Subgroups — #13252655

Order 1792.

502 conjugacy classes of subgroups.

The subgroup lattice has 264 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×C₈×C₇⋊C₈	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	C₈×C₇⋊C₈	—	2	2	no	—
20	#9656	—	7	1	no	maximal, Sylow
21	C₂²×C₇×Q₈	C₂×C₄	1	2	yes	—
22	C₂×C₇×Q₁₆	—	2	2	no	—
23	C₂×C₇×Q₁₆	—	2	2	no	—
24	C₂×C₇×Q₁₆	C₈	1	4	yes	complemented
25	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
27	C₄×C₇×Q₈	—	2	2	no	—
28	C₄×C₇×Q₈	—	2	2	no	—
29	C₄×C₇×Q₈	—	2	2	no	—
30	C₄×C₇×Q₈	—	2	2	no	—
31	C₂×C₇×C₄⋊C₄	—	2	2	no	—
32	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₇	—	2	2	no	—
34	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
35	C₂²×C₇⋊C₈	—	2	1	no	—
36	C₂²×C₇⋊C₈	—	2	2	no	—
37	C₂²×C₇⋊C₈	D₄	1	2	yes	—
38	C₇×#107	—	2	2	no	—
39	C₇×#103	—	2	4	no	—
40	C₄×C₇×C₈	D₄	1	2	yes	—
41	C₄×C₇×C₈	D₄	1	2	yes	—
42	C₄⋊(C₇⋊C₈)	—	2	2	no	—
43	C₄⋊(C₇⋊C₈)	—	2	2	no	—
44	C₄×C₇⋊C₈	—	2	1	no	—
45	C₄×C₇⋊C₈	—	2	1	no	—
46	C₄×C₇⋊C₈	—	2	1	no	—
47	C₄×C₇⋊C₈	D₄	1	2	yes	—
48	C₂×C₄×Q₁₆	D₇	1	1	yes	characteristic
49	#1984	—	7	2	no	—
50	C₂×#335	—	7	1	no	—
51	C₂×Q₈⋊C₈	—	7	2	no	—
52	C₂×C₈²	—	7	1	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	—	2	2	no	—
58	C₂×C₇×Q₈	C₂×C₈	1	4	yes	—
59	C₂²×C₄×C₇	—	2	2	no	—
60	C₂²×C₄×C₇	—	2	2	no	—
61	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₇	#41	1	1	yes	characteristic
64	C₇×Q₁₆	—	2	8	no	—
65	C₂×C₇×C₈	—	2	1	no	—
66	C₂×C₇×C₈	—	2	1	no	—
67	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
68	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
69	C₂×C₇×C₈	C₂×C₈	1	2	yes	—
70	C₂×C₇×C₈	M₁₆	1	2	yes	—
71	C₇×C₄⋊C₄	—	2	2	no	—
72	C₇×C₄⋊C₄	—	2	2	no	—
73	C₇×C₄⋊C₄	—	4	2	no	—
74	C₇×C₄⋊C₄	—	4	2	no	—
75	C₄²×C₇	—	4	2	no	—
76	C₄²×C₇	—	4	2	no	—
77	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₇	#41	1	1	yes	characteristic
81	C₂×C₇⋊C₈	—	2	4	no	—
82	C₂×C₇⋊C₈	—	2	2	no	—
83	C₂×C₇⋊C₈	—	2	2	no	—
84	C₂×C₇⋊C₈	—	2	2	no	—
85	C₂×C₇⋊C₈	—	4	2	no	—
86	C₂×C₇⋊C₈	—	4	4	no	—
87	C₂×C₇⋊C₈	Q₁₆	1	4	yes	complemented
88	C₂²×Q₁₆	Dic₇	1	1	yes	characteristic
89	C₂×C₄×Q₈	C₂×D₇	1	2	yes	—
90	C₄×Q₁₆	—	2	2	no	—
91	C₄×Q₁₆	—	2	2	no	—
92	C₂×#107	Dic₇	1	1	yes	characteristic
93	C₂×C₄⋊C₈	—	7	2	no	—
94	C₂×#103	Dic₇	1	2	yes	—
95	C₂×C₄×C₈	—	7	1	no	—
96	C₂×C₄×C₈	—	7	1	no	—
97	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
98	#336	—	14	4	no	—
99	#335	—	14	2	no	—
100	Q₈⋊C₈	—	14	4	no	—
101	C₈²	—	14	2	no	—
102	C₂³×C₇	C₄×D₄	1	1	yes	characteristic
103	C₇×Q₈	—	2	4	no	—
104	C₇×Q₈	—	2	4	no	—
105	C₇×Q₈	—	4	2	no	—
106	C₇×Q₈	—	4	2	no	—
107	C₂×C₄×C₇	—	2	2	no	—
108	C₂×C₄×C₇	—	2	4	no	—
109	C₂×C₄×C₇	—	4	2	no	—
110	C₂×C₄×C₇	—	4	2	no	—
111	C₂×C₄×C₇	—	4	2	no	—
112	C₂×C₄×C₇	—	4	4	no	—
113	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₇	#121	1	1	yes	characteristic
116	C₂×C₄×C₇	#121	1	1	yes	characteristic
117	C₂×C₄×C₇	#124	1	1	yes	characteristic
118	C₂×C₄×C₇	#127	1	1	yes	characteristic
119	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
120	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
121	C₂×C₄×C₇	#131	1	1	yes	characteristic
122	C₂×C₄×C₇	#131	1	1	yes	characteristic
123	C₂×C₄×C₇	C₂×Q₁₆	1	2	yes	—
124	C₂×C₄×C₇	#135	1	2	yes	—
125	C₇×C₈	—	2	2	no	—
126	C₇×C₈	C₂²⋊C₈	1	4	yes	—
127	C₇⋊C₈	—	2	4	no	—
128	C₇⋊C₈	—	2	4	no	—
129	C₇⋊C₈	—	8	2	no	—
130	C₂²×Q₈	C₂×Dic₇	1	2	yes	—
131	C₂×Q₁₆	—	2	2	no	—
132	C₂×Q₁₆	—	2	2	no	—
133	C₂×Q₁₆	C₇⋊C₈	1	4	yes	complemented
134	C₂²×C₈	—	7	2	no	—
135	C₂²×C₈	—	14	1	no	—
136	C₂²×C₈	—	14	2	no	—
137	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
138	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
139	C₄×Q₈	—	2	2	no	—
140	C₄×Q₈	—	2	2	no	—
141	C₄×Q₈	—	2	2	no	—
142	C₄×Q₈	—	2	2	no	—
143	C₂×C₄⋊C₄	—	2	2	no	—
144	C₂×C₄⋊C₄	C₂×Dic₇	1	2	yes	—
145	C₂×C₄²	—	2	2	no	—
146	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
147	#107	—	2	2	no	—
148	C₄⋊C₈	—	14	2	no	—
149	C₄⋊C₈	—	14	2	no	—
150	#103	—	2	4	no	—
151	C₄×C₈	—	7	2	no	—
152	C₄×C₈	—	14	1	no	—
153	C₄×C₈	—	14	1	no	—
154	C₄×C₈	—	14	1	no	—
155	C₄×C₈	C₂²⋊D₇	1	2	yes	—
156	C₄×C₈	C₂²⋊D₇	1	2	yes	—
157	C₂²×C₇	#390	1	1	yes	characteristic
158	C₂²×C₇	C₈×D₄	1	1	yes	characteristic
159	C₂²×C₇	#436	1	1	yes	characteristic
160	C₂²×C₇	C₄×Q₁₆	1	2	yes	—
161	C₂²×C₇	#443	1	2	yes	—
162	C₄×C₇	—	2	1	no	—
163	C₄×C₇	—	2	1	no	—
164	C₄×C₇	—	4	4	no	—
165	C₄×C₇	—	8	2	no	—
166	C₄×C₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
167	C₄×C₇	#408	1	1	yes	characteristic
168	C₄×C₇	C₈×D₄	1	1	yes	characteristic
169	C₄×C₇	#436	1	1	yes	characteristic
170	C₄×C₇	#467	1	2	yes	—
171	C₄×C₇	#494	1	1	yes	characteristic
172	C₄×C₇	#495	1	1	yes	characteristic
173	C₂×Q₈	—	2	2	no	—
174	C₂×Q₈	—	2	2	no	—
175	C₂×Q₈	—	2	2	no	—
176	C₂×Q₈	—	2	2	no	—
177	C₂×Q₈	—	2	2	no	—
178	C₂×Q₈	C₂×C₇⋊C₈	1	4	yes	—
179	C₂²×C₄	—	2	2	no	—
180	C₂²×C₄	—	2	2	no	—
181	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
182	C₂²×C₄	#1159	1	1	yes	characteristic
183	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
184	Q₁₆	—	2	8	no	—
185	C₂×C₈	—	2	1	no	—
186	C₂×C₈	—	2	1	no	—
187	C₂×C₈	—	7	4	no	—
188	C₂×C₈	—	14	4	no	—
189	C₂×C₈	—	14	2	no	—
190	C₂×C₈	—	14	2	no	—
191	C₂×C₈	—	14	2	no	—
192	C₂×C₈	—	28	4	no	—
193	C₂×C₈	—	28	2	no	—
194	C₂×C₈	C₂×C₇⋊C₈	1	2	yes	—
195	C₂×C₈	#1136	1	2	yes	—
196	C₂×C₈	C₂²⋊Dic₇	1	2	yes	—
197	C₂×C₈	C₂²⋊Dic₇	1	2	yes	—
198	C₄⋊C₄	—	2	2	no	—
199	C₄⋊C₄	—	2	2	no	—
200	C₄⋊C₄	—	4	2	no	—
201	C₄⋊C₄	—	4	2	no	—
202	C₄²	—	4	2	no	—
203	C₄²	—	4	2	no	—
204	C₄²	D₄×D₇	1	1	yes	characteristic
205	C₄²	#1159	1	1	yes	characteristic
206	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
207	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
208	C₂×C₇	C₈×Q₁₆	1	2	yes	—
209	C₂×C₇	#1591	1	2	yes	—
210	C₂×C₇	#1954	1	1	yes	characteristic
211	C₂×C₇	#1969	1	1	yes	characteristic
212	C₂×C₇	#1974	1	1	yes	characteristic
213	C₂³	D₄×Dic₇	1	1	yes	characteristic
214	Q₈	—	2	4	no	—
215	Q₈	—	2	4	no	—
216	Q₈	—	4	2	no	—
217	Q₈	—	4	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	4	4	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	D₇×Q₁₆	1	2	yes	—
225	C₂×C₄	#6482	1	2	yes	—
226	C₂×C₄	C₂²×C₇⋊C₈	1	1	yes	characteristic
227	C₂×C₄	C₂×#1136	1	1	yes	characteristic
228	C₂×C₄	D₄×Dic₇	1	1	yes	characteristic
229	C₂×C₄	#6499	1	1	yes	characteristic
230	C₂×C₄	#6501	1	1	yes	characteristic
231	C₂×C₄	#6501	1	1	yes	characteristic
232	C₂×C₄	#6503	1	1	yes	characteristic
233	C₂×C₄	#6511	1	1	yes	characteristic
234	C₂×C₄	#6511	1	1	yes	characteristic
235	C₂×C₄	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
236	C₈	—	2	2	no	—
237	C₈	—	14	4	no	—
238	C₈	—	14	4	no	—
239	C₈	—	56	2	no	—
240	C₈	C₂²⋊(C₇⋊C₈)	1	4	yes	—
241	C₇	#9656	1	1	yes	characteristic, complemented, Sylow
242	C₂²	D₄×C₇⋊C₈	1	1	yes	characteristic
243	C₂²	#89784	1	1	yes	characteristic
244	C₂²	Q₁₆×Dic₇	1	2	yes	—
245	C₂²	#89968	1	2	yes	—
246	C₂²	#89993	1	1	yes	characteristic
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	4	4	no	—
250	C₄	—	8	2	no	—
251	C₄	D₄×C₇⋊C₈	1	1	yes	characteristic
252	C₄	#89784	1	1	yes	characteristic
253	C₄	#89953	1	1	yes	characteristic
254	C₄	#89960	1	2	yes	—
255	C₄	#89963	1	1	yes	characteristic
256	C₄	C₂×C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic
257	C₄	#90006	1	1	yes	characteristic
258	C₂	Q₁₆×C₇⋊C₈	1	2	yes	—
259	C₂	#11728659	1	2	yes	—
260	C₂	#11729368	1	1	yes	characteristic
261	C₂	#11729820	1	1	yes	characteristic
262	C₂	#11729838	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₇
commutator	C₂×C₄×C₇	C₂²×C₈
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	#9656
LCS	G ▷ C₂×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
24. C₂×C₇×Q₁₆	C₈	semidirect
87. C₂×C₇⋊C₈	Q₁₆	semidirect
133. C₂×Q₁₆	C₇⋊C₈	semidirect
241. C₇	#9656	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×C₈×C₇⋊C₈	2
#9656	7