Subgroups — #13268685

Order 1792.

554 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×#108	C₂	1	1	yes	maximal, characteristic
2	C₂²×#6393	C₂	1	1	yes	maximal, characteristic
3	C₂²×#6386	C₂	1	1	yes	maximal, characteristic
4	#11725534	C₂	1	4	yes	maximal, complemented
5	C₂²×C₇×M₁₆	—	2	1	no	—
6	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
7	C₂²×#1136	—	2	1	no	—
8	C₂×C₇×#108	—	2	1	no	—
9	C₂×C₇×#108	—	2	2	no	—
10	C₂×C₇×#108	C₄	1	4	yes	—
11	C₂×C₇×#108	C₂²	1	2	yes	—
12	C₂×#6393	—	2	2	no	—
13	C₂×#6393	—	2	1	no	—
14	C₂×#6393	C₄	1	4	yes	—
15	C₂×#6393	C₂²	1	2	yes	—
16	C₂×#6386	—	2	2	no	—
17	C₂×#6386	—	2	2	no	—
18	C₂×#6386	C₄	1	2	yes	—
19	C₂×#6386	C₄	1	2	yes	—
20	C₂²×C₇⋊C₁₆	C₂²	1	2	yes	—
21	#12559	—	7	1	no	maximal, Sylow
22	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
23	C₂×C₇×M₁₆	—	2	4	no	—
24	C₂×C₇×M₁₆	—	2	2	no	—
25	C₂×C₇×M₁₆	—	4	1	no	—
26	C₂×C₇×M₁₆	—	4	2	no	—
27	C₂²×C₇×C₈	—	4	1	no	—
28	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
31	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
32	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
33	C₂²×C₇×C₈	D₄	1	2	yes	—
34	C₂²×C₇×C₈	D₄	1	2	yes	—
35	C₂²×C₇×C₈	Q₈	1	1	yes	characteristic
36	C₂²×C₇×C₈	Q₈	1	1	yes	characteristic
37	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
38	C₂×#1136	—	2	4	no	—
39	C₂×#1136	—	2	2	no	—
40	C₂×#1136	—	4	1	no	—
41	C₂×#1136	—	4	2	no	—
42	C₂²×C₇⋊C₈	—	4	1	no	—
43	C₇×#108	—	2	4	no	—
44	C₇×#108	—	2	2	no	—
45	C₇×#108	C₂×C₄	1	4	yes	—
46	#6393	—	2	4	no	—
47	#6393	—	2	2	no	—
48	#6393	C₂×C₄	1	4	yes	—
49	#6386	—	2	4	no	—
50	#6386	—	2	4	no	—
51	C₂×C₇⋊C₁₆	—	2	2	no	—
52	C₂×C₇⋊C₁₆	—	2	2	no	—
53	C₂×C₇⋊C₁₆	C₂×C₄	1	4	yes	—
54	C₂²×M₃₂	—	7	1	no	—
55	C₂²×#108	—	7	1	no	—
56	C₂²×#108	D₇	1	1	yes	characteristic
57	#1381	—	7	4	no	—
58	C₂⁴×C₇	Q₁₆	1	1	yes	characteristic
59	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
64	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
65	C₂²×C₄×C₇	SD₁₆	1	2	yes	—
66	C₂²×C₄×C₇	SD₁₆	1	2	yes	—
67	C₂²×C₄×C₇	Q₁₆	1	1	yes	characteristic
68	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
69	C₇×M₁₆	—	2	4	no	—
70	C₇×M₁₆	—	4	4	no	—
71	C₇×M₁₆	—	4	2	no	—
72	C₂×C₇×C₈	—	4	4	no	—
73	C₂×C₇×C₈	—	4	2	no	—
74	C₂×C₇×C₈	C₄²	1	2	yes	—
75	C₂×C₇×C₈	C₄²	1	2	yes	—
76	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
77	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
78	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
79	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
80	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
83	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
84	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
85	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
86	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
90	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
91	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
92	#1136	—	2	4	no	—
93	#1136	—	4	4	no	—
94	#1136	—	4	2	no	—
95	C₂×C₇⋊C₈	—	4	4	no	—
96	C₂×C₇⋊C₈	—	4	2	no	—
97	C₇⋊C₁₆	—	2	4	no	—
98	C₂²×M₁₆	—	2	1	no	—
99	C₂²×M₁₆	—	14	1	no	—
100	C₂³×C₈	C₂×D₇	1	1	yes	characteristic
101	C₂×M₃₂	—	7	2	no	—
102	C₂×M₃₂	—	7	2	no	—
103	C₂×M₃₂	—	14	2	no	—
104	C₂×M₃₂	—	14	2	no	—
105	C₂²×C₁₆	—	7	2	no	—
106	C₂×#108	—	2	1	no	—
107	C₂×#108	—	2	2	no	—
108	C₂×#108	—	7	4	no	—
109	C₂×#108	—	7	2	no	—
110	C₂×#108	—	14	2	no	—
111	C₂×#108	—	14	1	no	—
112	C₂×#108	Dic₇	1	4	yes	—
113	C₂×#108	C₂×D₇	1	2	yes	—
114	C₂³×C₇	—	4	2	no	—
115	C₂³×C₇	#103	1	2	yes	—
116	C₂³×C₇	#103	1	2	yes	—
117	C₂³×C₇	#107	1	1	yes	characteristic
118	C₂³×C₇	#107	1	1	yes	characteristic
119	C₂³×C₇	C₂×Q₁₆	1	1	yes	characteristic
120	C₂×C₄×C₇	#95	1	2	yes	—
121	C₂×C₄×C₇	#95	1	2	yes	—
122	C₂×C₄×C₇	#102	1	2	yes	—
123	C₂×C₄×C₇	#102	1	2	yes	—
124	C₂×C₄×C₇	#102	1	2	yes	—
125	C₂×C₄×C₇	#102	1	2	yes	—
126	C₂×C₄×C₇	#103	1	2	yes	—
127	C₂×C₄×C₇	#103	1	2	yes	—
128	C₂×C₄×C₇	#106	1	2	yes	—
129	C₂×C₄×C₇	#106	1	2	yes	—
130	C₂×C₄×C₇	#107	1	1	yes	characteristic
131	C₂×C₄×C₇	#107	1	1	yes	characteristic
132	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
137	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
138	C₇×C₈	—	4	4	no	—
139	C₇×C₈	#95	1	2	yes	—
140	C₇×C₈	#95	1	2	yes	—
141	C₇×C₈	C₂×C₄²	1	1	yes	characteristic
142	C₇×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
143	C₇×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
144	C₇×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₇⋊C₈	—	4	4	no	—
146	C₂³×C₄	C₂²⋊D₇	1	1	yes	characteristic
147	C₂×M₁₆	—	2	4	no	—
148	C₂×M₁₆	—	2	2	no	—
149	C₂×M₁₆	—	4	1	no	—
150	C₂×M₁₆	—	4	2	no	—
151	C₂×M₁₆	—	14	4	no	—
152	C₂×M₁₆	—	14	2	no	—
153	C₂×M₁₆	—	28	1	no	—
154	C₂×M₁₆	—	28	2	no	—
155	C₂²×C₈	—	4	1	no	—
156	C₂²×C₈	—	28	1	no	—
157	C₂²×C₈	Dic₁₄	1	1	yes	characteristic
158	C₂²×C₈	Dic₁₄	1	1	yes	characteristic
159	C₂²×C₈	C₄×D₇	1	2	yes	—
160	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
161	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
162	C₂²×C₈	D₂₈	1	2	yes	—
163	C₂²×C₈	C₂×Dic₇	1	2	yes	—
164	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
165	C₂²×C₈	C₂²⋊D₇	1	2	yes	—
166	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
167	M₃₂	—	14	4	no	—
168	M₃₂	—	14	4	no	—
169	C₂×C₁₆	—	7	4	no	—
170	C₂×C₁₆	—	14	2	no	—
171	C₂×C₁₆	—	14	2	no	—
172	#108	—	2	4	no	—
173	#108	—	2	2	no	—
174	#108	—	7	4	no	—
175	#108	—	14	4	no	—
176	#108	—	14	2	no	—
177	#108	C₂×Dic₇	1	4	yes	—
178	C₂²×C₇	—	2	2	no	—
179	C₂²×C₇	—	2	2	no	—
180	C₂²×C₇	—	2	2	no	—
181	C₂²×C₇	—	4	2	no	—
182	C₂²×C₇	—	4	2	no	—
183	C₂²×C₇	#340	1	2	yes	—
184	C₂²×C₇	#340	1	2	yes	—
185	C₂²×C₇	C₂×#103	1	1	yes	characteristic
186	C₂²×C₇	C₂×#103	1	1	yes	characteristic
187	C₂²×C₇	C₂×#107	1	1	yes	characteristic
188	C₄×C₇	#340	1	2	yes	—
189	C₄×C₇	#340	1	2	yes	—
190	C₄×C₇	C₂×#95	1	1	yes	characteristic
191	C₄×C₇	C₂×#102	1	1	yes	characteristic
192	C₄×C₇	C₂×#102	1	1	yes	characteristic
193	C₄×C₇	C₂×#106	1	1	yes	characteristic
194	C₂⁴	#1144	1	1	yes	characteristic
195	C₂²×C₄	Dic₇⋊C₄	1	1	yes	characteristic
196	C₂²×C₄	Dic₇⋊C₄	1	1	yes	characteristic
197	C₂²×C₄	(C₂×D₇)⋊C₄	1	2	yes	—
198	C₂²×C₄	#1141	1	1	yes	characteristic
199	C₂²×C₄	#1141	1	1	yes	characteristic
200	C₂²×C₄	#1142	1	2	yes	—
201	C₂²×C₄	#1143	1	2	yes	—
202	C₂²×C₄	#1144	1	1	yes	characteristic
203	C₂²×C₄	C₂²⋊Dic₇	1	2	yes	—
204	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
205	M₁₆	—	2	4	no	—
206	M₁₆	—	4	4	no	—
207	M₁₆	—	4	2	no	—
208	M₁₆	—	14	4	no	—
209	M₁₆	—	28	2	no	—
210	M₁₆	—	28	4	no	—
211	C₂×C₈	—	4	4	no	—
212	C₂×C₈	—	4	2	no	—
213	C₂×C₈	—	28	2	no	—
214	C₂×C₈	—	28	4	no	—
215	C₂×C₈	C₄×Dic₇	1	2	yes	—
216	C₂×C₈	C₄×Dic₇	1	2	yes	—
217	C₂×C₈	Dic₇⋊C₄	1	1	yes	characteristic
218	C₂×C₈	Dic₇⋊C₄	1	1	yes	characteristic
219	C₂×C₈	Dic₇⋊C₄	1	2	yes	—
220	C₂×C₈	Dic₇⋊C₄	1	2	yes	—
221	C₂×C₈	C₄⋊Dic₇	1	2	yes	—
222	C₂×C₈	C₄⋊Dic₇	1	2	yes	—
223	C₂×C₈	(C₂×D₇)⋊C₄	1	2	yes	—
224	C₂×C₈	(C₂×D₇)⋊C₄	1	2	yes	—
225	C₂×C₈	(C₂×D₇)⋊C₄	1	2	yes	—
226	C₂×C₈	C₂²⋊Dic₇	1	2	yes	—
227	C₂×C₈	C₂×Dic₁₄	1	1	yes	characteristic
228	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
229	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
230	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
231	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
232	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
233	C₁₆	—	14	4	no	—
234	C₂×C₇	—	2	2	no	—
235	C₂×C₇	—	2	2	no	—
236	C₂×C₇	—	2	1	no	—
237	C₂×C₇	—	2	1	no	—
238	C₂×C₇	C₂×#340	1	1	yes	characteristic
239	C₂×C₇	#2161	1	2	yes	—
240	C₂³	—	4	2	no	—
241	C₂³	#6381	1	1	yes	characteristic
242	C₂³	#6381	1	1	yes	characteristic
243	C₂³	#6384	1	2	yes	—
244	C₂³	#6409	1	2	yes	—
245	C₂³	C₂×#1144	1	1	yes	characteristic
246	C₂×C₄	#6381	1	1	yes	characteristic
247	C₂×C₄	#6381	1	1	yes	characteristic
248	C₂×C₄	#6382	1	2	yes	—
249	C₂×C₄	#6382	1	2	yes	—
250	C₂×C₄	#6383	1	2	yes	—
251	C₂×C₄	#6383	1	2	yes	—
252	C₂×C₄	#6384	1	2	yes	—
253	C₂×C₄	C₇⋊#95	1	2	yes	—
254	C₂×C₄	C₇⋊#95	1	2	yes	—
255	C₂×C₄	#6406	1	2	yes	—
256	C₂×C₄	#6406	1	2	yes	—
257	C₂×C₄	#6409	1	2	yes	—
258	C₂×C₄	C₂×Dic₇⋊C₄	1	1	yes	characteristic
259	C₂×C₄	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
260	C₂×C₄	C₂×#1141	1	1	yes	characteristic
261	C₂×C₄	C₂×#1142	1	1	yes	characteristic
262	C₂×C₄	C₂×#1143	1	1	yes	characteristic
263	C₂×C₄	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
264	C₈	—	4	4	no	—
265	C₈	—	28	4	no	—
266	C₈	C₇⋊#95	1	2	yes	—
267	C₈	C₇⋊#95	1	2	yes	—
268	C₈	C₂×C₄×Dic₇	1	1	yes	characteristic
269	C₈	C₂×Dic₇⋊C₄	1	1	yes	characteristic
270	C₈	C₂×C₄⋊Dic₇	1	1	yes	characteristic
271	C₈	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
272	C₇	#12559	1	1	yes	characteristic, complemented, Sylow
273	C₂²	—	2	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	4	2	no	—
277	C₂²	—	4	2	no	—
278	C₂²	#89324	1	2	yes	—
279	C₂²	#89324	1	2	yes	—
280	C₂²	C₂×#6381	1	1	yes	characteristic
281	C₂²	C₂×#6384	1	1	yes	characteristic
282	C₂²	C₂×#6409	1	1	yes	characteristic
283	C₄	#89324	1	2	yes	—
284	C₄	#89324	1	2	yes	—
285	C₄	C₂×#6382	1	1	yes	characteristic
286	C₄	C₂×#6383	1	1	yes	characteristic
287	C₄	C₂×C₇⋊#95	1	1	yes	characteristic
288	C₄	C₂×#6406	1	1	yes	characteristic
289	C₂	—	2	1	no	—
290	C₂	—	2	1	no	—
291	C₂	—	2	2	no	—
292	C₂	—	2	2	no	—
293	C₂	#11728437	1	2	yes	—
294	C₂	C₂×#89324	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#1141
commutator	C₇×C₈	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₇
Fitting	C₂²×C₇×#108	C₂
socle	C₂²×C₇	C₂×#107
prime radical	C₂⁴×C₇	Q₁₆
cosocle	C₂²×C₇×C₈	C₂³
hypercenter	C₂²×#108	D₇
hypocenter	C₇	#12559
LCS	G ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×#108

Complements

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

N	H ≅ G/N	type
4. #11725534	C₂	semidirect
272. C₇	#12559	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₇×#108	2
C₂²×#6393	2
C₂²×#6386	2
#11725534	2
#12559	7