Subgroups — #11731137

Order 896.

496 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#6461	C₂	1	2	yes	maximal, complemented
2	C₄×C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#89766	C₂	1	2	yes	maximal, complemented
4	#89711	C₂	1	1	yes	maximal, characteristic
5	#89711	C₂	1	1	yes	maximal, characteristic
6	#89711	C₂	1	1	yes	maximal, characteristic
7	#89707	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₄⋊Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#89700	C₂	1	1	yes	maximal, characteristic, complemented
10	#89446	C₂	1	2	yes	maximal, complemented
11	#89442	C₂	1	2	yes	maximal, complemented
12	C₂²×D₂₈	C₂²	1	1	yes	characteristic
13	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
17	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
18	C₂×(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
19	C₂×(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
20	C₂×(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
21	C₂×(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
22	C₂×(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
23	C₂×(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
24	C₂×C₄⋊Dic₇	C₂²	1	2	yes	—
25	C₂×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊Dic₇	C₂²	1	2	yes	—
29	C₂×Dic₇⋊C₄	C₂²	1	2	yes	—
30	C₂×C₄×Dic₇	C₂²	1	2	yes	—
31	#6461	—	2	8	no	—
32	C₇×#95	C₂²	1	1	yes	characteristic
33	C₇×#95	C₂²	1	1	yes	characteristic
34	C₇⋊#95	C₂²	1	2	yes	—
35	C₇⋊#95	C₂²	1	2	yes	—
36	#2512	—	7	1	no	maximal, Sylow
37	C₂³×D₇	C₂³	1	2	yes	—
38	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
45	C₂²×Dic₇	C₂³	1	2	yes	—
46	C₂²×Dic₇	C₂³	1	2	yes	—
47	C₂²×Dic₇	C₂³	1	2	yes	—
48	C₂×D₂₈	—	2	2	no	—
49	C₂×D₂₈	—	2	2	no	—
50	C₂×D₂₈	—	2	2	no	—
51	C₇×C₄⋊C₄	—	2	2	no	—
52	C₇×C₄⋊C₄	D₄	1	4	yes	complemented
53	C₄²×C₇	—	2	2	no	—
54	C₄²×C₇	—	2	2	no	—
55	C₄²×C₇	—	2	2	no	—
56	(C₂×D₇)⋊C₄	—	2	4	no	—
57	(C₂×D₇)⋊C₄	—	2	4	no	—
58	(C₂×D₇)⋊C₄	—	2	4	no	—
59	(C₂×D₇)⋊C₄	—	2	4	no	—
60	(C₂×D₇)⋊C₄	—	4	2	no	—
61	(C₂×D₇)⋊C₄	—	4	2	no	—
62	C₄⋊Dic₇	—	2	4	no	—
63	C₄⋊Dic₇	—	2	2	no	—
64	C₄⋊Dic₇	—	2	2	no	—
65	C₄⋊Dic₇	—	2	2	no	—
66	C₄⋊Dic₇	—	4	2	no	—
67	Dic₇⋊C₄	—	2	4	no	—
68	C₄×Dic₇	—	2	4	no	—
69	C₂×#126	—	7	2	no	—
70	#397	—	7	2	no	—
71	#396	—	7	2	no	—
72	#390	—	7	1	no	—
73	#390	—	7	1	no	—
74	#390	—	7	1	no	—
75	#389	—	7	1	no	—
76	#388	—	7	2	no	—
77	#384	—	7	1	no	—
78	C₄×C₄⋊C₄	—	7	1	no	—
79	C₄×C₄⋊C₄	D₇	1	1	yes	characteristic, complemented
80	C₂³×C₇	C₂⁴	1	1	yes	characteristic
81	C₂²×D₇	—	2	2	no	—
82	C₂²×D₇	—	2	2	no	—
83	C₂²×D₇	—	2	2	no	—
84	C₂²×D₇	—	2	2	no	—
85	C₂²×D₇	—	2	2	no	—
86	C₂²×D₇	—	2	2	no	—
87	C₂²×D₇	—	2	2	no	—
88	C₂×C₄×C₇	—	2	1	no	—
89	C₂×C₄×C₇	—	2	1	no	—
90	C₂×C₄×C₇	—	2	1	no	—
91	C₂×C₄×C₇	—	2	1	no	—
92	C₂×C₄×C₇	—	2	1	no	—
93	C₂×C₄×C₇	—	2	1	no	—
94	C₂×C₄×C₇	—	2	1	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	C₂×D₄	1	2	yes	—
103	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
104	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
105	C₂×C₄×C₇	#41	1	2	yes	—
106	C₂×C₄×C₇	#41	1	2	yes	—
107	C₂×C₄×C₇	#41	1	2	yes	—
108	C₂×C₄×C₇	#41	1	2	yes	—
109	C₂×Dic₇	—	2	2	no	—
110	C₂×Dic₇	—	2	2	no	—
111	C₂×Dic₇	—	2	2	no	—
112	C₂×Dic₇	—	2	2	no	—
113	C₂×Dic₇	—	2	2	no	—
114	C₂×Dic₇	—	2	2	no	—
115	C₂×Dic₇	—	2	2	no	—
116	C₂×Dic₇	—	2	2	no	—
117	C₂×Dic₇	—	2	2	no	—
118	D₂₈	—	4	4	no	—
119	C₂²×D₄	—	7	1	no	—
120	#126	—	14	8	no	—
121	C₂×C₄⋊C₄	—	7	2	no	—
122	C₂×C₄⋊C₄	—	7	2	no	—
123	C₂×C₄⋊C₄	—	7	1	no	—
124	C₂×C₄⋊C₄	—	7	1	no	—
125	C₂×C₄⋊C₄	—	7	1	no	—
126	C₂×C₄⋊C₄	—	7	2	no	—
127	C₂×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
128	C₂×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
129	C₂×C₂²⋊C₄	—	7	2	no	—
130	C₂×C₂²⋊C₄	—	7	2	no	—
131	C₂×C₂²⋊C₄	—	7	2	no	—
132	C₂×C₂²⋊C₄	—	7	2	no	—
133	C₂×C₂²⋊C₄	—	7	2	no	—
134	C₂×C₂²⋊C₄	—	7	2	no	—
135	C₂×C₄²	—	7	2	no	—
136	C₂×C₄²	C₂×D₇	1	1	yes	characteristic
137	C₂×C₄²	C₂×D₇	1	1	yes	characteristic
138	C₂×C₄²	C₂×D₇	1	1	yes	characteristic
139	#95	—	7	2	no	—
140	#95	—	7	2	no	—
141	#95	C₂×D₇	1	1	yes	characteristic
142	#95	C₂×D₇	1	1	yes	characteristic
143	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
144	C₂²×C₇	C₂×#41	1	1	yes	characteristic
145	C₂²×C₇	C₂×#41	1	1	yes	characteristic
146	C₂²×C₇	C₂×#41	1	1	yes	characteristic
147	C₂²×C₇	C₂×#41	1	1	yes	characteristic
148	C₂²×C₇	H(5,𝔽₂)	1	1	yes	characteristic
149	C₂²×C₇	2₋¹⁺⁴	1	1	yes	characteristic
150	C₂×D₇	—	4	2	no	—
151	C₂×D₇	—	4	2	no	—
152	C₂×D₇	—	4	2	no	—
153	C₂×D₇	—	4	2	no	—
154	C₂×D₇	—	4	2	no	—
155	C₂×D₇	—	4	2	no	—
156	C₂×D₇	—	4	2	no	—
157	C₄×C₇	—	2	2	no	—
158	C₄×C₇	—	2	2	no	—
159	C₄×C₇	—	2	2	no	—
160	C₄×C₇	—	2	2	no	—
161	C₄×C₇	—	4	1	no	—
162	C₄×C₇	—	4	1	no	—
163	C₄×C₇	#126	1	4	yes	complemented
164	Dic₇	—	4	2	no	—
165	Dic₇	—	4	2	no	—
166	Dic₇	—	4	2	no	—
167	C₂⁴	—	7	2	no	—
168	C₂×D₄	—	14	2	no	—
169	C₂×D₄	—	14	2	no	—
170	C₂×D₄	—	14	2	no	—
171	C₂²×C₄	—	7	2	no	—
172	C₂²×C₄	—	7	2	no	—
173	C₂²×C₄	—	7	2	no	—
174	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
175	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
176	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
177	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
178	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
179	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
180	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
181	C₄⋊C₄	—	2	2	no	—
182	C₄⋊C₄	—	14	4	no	—
183	C₄⋊C₄	—	14	4	no	—
184	C₄⋊C₄	—	14	2	no	—
185	C₄⋊C₄	—	14	2	no	—
186	C₄⋊C₄	—	14	2	no	—
187	C₄⋊C₄	—	28	2	no	—
188	C₄⋊C₄	D₂₈	1	4	yes	complemented
189	C₂²⋊C₄	—	14	4	no	—
190	C₂²⋊C₄	—	14	4	no	—
191	C₂²⋊C₄	—	14	4	no	—
192	C₂²⋊C₄	—	14	4	no	—
193	C₂²⋊C₄	—	28	2	no	—
194	C₂²⋊C₄	—	28	2	no	—
195	C₄²	—	2	2	no	—
196	C₄²	—	2	2	no	—
197	C₄²	—	2	2	no	—
198	C₄²	—	14	4	no	—
199	C₂×C₇	C₂×#126	1	1	yes	characteristic
200	C₂×C₇	#532	1	1	yes	characteristic
201	C₂×C₇	#542	1	1	yes	characteristic
202	C₂×C₇	#547	1	1	yes	characteristic
203	C₂×C₇	#550	1	1	yes	characteristic
204	C₂×C₇	#553	1	1	yes	characteristic
205	C₂×C₇	#556	1	1	yes	characteristic
206	D₇	—	8	2	no	—
207	C₂³	—	14	2	no	—
208	C₂³	—	14	2	no	—
209	C₂³	—	14	2	no	—
210	C₂³	—	14	2	no	—
211	C₂³	—	14	2	no	—
212	C₂³	—	14	2	no	—
213	C₂³	—	14	2	no	—
214	C₂³	C₂³×D₇	1	1	yes	characteristic
215	D₄	—	28	4	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	14	2	no	—
231	C₂×C₄	—	14	2	no	—
232	C₂×C₄	—	14	2	no	—
233	C₂×C₄	—	14	2	no	—
234	C₂×C₄	—	14	2	no	—
235	C₂×C₄	—	14	2	no	—
236	C₂×C₄	—	14	2	no	—
237	C₂×C₄	—	14	2	no	—
238	C₂×C₄	—	14	2	no	—
239	C₂×C₄	C₂×D₂₈	1	2	yes	—
240	C₂×C₄	C₂×D₂₈	1	2	yes	—
241	C₂×C₄	C₂×D₂₈	1	2	yes	—
242	C₂×C₄	#1157	1	2	yes	—
243	C₂×C₄	#1157	1	2	yes	—
244	C₂×C₄	#1159	1	2	yes	—
245	C₂×C₄	#1161	1	2	yes	—
246	C₇	#2512	1	1	yes	characteristic, complemented, Sylow
247	C₂²	—	28	2	no	—
248	C₂²	—	28	2	no	—
249	C₂²	—	28	2	no	—
250	C₂²	—	28	2	no	—
251	C₂²	—	28	2	no	—
252	C₂²	—	28	2	no	—
253	C₂²	—	28	2	no	—
254	C₂²	C₂²×D₂₈	1	1	yes	characteristic
255	C₂²	C₂×#1157	1	1	yes	characteristic
256	C₂²	C₂×#1157	1	1	yes	characteristic
257	C₂²	C₂×#1159	1	1	yes	characteristic
258	C₂²	C₂×#1161	1	1	yes	characteristic
259	C₂²	#6553	1	1	yes	characteristic
260	C₂²	#6554	1	1	yes	characteristic
261	C₄	—	2	2	no	—
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	2	2	no	—
265	C₄	—	4	1	no	—
266	C₄	—	4	1	no	—
267	C₄	—	28	2	no	—
268	C₄	—	28	2	no	—
269	C₄	—	28	2	no	—
270	C₄	#6461	1	4	yes	complemented
271	C₂	—	56	2	no	—
272	C₂	#90168	1	1	yes	characteristic
273	C₂	C₂×#6461	1	1	yes	characteristic
274	C₂	#90225	1	1	yes	characteristic
275	C₂	#90246	1	1	yes	characteristic
276	C₂	#90256	1	1	yes	characteristic
277	C₂	#90268	1	1	yes	characteristic
278	C₂	#90275	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₇
commutator, socle, cosocle	C₂³×C₇	C₂⁴
Fitting	C₄×C₇×C₄⋊C₄	C₂
prime radical	C₂²×D₂₈	C₂²
hypercenter	C₄×C₄⋊C₄	D₇
hypocenter	C₇	#2512
LCS	G ▷ C₂³×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂×#6461	C₂	semidirect
2. C₄×C₇×C₄⋊C₄	C₂	semidirect
3. #89766	C₂	semidirect
7. #89707	C₂	semidirect
8. C₄×C₄⋊Dic₇	C₂	semidirect
9. #89700	C₂	semidirect
10. #89446	C₂	semidirect
11. #89442	C₂	semidirect
52. C₇×C₄⋊C₄	D₄	semidirect
79. C₄×C₄⋊C₄	D₇	semidirect
163. C₄×C₇	#126	semidirect
188. C₄⋊C₄	D₂₈	semidirect
246. C₇	#2512	semidirect
270. C₄	#6461	semidirect

Maximal subgroups

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

subgroup	index
C₂×#6461	2
C₄×C₇×C₄⋊C₄	2
#89766	2
#89711	2
#89707	2
C₄×C₄⋊Dic₇	2
#89700	2
#89446	2
#89442	2
#2512	7