Subgroups — #13507299

Order 1792.

498 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11735890	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#11729155	C₂	1	2	yes	maximal
4	#11729153	C₂	1	2	yes	maximal
5	C₄×#6386	C₂	1	1	yes	maximal, characteristic
6	#11727036	C₂	1	4	yes	maximal, complemented
7	#11727035	C₂	1	2	yes	maximal, complemented
8	#11727034	C₂	1	2	yes	maximal, complemented
9	#90161	C₄	1	1	yes	characteristic
10	C₂²×C₇×C₁₆	C₂²	1	2	yes	—
11	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
12	#89874	C₄	1	2	yes	—
13	#89873	C₂²	1	2	yes	—
14	C₂×#6386	C₂²	1	1	yes	characteristic
15	C₂×#6386	C₂²	1	1	yes	characteristic
16	#89696	C₄	1	1	yes	characteristic
17	#89451	C₄	1	2	yes	—
18	#89451	C₄	1	2	yes	—
19	C₈×Dic₁₄	C₂²	1	2	yes	—
20	C₈×Dic₁₄	C₂²	1	2	yes	—
21	C₇×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
22	C₇×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
23	C₇×C₄⋊C₁₆	C₂²	1	2	yes	—
24	#89297	C₂²	1	4	yes	—
25	Dic₇⋊C₁₆	C₂²	1	4	yes	—
26	Dic₇⋊C₁₆	C₂²	1	4	yes	—
27	C₁₆×Dic₇	C₂²	1	4	yes	—
28	#89256	C₂²	1	2	yes	—
29	C₄×C₇⋊C₁₆	C₂²	1	2	yes	—
30	#33476	—	7	1	no	maximal, Sylow
31	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
34	#6489	C₂×C₄	1	2	yes	—
35	#6487	C₈	1	2	yes	—
36	#6487	C₈	1	2	yes	—
37	C₂×#1136	C₂×C₄	1	2	yes	—
38	#6433	C₈	1	2	yes	—
39	#6432	C₈	1	1	yes	characteristic
40	#6432	C₈	1	1	yes	characteristic
41	C₄×Dic₁₄	C₂×C₄	1	2	yes	—
42	C₄×Dic₁₄	C₂×C₄	1	2	yes	—
43	C₂×C₇×C₁₆	—	2	4	no	—
44	C₂×C₇×C₁₆	C₂³	1	2	yes	—
45	C₂×C₇×C₁₆	C₂³	1	2	yes	—
46	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
47	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
48	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
49	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
50	#6389	C₂×C₄	1	4	yes	—
51	Dic₇⋊C₈	C₂×C₄	1	4	yes	—
52	Dic₇⋊C₈	C₂×C₄	1	4	yes	—
53	C₈×Dic₇	C₂³	1	4	yes	—
54	#6386	—	2	2	no	—
55	#6386	Q₈	1	4	yes	complemented
56	C₂×C₇⋊C₁₆	C₂³	1	2	yes	—
57	C₂×C₇⋊C₁₆	C₂³	1	2	yes	—
58	C₄⋊(C₇⋊C₈)	C₂×C₄	1	2	yes	—
59	C₄⋊(C₇⋊C₈)	C₂×C₄	1	2	yes	—
60	#2960	—	7	1	no	—
61	#2183	—	7	2	no	—
62	#2183	—	7	4	no	—
63	Q₈×C₁₆	—	7	2	no	—
64	#2151	—	7	2	no	—
65	C₂×C₄⋊C₁₆	D₇	1	1	yes	characteristic
66	#2108	—	7	2	no	—
67	C₄×M₃₂	—	7	1	no	—
68	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
69	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
70	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
71	C₂×Dic₁₄	C₂×C₈	1	2	yes	—
72	C₂×Dic₁₄	C₂×C₈	1	2	yes	—
73	C₂×C₇×C₈	—	2	2	no	—
74	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
78	C₂×C₇×C₈	#41	1	2	yes	—
79	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
80	C₄²×C₇	C₂×C₈	1	1	yes	characteristic
81	C₄²×C₇	C₂×C₈	1	1	yes	characteristic
82	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
84	C₂²⋊Dic₇	C₂×C₈	1	4	yes	—
85	C₄⋊Dic₇	C₂×C₈	1	2	yes	—
86	C₄⋊Dic₇	C₂×C₈	1	2	yes	—
87	C₄⋊Dic₇	C₂×C₈	1	2	yes	—
88	C₄⋊Dic₇	C₂×C₈	1	2	yes	—
89	Dic₇⋊C₄	C₂×C₈	1	4	yes	—
90	Dic₇⋊C₄	C₂×C₈	1	4	yes	—
91	C₄×Dic₇	C₂²×C₄	1	4	yes	—
92	#1136	—	2	4	no	—
93	C₂×C₇⋊C₈	C₂²×C₄	1	4	yes	—
94	C₇×C₁₆	—	2	2	no	—
95	C₇×C₁₆	—	2	2	no	—
96	C₇⋊C₁₆	—	2	2	no	—
97	C₇⋊C₁₆	C₂×Q₈	1	4	yes	—
98	#533	—	7	1	no	—
99	C₂×M₃₂	—	7	1	no	—
100	C₂×M₃₂	—	7	1	no	—
101	C₂²×C₁₆	C₂×D₇	1	2	yes	—
102	#446	—	7	2	no	—
103	#446	—	7	2	no	—
104	C₈×Q₈	—	7	2	no	—
105	C₈×Q₈	—	7	2	no	—
106	#424	—	7	2	no	—
107	#423	—	7	1	no	—
108	#405	—	7	2	no	—
109	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
110	C₄⋊C₁₆	—	7	4	no	—
111	C₄⋊C₁₆	—	7	4	no	—
112	C₄⋊C₁₆	C₂×D₇	1	1	yes	characteristic
113	C₄⋊C₁₆	C₂×D₇	1	1	yes	characteristic
114	C₄⋊C₁₆	C₂×D₇	1	2	yes	—
115	#346	—	7	4	no	—
116	#346	—	7	2	no	—
117	C₄×C₁₆	—	7	4	no	—
118	C₄×C₁₆	—	7	2	no	—
119	C₂³×C₇	C₂²×C₈	1	1	yes	characteristic
120	C₂×C₄×C₇	—	2	2	no	—
121	C₂×C₄×C₇	—	4	1	no	—
122	C₂×C₄×C₇	C₄×Q₈	1	2	yes	—
123	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
124	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
125	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
126	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
127	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
128	C₂×C₄×C₇	#131	1	2	yes	—
129	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
130	C₂×Dic₇	C₂²×C₈	1	4	yes	—
131	C₂×Dic₇	C₂²×C₈	1	4	yes	—
132	Dic₁₄	—	2	4	no	—
133	Dic₁₄	—	4	2	no	—
134	C₇×C₈	—	2	1	no	—
135	C₇×C₈	—	2	1	no	—
136	C₇×C₈	C₄×Q₈	1	2	yes	—
137	C₇×C₈	C₂²×Q₈	1	1	yes	characteristic
138	C₇×C₈	C₂×#41	1	1	yes	characteristic
139	C₇⋊C₈	—	2	4	no	—
140	C₂×M₁₆	—	7	2	no	—
141	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
142	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
143	C₄⋊Q₈	—	7	1	no	—
144	C₄⋊Q₈	—	7	1	no	—
145	#125	—	7	2	no	—
146	C₂²⋊Q₈	—	7	2	no	—
147	C₂²⋊Q₈	—	7	2	no	—
148	C₄×Q₈	—	7	2	no	—
149	C₄×Q₈	—	7	2	no	—
150	#117	—	7	2	no	—
151	C₂×C₄²	C₄×D₇	1	1	yes	characteristic
152	M₃₂	—	7	4	no	—
153	M₃₂	—	14	2	no	—
154	C₂×C₁₆	—	2	4	no	—
155	C₂×C₁₆	—	7	2	no	—
156	C₂×C₁₆	—	7	2	no	—
157	C₂×C₁₆	C₂²×D₇	1	2	yes	—
158	C₂×C₁₆	C₂²×D₇	1	2	yes	—
159	C₄⋊C₈	—	7	4	no	—
160	C₄⋊C₈	—	7	4	no	—
161	C₄⋊C₈	—	7	2	no	—
162	C₄⋊C₈	—	7	2	no	—
163	#97	—	7	4	no	—
164	C₄×C₈	—	7	4	no	—
165	C₄×C₈	C₄×D₇	1	1	yes	characteristic
166	C₄×C₈	C₄×D₇	1	1	yes	characteristic
167	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
168	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
169	C₂²×C₇	—	2	1	no	—
170	C₂²×C₇	—	2	1	no	—
171	C₂²×C₇	C₈×Q₈	1	2	yes	—
172	C₂²×C₇	C₂³×C₈	1	1	yes	characteristic
173	C₄×C₇	—	2	1	no	—
174	C₄×C₇	—	2	1	no	—
175	C₄×C₇	—	2	1	no	—
176	C₄×C₇	C₈×Q₈	1	2	yes	—
177	C₄×C₇	#504	1	2	yes	—
178	C₄×C₇	C₂×C₄×Q₈	1	1	yes	characteristic
179	C₄×C₇	C₂×#131	1	1	yes	characteristic
180	Dic₇	—	2	4	no	—
181	Dic₇	—	2	4	no	—
182	C₂×Q₈	—	7	2	no	—
183	C₂×Q₈	—	7	2	no	—
184	C₂²×C₄	C₈×D₇	1	1	yes	characteristic
185	C₂²×C₄	C₈×D₇	1	1	yes	characteristic
186	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
187	M₁₆	—	14	4	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	7	4	no	—
190	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
191	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
192	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
193	C₂×C₈	#1159	1	2	yes	—
194	C₂×C₈	Q₈×D₇	1	2	yes	—
195	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
196	C₄⋊C₄	—	7	4	no	—
197	C₄⋊C₄	—	7	4	no	—
198	C₄⋊C₄	—	7	2	no	—
199	C₄⋊C₄	—	7	2	no	—
200	C₄⋊C₄	—	7	2	no	—
201	C₄⋊C₄	—	7	2	no	—
202	C₂²⋊C₄	—	7	4	no	—
203	C₄²	—	7	4	no	—
204	C₄²	C₈×D₇	1	1	yes	characteristic
205	C₄²	C₈×D₇	1	1	yes	characteristic
206	C₄²	C₂×C₄×D₇	1	1	yes	characteristic
207	C₄²	C₂×C₄×D₇	1	1	yes	characteristic
208	C₁₆	—	2	2	no	—
209	C₁₆	—	2	2	no	—
210	C₁₆	—	7	4	no	—
211	C₁₆	—	14	2	no	—
212	C₂×C₇	—	2	2	no	—
213	C₂×C₇	C₂×C₈×Q₈	1	1	yes	characteristic
214	C₂×C₇	C₂×#504	1	1	yes	characteristic
215	C₂×C₇	#3407	1	1	yes	characteristic
216	C₂³	C₂×C₈×D₇	1	1	yes	characteristic
217	Q₈	—	14	4	no	—
218	Q₈	—	28	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	4	1	no	—
221	C₂×C₄	—	7	4	no	—
222	C₂×C₄	—	7	4	no	—
223	C₂×C₄	#6450	1	2	yes	—
224	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
225	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
226	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
227	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
228	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
229	C₂×C₄	#6470	1	2	yes	—
230	C₂×C₄	C₂²×C₄×D₇	1	1	yes	characteristic
231	C₈	—	2	1	no	—
232	C₈	—	2	1	no	—
233	C₈	—	14	4	no	—
234	C₈	#6450	1	2	yes	—
235	C₈	C₂×#1159	1	1	yes	characteristic
236	C₈	C₂×Q₈×D₇	1	1	yes	characteristic
237	C₇	#33476	1	1	yes	characteristic, complemented, Sylow
238	C₂²	—	2	1	no	—
239	C₂²	—	2	1	no	—
240	C₂²	#89602	1	2	yes	—
241	C₂²	C₂²×C₈×D₇	1	1	yes	characteristic
242	C₄	—	2	1	no	—
243	C₄	—	2	1	no	—
244	C₄	—	2	1	no	—
245	C₄	—	14	4	no	—
246	C₄	—	14	4	no	—
247	C₄	#89602	1	2	yes	—
248	C₄	#89673	1	2	yes	—
249	C₄	C₂×#6450	1	1	yes	characteristic
250	C₄	C₂×#6470	1	1	yes	characteristic
251	C₂	—	2	2	no	—
252	C₂	C₂×#89602	1	1	yes	characteristic
253	C₂	C₂×#89673	1	1	yes	characteristic
254	C₂	#11739452	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₇
commutator, socle	C₂²×C₇	C₂³×C₈
Fitting	C₂×C₇×C₄⋊C₁₆	C₂
prime radical	C₂³×C₇	C₂²×C₈
cosocle	C₂×C₇×C₈	C₂⁴
hypercenter	C₂×C₄⋊C₁₆	D₇
hypocenter	C₇	#33476
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
6. #11727036	C₂	semidirect
7. #11727035	C₂	semidirect
8. #11727034	C₂	semidirect
55. #6386	Q₈	semidirect
237. C₇	#33476	semidirect

Maximal subgroups

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

subgroup	index
#11735890	2
C₂×C₇×C₄⋊C₁₆	2
#11729155	2
#11729153	2
C₄×#6386	2
#11727036	2
#11727035	2
#11727034	2
#33476	7