Subgroups — #13268635

Order 1792.

678 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₈×Dic₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×#6386	C₂	1	1	yes	maximal, characteristic
4	#11725610	C₂	1	4	yes	maximal, complemented
5	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
6	C₂²×C₄×Dic₇	C₄	1	1	yes	characteristic
7	C₂³×C₇⋊C₈	C₄	1	1	yes	characteristic
8	C₂×C₇×M₃₂	—	2	2	no	—
9	C₂×C₇×M₃₂	—	2	2	no	—
10	C₂×C₇×M₃₂	C₄	1	4	yes	complemented
11	C₂²×C₇×C₁₆	C₂²	1	2	yes	—
12	C₂×C₈×Dic₇	—	2	2	no	—
13	C₂×C₈×Dic₇	—	2	2	no	—
14	C₂×C₈×Dic₇	C₄	1	2	yes	—
15	C₂×C₈×Dic₇	C₂²	1	2	yes	—
16	C₂×#6386	—	2	2	no	—
17	C₂×#6386	—	2	2	no	—
18	C₂×#6386	C₄	1	4	yes	complemented
19	C₂²×C₇⋊C₁₆	C₂²	1	2	yes	—
20	#12544	—	7	1	no	maximal, Sylow
21	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
22	C₂³×Dic₇	C₈	1	2	yes	—
23	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
25	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
26	C₂²×C₇×C₈	D₄	1	2	yes	—
27	C₂²×C₇×C₈	D₄	1	2	yes	—
28	C₂²×C₇×C₈	D₄	1	2	yes	—
29	C₂²×C₇×C₈	Q₈	1	2	yes	—
30	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
31	C₂×C₄×Dic₇	—	2	2	no	—
32	C₂×C₄×Dic₇	—	2	2	no	—
33	C₂×C₄×Dic₇	C₈	1	2	yes	—
34	C₂×C₄×Dic₇	C₂×C₄	1	2	yes	—
35	C₂²×C₇⋊C₈	—	2	2	no	—
36	C₂²×C₇⋊C₈	—	2	2	no	—
37	C₂²×C₇⋊C₈	C₈	1	4	yes	—
38	C₂²×C₇⋊C₈	C₂×C₄	1	2	yes	—
39	C₇×M₃₂	—	2	8	no	—
40	C₂×C₇×C₁₆	—	2	2	no	—
41	C₂×C₇×C₁₆	—	2	2	no	—
42	C₂×C₇×C₁₆	C₂×C₄	1	4	yes	complemented
43	C₈×Dic₇	—	2	4	no	—
44	C₈×Dic₇	—	2	2	no	—
45	C₈×Dic₇	—	2	2	no	—
46	#6386	—	2	8	no	—
47	C₂×C₇⋊C₁₆	—	2	2	no	—
48	C₂×C₇⋊C₁₆	—	2	2	no	—
49	C₂×C₇⋊C₁₆	C₂×C₄	1	4	yes	complemented
50	C₂²×M₃₂	—	7	1	no	—
51	C₂²×M₃₂	D₇	1	1	yes	characteristic
52	C₂²×C₄×C₈	—	7	1	no	—
53	#1374	—	7	4	no	—
54	C₂⁴×C₇	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₄×C₇	C₄²	1	2	yes	—
56	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
59	C₂²×C₄×C₇	C₂×C₈	1	2	yes	—
60	C₂²×C₄×C₇	M₁₆	1	2	yes	—
61	C₂²×C₄×C₇	M₁₆	1	2	yes	—
62	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
63	C₂²×Dic₇	—	2	4	no	—
64	C₂²×Dic₇	—	2	8	no	—
65	C₂²×Dic₇	C₂×C₈	1	4	yes	—
66	C₂×C₇×C₈	C₄²	1	2	yes	—
67	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
68	C₂×C₇×C₈	C₂²⋊C₄	1	4	yes	—
69	C₂×C₇×C₈	C₂²⋊C₄	1	4	yes	—
70	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
71	C₂×C₇×C₈	C₄⋊C₄	1	4	yes	—
72	C₂×C₇×C₈	C₄⋊C₄	1	4	yes	—
73	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
76	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
77	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
78	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
79	C₄×Dic₇	—	2	4	no	—
80	C₄×Dic₇	—	2	2	no	—
81	C₄×Dic₇	—	2	2	no	—
82	C₂×C₇⋊C₈	—	2	8	no	—
83	C₂×C₇⋊C₈	—	2	2	no	—
84	C₂×C₇⋊C₈	—	2	2	no	—
85	C₂×C₇⋊C₈	C₂×C₈	1	4	yes	—
86	C₇×C₁₆	—	2	4	no	—
87	C₇⋊C₁₆	—	2	4	no	—
88	C₂³×C₈	—	7	1	no	—
89	C₂³×C₈	C₂×D₇	1	1	yes	characteristic
90	C₂²×C₄²	—	7	1	no	—
91	C₂×M₃₂	—	2	2	no	—
92	C₂×M₃₂	—	2	2	no	—
93	C₂×M₃₂	—	7	4	no	—
94	C₂×M₃₂	—	14	2	no	—
95	C₂×M₃₂	—	14	2	no	—
96	C₂×M₃₂	Dic₇	1	4	yes	complemented
97	C₂²×C₁₆	—	7	2	no	—
98	C₂²×C₁₆	C₂×D₇	1	2	yes	—
99	C₂×C₄×C₈	—	7	2	no	—
100	C₂×C₄×C₈	—	7	2	no	—
101	C₂×C₄×C₈	—	14	2	no	—
102	C₂×C₄×C₈	—	14	2	no	—
103	C₂³×C₇	—	2	4	no	—
104	C₂³×C₇	C₄×C₈	1	2	yes	—
105	C₂³×C₇	C₂²⋊C₈	1	2	yes	—
106	C₂³×C₇	C₄⋊C₈	1	2	yes	—
107	C₂³×C₇	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₇	#95	1	4	yes	—
109	C₂×C₄×C₇	C₄×C₈	1	2	yes	—
110	C₂×C₄×C₇	#97	1	4	yes	—
111	C₂×C₄×C₇	C₂²⋊C₈	1	2	yes	—
112	C₂×C₄×C₇	C₂²⋊C₈	1	4	yes	—
113	C₂×C₄×C₇	C₄⋊C₈	1	2	yes	—
114	C₂×C₄×C₇	C₄⋊C₈	1	4	yes	—
115	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
116	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
119	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
120	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
121	C₂×Dic₇	—	2	16	no	—
122	C₂×Dic₇	—	2	4	no	—
123	C₂×Dic₇	—	2	4	no	—
124	C₂×Dic₇	M₃₂	1	8	yes	complemented
125	C₇×C₈	#95	1	4	yes	—
126	C₇×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₇×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₇×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₇×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₇⋊C₈	—	2	4	no	—
131	C₂³×C₄	—	7	2	no	—
132	C₂³×C₄	C₄×D₇	1	1	yes	characteristic
133	C₂²×C₈	—	7	4	no	—
134	C₂²×C₈	—	7	2	no	—
135	C₂²×C₈	—	14	2	no	—
136	C₂²×C₈	—	14	2	no	—
137	C₂²×C₈	Dic₁₄	1	2	yes	—
138	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
139	C₂²×C₈	C₄×D₇	1	2	yes	—
140	C₂²×C₈	D₂₈	1	2	yes	—
141	C₂²×C₈	C₂×Dic₇	1	2	yes	—
142	C₂²×C₈	C₂²⋊D₇	1	2	yes	—
143	C₂²×C₈	C₂²⋊D₇	1	2	yes	—
144	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
145	C₂×C₄²	—	7	2	no	—
146	C₂×C₄²	—	7	2	no	—
147	C₂×C₄²	—	14	2	no	—
148	C₂×C₄²	—	14	2	no	—
149	M₃₂	—	2	8	no	—
150	M₃₂	—	14	8	no	—
151	C₂×C₁₆	—	2	2	no	—
152	C₂×C₁₆	—	2	2	no	—
153	C₂×C₁₆	—	7	4	no	—
154	C₂×C₁₆	—	14	2	no	—
155	C₂×C₁₆	—	14	2	no	—
156	C₂×C₁₆	C₂×Dic₇	1	4	yes	complemented
157	C₄×C₈	—	14	4	no	—
158	C₄×C₈	—	14	2	no	—
159	C₄×C₈	—	14	2	no	—
160	C₂²×C₇	—	2	4	no	—
161	C₂²×C₇	—	2	4	no	—
162	C₂²×C₇	—	2	4	no	—
163	C₂²×C₇	#336	1	4	yes	—
164	C₂²×C₇	C₂×C₄×C₈	1	1	yes	characteristic
165	C₂²×C₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
166	C₂²×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
167	C₂²×C₇	C₂×M₃₂	1	4	yes	—
168	C₄×C₇	#336	1	4	yes	—
169	C₄×C₇	C₂×#95	1	1	yes	characteristic
170	C₄×C₇	C₂×#97	1	1	yes	characteristic
171	C₄×C₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
172	C₄×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
173	Dic₇	—	2	8	no	—
174	C₂⁴	C₈×D₇	1	1	yes	characteristic
175	C₂²×C₄	—	7	4	no	—
176	C₂²×C₄	—	14	8	no	—
177	C₂²×C₄	—	14	4	no	—
178	C₂²×C₄	C₈×D₇	1	2	yes	—
179	C₂²×C₄	C₈×D₇	1	1	yes	characteristic
180	C₂²×C₄	#1131	1	2	yes	—
181	C₂²×C₄	#1131	1	2	yes	—
182	C₂²×C₄	C₄×Dic₇	1	2	yes	—
183	C₂²×C₄	Dic₇⋊C₄	1	2	yes	—
184	C₂²×C₄	(C₂×D₇)⋊C₄	1	2	yes	—
185	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
186	C₂×C₈	—	7	4	no	—
187	C₂×C₈	—	14	2	no	—
188	C₂×C₈	—	14	2	no	—
189	C₂×C₈	—	14	8	no	—
190	C₂×C₈	C₄×Dic₇	1	2	yes	—
191	C₂×C₈	Dic₇⋊C₄	1	2	yes	—
192	C₂×C₈	Dic₇⋊C₄	1	4	yes	—
193	C₂×C₈	C₄⋊Dic₇	1	4	yes	—
194	C₂×C₈	(C₂×D₇)⋊C₄	1	2	yes	—
195	C₂×C₈	(C₂×D₇)⋊C₄	1	4	yes	—
196	C₂×C₈	C₂²⋊Dic₇	1	4	yes	—
197	C₂×C₈	C₂×Dic₁₄	1	1	yes	characteristic
198	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
199	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
200	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
201	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
202	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
203	C₄²	—	14	2	no	—
204	C₄²	—	14	2	no	—
205	C₄²	—	14	4	no	—
206	C₁₆	—	2	4	no	—
207	C₁₆	—	14	4	no	—
208	C₂×C₇	—	2	4	no	—
209	C₂×C₇	C₂×#336	1	1	yes	characteristic
210	C₂×C₇	C₄×M₃₂	1	2	yes	—
211	C₂×C₇	#2112	1	2	yes	—
212	C₂×C₇	#2150	1	2	yes	—
213	C₂³	—	2	4	no	—
214	C₂³	C₈×Dic₇	1	2	yes	—
215	C₂³	Dic₇⋊C₈	1	2	yes	—
216	C₂³	(C₂×D₇)⋊C₈	1	2	yes	—
217	C₂³	C₂×C₈×D₇	1	1	yes	characteristic
218	C₂×C₄	—	7	8	no	—
219	C₂×C₄	—	14	16	no	—
220	C₂×C₄	—	14	4	no	—
221	C₂×C₄	—	14	4	no	—
222	C₂×C₄	C₈×Dic₇	1	2	yes	—
223	C₂×C₄	Dic₇⋊C₈	1	2	yes	—
224	C₂×C₄	Dic₇⋊C₈	1	4	yes	—
225	C₂×C₄	#6389	1	4	yes	—
226	C₂×C₄	(C₂×D₇)⋊C₈	1	2	yes	—
227	C₂×C₄	(C₂×D₇)⋊C₈	1	4	yes	—
228	C₂×C₄	C₇⋊#95	1	4	yes	—
229	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
230	C₂×C₄	C₂×#1131	1	1	yes	characteristic
231	C₂×C₄	C₂×#1131	1	1	yes	characteristic
232	C₂×C₄	C₂×C₄×Dic₇	1	1	yes	characteristic
233	C₂×C₄	C₂×Dic₇⋊C₄	1	1	yes	characteristic
234	C₂×C₄	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
235	C₈	—	14	4	no	—
236	C₈	C₇⋊#95	1	4	yes	—
237	C₈	C₂×Dic₇⋊C₄	1	1	yes	characteristic
238	C₈	C₂×C₄⋊Dic₇	1	1	yes	characteristic
239	C₈	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
240	C₈	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
241	C₇	#12544	1	1	yes	characteristic, complemented, Sylow
242	C₂²	—	2	4	no	—
243	C₂²	—	2	4	no	—
244	C₂²	—	2	4	no	—
245	C₂²	#89345	1	4	yes	—
246	C₂²	D₇×M₃₂	1	4	yes	—
247	C₂²	C₂×C₈×Dic₇	1	1	yes	characteristic
248	C₂²	C₂×Dic₇⋊C₈	1	1	yes	characteristic
249	C₂²	C₂×(C₂×D₇)⋊C₈	1	1	yes	characteristic
250	C₄	—	14	8	no	—
251	C₄	#89345	1	4	yes	—
252	C₄	C₂×Dic₇⋊C₈	1	1	yes	characteristic
253	C₄	C₂×#6389	1	1	yes	characteristic
254	C₄	C₂×(C₂×D₇)⋊C₈	1	1	yes	characteristic
255	C₄	C₂×C₇⋊#95	1	1	yes	characteristic
256	C₂	—	2	4	no	—
257	C₂	Dic₇×M₃₂	1	2	yes	—
258	C₂	#11729174	1	2	yes	—
259	C₂	#11729185	1	2	yes	—
260	C₂	C₂×#89345	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₇
commutator	C₂²×C₇	C₂×C₄×C₈
Fitting	C₂²×C₇×M₃₂	C₂
socle	C₂³×C₇	C₂²×C₈
prime radical	C₂⁴×C₇	C₂×C₈
cosocle	C₂²×C₇×C₈	C₂³
hypercenter	C₂²×M₃₂	D₇
hypocenter	C₇	#12544
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
4. #11725610	C₂	semidirect
10. C₂×C₇×M₃₂	C₄	semidirect
18. C₂×#6386	C₄	semidirect
42. C₂×C₇×C₁₆	C₂×C₄	semidirect
49. C₂×C₇⋊C₁₆	C₂×C₄	semidirect
96. C₂×M₃₂	Dic₇	semidirect
124. C₂×Dic₇	M₃₂	semidirect
156. C₂×C₁₆	C₂×Dic₇	semidirect
241. C₇	#12544	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₇×M₃₂	2
C₂²×C₈×Dic₇	2
C₂²×#6386	2
#11725610	2
#12544	7