Subgroups — #13372732

Order 1792.

822 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×Q₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₄⋊Dic₇	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#11727328	C₂	1	3	yes	maximal
5	#11727322	C₂	1	3	yes	maximal
6	C₄×#6382	C₂	1	3	yes	maximal
7	#11727221	C₂	1	3	yes	maximal
8	C₂×C₄×C₇×Q₈	—	2	1	no	—
9	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
10	C₂×Q₈×Dic₇	—	2	1	no	—
11	C₇×#389	—	2	3	no	—
12	C₇×#384	C₂²	1	3	yes	—
13	C₄×C₇×C₄⋊C₄	C₂²	1	3	yes	—
14	Q₈×C₇⋊C₈	C₄	1	8	yes	complemented
15	#89757	—	2	3	no	—
16	C₂×#6382	C₂²	1	3	yes	—
17	C₂×#6382	C₂²	1	3	yes	—
18	#89707	C₂²	1	3	yes	—
19	C₄×C₄⋊Dic₇	C₂²	1	3	yes	—
20	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	3	yes	—
21	C₂×C₄×C₇⋊C₈	C₂²	1	3	yes	—
22	#89324	C₂²	1	6	yes	—
23	#89324	C₂²	1	1	yes	characteristic
24	#89324	C₂²	1	3	yes	—
25	#22738	—	7	1	no	maximal, Sylow
26	C₂²×C₇×Q₈	D₄	1	1	yes	characteristic
27	C₄×C₇×Q₈	—	4	4	no	—
28	C₄×C₇×Q₈	C₂×C₄	1	4	yes	—
29	C₄×C₇×Q₈	Q₈	1	4	yes	—
30	C₂×C₇×C₄⋊C₄	—	2	3	no	—
31	C₂×C₇×C₄⋊C₄	—	2	3	no	—
32	C₂×C₇×C₄⋊C₄	D₄	1	3	yes	—
33	C₂×C₇×C₄⋊C₄	C₂³	1	3	yes	—
34	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₇	—	2	3	no	—
36	C₂×C₄²×C₇	C₂³	1	3	yes	—
37	Q₈×Dic₇	—	4	4	no	—
38	C₂×C₄⋊Dic₇	—	2	3	no	—
39	C₂×C₄⋊Dic₇	—	2	3	no	—
40	C₂×C₄⋊Dic₇	C₂³	1	3	yes	—
41	C₂×C₄⋊Dic₇	C₂³	1	1	yes	characteristic
42	C₂×C₄×Dic₇	—	2	3	no	—
43	C₂²×C₇⋊C₈	C₂³	1	1	yes	characteristic
44	C₂²×C₇⋊C₈	C₂³	1	3	yes	—
45	C₇×#95	—	2	3	no	—
46	C₇⋊#95	—	2	3	no	—
47	#6382	—	2	6	no	—
48	#6382	—	2	6	no	—
49	C₄⋊(C₇⋊C₈)	C₂×C₄	1	12	yes	—
50	C₄×C₇⋊C₈	C₂×C₄	1	12	yes	—
51	C₂×C₈×Q₈	—	7	1	no	—
52	Q₈×C₄⋊C₄	—	7	1	no	—
53	Q₈×C₄⋊C₄	D₇	1	1	yes	characteristic
54	#1948	—	7	3	no	—
55	#1948	—	7	3	no	—
56	#1834	—	7	3	no	—
57	C₄×#106	—	7	3	no	—
58	C₂×C₇×Q₈	C₄⋊C₄	1	4	yes	—
59	C₂×C₇×Q₈	SD₁₆	1	4	yes	—
60	C₂×C₇×Q₈	SD₁₆	1	4	yes	—
61	C₂²×C₄×C₇	—	2	1	no	—
62	C₂²×C₄×C₇	—	2	3	no	—
63	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
64	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
65	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
66	C₂²×Dic₇	—	2	1	no	—
67	C₂²×Dic₇	—	2	3	no	—
68	C₇×C₄⋊C₄	—	2	1	no	—
69	C₇×C₄⋊C₄	—	2	3	no	—
70	C₇×C₄⋊C₄	—	2	3	no	—
71	C₇×C₄⋊C₄	—	4	6	no	—
72	C₇×C₄⋊C₄	—	8	3	no	—
73	C₇×C₄⋊C₄	C₄⋊C₄	1	12	yes	—
74	C₇×C₄⋊C₄	#41	1	2	yes	—
75	C₄²×C₇	—	4	6	no	—
76	C₄²×C₇	C₂²×C₄	1	6	yes	—
77	C₄²×C₇	C₂×Q₈	1	6	yes	—
78	C₄⋊Dic₇	—	2	1	no	—
79	C₄⋊Dic₇	—	2	3	no	—
80	C₄⋊Dic₇	—	2	3	no	—
81	C₄⋊Dic₇	—	4	6	no	—
82	C₄⋊Dic₇	—	8	3	no	—
83	C₄⋊Dic₇	#41	1	2	yes	—
84	C₄×Dic₇	—	4	6	no	—
85	C₂×C₇⋊C₈	—	2	3	no	—
86	C₂×C₇⋊C₈	—	2	3	no	—
87	C₂×C₇⋊C₈	—	2	1	no	—
88	C₂×C₇⋊C₈	—	2	1	no	—
89	C₂×C₇⋊C₈	C₂²×C₄	1	2	yes	—
90	C₂×C₇⋊C₈	C₂²×C₄	1	6	yes	—
91	C₂×C₄×Q₈	—	2	1	no	—
92	C₂×C₄×Q₈	—	14	1	no	—
93	C₂×C₄×Q₈	C₂×D₇	1	1	yes	characteristic
94	C₈×Q₈	—	7	8	no	—
95	C₂×#106	—	7	3	no	—
96	C₂×#106	—	7	3	no	—
97	C₂×C₄⋊C₈	—	7	3	no	—
98	C₂×C₄×C₈	—	7	3	no	—
99	#389	—	2	3	no	—
100	#389	—	14	3	no	—
101	#384	—	7	3	no	—
102	#384	C₂×D₇	1	3	yes	—
103	C₄×C₄⋊C₄	—	7	3	no	—
104	C₄×C₄⋊C₄	C₂×D₇	1	3	yes	—
105	#340	—	7	3	no	—
106	#340	—	7	6	no	—
107	#340	—	7	1	no	—
108	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
109	C₇×Q₈	#106	1	16	yes	complemented
110	C₂×C₄×C₇	—	2	3	no	—
111	C₂×C₄×C₇	—	2	3	no	—
112	C₂×C₄×C₇	—	2	2	no	—
113	C₂×C₄×C₇	—	4	1	no	—
114	C₂×C₄×C₇	—	4	1	no	—
115	C₂×C₄×C₇	—	4	3	no	—
116	C₂×C₄×C₇	—	4	3	no	—
117	C₂×C₄×C₇	—	4	3	no	—
118	C₂×C₄×C₇	C₂×C₄⋊C₄	1	6	yes	—
119	C₂×C₄×C₇	C₂×C₄⋊C₄	1	6	yes	—
120	C₂×C₄×C₇	C₂×SD₁₆	1	6	yes	—
121	C₂×C₄×C₇	C₂×SD₁₆	1	6	yes	—
122	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
124	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
125	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
126	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
127	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
128	C₂×Dic₇	—	2	2	no	—
129	C₂×Dic₇	—	4	1	no	—
130	C₂×Dic₇	—	4	1	no	—
131	C₂×Dic₇	—	4	3	no	—
132	C₂×Dic₇	—	4	3	no	—
133	C₂×Dic₇	—	4	3	no	—
134	C₇⋊C₈	—	2	6	no	—
135	C₇⋊C₈	—	2	2	no	—
136	C₂²×Q₈	C₂²⋊D₇	1	1	yes	characteristic
137	C₂²×C₈	—	7	1	no	—
138	C₂²×C₈	—	7	3	no	—
139	C₄×Q₈	—	4	4	no	—
140	C₄×Q₈	—	28	4	no	—
141	C₄×Q₈	Dic₁₄	1	4	yes	—
142	C₄×Q₈	C₄×D₇	1	4	yes	—
143	C₂×C₄⋊C₄	—	2	3	no	—
144	C₂×C₄⋊C₄	—	2	3	no	—
145	C₂×C₄⋊C₄	—	7	1	no	—
146	C₂×C₄⋊C₄	—	7	3	no	—
147	C₂×C₄⋊C₄	—	14	3	no	—
148	C₂×C₄⋊C₄	—	14	3	no	—
149	C₂×C₄⋊C₄	C₂²⋊D₇	1	3	yes	—
150	C₂×C₄⋊C₄	C₂²×D₇	1	3	yes	—
151	C₂×C₄⋊C₄	C₂²×D₇	1	1	yes	characteristic
152	C₂×C₄²	—	2	3	no	—
153	C₂×C₄²	—	14	3	no	—
154	C₂×C₄²	C₂²×D₇	1	3	yes	—
155	#106	—	14	6	no	—
156	#106	—	14	6	no	—
157	C₄⋊C₈	—	7	12	no	—
158	C₄×C₈	—	7	12	no	—
159	#95	—	2	3	no	—
160	#95	—	14	3	no	—
161	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
162	C₂²×C₇	#548	1	1	yes	characteristic
163	C₂²×C₇	#548	1	1	yes	characteristic
164	C₂²×C₇	C₂²×SD₁₆	1	1	yes	characteristic
165	C₂²×C₇	C₂²×SD₁₆	1	1	yes	characteristic
166	C₂²×C₇	#576	1	1	yes	characteristic
167	C₂²×C₇	#578	1	1	yes	characteristic
168	C₄×C₇	—	2	6	no	—
169	C₄×C₇	—	4	2	no	—
170	C₄×C₇	—	8	3	no	—
171	C₄×C₇	C₂×#106	1	12	yes	—
172	C₄×C₇	C₄×#41	1	1	yes	characteristic
173	C₄×C₇	#519	1	1	yes	characteristic
174	C₄×C₇	#557	1	1	yes	characteristic
175	C₄×C₇	#557	1	1	yes	characteristic
176	Dic₇	—	4	2	no	—
177	Dic₇	—	8	3	no	—
178	C₂×Q₈	Dic₇⋊C₄	1	4	yes	—
179	C₂×Q₈	#1142	1	4	yes	—
180	C₂×Q₈	#1143	1	4	yes	—
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	2	3	no	—
183	C₂²×C₄	—	14	1	no	—
184	C₂²×C₄	—	14	3	no	—
185	C₂²×C₄	C₂×C₂²⋊D₇	1	3	yes	—
186	C₂²×C₄	C₂×C₂²⋊D₇	1	3	yes	—
187	C₂²×C₄	C₂³×D₇	1	1	yes	characteristic
188	C₂×C₈	—	7	2	no	—
189	C₂×C₈	—	7	6	no	—
190	C₂×C₈	—	14	3	no	—
191	C₂×C₈	—	14	1	no	—
192	C₂×C₈	—	14	3	no	—
193	C₂×C₈	—	14	1	no	—
194	C₄⋊C₄	—	2	1	no	—
195	C₄⋊C₄	—	2	3	no	—
196	C₄⋊C₄	—	2	3	no	—
197	C₄⋊C₄	—	4	6	no	—
198	C₄⋊C₄	—	7	2	no	—
199	C₄⋊C₄	—	8	3	no	—
200	C₄⋊C₄	—	14	1	no	—
201	C₄⋊C₄	—	14	3	no	—
202	C₄⋊C₄	—	14	3	no	—
203	C₄⋊C₄	—	28	6	no	—
204	C₄⋊C₄	—	56	3	no	—
205	C₄⋊C₄	Dic₇⋊C₄	1	12	yes	—
206	C₄⋊C₄	#1161	1	2	yes	—
207	C₄²	—	4	6	no	—
208	C₄²	—	28	6	no	—
209	C₄²	C₂×Dic₁₄	1	6	yes	—
210	C₄²	C₂×C₄×D₇	1	6	yes	—
211	C₂×C₇	#2351	1	1	yes	characteristic
212	C₂×C₇	C₂²×#106	1	1	yes	characteristic
213	C₂×C₇	#2913	1	1	yes	characteristic
214	C₂×C₇	#3359	1	1	yes	characteristic
215	C₂×C₇	#3359	1	1	yes	characteristic
216	C₂×C₇	#3362	1	1	yes	characteristic
217	C₂×C₇	#3362	1	1	yes	characteristic
218	C₂³	C₂²×C₂²⋊D₇	1	1	yes	characteristic
219	Q₈	#6382	1	16	yes	complemented
220	C₂×C₄	—	2	3	no	—
221	C₂×C₄	—	2	3	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	4	1	no	—
224	C₂×C₄	—	4	1	no	—
225	C₂×C₄	—	4	3	no	—
226	C₂×C₄	—	4	3	no	—
227	C₂×C₄	—	4	3	no	—
228	C₂×C₄	—	14	2	no	—
229	C₂×C₄	—	28	1	no	—
230	C₂×C₄	—	28	1	no	—
231	C₂×C₄	—	28	3	no	—
232	C₂×C₄	—	28	3	no	—
233	C₂×C₄	—	28	3	no	—
234	C₂×C₄	C₂×Dic₇⋊C₄	1	6	yes	—
235	C₂×C₄	C₂×Dic₇⋊C₄	1	6	yes	—
236	C₂×C₄	C₂×#1142	1	6	yes	—
237	C₂×C₄	C₂×#1143	1	6	yes	—
238	C₂×C₄	C₂²×Dic₁₄	1	1	yes	characteristic
239	C₂×C₄	C₂²×C₄×D₇	1	1	yes	characteristic
240	C₂×C₄	C₂×#1161	1	1	yes	characteristic
241	C₂×C₄	#6551	1	1	yes	characteristic
242	C₂×C₄	D₇×#41	1	1	yes	characteristic
243	C₂×C₄	#6554	1	1	yes	characteristic
244	C₈	—	14	2	no	—
245	C₈	—	14	6	no	—
246	C₇	#22738	1	1	yes	characteristic, complemented, Sylow
247	C₂²	C₂²×Dic₇⋊C₄	1	1	yes	characteristic
248	C₂²	C₂²×#1142	1	1	yes	characteristic
249	C₂²	C₂²×#1143	1	1	yes	characteristic
250	C₂²	#90508	1	1	yes	characteristic
251	C₂²	#90522	1	1	yes	characteristic
252	C₂²	#90526	1	1	yes	characteristic
253	C₂²	#90529	1	1	yes	characteristic
254	C₄	—	2	6	no	—
255	C₄	—	4	2	no	—
256	C₄	—	8	3	no	—
257	C₄	—	28	2	no	—
258	C₄	—	56	3	no	—
259	C₄	C₂×#6382	1	12	yes	—
260	C₄	#90260	1	1	yes	characteristic
261	C₄	#90261	1	1	yes	characteristic
262	C₄	#90263	1	1	yes	characteristic
263	C₄	C₄×#1161	1	1	yes	characteristic
264	C₂	C₂²×#6382	1	1	yes	characteristic
265	C₂	#11739906	1	1	yes	characteristic
266	C₂	#11740007	1	1	yes	characteristic
267	C₂	#11741013	1	1	yes	characteristic
268	C₂	#11741031	1	1	yes	characteristic
269	C₂	#11741133	1	1	yes	characteristic
270	C₂	#11741148	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₇
commutator	C₂×C₄×C₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇
Fitting	C₇×Q₈×C₄⋊C₄	C₂
socle, prime radical	C₂³×C₇	C₂²×D₄
cosocle	C₂²×C₄×C₇	C₂⁴
hypercenter	Q₈×C₄⋊C₄	D₇
hypocenter	C₇	#22738
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ Q₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
14. Q₈×C₇⋊C₈	C₄	semidirect
109. C₇×Q₈	#106	semidirect
219. Q₈	#6382	semidirect
246. C₇	#22738	semidirect

Maximal subgroups

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

subgroup	index
C₇×Q₈×C₄⋊C₄	2
Q₈×C₄⋊Dic₇	2
C₂×Q₈×C₇⋊C₈	2
#11727328	2
#11727322	2
C₄×#6382	2
#11727221	2
#22738	7