Subgroups — #13319512

Order 1792.

806 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₇×M₁₆	C₂	1	1	yes	maximal, characteristic
2	#11739494	C₂	1	2	yes	maximal
3	#11727603	C₂	1	4	yes	maximal, complemented
4	#11727602	C₂	1	4	yes	maximal, complemented
5	#11727480	C₂	1	2	yes	maximal, complemented
6	#11727478	C₂	1	2	yes	maximal, complemented
7	C₂²×C₇×M₁₆	C₄	1	2	yes	—
8	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
9	C₄×C₇×M₁₆	C₄	1	2	yes	—
10	C₄×C₇×M₁₆	C₄	1	2	yes	—
11	C₄×C₇×M₁₆	C₂²	1	2	yes	—
12	C₄×C₇×M₁₆	C₂²	1	2	yes	—
13	C₂×C₇×#97	C₄	1	2	yes	—
14	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
16	#89979	C₄	1	4	yes	—
17	#89698	C₄	1	4	yes	—
18	#89698	C₄	1	4	yes	—
19	#89697	C₂²	1	4	yes	—
20	#89697	C₂²	1	4	yes	—
21	#89696	C₄	1	4	yes	—
22	C₄×#1136	C₂²	1	4	yes	—
23	#89250	C₂²	1	4	yes	complemented
24	#89249	C₂²	1	8	yes	complemented
25	C₈×C₇⋊C₈	C₂²	1	4	yes	complemented
26	#17334	—	7	1	no	maximal, Sylow
27	C₂³×C₄×C₇	C₂×C₄	1	2	yes	—
28	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₇×M₁₆	—	2	8	no	—
30	C₂×C₇×M₁₆	C₂×C₄	1	4	yes	—
31	C₂×C₇×M₁₆	C₂×C₄	1	4	yes	—
32	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
33	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
43	C₂×#1136	C₂×C₄	1	8	yes	—
44	C₇×#97	C₂×C₄	1	4	yes	—
45	C₇×#97	C₂×C₄	1	4	yes	—
46	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
47	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
48	C₄×C₇×C₈	C₂³	1	2	yes	—
49	C₄×C₇×C₈	C₂³	1	2	yes	—
50	C₂²⋊(C₇⋊C₈)	C₂×C₄	1	8	yes	—
51	C₂²⋊(C₇⋊C₈)	C₂×C₄	1	8	yes	—
52	C₄⋊(C₇⋊C₈)	C₂×C₄	1	8	yes	—
53	C₄⋊(C₇⋊C₈)	C₂×C₄	1	8	yes	—
54	#6377	C₂×C₄	1	8	yes	—
55	C₄×C₇⋊C₈	C₂³	1	8	yes	—
56	#2920	—	7	2	no	—
57	C₂×C₄×M₁₆	D₇	1	1	yes	characteristic
58	#1455	—	7	4	no	—
59	#1453	—	7	4	no	—
60	#1452	—	7	2	no	—
61	#1450	—	7	2	no	—
62	C₂⁴×C₇	C₄²	1	1	yes	characteristic
63	C₂²×C₄×C₇	—	2	4	no	—
64	C₂²×C₄×C₇	—	4	4	no	—
65	C₂²×C₄×C₇	C₄²	1	1	yes	characteristic
66	C₂²×C₄×C₇	C₄²	1	1	yes	characteristic
67	C₂²×C₄×C₇	C₄²	1	1	yes	characteristic
68	C₂²×C₄×C₇	C₄²	1	2	yes	—
69	C₂²×C₄×C₇	C₄²	1	2	yes	—
70	C₂²×C₄×C₇	C₄²	1	2	yes	—
71	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
77	C₇×M₁₆	—	2	8	no	—
78	C₇×M₁₆	—	2	8	no	—
79	C₂×C₇×C₈	—	2	4	no	—
80	C₂×C₇×C₈	M₁₆	1	8	yes	complemented
81	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
82	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
83	C₄²×C₇	C₄²	1	2	yes	—
84	C₄²×C₇	C₄²	1	2	yes	—
85	C₄²×C₇	C₄²	1	2	yes	—
86	C₄²×C₇	C₂²×C₄	1	2	yes	—
87	C₄²×C₇	C₂²×C₄	1	2	yes	—
88	C₄²×C₇	C₂²×C₄	1	2	yes	—
89	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
91	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₇	C₂⁴	1	1	yes	characteristic
93	#1136	—	4	8	no	—
94	C₂×C₇⋊C₈	C₂²×C₄	1	16	yes	—
95	C₂²×M₁₆	Dic₇	1	2	yes	—
96	C₂²×C₄²	C₂×D₇	1	1	yes	characteristic
97	#432	—	7	4	no	—
98	#432	—	7	4	no	—
99	#431	—	7	4	no	—
100	#431	—	7	4	no	—
101	#423	—	7	4	no	—
102	#407	—	7	4	no	—
103	C₄×M₁₆	—	7	4	no	—
104	C₄×M₁₆	Dic₇	1	2	yes	—
105	C₄×M₁₆	Dic₇	1	2	yes	—
106	C₄×M₁₆	C₂×D₇	1	2	yes	—
107	C₄×M₁₆	C₂×D₇	1	2	yes	—
108	C₂×#97	Dic₇	1	2	yes	—
109	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
110	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
111	#322	—	7	8	no	—
112	#322	—	7	4	no	—
113	C₈²	—	7	4	no	—
114	C₂³×C₇	—	2	2	no	—
115	C₂³×C₇	—	4	2	no	—
116	C₂³×C₇	C₂×C₄²	1	1	yes	characteristic
117	C₂³×C₇	C₂×C₄²	1	1	yes	characteristic
118	C₂³×C₇	C₂×C₄²	1	1	yes	characteristic
119	C₂×C₄×C₇	—	2	2	no	—
120	C₂×C₄×C₇	—	2	2	no	—
121	C₂×C₄×C₇	—	2	2	no	—
122	C₂×C₄×C₇	—	2	4	no	—
123	C₂×C₄×C₇	—	2	4	no	—
124	C₂×C₄×C₇	—	2	4	no	—
125	C₂×C₄×C₇	—	2	2	no	—
126	C₂×C₄×C₇	—	2	2	no	—
127	C₂×C₄×C₇	—	4	2	no	—
128	C₂×C₄×C₇	—	4	2	no	—
129	C₂×C₄×C₇	—	4	2	no	—
130	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
131	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
132	C₂×C₄×C₇	C₂×C₄²	1	2	yes	—
133	C₂×C₄×C₇	C₂×C₄²	1	2	yes	—
134	C₂×C₄×C₇	C₂×C₄²	1	2	yes	—
135	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
136	C₂×C₄×C₇	C₂×M₁₆	1	4	yes	—
137	C₂×C₄×C₇	C₂³×C₄	1	2	yes	—
138	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
139	C₇×C₈	—	2	4	no	—
140	C₇×C₈	C₂×M₁₆	1	8	yes	—
141	C₇⋊C₈	—	2	16	no	—
142	C₂³×C₄	C₄×D₇	1	2	yes	—
143	C₂³×C₄	C₂×Dic₇	1	1	yes	characteristic
144	C₂×M₁₆	—	2	8	no	—
145	C₂×M₁₆	—	7	8	no	—
146	C₂×M₁₆	C₂×Dic₇	1	4	yes	—
147	C₂×M₁₆	C₂×Dic₇	1	4	yes	—
148	C₂²×C₈	C₂×Dic₇	1	2	yes	—
149	C₂²×C₈	C₂×Dic₇	1	2	yes	—
150	C₂×C₄²	C₄×D₇	1	2	yes	—
151	C₂×C₄²	C₄×D₇	1	2	yes	—
152	C₂×C₄²	C₄×D₇	1	2	yes	—
153	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
154	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
155	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
156	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
157	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
158	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
159	C₄⋊C₈	—	7	8	no	—
160	C₄⋊C₈	—	7	8	no	—
161	C₂²⋊C₈	—	7	8	no	—
162	C₂²⋊C₈	—	7	8	no	—
163	#97	—	7	8	no	—
164	#97	C₂×Dic₇	1	4	yes	—
165	#97	C₂×Dic₇	1	4	yes	—
166	C₄×C₈	—	7	8	no	—
167	C₄×C₈	C₂×Dic₇	1	2	yes	—
168	C₄×C₈	C₂×Dic₇	1	2	yes	—
169	C₄×C₈	C₂²×D₇	1	2	yes	—
170	C₄×C₈	C₂²×D₇	1	2	yes	—
171	C₂²×C₇	—	2	2	no	—
172	C₂²×C₇	—	2	1	no	—
173	C₂²×C₇	—	2	1	no	—
174	C₂²×C₇	—	2	2	no	—
175	C₂²×C₇	—	2	2	no	—
176	C₂²×C₇	—	4	4	no	—
177	C₂²×C₇	C₄×M₁₆	1	2	yes	—
178	C₂²×C₇	C₂²×C₄²	1	1	yes	characteristic
179	C₄×C₇	—	2	1	no	—
180	C₄×C₇	—	2	1	no	—
181	C₄×C₇	—	2	2	no	—
182	C₄×C₇	—	2	2	no	—
183	C₄×C₇	—	2	2	no	—
184	C₄×C₇	C₄×M₁₆	1	2	yes	—
185	C₄×C₇	C₂²×M₁₆	1	2	yes	—
186	C₄×C₇	#568	1	4	yes	—
187	C₂⁴	C₄×Dic₇	1	1	yes	characteristic
188	C₂²×C₄	—	2	4	no	—
189	C₂²×C₄	—	4	4	no	—
190	C₂²×C₄	C₄×Dic₇	1	1	yes	characteristic
191	C₂²×C₄	C₄×Dic₇	1	1	yes	characteristic
192	C₂²×C₄	C₄×Dic₇	1	1	yes	characteristic
193	C₂²×C₄	C₄×Dic₇	1	2	yes	—
194	C₂²×C₄	C₄×Dic₇	1	2	yes	—
195	C₂²×C₄	C₄×Dic₇	1	2	yes	—
196	C₂²×C₄	C₂×C₄×D₇	1	2	yes	—
197	C₂²×C₄	C₂×C₄×D₇	1	2	yes	—
198	C₂²×C₄	C₂×C₄×D₇	1	2	yes	—
199	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
200	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
201	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
202	M₁₆	—	2	8	no	—
203	M₁₆	—	2	8	no	—
204	M₁₆	—	28	8	no	—
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	7	16	no	—
207	C₂×C₈	#1136	1	8	yes	complemented
208	C₂×C₈	C₂²×Dic₇	1	4	yes	—
209	C₂×C₈	C₂²×Dic₇	1	4	yes	—
210	C₄²	C₄×Dic₇	1	2	yes	—
211	C₄²	C₄×Dic₇	1	2	yes	—
212	C₄²	C₄×Dic₇	1	2	yes	—
213	C₄²	C₂×C₄×D₇	1	2	yes	—
214	C₄²	C₂×C₄×D₇	1	2	yes	—
215	C₄²	C₂×C₄×D₇	1	2	yes	—
216	C₄²	C₂²×Dic₇	1	1	yes	characteristic
217	C₄²	C₂²×Dic₇	1	1	yes	characteristic
218	C₄²	C₂²×Dic₇	1	1	yes	characteristic
219	C₄²	C₂³×D₇	1	1	yes	characteristic
220	C₂×C₇	—	2	2	no	—
221	C₂×C₇	—	4	1	no	—
222	C₂×C₇	—	4	1	no	—
223	C₂×C₇	C₂×C₄×M₁₆	1	1	yes	characteristic
224	C₂×C₇	#2873	1	2	yes	—
225	C₂³	—	2	2	no	—
226	C₂³	—	4	2	no	—
227	C₂³	C₂×C₄×Dic₇	1	1	yes	characteristic
228	C₂³	C₂×C₄×Dic₇	1	1	yes	characteristic
229	C₂³	C₂×C₄×Dic₇	1	1	yes	characteristic
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	C₂×#1136	1	4	yes	—
242	C₂×C₄	C₂×C₄×Dic₇	1	1	yes	characteristic
243	C₂×C₄	C₂×C₄×Dic₇	1	1	yes	characteristic
244	C₂×C₄	C₂×C₄×Dic₇	1	2	yes	—
245	C₂×C₄	C₂×C₄×Dic₇	1	2	yes	—
246	C₂×C₄	C₂×C₄×Dic₇	1	2	yes	—
247	C₂×C₄	C₂×C₄×Dic₇	1	1	yes	characteristic
248	C₂×C₄	C₂²×C₄×D₇	1	2	yes	—
249	C₂×C₄	C₂³×Dic₇	1	1	yes	characteristic
250	C₈	—	2	4	no	—
251	C₈	—	14	16	no	—
252	C₈	C₂×#1136	1	8	yes	—
253	C₇	#17334	1	1	yes	characteristic, complemented, Sylow
254	C₂²	—	2	2	no	—
255	C₂²	—	2	2	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	2	1	no	—
258	C₂²	—	2	1	no	—
259	C₂²	—	4	4	no	—
260	C₂²	C₄×#1136	1	2	yes	—
261	C₂²	C₂²×C₄×Dic₇	1	1	yes	characteristic
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	2	2	no	—
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	C₄×#1136	1	2	yes	—
268	C₄	#90436	1	4	yes	—
269	C₄	C₂²×#1136	1	2	yes	—
270	C₂	—	2	2	no	—
271	C₂	—	4	1	no	—
272	C₂	—	4	1	no	—
273	C₂	C₂×C₄×#1136	1	1	yes	characteristic
274	C₂	#11741192	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₇
commutator, socle	C₂²×C₇	C₂²×C₄²
Fitting	C₂×C₄×C₇×M₁₆	C₂
prime radical	C₂⁴×C₇	C₄²
cosocle	C₄²×C₇	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₇
hypocenter	C₇	#17334
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. #11727603	C₂	semidirect
4. #11727602	C₂	semidirect
5. #11727480	C₂	semidirect
6. #11727478	C₂	semidirect
23. #89250	C₂²	semidirect
24. #89249	C₂²	semidirect
25. C₈×C₇⋊C₈	C₂²	semidirect
80. C₂×C₇×C₈	M₁₆	semidirect
207. C₂×C₈	#1136	semidirect
253. C₇	#17334	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₇×M₁₆	2
#11739494	2
#11727603	2
#11727602	2
#11727480	2
#11727478	2
#17334	7