Subgroups — #13508548

Order 1792.

514 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11741355	C₂	1	1	yes	maximal, characteristic
2	#11736617	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#2145	C₂	1	1	yes	maximal, characteristic
4	C₂×#89361	C₂	1	1	yes	maximal, characteristic
5	C₂×#89308	C₂	1	1	yes	maximal, characteristic, complemented
6	#11729177	C₂	1	1	yes	maximal, characteristic, complemented
7	#11728404	C₂	1	1	yes	maximal, characteristic, complemented
8	#11727001	C₂	1	4	yes	maximal, complemented
9	#11727000	C₂	1	4	yes	maximal, complemented
10	C₂×C₇×#135	C₂²	1	1	yes	characteristic
11	#90510	—	2	1	no	—
12	#90210	—	2	1	no	—
13	C₂×C₇×M₃₂	C₂²	1	1	yes	characteristic
14	C₇×#427	C₂²	1	1	yes	characteristic
15	#90008	—	2	1	no	—
16	#89969	—	2	2	no	—
17	#89968	C₂²	1	4	yes	—
18	C₂×#6393	C₂²	1	1	yes	characteristic
19	C₂×#6391	C₂²	1	1	yes	characteristic
20	#89873	C₂²	1	1	yes	characteristic
21	C₂×#6386	C₂²	1	1	yes	characteristic
22	#89767	—	2	1	no	—
23	#89648	—	2	2	no	—
24	#89630	C₂²	1	2	yes	complemented
25	#89627	C₂²	1	2	yes	complemented
26	C₇×#360	C₂²	1	2	yes	—
27	C₇×#360	C₂²	1	2	yes	—
28	#89361	C₂²	1	4	yes	—
29	#89308	C₂²	1	4	yes	complemented
30	#89306	C₂²	1	4	yes	complemented
31	#89282	C₂²	1	2	yes	complemented
32	#89282	C₂²	1	2	yes	complemented
33	#33638	—	7	1	no	maximal, Sylow
34	C₂×C₇×#41	—	2	1	no	—
35	C₇×#135	—	2	1	no	—
36	C₇×#135	—	2	1	no	—
37	C₇×#135	Q₈	1	2	yes	complemented
38	C₇×#135	C₂³	1	2	yes	—
39	C₂×C₇×Q₁₆	Q₈	1	1	yes	characteristic, complemented
40	C₂×C₇×SD₁₆	—	2	1	no	—
41	C₂×C₇×D₈	Q₈	1	1	yes	characteristic, complemented
42	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
43	C₇×#117	—	2	1	no	—
44	#6511	—	2	4	no	—
45	#6511	—	4	2	no	—
46	#6489	D₄	1	1	yes	characteristic
47	C₂×C₄⋊Dic₇	—	2	1	no	—
48	C₂×#1136	—	2	1	no	—
49	C₂×#1136	—	4	2	no	—
50	C₂×#1136	D₄	1	1	yes	characteristic
51	#6453	—	2	2	no	—
52	#6451	—	2	2	no	—
53	#6441	—	4	2	no	—
54	#6433	—	4	1	no	—
55	#6432	—	4	1	no	—
56	C₇×M₃₂	—	2	1	no	—
57	C₇×M₃₂	C₂³	1	2	yes	—
58	C₂×C₇×C₁₆	—	2	1	no	—
59	C₇×#107	—	2	1	no	—
60	C₇×#106	C₂³	1	1	yes	characteristic
61	C₇×#106	C₂³	1	1	yes	characteristic
62	#6411	—	2	4	no	—
63	#6393	—	2	1	no	—
64	#6393	C₂³	1	2	yes	—
65	#6391	—	2	1	no	—
66	#6391	C₂³	1	2	yes	—
67	#6389	D₄	1	2	yes	—
68	C₈×Dic₇	C₂³	1	2	yes	—
69	#6386	—	2	1	no	—
70	#6386	C₂³	1	2	yes	—
71	C₂×C₇⋊C₁₆	—	2	1	no	—
72	#6382	—	2	2	no	—
73	#6381	—	2	2	no	—
74	#3163	—	7	1	no	—
75	#2957	—	7	1	no	—
76	#2231	—	7	4	no	—
77	#2230	—	7	4	no	—
78	#2155	—	7	1	no	—
79	C₂×#365	—	7	1	no	—
80	#2153	—	7	1	no	—
81	#2145	D₇	1	1	yes	characteristic
82	C₂×#360	—	7	1	no	—
83	C₇×#41	—	2	2	no	—
84	C₇×#41	—	2	2	no	—
85	C₇×#41	—	4	1	no	—
86	C₇×#41	—	4	1	no	—
87	C₂×C₇×Q₈	—	2	1	no	—
88	C₂×C₇×D₄	—	2	1	no	—
89	C₂×C₇×D₄	—	4	1	no	—
90	C₂²×C₄×C₇	—	4	1	no	—
91	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
92	C₂²×Dic₇	—	4	1	no	—
93	C₂×Dic₁₄	—	4	2	no	—
94	C₇×Q₁₆	—	2	1	no	—
95	C₇×Q₁₆	C₂×Q₈	1	2	yes	—
96	C₇×SD₁₆	—	2	2	no	—
97	C₇×SD₁₆	—	2	2	no	—
98	C₇×D₈	—	2	1	no	—
99	C₇×D₈	C₂×Q₈	1	2	yes	—
100	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
101	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
102	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
103	C₂×C₇×C₈	#41	1	1	yes	characteristic
104	C₂×C₇×C₈	#41	1	1	yes	characteristic
105	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
106	C₇×C₄⋊C₄	—	2	1	no	—
107	C₇×C₄⋊C₄	—	2	1	no	—
108	C₇×C₂²⋊C₄	—	4	1	no	—
109	C₄²×C₇	—	4	1	no	—
110	C₂²⋊Dic₇	—	2	1	no	—
111	C₄⋊Dic₇	—	2	1	no	—
112	C₄⋊Dic₇	—	2	2	no	—
113	C₄⋊Dic₇	—	4	2	no	—
114	C₄⋊Dic₇	—	4	1	no	—
115	Dic₇⋊C₄	—	4	2	no	—
116	Dic₇⋊C₄	—	4	2	no	—
117	C₄×Dic₇	C₂×D₄	1	2	yes	—
118	#1136	—	2	2	no	—
119	#1136	—	4	1	no	—
120	#1136	—	4	4	no	—
121	#1136	SD₁₆	1	2	yes	—
122	#1136	SD₁₆	1	2	yes	—
123	C₂×C₇⋊C₈	—	4	1	no	—
124	C₂×C₇⋊C₈	—	4	4	no	—
125	C₂×C₇⋊C₈	C₂×D₄	1	2	yes	—
126	C₇×C₁₆	—	2	2	no	—
127	C₇⋊C₁₆	—	2	2	no	—
128	C₂×#135	C₂×D₇	1	1	yes	characteristic
129	#568	—	14	1	no	—
130	#544	—	14	1	no	—
131	C₂×M₃₂	—	7	1	no	—
132	C₂×M₃₂	C₂×D₇	1	1	yes	characteristic
133	#501	—	14	2	no	—
134	#499	—	7	2	no	—
135	#498	—	7	2	no	—
136	#444	—	14	2	no	—
137	#443	—	7	4	no	—
138	C₂×#108	—	7	1	no	—
139	#428	—	14	1	no	—
140	#427	C₂×D₇	1	1	yes	characteristic
141	C₂×#106	—	7	1	no	—
142	#421	—	14	1	no	—
143	#405	—	7	1	no	—
144	#365	—	7	4	no	—
145	#365	—	7	2	no	—
146	#365	—	7	2	no	—
147	#364	—	7	4	no	—
148	#360	—	7	4	no	—
149	#360	C₂×D₇	1	2	yes	—
150	#360	C₂×D₇	1	2	yes	—
151	C₂³×C₇	—	4	1	no	—
152	C₂³×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
153	C₇×Q₈	—	2	2	no	—
154	C₇×Q₈	—	4	1	no	—
155	C₇×D₄	—	2	2	no	—
156	C₇×D₄	—	4	2	no	—
157	C₇×D₄	—	4	2	no	—
158	C₇×D₄	—	4	1	no	—
159	C₂×C₄×C₇	—	4	1	no	—
160	C₂×C₄×C₇	—	4	2	no	—
161	C₂×C₄×C₇	—	4	2	no	—
162	C₂×C₄×C₇	—	4	1	no	—
163	C₂×C₄×C₇	—	4	1	no	—
164	C₂×C₄×C₇	C₂²⋊Q₈	1	1	yes	characteristic
165	C₂×C₄×C₇	C₂²⋊Q₈	1	1	yes	characteristic
166	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
167	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
168	C₂×C₄×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
169	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
170	C₂×Dic₇	—	2	2	no	—
171	C₂×Dic₇	—	4	2	no	—
172	C₂×Dic₇	—	16	1	no	—
173	Dic₁₄	—	16	2	no	—
174	C₇×C₈	C₂²⋊Q₈	1	1	yes	characteristic
175	C₇×C₈	C₂²⋊Q₈	1	1	yes	characteristic
176	C₇×C₈	C₂²×Q₈	1	1	yes	characteristic
177	C₇×C₈	C₂×#41	1	1	yes	characteristic
178	C₇⋊C₈	—	4	2	no	—
179	C₇⋊C₈	C₂×SD₁₆	1	4	yes	—
180	C₂×#41	—	2	1	no	—
181	#135	—	2	1	no	—
182	#135	—	2	1	no	—
183	#135	Dic₁₄	1	2	yes	complemented
184	#135	C₂²×D₇	1	2	yes	—
185	C₂×Q₁₆	Dic₁₄	1	1	yes	characteristic, complemented
186	C₂×SD₁₆	—	2	1	no	—
187	C₂×D₈	Dic₁₄	1	1	yes	characteristic, complemented
188	#131	—	14	4	no	—
189	#131	—	28	2	no	—
190	C₂×M₁₆	—	7	1	no	—
191	C₂×M₁₆	—	14	1	no	—
192	C₂×M₁₆	—	28	2	no	—
193	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
194	C₄⋊Q₈	—	14	2	no	—
195	C₄⋊Q₈	—	28	1	no	—
196	#125	—	14	2	no	—
197	#125	—	28	1	no	—
198	C₂²⋊Q₈	—	28	2	no	—
199	#117	—	2	1	no	—
200	#117	—	7	1	no	—
201	C₂×C₄⋊C₄	—	14	1	no	—
202	M₃₂	—	2	1	no	—
203	M₃₂	—	7	2	no	—
204	M₃₂	—	14	1	no	—
205	M₃₂	C₂²×D₇	1	2	yes	—
206	C₂×C₁₆	—	2	1	no	—
207	C₂×C₁₆	—	14	1	no	—
208	#108	—	7	2	no	—
209	#108	—	14	1	no	—
210	#107	—	2	1	no	—
211	#107	—	14	2	no	—
212	#106	—	7	2	no	—
213	#106	—	14	2	no	—
214	#106	—	14	1	no	—
215	#106	C₂²×D₇	1	1	yes	characteristic
216	#106	C₂²×D₇	1	1	yes	characteristic
217	#104	—	14	4	no	—
218	#97	—	7	2	no	—
219	C₄×C₈	—	7	2	no	—
220	C₂²×C₇	—	4	1	no	—
221	C₂²×C₇	—	4	2	no	—
222	C₂²×C₇	—	16	1	no	—
223	C₂²×C₇	#476	1	1	yes	characteristic
224	C₂²×C₇	#476	1	1	yes	characteristic
225	C₂²×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
226	C₄×C₇	—	4	2	no	—
227	C₄×C₇	—	8	1	no	—
228	C₄×C₇	—	8	1	no	—
229	C₄×C₇	#476	1	1	yes	characteristic
230	C₄×C₇	#476	1	1	yes	characteristic
231	C₄×C₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
232	C₄×C₇	C₂²×SD₁₆	1	1	yes	characteristic
233	Dic₇	—	4	2	no	—
234	Dic₇	—	8	2	no	—
235	#41	—	2	2	no	—
236	#41	—	2	2	no	—
237	#41	—	4	1	no	—
238	#41	—	4	1	no	—
239	C₂×Q₈	—	2	1	no	—
240	C₂×Q₈	—	28	2	no	—
241	C₂×D₄	—	2	1	no	—
242	C₂×D₄	—	4	1	no	—
243	C₂²×C₄	—	4	1	no	—
244	C₂²×C₄	—	28	1	no	—
245	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
246	Q₁₆	—	2	1	no	—
247	Q₁₆	C₂×Dic₁₄	1	2	yes	—
248	SD₁₆	—	2	2	no	—
249	SD₁₆	—	2	2	no	—
250	D₈	—	2	1	no	—
251	D₈	C₂×Dic₁₄	1	2	yes	—
252	M₁₆	—	7	2	no	—
253	M₁₆	—	7	2	no	—
254	M₁₆	—	14	2	no	—
255	M₁₆	—	28	4	no	—
256	M₁₆	—	28	1	no	—
257	C₂×C₈	—	7	2	no	—
258	C₂×C₈	—	28	4	no	—
259	C₂×C₈	—	28	1	no	—
260	C₂×C₈	C₂×Dic₁₄	1	1	yes	characteristic
261	C₂×C₈	C₂×Dic₁₄	1	1	yes	characteristic
262	C₂×C₈	D₄×D₇	1	1	yes	characteristic
263	C₂×C₈	#1159	1	1	yes	characteristic
264	C₂×C₈	#1159	1	1	yes	characteristic
265	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
266	C₄⋊C₄	—	2	1	no	—
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	14	1	no	—
269	C₄⋊C₄	—	14	2	no	—
270	C₄⋊C₄	—	28	2	no	—
271	C₄⋊C₄	—	28	2	no	—
272	C₄⋊C₄	—	28	2	no	—
273	C₄⋊C₄	—	28	1	no	—
274	C₂²⋊C₄	—	4	1	no	—
275	C₂²⋊C₄	—	14	1	no	—
276	C₄²	—	4	1	no	—
277	C₄²	—	7	2	no	—
278	C₁₆	—	2	2	no	—
279	C₁₆	—	14	2	no	—
280	C₂×C₇	—	2	1	no	—
281	C₂×C₇	—	2	1	no	—
282	C₂×C₇	—	2	1	no	—
283	C₂×C₇	—	8	2	no	—
284	C₂×C₇	C₂×#476	1	1	yes	characteristic
285	C₂³	—	4	1	no	—
286	C₂³	#6474	1	1	yes	characteristic
287	Q₈	—	2	2	no	—
288	Q₈	—	4	1	no	—
289	Q₈	—	112	2	no	—
290	D₄	—	2	2	no	—
291	D₄	—	4	2	no	—
292	D₄	—	4	2	no	—
293	D₄	—	4	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	14	2	no	—
300	C₂×C₄	—	28	2	no	—
301	C₂×C₄	—	112	1	no	—
302	C₂×C₄	#6441	1	1	yes	characteristic
303	C₂×C₄	#6441	1	1	yes	characteristic
304	C₂×C₄	#6474	1	1	yes	characteristic
305	C₂×C₄	D₇×SD₁₆	1	1	yes	characteristic
306	C₂×C₄	D₇×SD₁₆	1	1	yes	characteristic
307	C₂×C₄	C₂×D₄×D₇	1	1	yes	characteristic
308	C₈	—	7	4	no	—
309	C₈	—	28	2	no	—
310	C₈	#6441	1	1	yes	characteristic
311	C₈	#6441	1	1	yes	characteristic
312	C₈	C₂²×Dic₁₄	1	1	yes	characteristic
313	C₈	C₂×#1159	1	1	yes	characteristic
314	C₇	#33638	1	1	yes	characteristic, complemented, Sylow
315	C₂²	—	4	1	no	—
316	C₂²	—	4	2	no	—
317	C₂²	—	16	1	no	—
318	C₂²	#89533	1	1	yes	characteristic
319	C₂²	#89533	1	1	yes	characteristic
320	C₂²	C₂×#6474	1	1	yes	characteristic
321	C₄	—	4	2	no	—
322	C₄	—	8	1	no	—
323	C₄	—	8	1	no	—
324	C₄	—	28	2	no	—
325	C₄	—	56	2	no	—
326	C₄	#89533	1	1	yes	characteristic
327	C₄	#89533	1	1	yes	characteristic
328	C₄	C₂×#6441	1	1	yes	characteristic
329	C₄	C₂×D₇×SD₁₆	1	1	yes	characteristic
330	C₂	—	2	1	no	—
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	—	8	2	no	—
334	C₂	C₂×#89533	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#89533
commutator, cosocle	C₂×C₇×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₇
Fitting	C₇×#2145	C₂
socle	C₂×C₇	C₂×#476
prime radical	C₂×C₇×#135	C₂²
hypercenter	#2145	D₇
hypocenter	C₇	#33638
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2145

Complements

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

N	H ≅ G/N	type
2. #11736617	C₂	semidirect
5. C₂×#89308	C₂	semidirect
6. #11729177	C₂	semidirect
7. #11728404	C₂	semidirect
8. #11727001	C₂	semidirect
9. #11727000	C₂	semidirect
24. #89630	C₂²	semidirect
25. #89627	C₂²	semidirect
29. #89308	C₂²	semidirect
30. #89306	C₂²	semidirect
31. #89282	C₂²	semidirect
32. #89282	C₂²	semidirect
37. C₇×#135	Q₈	semidirect
39. C₂×C₇×Q₁₆	Q₈	semidirect
41. C₂×C₇×D₈	Q₈	semidirect
183. #135	Dic₁₄	semidirect
185. C₂×Q₁₆	Dic₁₄	semidirect
187. C₂×D₈	Dic₁₄	semidirect
314. C₇	#33638	semidirect

Maximal subgroups

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

subgroup	index
#11741355	2
#11736617	2
C₇×#2145	2
C₂×#89361	2
C₂×#89308	2
#11729177	2
#11728404	2
#11727001	2
#11727000	2
#33638	7