Subgroups — #13337006

Order 1792.

646 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#494	C₂	1	1	yes	maximal, characteristic
2	#11740668	C₂	1	2	yes	maximal
3	#11735890	C₂	1	1	yes	maximal, characteristic
4	#11729527	C₂	1	2	yes	maximal
5	#11729063	C₂	1	4	yes	maximal, complemented
6	#11729062	C₂	1	4	yes	maximal, complemented
7	#11727483	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₇×Q₁₆	—	2	1	no	—
9	C₂²×C₇×Q₁₆	—	2	1	no	—
10	C₂×C₇×C₄⋊Q₈	C₂²	1	2	yes	—
11	#90161	C₂²	1	1	yes	characteristic
12	C₇×#494	—	2	1	no	—
13	C₇×#494	—	2	1	no	—
14	C₇×#494	C₂²	1	2	yes	—
15	C₇×#494	C₂²	1	2	yes	—
16	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
17	#89874	—	2	1	no	—
18	#89873	—	2	1	no	—
19	#89866	C₂²	1	8	yes	—
20	#89865	—	2	2	no	—
21	#89851	—	2	2	no	—
22	#89743	—	2	4	no	—
23	#89697	C₂²	1	2	yes	—
24	#89696	C₂²	1	1	yes	characteristic
25	#89451	—	2	1	no	—
26	#89451	C₂²	1	2	yes	—
27	C₈×Dic₁₄	—	2	1	no	—
28	C₈×Dic₁₄	C₂²	1	2	yes	—
29	#89342	C₂²	1	8	yes	—
30	#89253	C₂²	1	2	yes	—
31	#89252	C₂²	1	2	yes	—
32	#19204	—	7	1	no	maximal, Sylow
33	C₂²×C₇×Q₈	—	2	2	no	—
34	C₂×C₇×Q₁₆	—	2	2	no	—
35	C₂×C₇×Q₁₆	—	2	2	no	—
36	C₂×C₇×Q₁₆	—	2	2	no	—
37	C₂×C₇×Q₁₆	—	2	2	no	—
38	C₂×C₇×Q₁₆	—	4	4	no	—
39	C₂×C₇×Q₁₆	—	4	4	no	—
40	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
41	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
42	C₇×C₄⋊Q₈	—	2	2	no	—
43	C₇×C₄⋊Q₈	—	2	2	no	—
44	C₇×C₄⋊Q₈	—	2	2	no	—
45	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
46	C₂×C₇×C₄⋊C₄	—	2	2	no	—
47	C₂×C₇×C₄⋊C₄	—	2	2	no	—
48	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
49	#6489	—	2	1	no	—
50	#6487	—	2	1	no	—
51	#6487	D₄	1	2	yes	—
52	C₂×#1136	—	2	1	no	—
53	#6433	—	2	1	no	—
54	#6432	D₄	1	1	yes	characteristic
55	#6432	D₄	1	1	yes	characteristic
56	C₄×Dic₁₄	—	2	1	no	—
57	C₄×Dic₁₄	C₂³	1	2	yes	—
58	C₄×C₇×C₈	D₄	1	1	yes	characteristic
59	C₄×C₇×C₈	D₄	1	1	yes	characteristic
60	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
61	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
62	C₂²⋊(C₇⋊C₈)	—	2	2	no	—
63	#6389	—	2	2	no	—
64	Dic₇⋊C₈	—	2	2	no	—
65	Dic₇⋊C₈	—	2	2	no	—
66	C₈×Dic₇	—	2	2	no	—
67	#6384	—	2	8	no	—
68	#6382	—	2	8	no	—
69	C₄⋊(C₇⋊C₈)	—	2	1	no	—
70	C₄⋊(C₇⋊C₈)	C₂³	1	4	yes	—
71	C₄⋊(C₇⋊C₈)	C₂³	1	2	yes	—
72	C₄×C₇⋊C₈	—	2	2	no	—
73	#3229	—	7	2	no	—
74	C₂×#494	D₇	1	1	yes	characteristic
75	#2960	—	7	1	no	—
76	#1690	—	7	4	no	—
77	#1670	—	7	4	no	—
78	#1564	—	7	1	no	—
79	#1546	—	7	2	no	—
80	C₂×C₇×Q₈	—	2	4	no	—
81	C₂×C₇×Q₈	—	4	2	no	—
82	C₂×C₇×Q₈	—	4	4	no	—
83	C₂×C₇×Q₈	—	4	4	no	—
84	C₂²×C₄×C₇	—	4	2	no	—
85	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₇	#41	1	1	yes	characteristic
88	C₂×Dic₁₄	—	2	1	no	—
89	C₂×Dic₁₄	C₂×D₄	1	2	yes	—
90	C₇×Q₁₆	—	4	4	no	—
91	C₇×Q₁₆	—	4	4	no	—
92	C₇×Q₁₆	—	4	4	no	—
93	C₇×Q₁₆	—	4	4	no	—
94	C₂×C₇×C₈	—	2	2	no	—
95	C₂×C₇×C₈	SD₁₆	1	4	yes	—
96	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
97	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
98	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
99	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
100	C₇×C₄⋊C₄	—	2	4	no	—
101	C₇×C₄⋊C₄	—	2	4	no	—
102	C₇×C₄⋊C₄	—	4	2	no	—
103	C₇×C₄⋊C₄	—	4	2	no	—
104	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₇	#41	1	1	yes	characteristic
107	C₄²×C₇	C₂⁴	1	1	yes	characteristic
108	C₂²⋊Dic₇	—	2	2	no	—
109	C₄⋊Dic₇	—	2	1	no	—
110	C₄⋊Dic₇	—	2	1	no	—
111	C₄⋊Dic₇	—	2	1	no	—
112	C₄⋊Dic₇	C₂×D₄	1	2	yes	—
113	Dic₇⋊C₄	—	2	2	no	—
114	Dic₇⋊C₄	—	2	2	no	—
115	C₄×Dic₇	—	2	2	no	—
116	#1136	—	4	2	no	—
117	C₂×C₇⋊C₈	—	2	4	no	—
118	C₂×C₇⋊C₈	—	2	2	no	—
119	C₂²×Q₁₆	—	2	1	no	—
120	C₂²×Q₁₆	—	2	1	no	—
121	#533	—	7	1	no	—
122	C₂×C₄⋊Q₈	C₂×D₇	1	2	yes	—
123	#498	—	14	2	no	—
124	#494	—	2	1	no	—
125	#494	—	2	1	no	—
126	#494	C₂×D₇	1	2	yes	—
127	#494	C₂×D₇	1	2	yes	—
128	#487	—	14	2	no	—
129	#483	—	14	4	no	—
130	#475	—	7	8	no	—
131	#446	—	7	2	no	—
132	#446	—	14	1	no	—
133	C₈×Q₈	—	7	2	no	—
134	C₈×Q₈	—	14	1	no	—
135	#431	—	7	2	no	—
136	#424	—	14	1	no	—
137	#423	—	7	1	no	—
138	#405	—	14	1	no	—
139	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
140	#335	—	7	2	no	—
141	#334	—	7	2	no	—
142	#332	—	7	8	no	—
143	C₂³×C₇	#121	1	1	yes	characteristic
144	C₇×Q₈	—	4	4	no	—
145	C₇×Q₈	—	4	4	no	—
146	C₇×Q₈	—	8	4	no	—
147	C₂×C₄×C₇	—	2	2	no	—
148	C₂×C₄×C₇	—	2	2	no	—
149	C₂×C₄×C₇	—	4	4	no	—
150	C₂×C₄×C₇	—	8	4	no	—
151	C₂×C₄×C₇	#121	1	1	yes	characteristic
152	C₂×C₄×C₇	#121	1	1	yes	characteristic
153	C₂×C₄×C₇	#121	1	1	yes	characteristic
154	C₂×C₄×C₇	C₂×SD₁₆	1	2	yes	—
155	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
156	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
157	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
158	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
159	C₂×Dic₇	—	2	2	no	—
160	C₂×Dic₇	—	2	2	no	—
161	Dic₁₄	—	2	4	no	—
162	Dic₁₄	—	4	2	no	—
163	C₇×C₈	—	2	1	no	—
164	C₇×C₈	—	2	1	no	—
165	C₇×C₈	C₂×SD₁₆	1	2	yes	—
166	C₇×C₈	C₂×SD₁₆	1	2	yes	—
167	C₇⋊C₈	—	4	2	no	—
168	C₇⋊C₈	—	4	4	no	—
169	C₂²×Q₈	—	2	2	no	—
170	C₂×Q₁₆	—	2	2	no	—
171	C₂×Q₁₆	—	2	2	no	—
172	C₂×Q₁₆	—	2	2	no	—
173	C₂×Q₁₆	—	2	2	no	—
174	C₂×Q₁₆	—	4	4	no	—
175	C₂×Q₁₆	—	4	4	no	—
176	C₂×M₁₆	—	14	1	no	—
177	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
178	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
179	C₄⋊Q₈	—	2	2	no	—
180	C₄⋊Q₈	—	2	2	no	—
181	C₄⋊Q₈	—	2	2	no	—
182	C₄⋊Q₈	—	7	1	no	—
183	C₄⋊Q₈	—	7	1	no	—
184	C₄⋊Q₈	C₂²×D₇	1	4	yes	—
185	#125	—	14	1	no	—
186	C₂²⋊Q₈	—	7	2	no	—
187	C₂²⋊Q₈	—	14	1	no	—
188	C₄×Q₈	—	7	2	no	—
189	C₄×Q₈	—	14	1	no	—
190	#117	—	14	1	no	—
191	C₂×C₄⋊C₄	—	2	2	no	—
192	C₂×C₄⋊C₄	—	2	2	no	—
193	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
194	#106	—	14	8	no	—
195	C₄⋊C₈	—	7	2	no	—
196	C₄⋊C₈	—	7	4	no	—
197	C₄⋊C₈	—	14	2	no	—
198	C₄⋊C₈	—	14	2	no	—
199	C₄⋊C₈	—	14	1	no	—
200	#103	—	14	8	no	—
201	C₂²⋊C₈	—	14	2	no	—
202	#97	—	14	2	no	—
203	C₄×C₈	—	14	2	no	—
204	C₄×C₈	—	14	2	no	—
205	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
206	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
207	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
208	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
209	C₂²×C₇	—	2	1	no	—
210	C₂²×C₇	—	2	1	no	—
211	C₂²×C₇	#460	1	2	yes	—
212	C₂²×C₇	C₂×#121	1	1	yes	characteristic
213	C₄×C₇	—	2	1	no	—
214	C₄×C₇	—	2	1	no	—
215	C₄×C₇	—	8	4	no	—
216	C₄×C₇	#460	1	2	yes	—
217	C₄×C₇	#473	1	2	yes	—
218	C₄×C₇	C₂²×SD₁₆	1	1	yes	characteristic
219	C₄×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
220	C₄×C₇	#578	1	2	yes	—
221	Dic₇	—	4	2	no	—
222	Dic₇	—	4	2	no	—
223	C₂×Q₈	—	2	4	no	—
224	C₂×Q₈	—	4	2	no	—
225	C₂×Q₈	—	4	4	no	—
226	C₂×Q₈	—	4	4	no	—
227	C₂×Q₈	—	7	2	no	—
228	C₂×Q₈	—	14	1	no	—
229	C₂²×C₄	—	4	2	no	—
230	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
231	C₂²×C₄	#1159	1	1	yes	characteristic
232	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
233	Q₁₆	—	4	4	no	—
234	Q₁₆	—	4	4	no	—
235	Q₁₆	—	4	4	no	—
236	Q₁₆	—	4	4	no	—
237	M₁₆	—	28	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	14	4	no	—
240	C₂×C₈	—	14	2	no	—
241	C₂×C₈	#1142	1	4	yes	—
242	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
243	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
244	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
245	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
246	C₄⋊C₄	—	2	4	no	—
247	C₄⋊C₄	—	2	4	no	—
248	C₄⋊C₄	—	4	2	no	—
249	C₄⋊C₄	—	4	2	no	—
250	C₄⋊C₄	—	7	2	no	—
251	C₄⋊C₄	—	14	2	no	—
252	C₄⋊C₄	—	14	1	no	—
253	C₄⋊C₄	—	14	1	no	—
254	C₄⋊C₄	—	14	1	no	—
255	C₄⋊C₄	—	14	2	no	—
256	C₂²⋊C₄	—	14	2	no	—
257	C₄²	—	14	2	no	—
258	C₄²	D₄×D₇	1	1	yes	characteristic
259	C₄²	#1159	1	1	yes	characteristic
260	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
261	C₄²	C₂³×D₇	1	1	yes	characteristic
262	C₂×C₇	—	2	2	no	—
263	C₂×C₇	C₂×#460	1	1	yes	characteristic
264	C₂×C₇	#3059	1	1	yes	characteristic
265	C₂×C₇	C₂×#473	1	1	yes	characteristic
266	C₂³	#6501	1	1	yes	characteristic
267	Q₈	—	4	4	no	—
268	Q₈	—	4	4	no	—
269	Q₈	—	8	4	no	—
270	Q₈	—	14	4	no	—
271	Q₈	—	28	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	8	4	no	—
276	C₂×C₄	—	14	2	no	—
277	C₂×C₄	—	14	2	no	—
278	C₂×C₄	#6495	1	2	yes	—
279	C₂×C₄	C₂×#1142	1	2	yes	—
280	C₂×C₄	#6501	1	1	yes	characteristic
281	C₂×C₄	#6501	1	1	yes	characteristic
282	C₂×C₄	#6501	1	1	yes	characteristic
283	C₂×C₄	C₂×D₄×D₇	1	1	yes	characteristic
284	C₂×C₄	C₂×#1159	1	1	yes	characteristic
285	C₂×C₄	C₂²×C₂²⋊D₇	1	1	yes	characteristic
286	C₈	—	2	1	no	—
287	C₈	—	2	1	no	—
288	C₈	—	28	4	no	—
289	C₈	—	28	2	no	—
290	C₈	C₂×#1142	1	2	yes	—
291	C₈	C₂×#1142	1	2	yes	—
292	C₇	#19204	1	1	yes	characteristic, complemented, Sylow
293	C₂²	—	2	1	no	—
294	C₂²	—	2	1	no	—
295	C₂²	#89847	1	2	yes	—
296	C₂²	C₂×#6501	1	1	yes	characteristic
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	8	4	no	—
300	C₄	—	28	2	no	—
301	C₄	—	28	2	no	—
302	C₄	#89847	1	2	yes	—
303	C₄	#89964	1	2	yes	—
304	C₄	#90457	1	2	yes	—
305	C₄	C₂×#6495	1	1	yes	characteristic
306	C₄	C₂²×#1142	1	1	yes	characteristic
307	C₂	—	2	2	no	—
308	C₂	C₂×#89847	1	1	yes	characteristic
309	C₂	#11741332	1	1	yes	characteristic
310	C₂	C₂×#89964	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#6501
commutator, cosocle	C₄²×C₇	C₂⁴
Frattini	C₄²	C₂³×D₇
Fitting	C₂×C₇×#494	C₂
socle	C₂²×C₇	C₂×#121
prime radical	C₂³×C₇	#121
hypercenter	C₂×#494	D₇
hypocenter	C₇	#19204
LCS	G ▷ C₄²×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
5. #11729063	C₂	semidirect
6. #11729062	C₂	semidirect
292. C₇	#19204	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×#494	2
#11740668	2
#11735890	2
#11729527	2
#11729063	2
#11729062	2
#11727483	2
#19204	7