Subgroups — #13263519

Order 1792.

414 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×C₇×#436	C₂	1	1	yes	maximal, characteristic
2	#11727497	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#6386	C₂	1	1	yes	maximal, characteristic, complemented
4	#11725567	C₂	1	4	yes	maximal, complemented
5	C₂²×C₇×M₁₆	—	2	1	no	—
6	C₂×C₄×C₇×D₄	C₄	1	1	yes	characteristic
7	C₇×#436	—	2	1	no	—
8	C₇×#436	—	2	1	no	—
9	C₇×#436	C₄	1	2	yes	—
10	C₇×#436	C₂²	1	2	yes	—
11	C₂×C₇×C₄⋊C₈	C₄	1	1	yes	characteristic
12	C₂×C₇×C₂²⋊C₈	—	2	1	no	—
13	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
14	C₂×#6386	—	2	1	no	—
15	C₂×#6386	—	2	1	no	—
16	C₄⋊(C₇⋊C₁₆)	—	2	1	no	—
17	C₄⋊(C₇⋊C₁₆)	C₂²	1	2	yes	complemented
18	#89256	C₂²	1	2	yes	complemented
19	C₄×C₇⋊C₁₆	—	2	1	no	—
20	#11587	—	7	1	no	maximal, Sylow
21	C₂²×C₇×D₄	C₈	1	1	yes	characteristic
22	C₂³×C₄×C₇	—	2	1	no	—
23	C₂×C₇×M₁₆	—	2	2	no	—
24	C₂×C₇×M₁₆	—	2	2	no	—
25	C₂×C₇×M₁₆	—	4	2	no	—
26	C₂×C₇×M₁₆	—	4	2	no	—
27	C₂²×C₇×C₈	—	2	1	no	—
28	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
29	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
30	C₄×C₇×D₄	—	2	1	no	—
31	C₄×C₇×D₄	—	2	1	no	—
32	C₄×C₇×D₄	C₈	1	2	yes	—
33	C₄×C₇×D₄	C₂×C₄	1	2	yes	—
34	C₂×C₇×C₄⋊C₄	C₈	1	1	yes	characteristic
35	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
36	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
37	C₇×C₄⋊C₈	—	2	1	no	—
38	C₇×C₄⋊C₈	C₂×C₄	1	2	yes	—
39	C₇×C₂²⋊C₈	—	2	2	no	—
40	C₇×C₂²⋊C₈	—	2	2	no	—
41	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
42	C₄×C₇×C₈	D₄	1	1	yes	characteristic
43	C₄×C₇×C₈	D₄	1	1	yes	characteristic
44	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
45	#6386	—	2	4	no	—
46	#6386	—	4	2	no	—
47	C₂×C₇⋊C₁₆	—	2	2	no	—
48	C₂×C₇⋊C₁₆	—	2	2	no	—
49	C₂×#436	D₇	1	1	yes	characteristic
50	#2150	—	7	1	no	—
51	C₄×M₃₂	—	7	1	no	—
52	#1330	—	7	4	no	—
53	C₂⁴×C₇	—	2	1	no	—
54	C₂×C₇×D₄	—	2	1	no	—
55	C₂×C₇×D₄	—	4	2	no	—
56	C₂×C₇×D₄	C₂×C₈	1	2	yes	—
57	C₂²×C₄×C₇	—	2	1	no	—
58	C₂²×C₄×C₇	—	2	2	no	—
59	C₂²×C₄×C₇	—	2	2	no	—
60	C₂²×C₄×C₇	—	4	2	no	—
61	C₂²×C₄×C₇	—	4	2	no	—
62	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
64	C₂²×C₄×C₇	M₁₆	1	1	yes	characteristic
65	C₇×M₁₆	—	4	4	no	—
66	C₇×M₁₆	—	4	2	no	—
67	C₇×M₁₆	—	4	2	no	—
68	C₂×C₇×C₈	—	2	2	no	—
69	C₂×C₇×C₈	—	2	2	no	—
70	C₂×C₇×C₈	—	4	2	no	—
71	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₇×C₈	D₈	1	2	yes	—
74	C₂×C₇×C₈	SD₁₆	1	2	yes	—
75	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
76	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
77	C₇×C₄⋊C₄	—	2	1	no	—
78	C₇×C₄⋊C₄	C₂×C₈	1	2	yes	—
79	C₇×C₂²⋊C₄	—	2	2	no	—
80	C₇×C₂²⋊C₄	—	2	2	no	—
81	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
82	C₄²×C₇	C₂×C₈	1	1	yes	characteristic
83	C₄²×C₇	M₁₆	1	1	yes	characteristic
84	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
85	C₇⋊C₁₆	—	2	4	no	—
86	C₇⋊C₁₆	—	4	2	no	—
87	C₂²×M₁₆	—	2	1	no	—
88	C₂×C₄×D₄	Dic₇	1	1	yes	characteristic
89	C₂×M₃₂	—	14	1	no	—
90	C₂×M₃₂	—	14	1	no	—
91	#436	—	2	1	no	—
92	#436	—	2	1	no	—
93	#436	Dic₇	1	2	yes	—
94	#436	C₂×D₇	1	2	yes	—
95	C₂×C₄⋊C₈	Dic₇	1	1	yes	characteristic
96	C₂×C₂²⋊C₈	—	2	1	no	—
97	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
98	C₄⋊C₁₆	—	7	2	no	—
99	C₄⋊C₁₆	—	14	1	no	—
100	#346	—	7	2	no	—
101	C₄×C₁₆	—	14	1	no	—
102	C₂³×C₇	—	2	2	no	—
103	C₂³×C₇	—	4	2	no	—
104	C₂³×C₇	—	4	2	no	—
105	C₂³×C₇	—	4	2	no	—
106	C₂³×C₇	C₂²⋊C₈	1	1	yes	characteristic
107	C₇×D₄	—	4	2	no	—
108	C₇×D₄	—	4	2	no	—
109	C₂×C₄×C₇	—	2	2	no	—
110	C₂×C₄×C₇	—	2	2	no	—
111	C₂×C₄×C₇	—	4	1	no	—
112	C₂×C₄×C₇	—	4	2	no	—
113	C₂×C₄×C₇	—	4	4	no	—
114	C₂×C₄×C₇	—	4	2	no	—
115	C₂×C₄×C₇	—	4	2	no	—
116	C₂×C₄×C₇	C₂²⋊C₈	1	1	yes	characteristic
117	C₂×C₄×C₇	C₂²⋊C₈	1	1	yes	characteristic
118	C₂×C₄×C₇	C₂²⋊C₈	1	1	yes	characteristic
119	C₂×C₄×C₇	#102	1	2	yes	—
120	C₂×C₄×C₇	#104	1	2	yes	—
121	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
123	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
124	C₇×C₈	—	2	1	no	—
125	C₇×C₈	—	2	1	no	—
126	C₇×C₈	—	4	2	no	—
127	C₇×C₈	#102	1	2	yes	—
128	C₇×C₈	C₂×D₈	1	1	yes	characteristic
129	C₇×C₈	C₂×SD₁₆	1	1	yes	characteristic
130	C₂²×D₄	C₇⋊C₈	1	1	yes	characteristic
131	C₂³×C₄	—	2	1	no	—
132	C₂×M₁₆	—	2	2	no	—
133	C₂×M₁₆	—	2	2	no	—
134	C₂×M₁₆	—	4	2	no	—
135	C₂×M₁₆	—	4	2	no	—
136	C₂²×C₈	—	2	1	no	—
137	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
138	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
139	C₄×D₄	—	2	1	no	—
140	C₄×D₄	—	2	1	no	—
141	C₄×D₄	C₇⋊C₈	1	2	yes	—
142	C₄×D₄	C₂×Dic₇	1	2	yes	—
143	C₂×C₄⋊C₄	C₇⋊C₈	1	1	yes	characteristic
144	C₂×C₂²⋊C₄	—	2	1	no	—
145	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
146	M₃₂	—	14	4	no	—
147	M₃₂	—	28	2	no	—
148	C₂×C₁₆	—	14	2	no	—
149	C₂×C₁₆	—	14	2	no	—
150	C₄⋊C₈	—	2	1	no	—
151	C₄⋊C₈	C₂×Dic₇	1	2	yes	—
152	C₂²⋊C₈	—	2	2	no	—
153	C₂²⋊C₈	—	2	2	no	—
154	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
155	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
156	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
157	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
158	C₂²×C₇	—	2	1	no	—
159	C₂²×C₇	—	2	1	no	—
160	C₂²×C₇	—	4	2	no	—
161	C₂²×C₇	—	4	2	no	—
162	C₂²×C₇	—	4	2	no	—
163	C₂²×C₇	—	8	4	no	—
164	C₂²×C₇	D₄⋊C₈	1	2	yes	—
165	C₂²×C₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
166	C₄×C₇	—	2	1	no	—
167	C₄×C₇	—	2	1	no	—
168	C₄×C₇	—	2	1	no	—
169	C₄×C₇	—	4	2	no	—
170	C₄×C₇	D₄⋊C₈	1	2	yes	—
171	C₄×C₇	#349	1	2	yes	—
172	C₄×C₇	C₂×#102	1	1	yes	characteristic
173	C₄×C₇	C₂×#104	1	1	yes	characteristic
174	C₂⁴	—	2	1	no	—
175	C₂×D₄	—	2	1	no	—
176	C₂×D₄	—	4	2	no	—
177	C₂×D₄	C₂×C₇⋊C₈	1	2	yes	—
178	C₂²×C₄	—	2	1	no	—
179	C₂²×C₄	—	2	2	no	—
180	C₂²×C₄	—	2	2	no	—
181	C₂²×C₄	—	4	2	no	—
182	C₂²×C₄	—	4	2	no	—
183	C₂²×C₄	C₂×C₇⋊C₈	1	1	yes	characteristic
184	C₂²×C₄	#1136	1	1	yes	characteristic
185	C₂²×C₄	C₂²⋊Dic₇	1	1	yes	characteristic
186	M₁₆	—	4	4	no	—
187	M₁₆	—	4	2	no	—
188	M₁₆	—	4	2	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	4	2	no	—
192	C₂×C₈	#1141	1	2	yes	—
193	C₂×C₈	#1142	1	2	yes	—
194	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
195	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
196	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
197	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
198	C₄⋊C₄	—	2	1	no	—
199	C₄⋊C₄	C₂×C₇⋊C₈	1	2	yes	—
200	C₂²⋊C₄	—	2	2	no	—
201	C₂²⋊C₄	—	2	2	no	—
202	C₄²	C₂×C₇⋊C₈	1	1	yes	characteristic
203	C₄²	#1136	1	1	yes	characteristic
204	C₄²	C₂²⋊Dic₇	1	1	yes	characteristic
205	C₄²	C₂²×Dic₇	1	1	yes	characteristic
206	C₁₆	—	14	4	no	—
207	C₁₆	—	28	2	no	—
208	C₂×C₇	—	2	2	no	—
209	C₂×C₇	—	8	2	no	—
210	C₂×C₇	C₂×D₄⋊C₈	1	1	yes	characteristic
211	C₂×C₇	C₂×#349	1	1	yes	characteristic
212	C₂×C₇	#2117	1	1	yes	characteristic
213	C₂³	—	2	2	no	—
214	C₂³	—	4	2	no	—
215	C₂³	—	4	2	no	—
216	C₂³	—	4	2	no	—
217	C₂³	C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic
218	D₄	—	4	2	no	—
219	D₄	—	4	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	4	1	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	4	4	no	—
227	C₂×C₄	C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic
228	C₂×C₄	C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic
229	C₂×C₄	C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic
230	C₂×C₄	#6406	1	2	yes	—
231	C₂×C₄	#6411	1	2	yes	—
232	C₂×C₄	C₂²×C₇⋊C₈	1	1	yes	characteristic
233	C₂×C₄	C₂×#1136	1	1	yes	characteristic
234	C₂×C₄	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
235	C₈	—	2	1	no	—
236	C₈	—	2	1	no	—
237	C₈	—	4	2	no	—
238	C₈	#6406	1	2	yes	—
239	C₈	C₂×#1141	1	1	yes	characteristic
240	C₈	C₂×#1142	1	1	yes	characteristic
241	C₇	#11587	1	1	yes	characteristic, complemented, Sylow
242	C₂²	—	2	1	no	—
243	C₂²	—	2	1	no	—
244	C₂²	—	4	2	no	—
245	C₂²	—	4	2	no	—
246	C₂²	—	4	2	no	—
247	C₂²	—	8	4	no	—
248	C₂²	D₄⋊(C₇⋊C₈)	1	2	yes	—
249	C₂²	C₂×C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	4	2	no	—
254	C₄	D₄⋊(C₇⋊C₈)	1	2	yes	—
255	C₄	#89348	1	2	yes	—
256	C₄	C₂×#6406	1	1	yes	characteristic
257	C₄	C₂×#6411	1	1	yes	characteristic
258	C₂	—	2	2	no	—
259	C₂	—	8	2	no	—
260	C₂	C₂×D₄⋊(C₇⋊C₈)	1	1	yes	characteristic
261	C₂	C₂×#89348	1	1	yes	characteristic
262	C₂	#11729760	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₇
commutator	C₂×C₄×C₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₇
Fitting	C₂×C₇×#436	C₂
socle	C₂²×C₇	C₂×C₂²⋊C₈
prime radical	C₂²×C₇×D₄	C₈
cosocle	C₄×C₇×C₈	C₂³
hypercenter	C₂×#436	D₇
hypocenter	C₇	#11587
LCS	G ▷ C₂×C₄×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#436

Complements

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

N	H ≅ G/N	type
2. #11727497	C₂	semidirect
3. C₄×#6386	C₂	semidirect
4. #11725567	C₂	semidirect
17. C₄⋊(C₇⋊C₁₆)	C₂²	semidirect
18. #89256	C₂²	semidirect
241. C₇	#11587	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×#436	2
#11727497	2
C₄×#6386	2
#11725567	2
#11587	7