Subgroups — #13247539

Order 1792.

422 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#1831	C₂	1	1	yes	maximal, characteristic
2	C₂×#89274	C₂	1	2	yes	maximal
3	C₂×#89275	C₂	1	2	yes	maximal
4	#11727146	C₂	1	1	yes	maximal, characteristic
5	C₄²×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
7	C₄³×C₇	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
9	C₂×C₄×C₇⋊C₈	C₄	1	1	yes	characteristic
10	C₂×C₄×C₇⋊C₈	C₄	1	1	yes	characteristic
11	C₂×C₄×C₇⋊C₈	C₄	1	1	yes	characteristic
12	C₂×C₄×C₇⋊C₈	C₄	1	1	yes	characteristic
13	C₂×C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
15	C₇×#336	—	2	2	no	—
16	#89319	—	2	2	no	—
17	#89275	—	2	4	no	—
18	#89274	—	2	4	no	—
19	#8807	—	7	1	no	maximal, Sylow
20	C₂²×C₇×C₈	—	2	2	no	—
21	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₇	D₄	1	2	yes	—
24	C₂×C₄²×C₇	Q₈	1	2	yes	—
25	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
26	C₂²×C₇⋊C₈	—	2	2	no	—
27	C₂²×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
31	C₇×C₄⋊C₈	—	2	2	no	—
32	C₇×C₄⋊C₈	—	2	2	no	—
33	C₄⋊(C₇⋊C₈)	—	2	2	no	—
34	C₄⋊(C₇⋊C₈)	—	2	2	no	—
35	C₄×C₇⋊C₈	—	2	2	no	—
36	C₄×C₇⋊C₈	—	2	2	no	—
37	C₄×C₇⋊C₈	—	2	2	no	—
38	C₄×C₇⋊C₈	C₈	1	4	yes	complemented
39	C₄×C₇⋊C₈	D₄	1	2	yes	—
40	C₄×C₇⋊C₈	D₄	1	2	yes	—
41	C₄×C₇⋊C₈	Q₈	1	2	yes	—
42	C₄×C₇⋊C₈	Q₈	1	2	yes	—
43	#1831	—	7	1	no	—
44	#1831	D₇	1	1	yes	characteristic
45	C₄²×C₈	—	7	1	no	—
46	C₂×#335	—	7	2	no	—
47	C₂×#334	—	7	2	no	—
48	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
50	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₇×C₈	—	4	2	no	—
54	C₂×C₇×C₈	—	4	4	no	—
55	C₄²×C₇	—	2	4	no	—
56	C₄²×C₇	—	2	1	no	—
57	C₄²×C₇	—	2	1	no	—
58	C₄²×C₇	—	2	4	no	—
59	C₄²×C₇	C₂×C₈	1	2	yes	—
60	C₄²×C₇	M₁₆	1	2	yes	—
61	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
62	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
63	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
64	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
65	C₂×C₇⋊C₈	—	2	1	no	—
66	C₂×C₇⋊C₈	—	2	1	no	—
67	C₂×C₇⋊C₈	—	2	1	no	—
68	C₂×C₇⋊C₈	—	2	1	no	—
69	C₂×C₇⋊C₈	—	4	2	no	—
70	C₂×C₇⋊C₈	—	4	4	no	—
71	C₂×C₇⋊C₈	C₄⋊C₄	1	2	yes	—
72	C₂×C₇⋊C₈	C₄⋊C₄	1	2	yes	—
73	C₂×C₇⋊C₈	C₄⋊C₄	1	2	yes	—
74	C₂×C₇⋊C₈	C₄⋊C₄	1	2	yes	—
75	C₂×C₇⋊C₈	C₂×C₈	1	2	yes	—
76	C₂×C₇⋊C₈	C₂×C₈	1	2	yes	—
77	C₂×C₇⋊C₈	M₁₆	1	2	yes	—
78	C₂×C₇⋊C₈	M₁₆	1	2	yes	—
79	C₂×C₄⋊C₈	—	7	2	no	—
80	C₂×C₄⋊C₈	C₂×D₇	1	2	yes	—
81	C₂×C₄×C₈	—	7	1	no	—
82	C₂×C₄×C₈	—	7	1	no	—
83	C₂×C₄×C₈	—	7	1	no	—
84	C₂×C₄×C₈	—	7	1	no	—
85	C₂×C₄×C₈	—	7	1	no	—
86	C₂×C₄×C₈	—	7	1	no	—
87	C₄³	C₂×D₇	1	1	yes	characteristic
88	#336	—	2	2	no	—
89	#336	—	14	2	no	—
90	#335	—	14	4	no	—
91	#334	—	14	4	no	—
92	C₂³×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₇	—	2	2	no	—
94	C₂×C₄×C₇	—	2	2	no	—
95	C₂×C₄×C₇	—	2	2	no	—
96	C₂×C₄×C₇	—	2	2	no	—
97	C₂×C₄×C₇	—	2	2	no	—
98	C₂×C₄×C₇	C₄⋊C₈	1	4	yes	—
99	C₂×C₄×C₇	#108	1	4	yes	—
100	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₇	#127	1	1	yes	characteristic
103	C₂×C₄×C₇	C₄⋊Q₈	1	1	yes	characteristic
104	C₂×C₄×C₇	C₄⋊Q₈	1	1	yes	characteristic
105	C₂×C₄×C₇	C₄⋊Q₈	1	1	yes	characteristic
106	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
107	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
108	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
109	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
110	C₂×C₄×C₇	#135	1	4	yes	—
111	C₇×C₈	—	8	2	no	—
112	C₇⋊C₈	—	2	2	no	—
113	C₇⋊C₈	—	2	2	no	—
114	C₇⋊C₈	—	8	2	no	—
115	C₇⋊C₈	C₄⋊C₈	1	4	yes	—
116	C₇⋊C₈	C₄⋊C₈	1	4	yes	—
117	C₂²×C₈	—	2	2	no	—
118	C₂²×C₈	—	7	1	no	—
119	C₂²×C₈	—	7	1	no	—
120	C₂²×C₈	—	7	1	no	—
121	C₂²×C₈	—	7	1	no	—
122	C₂²×C₈	—	14	2	no	—
123	C₂×C₄²	Dic₁₄	1	2	yes	—
124	C₂×C₄²	C₄×D₇	1	1	yes	characteristic
125	C₂×C₄²	C₄×D₇	1	1	yes	characteristic
126	C₂×C₄²	C₂²⋊D₇	1	2	yes	—
127	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
128	C₄⋊C₈	—	2	2	no	—
129	C₄⋊C₈	—	2	2	no	—
130	C₄⋊C₈	—	14	2	no	—
131	C₄⋊C₈	—	14	2	no	—
132	C₄×C₈	—	7	4	no	—
133	C₄×C₈	—	7	2	no	—
134	C₄×C₈	—	7	2	no	—
135	C₄×C₈	—	7	2	no	—
136	C₄×C₈	—	7	2	no	—
137	C₄×C₈	—	14	2	no	—
138	C₄×C₈	—	14	2	no	—
139	C₄×C₈	—	14	2	no	—
140	C₂²×C₇	#384	1	1	yes	characteristic
141	C₂²×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
142	C₂²×C₇	#423	1	1	yes	characteristic
143	C₂²×C₇	#427	1	2	yes	—
144	C₂²×C₇	C₂×#108	1	2	yes	—
145	C₄×C₇	—	2	4	no	—
146	C₄×C₇	—	2	4	no	—
147	C₄×C₇	—	2	1	no	—
148	C₄×C₇	—	2	1	no	—
149	C₄×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
150	C₄×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
151	C₄×C₇	#423	1	1	yes	characteristic
152	C₄×C₇	#423	1	1	yes	characteristic
153	C₄×C₇	#495	1	2	yes	—
154	C₄×C₇	#499	1	2	yes	—
155	C₂²×C₄	Dic₇⋊C₄	1	2	yes	—
156	C₂²×C₄	Dic₇⋊C₄	1	2	yes	—
157	C₂²×C₄	C₂×Dic₁₄	1	1	yes	characteristic
158	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
159	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
160	C₂×C₈	—	4	2	no	—
161	C₂×C₈	—	4	4	no	—
162	C₂×C₈	—	7	2	no	—
163	C₂×C₈	—	7	2	no	—
164	C₂×C₈	—	7	2	no	—
165	C₂×C₈	—	7	2	no	—
166	C₂×C₈	—	7	2	no	—
167	C₂×C₈	—	7	2	no	—
168	C₂×C₈	—	7	2	no	—
169	C₂×C₈	—	7	2	no	—
170	C₂×C₈	—	14	1	no	—
171	C₂×C₈	—	14	1	no	—
172	C₂×C₈	—	14	1	no	—
173	C₂×C₈	—	14	1	no	—
174	C₂×C₈	—	28	2	no	—
175	C₂×C₈	—	28	4	no	—
176	C₄²	—	2	1	no	—
177	C₄²	—	2	1	no	—
178	C₄²	—	2	4	no	—
179	C₄²	—	2	4	no	—
180	C₄²	C₈×D₇	1	2	yes	—
181	C₄²	#1131	1	2	yes	—
182	C₄²	D₄×D₇	1	1	yes	characteristic
183	C₄²	D₄×D₇	1	1	yes	characteristic
184	C₄²	Q₈×D₇	1	1	yes	characteristic
185	C₄²	Q₈×D₇	1	1	yes	characteristic
186	C₂×C₇	#1564	1	2	yes	—
187	C₂×C₇	#1568	1	2	yes	—
188	C₂×C₇	#1842	1	1	yes	characteristic
189	C₂×C₇	#1846	1	1	yes	characteristic
190	C₂×C₇	#1848	1	1	yes	characteristic
191	C₂³	C₂×Dic₇⋊C₄	1	1	yes	characteristic
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	Dic₇⋊C₈	1	4	yes	—
198	C₂×C₄	#6396	1	4	yes	—
199	C₂×C₄	#6451	1	1	yes	characteristic
200	C₂×C₄	#6451	1	1	yes	characteristic
201	C₂×C₄	D₇×C₄⋊C₄	1	1	yes	characteristic
202	C₂×C₄	D₇×C₄⋊C₄	1	1	yes	characteristic
203	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
204	C₂×C₄	C₂×#1131	1	1	yes	characteristic
205	C₂×C₄	D₇×M₁₆	1	1	yes	characteristic
206	C₂×C₄	D₇×M₁₆	1	1	yes	characteristic
207	C₂×C₄	#6503	1	1	yes	characteristic
208	C₂×C₄	#6507	1	1	yes	characteristic
209	C₂×C₄	#6513	1	4	yes	—
210	C₈	—	7	4	no	—
211	C₈	—	7	4	no	—
212	C₈	—	8	2	no	—
213	C₈	—	14	2	no	—
214	C₈	—	14	2	no	—
215	C₈	—	56	2	no	—
216	C₇	#8807	1	1	yes	characteristic, complemented, Sylow
217	C₂²	#89748	1	1	yes	characteristic
218	C₂²	#89770	1	2	yes	—
219	C₂²	C₂×Dic₇⋊C₈	1	1	yes	characteristic
220	C₂²	#89890	1	1	yes	characteristic
221	C₂²	C₂×#6396	1	2	yes	—
222	C₄	—	2	4	no	—
223	C₄	—	2	1	no	—
224	C₄	—	2	1	no	—
225	C₄	—	2	4	no	—
226	C₄	D₇×C₄⋊C₈	1	1	yes	characteristic
227	C₄	D₇×C₄⋊C₈	1	1	yes	characteristic
228	C₄	#89609	1	1	yes	characteristic
229	C₄	#89609	1	1	yes	characteristic
230	C₄	#89831	1	2	yes	—
231	C₄	#89836	1	2	yes	—
232	C₂	#11727303	1	1	yes	characteristic
233	C₂	#11728199	1	1	yes	characteristic
234	C₂	#11728272	1	1	yes	characteristic
235	C₂	#11728499	1	2	yes	—
236	C₂	#11728548	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₇
commutator	C₂×C₄×C₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₇×#1831	C₂
socle, prime radical	C₂³×C₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	#1831	D₇
hypocenter	C₇	#8807
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1831

Complements

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

N	H ≅ G/N	type
38. C₄×C₇⋊C₈	C₈	semidirect
216. C₇	#8807	semidirect

Maximal subgroups

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

subgroup	index
C₇×#1831	2
C₂×#89274	2
C₂×#89275	2
#11727146	2
C₄²×C₇⋊C₈	2
#8807	7