Subgroups — #13508538

Order 1792.

524 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#89667	C₂	1	1	yes	maximal, characteristic
2	#11736615	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#2155	C₂	1	1	yes	maximal, characteristic, complemented
4	#11729204	C₂	1	1	yes	maximal, characteristic
5	C₂×#89306	C₂	1	1	yes	maximal, characteristic, complemented
6	#11728439	C₂	1	1	yes	maximal, characteristic, complemented
7	#11728410	C₂	1	1	yes	maximal, characteristic
8	#11727070	C₂	1	4	yes	maximal, complemented
9	#11727069	C₂	1	4	yes	maximal, complemented
10	C₂×#6470	—	2	1	no	—
11	C₂×#6467	C₂²	1	1	yes	characteristic
12	#90208	—	2	1	no	—
13	C₂×C₇×M₃₂	C₂²	1	1	yes	characteristic
14	C₂×C₇×#108	C₂²	1	1	yes	characteristic
15	C₇×#427	C₂²	1	1	yes	characteristic
16	C₂×#6399	—	2	1	no	—
17	#89878	C₂²	1	1	yes	characteristic
18	C₂×C₈×Dic₇	C₂²	1	1	yes	characteristic
19	C₂×#6386	C₂²	1	1	yes	characteristic
20	#89770	—	2	1	no	—
21	#89667	—	2	1	no	—
22	#89667	—	2	1	no	—
23	#89667	C₂²	1	2	yes	—
24	#89667	C₂²	1	2	yes	—
25	#89650	—	2	1	no	—
26	#89646	—	2	1	no	—
27	#89630	C₂²	1	2	yes	complemented
28	#89627	C₂²	1	2	yes	complemented
29	C₇×#365	C₂²	1	4	yes	complemented
30	#89313	C₂²	1	4	yes	—
31	#89306	C₂²	1	2	yes	complemented
32	#89306	C₂²	1	2	yes	complemented
33	#89289	C₂²	1	4	yes	complemented
34	#89283	C₂²	1	4	yes	—
35	#33637	—	7	1	no	maximal, Sylow
36	C₂×#1157	—	2	1	no	—
37	C₂×C₇×M₁₆	—	2	1	no	—
38	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
39	C₇×#117	—	2	1	no	—
40	#6489	—	2	1	no	—
41	C₂×C₄×Dic₇	D₄	1	1	yes	characteristic
42	C₂²×C₇⋊C₈	D₄	1	1	yes	characteristic
43	#6470	—	2	2	no	—
44	#6470	—	2	2	no	—
45	#6470	—	4	1	no	—
46	#6470	—	4	1	no	—
47	C₂×Dic₂₈	Q₈	1	1	yes	characteristic, complemented
48	#6467	—	2	1	no	—
49	#6467	—	2	1	no	—
50	#6467	Q₈	1	2	yes	complemented
51	#6467	C₂³	1	2	yes	—
52	C₂×D₅₆	Q₈	1	1	yes	characteristic, complemented
53	C₂×#1132	—	2	1	no	—
54	C₂×#1131	—	4	1	no	—
55	C₂×C₈×D₇	—	4	1	no	—
56	#6453	—	2	2	no	—
57	#6451	—	2	2	no	—
58	#6441	—	4	2	no	—
59	C₄×Dic₁₄	—	4	2	no	—
60	C₇×M₃₂	—	2	1	no	—
61	C₇×M₃₂	C₂³	1	2	yes	—
62	C₂×C₇×C₁₆	—	2	1	no	—
63	C₇×#108	—	2	1	no	—
64	C₇×#108	C₂³	1	2	yes	—
65	C₇×#107	—	2	1	no	—
66	C₇×#106	C₂³	1	2	yes	—
67	#6399	—	2	2	no	—
68	#6399	—	2	2	no	—
69	#6392	—	2	1	no	—
70	#6391	C₂³	1	2	yes	—
71	C₈×Dic₇	D₄	1	1	yes	characteristic
72	C₈×Dic₇	D₄	1	1	yes	characteristic
73	C₈×Dic₇	C₂³	1	1	yes	characteristic
74	C₈×Dic₇	C₂³	1	1	yes	characteristic
75	#6386	—	2	1	no	—
76	#6386	C₂³	1	2	yes	—
77	C₂×C₇⋊C₁₆	—	2	1	no	—
78	#6382	—	2	2	no	—
79	#6381	—	2	2	no	—
80	#3160	—	7	1	no	—
81	C₂×#443	—	7	1	no	—
82	#2231	—	7	4	no	—
83	#2230	—	7	4	no	—
84	#2155	—	7	1	no	—
85	#2155	D₇	1	1	yes	characteristic, complemented
86	C₂×#364	—	7	1	no	—
87	#2145	—	7	1	no	—
88	#2145	—	7	1	no	—
89	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
90	C₂×C₂²⋊D₇	—	4	1	no	—
91	C₂²×Dic₇	—	2	1	no	—
92	#1157	—	2	2	no	—
93	#1157	—	2	2	no	—
94	#1157	—	4	1	no	—
95	#1157	—	4	1	no	—
96	C₂×D₂₈	—	2	1	no	—
97	C₂×C₄×D₇	—	4	1	no	—
98	C₂×Dic₁₄	—	2	1	no	—
99	C₂×Dic₁₄	—	4	2	no	—
100	C₇×M₁₆	—	2	2	no	—
101	C₇×M₁₆	—	4	1	no	—
102	C₂×C₇×C₈	—	4	1	no	—
103	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
104	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
105	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
106	C₂×C₇×C₈	#41	1	1	yes	characteristic
107	C₂×C₇×C₈	#41	1	1	yes	characteristic
108	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
109	C₇×C₄⋊C₄	—	2	2	no	—
110	C₇×C₂²⋊C₄	—	4	1	no	—
111	C₄²×C₇	—	4	1	no	—
112	C₂²⋊Dic₇	—	4	1	no	—
113	C₄⋊Dic₇	—	2	2	no	—
114	C₄⋊Dic₇	—	4	2	no	—
115	Dic₇⋊C₄	—	4	2	no	—
116	Dic₇⋊C₄	—	4	2	no	—
117	C₄×Dic₇	—	2	1	no	—
118	C₄×Dic₇	—	4	1	no	—
119	C₄×Dic₇	C₂×D₄	1	1	yes	characteristic
120	C₄×Dic₇	C₂×D₄	1	1	yes	characteristic
121	C₂×C₇⋊C₈	SD₁₆	1	1	yes	characteristic
122	C₂×C₇⋊C₈	SD₁₆	1	1	yes	characteristic
123	C₂×C₇⋊C₈	SD₁₆	1	1	yes	characteristic
124	C₂×C₇⋊C₈	SD₁₆	1	1	yes	characteristic
125	C₂×C₇⋊C₈	C₂×D₄	1	1	yes	characteristic
126	C₂×C₇⋊C₈	C₂×D₄	1	1	yes	characteristic
127	Dic₂₈	—	2	1	no	—
128	Dic₂₈	C₂×Q₈	1	2	yes	—
129	D₅₆	—	2	1	no	—
130	D₅₆	C₂×Q₈	1	2	yes	—
131	#1132	—	2	2	no	—
132	#1132	—	2	2	no	—
133	#1131	—	4	2	no	—
134	#1131	—	4	2	no	—
135	C₈×D₇	—	4	2	no	—
136	C₈×D₇	—	4	2	no	—
137	C₇×C₁₆	—	2	2	no	—
138	C₇⋊C₁₆	—	2	2	no	—
139	C₂×#135	—	7	1	no	—
140	C₂×#131	—	14	1	no	—
141	#533	—	14	1	no	—
142	C₂×M₃₂	—	7	1	no	—
143	C₂×M₃₂	C₂×D₇	1	1	yes	characteristic
144	#500	—	14	1	no	—
145	#499	—	7	2	no	—
146	#499	—	14	1	no	—
147	#498	—	7	2	no	—
148	#443	—	7	2	no	—
149	#443	—	7	2	no	—
150	#443	—	14	1	no	—
151	#443	—	14	1	no	—
152	C₂×#108	C₂×D₇	1	1	yes	characteristic
153	#427	—	7	1	no	—
154	#427	—	14	1	no	—
155	#427	C₂×D₇	1	1	yes	characteristic
156	C₂×#104	—	14	1	no	—
157	C₂×C₄×C₈	—	7	1	no	—
158	#365	—	7	4	no	—
159	#365	C₂×D₇	1	4	yes	complemented
160	#364	—	7	2	no	—
161	#364	—	7	2	no	—
162	#360	—	7	4	no	—
163	#360	—	7	4	no	—
164	C₂³×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
165	C₂²×D₇	—	4	1	no	—
166	C₂×C₄×C₇	—	4	2	no	—
167	C₂×C₄×C₇	C₂²⋊Q₈	1	1	yes	characteristic
168	C₂×C₄×C₇	C₂²⋊Q₈	1	1	yes	characteristic
169	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
170	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
171	C₂×C₄×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
172	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
173	C₂²⋊D₇	—	4	2	no	—
174	C₂²⋊D₇	—	4	2	no	—
175	C₂×Dic₇	—	2	1	no	—
176	C₂×Dic₇	—	2	1	no	—
177	C₂×Dic₇	—	4	2	no	—
178	C₂×Dic₇	—	4	1	no	—
179	C₂×Dic₇	—	4	2	no	—
180	D₂₈	—	2	2	no	—
181	D₂₈	—	4	1	no	—
182	C₄×D₇	—	4	2	no	—
183	C₄×D₇	—	4	2	no	—
184	Dic₁₄	—	2	2	no	—
185	Dic₁₄	—	4	1	no	—
186	Dic₁₄	—	8	4	no	—
187	C₇×C₈	—	4	2	no	—
188	C₇×C₈	C₂²⋊Q₈	1	1	yes	characteristic
189	C₇×C₈	C₂²⋊Q₈	1	1	yes	characteristic
190	C₇×C₈	C₂²×Q₈	1	1	yes	characteristic
191	C₇×C₈	C₂×#41	1	1	yes	characteristic
192	C₇⋊C₈	C₂×SD₁₆	1	1	yes	characteristic
193	C₇⋊C₈	C₂×SD₁₆	1	1	yes	characteristic
194	C₇⋊C₈	C₂×SD₁₆	1	1	yes	characteristic
195	C₇⋊C₈	C₂×SD₁₆	1	1	yes	characteristic
196	C₂×#41	—	14	1	no	—
197	#135	—	7	2	no	—
198	#135	—	7	2	no	—
199	#135	—	14	1	no	—
200	#135	—	14	1	no	—
201	C₂×Q₁₆	—	7	1	no	—
202	C₂×SD₁₆	—	14	1	no	—
203	C₂×D₈	—	7	1	no	—
204	#131	—	14	2	no	—
205	#131	—	14	2	no	—
206	#131	—	28	1	no	—
207	#131	—	28	1	no	—
208	C₂×M₁₆	—	2	1	no	—
209	C₂×M₁₆	—	28	1	no	—
210	C₂²×C₈	—	7	1	no	—
211	C₂²×C₈	—	28	1	no	—
212	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
213	C₄⋊Q₈	—	14	2	no	—
214	#125	—	14	2	no	—
215	C₂²⋊Q₈	—	28	2	no	—
216	C₄×Q₈	—	28	2	no	—
217	#117	—	2	1	no	—
218	#117	—	14	1	no	—
219	C₂×C₄²	—	7	1	no	—
220	M₃₂	—	2	1	no	—
221	M₃₂	—	7	2	no	—
222	M₃₂	—	14	1	no	—
223	M₃₂	C₂²×D₇	1	2	yes	—
224	C₂×C₁₆	—	2	1	no	—
225	C₂×C₁₆	—	14	1	no	—
226	#108	—	2	1	no	—
227	#108	C₂²×D₇	1	2	yes	—
228	#107	—	2	1	no	—
229	#107	—	14	2	no	—
230	#107	—	14	1	no	—
231	#106	—	7	2	no	—
232	#106	—	14	2	no	—
233	#106	C₂²×D₇	1	2	yes	—
234	#104	—	14	2	no	—
235	#104	—	14	2	no	—
236	C₄×C₈	—	7	1	no	—
237	C₄×C₈	—	7	1	no	—
238	C₄×C₈	—	7	1	no	—
239	C₄×C₈	—	7	1	no	—
240	C₂²×C₇	—	4	1	no	—
241	C₂²×C₇	#476	1	1	yes	characteristic
242	C₂²×C₇	#476	1	1	yes	characteristic
243	C₂²×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
244	C₂×D₇	—	4	2	no	—
245	C₂×D₇	—	8	2	no	—
246	C₄×C₇	—	8	2	no	—
247	C₄×C₇	#476	1	1	yes	characteristic
248	C₄×C₇	#476	1	1	yes	characteristic
249	C₄×C₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
250	C₄×C₇	C₂²×SD₁₆	1	1	yes	characteristic
251	Dic₇	—	2	2	no	—
252	Dic₇	—	4	1	no	—
253	Dic₇	—	4	2	no	—
254	Dic₇	—	8	2	no	—
255	#41	—	14	2	no	—
256	#41	—	14	2	no	—
257	#41	—	28	1	no	—
258	#41	—	28	1	no	—
259	C₂×Q₈	—	14	1	no	—
260	C₂×Q₈	—	28	2	no	—
261	C₂×D₄	—	14	1	no	—
262	C₂×D₄	—	28	1	no	—
263	C₂²×C₄	—	14	1	no	—
264	C₂²×C₄	—	28	1	no	—
265	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
266	Q₁₆	—	7	2	no	—
267	Q₁₆	—	14	1	no	—
268	SD₁₆	—	14	2	no	—
269	SD₁₆	—	14	2	no	—
270	D₈	—	7	2	no	—
271	D₈	—	14	1	no	—
272	M₁₆	—	2	2	no	—
273	M₁₆	—	4	1	no	—
274	M₁₆	—	28	2	no	—
275	M₁₆	—	28	2	no	—
276	C₂×C₈	—	4	1	no	—
277	C₂×C₈	—	7	1	no	—
278	C₂×C₈	—	7	1	no	—
279	C₂×C₈	—	7	1	no	—
280	C₂×C₈	—	7	1	no	—
281	C₂×C₈	—	7	1	no	—
282	C₂×C₈	—	7	1	no	—
283	C₂×C₈	—	28	2	no	—
284	C₂×C₈	—	28	2	no	—
285	C₂×C₈	#1157	1	1	yes	characteristic
286	C₂×C₈	#1157	1	1	yes	characteristic
287	C₂×C₈	D₄×D₇	1	1	yes	characteristic
288	C₂×C₈	Q₈×D₇	1	1	yes	characteristic
289	C₂×C₈	Q₈×D₇	1	1	yes	characteristic
290	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	14	2	no	—
293	C₄⋊C₄	—	28	2	no	—
294	C₄⋊C₄	—	28	2	no	—
295	C₄⋊C₄	—	28	2	no	—
296	C₂²⋊C₄	—	4	1	no	—
297	C₂²⋊C₄	—	28	1	no	—
298	C₄²	—	4	1	no	—
299	C₄²	—	7	1	no	—
300	C₄²	—	7	1	no	—
301	C₄²	—	14	1	no	—
302	C₄²	—	28	1	no	—
303	C₁₆	—	2	2	no	—
304	C₁₆	—	14	2	no	—
305	C₂×C₇	—	2	1	no	—
306	C₂×C₇	—	2	1	no	—
307	C₂×C₇	—	2	1	no	—
308	C₂×C₇	C₂×#476	1	1	yes	characteristic
309	D₇	—	8	2	no	—
310	C₂³	—	28	1	no	—
311	C₂³	#6477	1	1	yes	characteristic
312	Q₈	—	14	2	no	—
313	Q₈	—	28	1	no	—
314	Q₈	—	56	4	no	—
315	D₄	—	14	2	no	—
316	D₄	—	28	2	no	—
317	D₄	—	28	2	no	—
318	D₄	—	28	1	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	14	1	no	—
321	C₂×C₄	—	14	1	no	—
322	C₂×C₄	—	28	2	no	—
323	C₂×C₄	—	28	1	no	—
324	C₂×C₄	—	28	2	no	—
325	C₂×C₄	—	28	2	no	—
326	C₂×C₄	—	28	2	no	—
327	C₂×C₄	#6459	1	1	yes	characteristic
328	C₂×C₄	#6459	1	1	yes	characteristic
329	C₂×C₄	D₇×SD₁₆	1	1	yes	characteristic
330	C₂×C₄	D₇×SD₁₆	1	1	yes	characteristic
331	C₂×C₄	#6477	1	1	yes	characteristic
332	C₂×C₄	C₂×D₄×D₇	1	1	yes	characteristic
333	C₈	—	4	2	no	—
334	C₈	—	7	1	no	—
335	C₈	—	7	1	no	—
336	C₈	—	7	1	no	—
337	C₈	—	7	1	no	—
338	C₈	#6459	1	1	yes	characteristic
339	C₈	#6459	1	1	yes	characteristic
340	C₈	C₂×#1157	1	1	yes	characteristic
341	C₈	C₂×Q₈×D₇	1	1	yes	characteristic
342	C₇	#33637	1	1	yes	characteristic, complemented, Sylow
343	C₂²	—	4	1	no	—
344	C₂²	—	28	2	no	—
345	C₂²	—	56	2	no	—
346	C₂²	#89642	1	1	yes	characteristic
347	C₂²	#89642	1	1	yes	characteristic
348	C₂²	C₂×#6477	1	1	yes	characteristic
349	C₄	—	8	2	no	—
350	C₄	—	14	2	no	—
351	C₄	—	28	1	no	—
352	C₄	—	28	2	no	—
353	C₄	—	56	2	no	—
354	C₄	#89642	1	1	yes	characteristic
355	C₄	#89642	1	1	yes	characteristic
356	C₄	C₂×#6459	1	1	yes	characteristic
357	C₄	C₂×D₇×SD₁₆	1	1	yes	characteristic
358	C₂	—	2	1	no	—
359	C₂	—	2	1	no	—
360	C₂	—	2	1	no	—
361	C₂	—	56	2	no	—
362	C₂	C₂×#89642	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#89642
commutator, cosocle	C₂×C₇×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₇
Fitting	C₇×#2155	C₂
socle	C₂×C₇	C₂×#476
prime radical	C₂×#6467	C₂²
hypercenter	#2155	D₇
hypocenter	C₇	#33637
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2155

Complements

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

N	H ≅ G/N	type
2. #11736615	C₂	semidirect
3. C₇×#2155	C₂	semidirect
5. C₂×#89306	C₂	semidirect
6. #11728439	C₂	semidirect
8. #11727070	C₂	semidirect
9. #11727069	C₂	semidirect
27. #89630	C₂²	semidirect
28. #89627	C₂²	semidirect
29. C₇×#365	C₂²	semidirect
31. #89306	C₂²	semidirect
32. #89306	C₂²	semidirect
33. #89289	C₂²	semidirect
47. C₂×Dic₂₈	Q₈	semidirect
50. #6467	Q₈	semidirect
52. C₂×D₅₆	Q₈	semidirect
85. #2155	D₇	semidirect
159. #365	C₂×D₇	semidirect
342. C₇	#33637	semidirect

Maximal subgroups

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

subgroup	index
C₂×#89667	2
#11736615	2
C₇×#2155	2
#11729204	2
C₂×#89306	2
#11728439	2
#11728410	2
#11727070	2
#11727069	2
#33637	7