Subgroups — #13253944

Order 1792.

506 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2868	C₂	1	1	yes	maximal, characteristic
2	#11727146	C₂	1	1	yes	maximal, characteristic
3	#11727145	C₂	1	1	yes	maximal, characteristic
4	#11725440	C₂	1	4	yes	maximal
5	C₂×C₇×C₄⋊Q₈	—	2	1	no	—
6	C₂×C₇×C₄⋊Q₈	—	2	1	no	—
7	C₂×C₇×C₄⋊Q₈	C₄	1	2	yes	—
8	C₇×#384	—	2	1	no	—
9	C₇×#384	—	2	1	no	—
10	C₇×#384	C₄	1	2	yes	—
11	C₇×#384	C₂²	1	2	yes	—
12	C₄³×C₇	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
14	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
15	C₂×#6377	—	2	1	no	—
16	#89319	—	2	1	no	—
17	#89319	—	2	1	no	—
18	#89319	—	2	1	no	—
19	#9903	—	7	1	no	maximal, Sylow
20	C₂²×C₇×Q₈	—	2	2	no	—
21	C₇×C₄⋊Q₈	—	4	4	no	—
22	C₇×C₄⋊Q₈	—	4	4	no	—
23	C₇×C₄⋊Q₈	—	4	4	no	—
24	C₂×C₇×C₄⋊C₄	—	2	2	no	—
25	C₂×C₇×C₄⋊C₄	—	2	2	no	—
26	C₂×C₇×C₄⋊C₄	—	2	2	no	—
27	C₂×C₇×C₄⋊C₄	—	2	2	no	—
28	C₂×C₇×C₄⋊C₄	—	2	2	no	—
29	C₂×C₇×C₄⋊C₄	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
37	C₂²×C₇⋊C₈	—	2	2	no	—
38	C₂²×C₇⋊C₈	—	2	2	no	—
39	C₄⋊(C₇⋊C₈)	—	2	2	no	—
40	C₄⋊(C₇⋊C₈)	—	2	4	no	—
41	C₄⋊(C₇⋊C₈)	Q₈	1	4	yes	complemented
42	#6377	—	4	2	no	—
43	#2868	D₇	1	1	yes	characteristic
44	#1831	—	7	1	no	—
45	#1767	—	7	1	no	—
46	#1296	—	7	4	no	—
47	C₂×C₇×Q₈	—	4	4	no	—
48	C₂×C₇×Q₈	—	4	4	no	—
49	C₂×C₇×Q₈	—	4	4	no	—
50	C₂²×C₄×C₇	—	2	4	no	—
51	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
58	C₇×C₄⋊C₄	—	4	4	no	—
59	C₇×C₄⋊C₄	—	4	4	no	—
60	C₇×C₄⋊C₄	—	4	4	no	—
61	C₇×C₄⋊C₄	—	4	4	no	—
62	C₇×C₄⋊C₄	—	4	4	no	—
63	C₇×C₄⋊C₄	—	4	4	no	—
64	C₄²×C₇	—	2	2	no	—
65	C₄²×C₇	—	2	2	no	—
66	C₄²×C₇	—	2	2	no	—
67	C₄²×C₇	—	2	2	no	—
68	C₄²×C₇	—	2	2	no	—
69	C₄²×C₇	SD₁₆	1	2	yes	—
70	C₄²×C₇	Q₁₆	1	2	yes	—
71	C₄²×C₇	C₂×Q₈	1	2	yes	—
72	C₄²×C₇	#41	1	2	yes	—
73	C₂×C₇⋊C₈	—	2	4	no	—
74	C₂×C₇⋊C₈	—	4	2	no	—
75	C₂×C₇⋊C₈	—	4	2	no	—
76	C₂×C₇⋊C₈	—	4	2	no	—
77	C₂×C₇⋊C₈	—	4	2	no	—
78	C₂×C₇⋊C₈	—	4	2	no	—
79	C₂×C₄⋊Q₈	—	2	1	no	—
80	C₂×C₄⋊Q₈	—	2	1	no	—
81	C₂×C₄⋊Q₈	Dic₇	1	2	yes	—
82	C₂×C₄⋊C₈	—	7	2	no	—
83	C₂×C₄⋊C₈	—	7	2	no	—
84	C₂×#97	—	14	1	no	—
85	#384	—	2	1	no	—
86	#384	—	2	1	no	—
87	#384	Dic₇	1	2	yes	—
88	#384	C₂×D₇	1	2	yes	—
89	C₄³	C₂×D₇	1	1	yes	characteristic
90	#336	—	14	1	no	—
91	#336	—	14	1	no	—
92	#336	—	14	1	no	—
93	C₂³×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₇×Q₈	—	8	8	no	—
95	C₂×C₄×C₇	—	2	1	no	—
96	C₂×C₄×C₇	—	2	1	no	—
97	C₂×C₄×C₇	—	2	1	no	—
98	C₂×C₄×C₇	—	2	1	no	—
99	C₂×C₄×C₇	—	2	1	no	—
100	C₂×C₄×C₇	—	2	1	no	—
101	C₂×C₄×C₇	—	2	1	no	—
102	C₂×C₄×C₇	—	2	1	no	—
103	C₂×C₄×C₇	—	2	1	no	—
104	C₂×C₄×C₇	—	2	1	no	—
105	C₂×C₄×C₇	—	4	4	no	—
106	C₂×C₄×C₇	—	4	4	no	—
107	C₂×C₄×C₇	—	4	4	no	—
108	C₂×C₄×C₇	#100	1	2	yes	—
109	C₂×C₄×C₇	#101	1	2	yes	—
110	C₂×C₄×C₇	#103	1	2	yes	—
111	C₂×C₄×C₇	#103	1	2	yes	—
112	C₂×C₄×C₇	C₄×Q₈	1	2	yes	—
113	C₂×C₄×C₇	C₂²⋊Q₈	1	2	yes	—
114	C₂×C₄×C₇	#124	1	1	yes	characteristic
115	C₂×C₄×C₇	C₄⋊Q₈	1	1	yes	characteristic
116	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
117	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
118	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
119	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
120	C₂×C₄×C₇	#137	1	2	yes	—
121	C₇⋊C₈	—	4	4	no	—
122	C₇⋊C₈	—	8	2	no	—
123	C₂²×Q₈	—	2	2	no	—
124	C₂²×C₈	—	14	2	no	—
125	C₂²×C₈	—	14	2	no	—
126	C₄⋊Q₈	—	4	4	no	—
127	C₄⋊Q₈	—	4	4	no	—
128	C₄⋊Q₈	—	4	4	no	—
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	—	2	2	no	—
131	C₂×C₄⋊C₄	—	2	2	no	—
132	C₂×C₄⋊C₄	—	2	2	no	—
133	C₂×C₄⋊C₄	—	2	2	no	—
134	C₂×C₄⋊C₄	C₂×Dic₇	1	4	yes	—
135	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
136	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
137	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
138	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
139	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
140	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
141	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
142	C₄⋊C₈	—	7	4	no	—
143	C₄⋊C₈	—	14	2	no	—
144	C₄⋊C₈	—	14	4	no	—
145	#97	—	28	2	no	—
146	C₂²×C₇	#391	1	1	yes	characteristic
147	C₂²×C₇	C₂×#100	1	1	yes	characteristic
148	C₂²×C₇	C₂×#101	1	1	yes	characteristic
149	C₂²×C₇	C₂×#103	1	1	yes	characteristic
150	C₂²×C₇	#418	1	1	yes	characteristic
151	C₂²×C₇	#418	1	1	yes	characteristic
152	C₂²×C₇	#419	1	1	yes	characteristic
153	C₄×C₇	—	2	2	no	—
154	C₄×C₇	—	2	2	no	—
155	C₄×C₇	—	2	2	no	—
156	C₄×C₇	—	2	2	no	—
157	C₄×C₇	—	2	2	no	—
158	C₄×C₇	—	8	4	no	—
159	C₄×C₇	#476	1	2	yes	—
160	C₄×C₇	#477	1	2	yes	—
161	C₄×C₇	#489	1	2	yes	—
162	C₄×C₇	#501	1	2	yes	—
163	C₂×Q₈	—	4	4	no	—
164	C₂×Q₈	—	4	4	no	—
165	C₂×Q₈	—	4	4	no	—
166	C₂²×C₄	—	2	4	no	—
167	C₂²×C₄	C₂²⋊Dic₇	1	1	yes	characteristic
168	C₂²×C₄	C₂²⋊Dic₇	1	1	yes	characteristic
169	C₂²×C₄	C₂²⋊Dic₇	1	1	yes	characteristic
170	C₂²×C₄	C₂²⋊Dic₇	1	1	yes	characteristic
171	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
172	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
173	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
174	C₂×C₈	—	14	4	no	—
175	C₂×C₈	—	28	2	no	—
176	C₂×C₈	—	28	2	no	—
177	C₂×C₈	—	28	2	no	—
178	C₂×C₈	—	28	2	no	—
179	C₂×C₈	—	28	2	no	—
180	C₄⋊C₄	—	4	4	no	—
181	C₄⋊C₄	—	4	4	no	—
182	C₄⋊C₄	—	4	4	no	—
183	C₄⋊C₄	—	4	4	no	—
184	C₄⋊C₄	—	4	4	no	—
185	C₄⋊C₄	—	4	4	no	—
186	C₄²	—	2	2	no	—
187	C₄²	—	2	2	no	—
188	C₄²	—	2	2	no	—
189	C₄²	—	2	2	no	—
190	C₄²	—	2	2	no	—
191	C₄²	#1143	1	2	yes	—
192	C₄²	#1144	1	2	yes	—
193	C₄²	Q₈×D₇	1	2	yes	—
194	C₄²	#1161	1	2	yes	—
195	C₂×C₇	#1548	1	1	yes	characteristic
196	C₂×C₇	#1549	1	1	yes	characteristic
197	C₂×C₇	#1555	1	1	yes	characteristic
198	C₂×C₇	#1556	1	1	yes	characteristic
199	C₂×C₇	#1987	1	1	yes	characteristic
200	C₂×C₇	#1993	1	1	yes	characteristic
201	C₂×C₇	#1997	1	1	yes	characteristic
202	C₂³	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
203	Q₈	—	8	8	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	4	4	no	—
215	C₂×C₄	—	4	4	no	—
216	C₂×C₄	—	4	4	no	—
217	C₂×C₄	#6407	1	2	yes	—
218	C₂×C₄	#6409	1	2	yes	—
219	C₂×C₄	#6409	1	2	yes	—
220	C₂×C₄	#6410	1	2	yes	—
221	C₂×C₄	#6495	1	2	yes	—
222	C₂×C₄	#6495	1	2	yes	—
223	C₂×C₄	C₂×#1143	1	1	yes	characteristic
224	C₂×C₄	#6505	1	2	yes	—
225	C₂×C₄	C₂×#1144	1	1	yes	characteristic
226	C₂×C₄	#6507	1	1	yes	characteristic
227	C₂×C₄	Q₈×Dic₇	1	2	yes	—
228	C₂×C₄	#6509	1	2	yes	—
229	C₂×C₄	#6510	1	1	yes	characteristic
230	C₈	—	28	4	no	—
231	C₈	—	56	2	no	—
232	C₇	#9903	1	1	yes	characteristic, complemented, Sylow
233	C₂²	#89987	1	1	yes	characteristic
234	C₂²	#89987	1	1	yes	characteristic
235	C₂²	C₂×#6407	1	1	yes	characteristic
236	C₂²	C₂×#6409	1	1	yes	characteristic
237	C₂²	#89997	1	1	yes	characteristic
238	C₂²	C₂×#6410	1	1	yes	characteristic
239	C₂²	#90002	1	1	yes	characteristic
240	C₄	—	2	2	no	—
241	C₄	—	2	2	no	—
242	C₄	—	2	2	no	—
243	C₄	—	2	2	no	—
244	C₄	—	2	2	no	—
245	C₄	—	8	4	no	—
246	C₄	#89852	1	2	yes	—
247	C₄	#89856	1	2	yes	—
248	C₄	#89861	1	2	yes	—
249	C₄	#89863	1	2	yes	—
250	C₂	#11729514	1	1	yes	characteristic
251	C₂	#11729528	1	1	yes	characteristic
252	C₂	#11729529	1	1	yes	characteristic
253	C₂	#11729530	1	1	yes	characteristic
254	C₂	#11729533	1	1	yes	characteristic
255	C₂	#11729542	1	1	yes	characteristic
256	C₂	#11729544	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₇
commutator	C₂²×C₄×C₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₇×#2868	C₂
socle, prime radical	C₂³×C₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	#2868	D₇
hypocenter	C₇	#9903
LCS	G ▷ C₂²×C₄×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2868

Complements

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

N	H ≅ G/N	type
41. C₄⋊(C₇⋊C₈)	Q₈	semidirect
232. C₇	#9903	semidirect

Maximal subgroups

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

subgroup	index
C₇×#2868	2
#11727146	2
#11727145	2
#11725440	2
#9903	7