Subgroups — #13272036

Order 1792.

490 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11735912	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₇×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#89305	C₂	1	4	yes	maximal, complemented
4	#11727546	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#6467	—	2	1	no	—
6	C₂×#6467	C₂²	1	2	yes	—
7	#90168	—	2	1	no	—
8	C₂²×C₇×C₁₆	C₂²	1	2	yes	—
9	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
10	C₂×#6393	—	2	1	no	—
11	C₂×#6393	C₂²	1	2	yes	—
12	#89696	—	2	1	no	—
13	#89471	C₄	1	1	yes	characteristic
14	#89468	—	2	1	no	—
15	#89468	C₄	1	2	yes	—
16	#89467	C₄	1	1	yes	characteristic
17	#89465	—	2	1	no	—
18	C₄×C₇×C₁₆	—	2	1	no	—
19	C₄×C₇×C₁₆	—	2	1	no	—
20	#89305	—	2	8	no	—
21	#13208	—	7	1	no	maximal, Sylow
22	C₂×#1157	—	2	2	no	—
23	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
24	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
25	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
26	#6493	—	4	2	no	—
27	C₂×#1136	—	2	2	no	—
28	C₂×Dic₂₈	—	2	1	no	—
29	C₂×Dic₂₈	C₂×C₄	1	2	yes	—
30	#6467	—	2	4	no	—
31	#6467	—	2	4	no	—
32	#6467	—	4	2	no	—
33	#6467	—	4	2	no	—
34	C₂×D₅₆	—	2	1	no	—
35	C₂×D₅₆	C₂×C₄	1	2	yes	—
36	C₂×#1132	—	2	2	no	—
37	C₂×#1132	—	2	2	no	—
38	#6438	—	2	2	no	—
39	#6437	—	2	1	no	—
40	C₄×D₂₈	—	4	2	no	—
41	#6432	—	2	1	no	—
42	C₂×C₇×C₁₆	—	2	2	no	—
43	C₂×C₇×C₁₆	—	2	2	no	—
44	C₂×C₇×C₁₆	D₄	1	4	yes	complemented
45	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
46	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
47	C₄×C₇×C₈	D₄	1	1	yes	characteristic
48	C₄×C₇×C₈	D₄	1	1	yes	characteristic
49	#6393	—	2	4	no	—
50	#6393	—	4	2	no	—
51	C₄⋊(C₇⋊C₈)	—	4	2	no	—
52	#3147	—	7	1	no	—
53	C₂×#359	—	7	4	no	—
54	C₂×C₄×C₁₆	D₇	1	1	yes	characteristic, complemented
55	#1848	—	7	1	no	—
56	C₂²×C₄×C₇	SD₁₆	1	2	yes	—
57	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
58	C₂×C₂²⋊D₇	—	4	2	no	—
59	#1157	—	4	2	no	—
60	#1157	—	4	2	no	—
61	#1157	—	4	2	no	—
62	#1157	—	4	2	no	—
63	C₂×D₂₈	—	2	2	no	—
64	C₂×D₂₈	—	4	2	no	—
65	C₂×C₄×D₇	—	4	2	no	—
66	C₂×Dic₁₄	—	2	2	no	—
67	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₇×C₈	D₈	1	2	yes	—
70	C₂×C₇×C₈	SD₁₆	1	2	yes	—
71	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
73	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
74	C₂×C₇×C₈	#41	1	2	yes	—
75	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
76	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
77	C₄²×C₇	D₈	1	2	yes	—
78	(C₂×D₇)⋊C₄	—	4	4	no	—
79	C₄⋊Dic₇	—	4	2	no	—
80	#1136	—	4	2	no	—
81	#1136	—	4	2	no	—
82	C₂×C₇⋊C₈	—	4	2	no	—
83	Dic₂₈	—	2	4	no	—
84	Dic₂₈	—	4	2	no	—
85	D₅₆	—	2	4	no	—
86	D₅₆	—	4	2	no	—
87	#1132	—	4	4	no	—
88	#1132	—	4	4	no	—
89	C₇×C₁₆	—	2	4	no	—
90	C₂×#135	—	7	2	no	—
91	C₂×#135	—	14	1	no	—
92	#532	—	14	1	no	—
93	C₂²×C₁₆	C₂×D₇	1	2	yes	—
94	#495	—	7	2	no	—
95	#495	—	14	1	no	—
96	#494	—	7	1	no	—
97	#493	—	7	1	no	—
98	#492	—	14	1	no	—
99	C₂×#108	—	7	2	no	—
100	C₂×#108	—	14	1	no	—
101	#423	—	14	1	no	—
102	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
103	#359	—	14	8	no	—
104	C₄×C₁₆	—	2	1	no	—
105	C₄×C₁₆	—	2	1	no	—
106	C₂³×C₇	C₂×SD₁₆	1	1	yes	characteristic
107	C₂²×D₇	—	4	2	no	—
108	C₂×C₄×C₇	—	2	4	no	—
109	C₂×C₄×C₇	#102	1	2	yes	—
110	C₂×C₄×C₇	#102	1	2	yes	—
111	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₇	#124	1	1	yes	characteristic
113	C₂×C₄×C₇	#127	1	1	yes	characteristic
114	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
115	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
116	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
117	C₂²⋊D₇	—	8	2	no	—
118	C₂²⋊D₇	—	8	2	no	—
119	C₂×Dic₇	—	4	2	no	—
120	D₂₈	—	4	2	no	—
121	D₂₈	—	4	2	no	—
122	D₂₈	—	8	4	no	—
123	C₄×D₇	—	8	2	no	—
124	C₄×D₇	—	8	2	no	—
125	Dic₁₄	—	4	2	no	—
126	Dic₁₄	—	4	2	no	—
127	C₇×C₈	#102	1	2	yes	—
128	C₇×C₈	#102	1	2	yes	—
129	C₇×C₈	C₄×D₄	1	2	yes	—
130	C₇×C₈	#121	1	2	yes	—
131	C₇⋊C₈	—	8	2	no	—
132	C₂×#41	—	14	2	no	—
133	#135	—	14	4	no	—
134	#135	—	14	4	no	—
135	#135	—	28	2	no	—
136	#135	—	28	2	no	—
137	C₂×Q₁₆	—	7	2	no	—
138	C₂×Q₁₆	—	14	1	no	—
139	C₂×SD₁₆	—	14	2	no	—
140	C₂×SD₁₆	—	14	2	no	—
141	C₂×D₈	—	7	2	no	—
142	C₂×D₈	—	14	1	no	—
143	C₂×M₁₆	—	14	2	no	—
144	C₂²×C₈	D₂₈	1	1	yes	characteristic
145	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
146	C₄⋊Q₈	—	14	1	no	—
147	#127	—	14	1	no	—
148	#124	—	14	2	no	—
149	#121	—	28	2	no	—
150	C₄×D₄	—	28	2	no	—
151	C₂×C₄²	D₂₈	1	1	yes	characteristic
152	C₂×C₁₆	—	2	2	no	—
153	C₂×C₁₆	—	2	2	no	—
154	C₂×C₁₆	D₂₈	1	4	yes	complemented
155	#108	—	14	4	no	—
156	#108	—	28	2	no	—
157	C₄⋊C₈	—	28	2	no	—
158	C₄×C₈	C₄×D₇	1	1	yes	characteristic
159	C₄×C₈	C₄×D₇	1	1	yes	characteristic
160	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
161	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
162	C₂²×C₇	—	2	2	no	—
163	C₂²×C₇	C₂×#102	1	1	yes	characteristic
164	C₂²×C₇	#486	1	1	yes	characteristic
165	C₂²×C₇	#492	1	1	yes	characteristic
166	C₂×D₇	—	8	4	no	—
167	C₂×D₇	—	8	2	no	—
168	C₄×C₇	—	2	2	no	—
169	C₄×C₇	#359	1	4	yes	complemented
170	C₄×C₇	#390	1	1	yes	characteristic
171	C₄×C₇	C₂×#102	1	1	yes	characteristic
172	C₄×C₇	#486	1	1	yes	characteristic
173	C₄×C₇	#493	1	1	yes	characteristic
174	Dic₇	—	8	2	no	—
175	#41	—	28	2	no	—
176	#41	—	28	2	no	—
177	#41	—	28	2	no	—
178	#41	—	28	2	no	—
179	C₂×Q₈	—	14	2	no	—
180	C₂×D₄	—	14	2	no	—
181	C₂×D₄	—	28	2	no	—
182	C₂×D₄	—	28	2	no	—
183	C₂²×C₄	—	28	2	no	—
184	C₂²×C₄	#1132	1	2	yes	—
185	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
186	Q₁₆	—	14	4	no	—
187	Q₁₆	—	28	2	no	—
188	SD₁₆	—	28	4	no	—
189	SD₁₆	—	28	4	no	—
190	D₈	—	14	4	no	—
191	D₈	—	28	2	no	—
192	M₁₆	—	28	2	no	—
193	M₁₆	—	28	2	no	—
194	C₂×C₈	—	28	2	no	—
195	C₂×C₈	#1132	1	2	yes	—
196	C₂×C₈	D₅₆	1	2	yes	—
197	C₂×C₈	(C₂×D₇)⋊C₄	1	1	yes	characteristic
198	C₂×C₈	(C₂×D₇)⋊C₄	1	1	yes	characteristic
199	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
200	C₂×C₈	C₂×D₂₈	1	2	yes	—
201	C₂×C₈	#1157	1	2	yes	—
202	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
203	C₄⋊C₄	—	28	2	no	—
204	C₂²⋊C₄	—	28	4	no	—
205	C₄²	D₅₆	1	2	yes	—
206	C₄²	(C₂×D₇)⋊C₄	1	1	yes	characteristic
207	C₄²	(C₂×D₇)⋊C₄	1	1	yes	characteristic
208	C₁₆	—	2	4	no	—
209	C₂×C₇	—	2	1	no	—
210	C₂×C₇	—	2	1	no	—
211	C₂×C₇	#1957	1	1	yes	characteristic
212	C₂×C₇	C₂×#359	1	2	yes	—
213	D₇	—	16	2	no	—
214	C₂³	—	28	2	no	—
215	C₂³	C₂×#1132	1	1	yes	characteristic
216	Q₈	—	28	2	no	—
217	Q₈	—	28	2	no	—
218	D₄	—	28	2	no	—
219	D₄	—	28	2	no	—
220	D₄	—	56	2	no	—
221	D₄	—	56	2	no	—
222	D₄	—	56	4	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	28	2	no	—
225	C₂×C₄	—	56	2	no	—
226	C₂×C₄	—	56	2	no	—
227	C₂×C₄	#6395	1	2	yes	—
228	C₂×C₄	#6395	1	2	yes	—
229	C₂×C₄	#6437	1	1	yes	characteristic
230	C₂×C₄	#6438	1	1	yes	characteristic
231	C₂×C₄	C₂×#1132	1	1	yes	characteristic
232	C₂×C₄	C₂×D₅₆	1	1	yes	characteristic
233	C₂×C₄	C₂×D₅₆	1	1	yes	characteristic
234	C₂×C₄	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
235	C₈	—	56	2	no	—
236	C₈	#6395	1	2	yes	—
237	C₈	#6395	1	2	yes	—
238	C₈	C₄×D₂₈	1	2	yes	—
239	C₈	#6493	1	2	yes	—
240	C₇	#13208	1	1	yes	characteristic, complemented, Sylow
241	C₂²	—	2	2	no	—
242	C₂²	—	56	4	no	—
243	C₂²	—	56	2	no	—
244	C₂²	#89465	1	1	yes	characteristic
245	C₂²	#89466	1	1	yes	characteristic
246	C₂²	C₂×#6395	1	1	yes	characteristic
247	C₄	—	2	2	no	—
248	C₄	—	56	2	no	—
249	C₄	#89305	1	4	yes	complemented
250	C₄	#89466	1	1	yes	characteristic
251	C₄	#89467	1	1	yes	characteristic
252	C₄	#89711	1	1	yes	characteristic
253	C₄	C₂×#6395	1	1	yes	characteristic
254	C₂	—	2	1	no	—
255	C₂	—	2	1	no	—
256	C₂	—	112	2	no	—
257	C₂	#11727526	1	1	yes	characteristic
258	C₂	C₂×#89305	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₅₆
commutator	C₂×C₇×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇
Fitting	C₂×C₄×C₇×C₁₆	C₂
socle	C₂²×C₇	C₂×#102
prime radical	#11735912	C₂
cosocle	C₂²×C₇×C₈	C₂³
hypercenter	C₂×C₄×C₁₆	D₇
hypocenter	C₇	#13208
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×C₇×C₁₆	C₂	semidirect
3. C₂×#89305	C₂	semidirect
4. #11727546	C₂	semidirect
44. C₂×C₇×C₁₆	D₄	semidirect
54. C₂×C₄×C₁₆	D₇	semidirect
154. C₂×C₁₆	D₂₈	semidirect
169. C₄×C₇	#359	semidirect
240. C₇	#13208	semidirect
249. C₄	#89305	semidirect

Maximal subgroups

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

subgroup	index
#11735912	2
C₂×C₄×C₇×C₁₆	2
C₂×#89305	2
#11727546	2
#13208	7