—
27	C₄×C₇×Q₈	—	2	2	no	—
28	C₄×C₇×Q₈	—	2	2	no	—
29	C₄×C₇×Q₈	—	2	2	no	—
30	C₂×C₇×C₄⋊C₄	—	2	2	no	—
31	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₇	—	2	2	no	—
33	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
34	C₂²×C₇⋊C₈	—	2	1	no	—
35	C₂²×C₇⋊C₈	—	2	2	no	—
36	C₂²×C₇⋊C₈	D₄	1	2	yes	—
37	C₇×#107	—	2	2	no	—
38	C₇×#103	—	2	4	no	—
39	C₄×C₇×C₈	D₄	1	2	yes	—
40	C₄×C₇×C₈	D₄	1	2	yes	—
41	C₄⋊(C₇⋊C₈)	—	2	2	no	—
42	C₄⋊(C₇⋊C₈)	—	2	2	no	—
43	C₄×C₇⋊C₈	—	2	1	no	—
44	C₄×C₇⋊C₈	—	2	1	no	—
45	C₄×C₇⋊C₈	—	2	1	no	—
46	C₄×C₇⋊C₈	D₄	1	2	yes	—
47	C₂×C₄×Q₁₆	D₇	1	1	yes	characteristic
48	#1984	—	7	2	no	—
49	C₂×#335	—	7	1	no	—
50	C₂×Q₈⋊C₈	—	7	2	no	—
51	C₂×#322	—	7	1	no	—
52	C₂×C₇×Q₈	—	2	2	no	—
53	C₂×C₇×Q₈	—	2	2	no	—
54	C₂×C₇×Q₈	—	2	2	no	—
55	C₂×C₇×Q₈	—	2	2	no	—
56	C₂×C₇×Q₈	—	2	2	no	—
57	C₂×C₇×Q₈	M₁₆	1	4	yes	—
58	C₂²×C₄×C₇	—	2	2	no	—
59	C₂²×C₄×C₇	—	2	2	no	—
60	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₇	#41	1	1	yes	characteristic
63	C₇×Q₁₆	—	2	8	no	—
64	C₂×C₇×C₈	—	2	2	no	—
65	C₂×C₇×C₈	—	2	2	no	—
66	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
67	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
68	C₇×C₄⋊C₄	—	2	2	no	—
69	C₇×C₄⋊C₄	—	2	2	no	—
70	C₇×C₄⋊C₄	—	4	2	no	—
71	C₇×C₄⋊C₄	—	4	2	no	—
72	C₄²×C₇	—	4	2	no	—
73	C₄²×C₇	—	4	2	no	—
74	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₇	#41	1	1	yes	characteristic
78	C₂×C₇⋊C₈	—	2	2	no	—
79	C₂×C₇⋊C₈	—	2	2	no	—
80	C₂×C₇⋊C₈	—	2	2	no	—
81	C₂×C₇⋊C₈	—	4	2	no	—
82	C₂×C₇⋊C₈	—	4	2	no	—
83	C₂×C₇⋊C₈	—	4	4	no	—
84	C₂×C₇⋊C₈	Q₁₆	1	4	yes	complemented
85	C₂²×Q₁₆	Dic₇	1	1	yes	characteristic
86	C₂×C₄×Q₈	C₂×D₇	1	2	yes	—
87	C₄×Q₁₆	—	2	2	no	—
88	C₄×Q₁₆	—	2	2	no	—
89	C₂×#107	Dic₇	1	1	yes	characteristic
90	C₂×C₄⋊C₈	—	7	2	no	—
91	C₂×#103	Dic₇	1	2	yes	—
92	C₂×C₄×C₈	—	7	1	no	—
93	C₂×C₄×C₈	—	7	1	no	—
94	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
95	#336	—	14	4	no	—
96	#335	—	14	2	no	—
97	Q₈⋊C₈	—	14	4	no	—
98	#322	—	14	2	no	—
99	C₂³×C₇	C₄×D₄	1	1	yes	characteristic
100	C₇×Q₈	—	2	4	no	—
101	C₇×Q₈	—	2	4	no	—
102	C₇×Q₈	—	4	2	no	—
103	C₇×Q₈	—	4	2	no	—
104	C₂×C₄×C₇	—	2	2	no	—
105	C₂×C₄×C₇	—	2	4	no	—
106	C₂×C₄×C₇	—	4	2	no	—
107	C₂×C₄×C₇	—	4	2	no	—
108	C₂×C₄×C₇	—	4	2	no	—
109	C₂×C₄×C₇	—	4	4	no	—
110	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₇	#121	1	1	yes	characteristic
113	C₂×C₄×C₇	#121	1	1	yes	characteristic
114	C₂×C₄×C₇	#124	1	1	yes	characteristic
115	C₂×C₄×C₇	#127	1	1	yes	characteristic
116	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
117	C₂×C₄×C₇	#131	1	2	yes	—
118	C₂×C₄×C₇	C₂×Q₁₆	1	2	yes	—
119	C₂×C₄×C₇	#135	1	2	yes	—
120	C₇×C₈	—	2	2	no	—
121	C₇×C₈	—	2	2	no	—
122	C₇⋊C₈	—	2	4	no	—
123	C₇⋊C₈	—	4	2	no	—
124	C₇⋊C₈	—	8	2	no	—
125	C₂²×Q₈	C₂×Dic₇	1	2	yes	—
126	C₂×Q₁₆	—	2	4	no	—
127	C₂×Q₁₆	—	2	2	no	—
128	C₂²×C₈	—	7	2	no	—
129	C₂²×C₈	—	14	1	no	—
130	C₂²×C₈	—	14	2	no	—
131	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
132	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
133	C₄×Q₈	—	2	2	no	—
134	C₄×Q₈	—	2	2	no	—
135	C₄×Q₈	—	2	2	no	—
136	C₄×Q₈	—	2	2	no	—
137	C₂×C₄⋊C₄	—	2	2	no	—
138	C₂×C₄⋊C₄	C₂×Dic₇	1	2	yes	—
139	C₂×C₄²	—	2	2	no	—
140	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
141	#107	—	2	2	no	—
142	C₄⋊C₈	—	14	2	no	—
143	C₄⋊C₈	—	14	2	no	—
144	#103	—	2	4	no	—
145	C₄×C₈	—	7	2	no	—
146	C₄×C₈	—	14	1	no	—
147	C₄×C₈	—	14	1	no	—
148	C₄×C₈	—	14	1	no	—
149	C₄×C₈	C₂²⋊D₇	1	2	yes	—
150	C₄×C₈	C₂²⋊D₇	1	2	yes	—
151	C₂²×C₇	#390	1	1	yes	characteristic
152	C₂²×C₇	#435	1	2	yes	—
153	C₂²×C₇	C₄×Q₁₆	1	2	yes	—
154	C₂²×C₇	#444	1	2	yes	—
155	C₄×C₇	—	2	1	no	—
156	C₄×C₇	—	2	1	no	—
157	C₄×C₇	—	4	4	no	—
158	C₄×C₇	—	8	2	no	—
159	C₄×C₇	#407	1	1	yes	characteristic
160	C₄×C₇	#408	1	1	yes	characteristic
161	C₄×C₇	#435	1	2	yes	—
162	C₄×C₇	#467	1	2	yes	—
163	C₄×C₇	#494	1	1	yes	characteristic
164	C₄×C₇	#495	1	1	yes	characteristic
165	C₂×Q₈	—	2	2	no	—
166	C₂×Q₈	—	2	2	no	—
167	C₂×Q₈	—	2	2	no	—
168	C₂×Q₈	—	2	2	no	—
169	C₂×Q₈	—	2	2	no	—
170	C₂×Q₈	#1136	1	4	yes	—
171	C₂²×C₄	—	2	2	no	—
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
174	C₂²×C₄	#1159	1	1	yes	characteristic
175	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
176	Q₁₆	—	2	8	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	7	4	no	—
180	C₂×C₈	—	14	2	no	—
181	C₂×C₈	—	14	2	no	—
182	C₂×C₈	—	14	2	no	—
183	C₂×C₈	—	28	2	no	—
184	C₂×C₈	—	28	4	no	—
185	C₂×C₈	—	28	2	no	—
186	C₂×C₈	C₂²⋊Dic₇	1	2	yes	—
187	C₂×C₈	C₂²⋊Dic₇	1	2	yes	—
188	C₄⋊C₄	—	2	2	no	—
189	C₄⋊C₄	—	2	2	no	—
190	C₄⋊C₄	—	4	2	no	—
191	C₄⋊C₄	—	4	2	no	—
192	C₄²	—	4	2	no	—
193	C₄²	—	4	2	no	—
194	C₄²	D₄×D₇	1	1	yes	characteristic
195	C₄²	#1159	1	1	yes	characteristic
196	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
197	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
198	C₂×C₇	#1584	1	4	yes	—
199	C₂×C₇	#1955	1	1	yes	characteristic
200	C₂×C₇	#1969	1	1	yes	characteristic
201	C₂×C₇	#1975	1	1	yes	characteristic
202	C₂³	D₄×Dic₇	1	1	yes	characteristic
203	Q₈	—	2	4	no	—
204	Q₈	—	2	4	no	—
205	Q₈	—	4	2	no	—
206	Q₈	—	4	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	4	4	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	D₇×Q₁₆	1	2	yes	—
214	C₂×C₄	#6482	1	2	yes	—
215	C₂×C₄	C₂×#1136	1	2	yes	—
216	C₂×C₄	D₄×Dic₇	1	1	yes	characteristic
217	C₂×C₄	#6499	1	1	yes	characteristic
218	C₂×C₄	#6501	1	1	yes	characteristic
219	C₂×C₄	#6501	1	1	yes	characteristic
220	C₂×C₄	#6503	1	1	yes	characteristic
221	C₂×C₄	#6511	1	2	yes	—
222	C₂×C₄	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
223	C₈	—	2	2	no	—
224	C₈	—	2	2	no	—
225	C₈	—	14	4	no	—
226	C₈	—	28	2	no	—
227	C₈	—	56	2	no	—
228	C₇	#9662	1	1	yes	characteristic, complemented, Sylow
229	C₂²	#89783	1	2	yes	—
230	C₂²	Q₁₆×Dic₇	1	2	yes	—
231	C₂²	#89969	1	2	yes	—
232	C₂²	#89993	1	1	yes	characteristic
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	4	4	no	—
236	C₄	—	8	2	no	—
237	C₄	#89783	1	2	yes	—
238	C₄	#89953	1	1	yes	characteristic
239	C₄	#89960	1	2	yes	—
240	C₄	#89963	1	1	yes	characteristic
241	C₄	#89979	1	1	yes	characteristic
242	C₄	#90006	1	1	yes	characteristic
243	C₂	#11728657	1	4	yes	—
244	C₂	#11729369	1	1	yes	characteristic
245	C₂	#11729820	1	1	yes	characteristic
246	C₂	#11729839	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₇
commutator	C₂²×C₄×C₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₂×C₄×C₇×Q₁₆	C₂
socle, prime radical	C₂³×C₇	C₄×D₄
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	C₂×C₄×Q₁₆	D₇
hypocenter	C₇	#9662
LCS	G ▷ C₂²×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
84. C₂×C₇⋊C₈	Q₁₆	semidirect
228. C₇	#9662	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₇×Q₁₆	2
#11729388	2
C₂×Q₈⋊(C₇⋊C₈)	2
C₂×#89253	2
C₂×#89249	2
#9662	7