Subgroups — #13508539

Order 1792.

524 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#443	C₂	1	1	yes	maximal, characteristic
2	#11735887	C₂	1	1	yes	maximal, characteristic, complemented
3	#11729830	C₂	1	2	yes	maximal
4	#11728674	C₂	1	4	yes	maximal, complemented
5	#11728673	C₂	1	4	yes	maximal, complemented
6	#11728439	C₂	1	2	yes	maximal, complemented
7	C₂×#89258	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₇×#135	C₂²	1	1	yes	characteristic
9	C₂×C₇×#131	—	2	1	no	—
10	#90161	—	2	1	no	—
11	C₇×#443	—	2	1	no	—
12	C₇×#443	—	2	1	no	—
13	C₇×#443	C₂²	1	2	yes	—
14	C₇×#443	C₂²	1	2	yes	—
15	C₂×C₇×#108	C₂²	1	1	yes	characteristic
16	C₂×C₇×#104	—	2	1	no	—
17	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
18	#89878	—	2	1	no	—
19	#89878	C₂²	1	2	yes	—
20	C₂×#6386	C₂²	1	2	yes	—
21	#89455	—	2	1	no	—
22	#89455	C₂²	1	2	yes	complemented
23	#89454	—	2	1	no	—
24	#89452	C₂²	1	2	yes	complemented
25	#89361	C₂²	1	8	yes	—
26	#89289	C₂²	1	8	yes	complemented
27	#89258	C₂²	1	2	yes	complemented
28	#89258	C₂²	1	2	yes	complemented
29	#33637	—	7	1	no	maximal, Sylow
30	C₂×C₇×#41	—	2	1	no	—
31	C₇×#135	—	2	1	no	—
32	C₇×#135	—	2	1	no	—
33	C₇×#135	Q₈	1	2	yes	complemented
34	C₇×#135	C₂³	1	2	yes	—
35	C₂×C₇×Q₁₆	Q₈	1	1	yes	characteristic, complemented
36	C₂×C₇×SD₁₆	—	2	1	no	—
37	C₂×C₇×D₈	Q₈	1	1	yes	characteristic, complemented
38	C₇×#131	—	2	2	no	—
39	C₇×#131	—	2	2	no	—
40	C₇×#131	—	4	1	no	—
41	C₇×#131	—	4	1	no	—
42	C₂×C₇×M₁₆	—	2	1	no	—
43	C₂×C₇×M₁₆	—	4	1	no	—
44	C₂²×C₇×C₈	—	4	1	no	—
45	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
46	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
48	#6489	—	2	2	no	—
49	#6487	—	4	2	no	—
50	#6433	—	2	2	no	—
51	#6432	—	2	2	no	—
52	C₄×Dic₁₄	—	4	2	no	—
53	C₇×#108	—	2	1	no	—
54	C₇×#108	C₂³	1	2	yes	—
55	C₇×#104	—	2	2	no	—
56	C₇×#104	—	2	2	no	—
57	C₄×C₇×C₈	D₄	1	1	yes	characteristic
58	C₄×C₇×C₈	D₄	1	1	yes	characteristic
59	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
60	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
61	#6392	—	2	2	no	—
62	#6392	—	2	2	no	—
63	#6391	—	2	2	no	—
64	#6391	C₂³	1	4	yes	—
65	#6386	—	2	2	no	—
66	#6386	C₂³	1	4	yes	—
67	C₂×C₇⋊C₁₆	—	2	2	no	—
68	#3160	—	7	1	no	—
69	C₂×#443	D₇	1	1	yes	characteristic
70	#2231	—	7	4	no	—
71	#2230	—	7	4	no	—
72	#2155	—	7	2	no	—
73	C₂×#364	—	7	1	no	—
74	#2145	—	7	2	no	—
75	C₇×#41	—	2	2	no	—
76	C₇×#41	—	2	2	no	—
77	C₇×#41	—	4	1	no	—
78	C₇×#41	—	4	1	no	—
79	C₂×C₇×Q₈	—	2	1	no	—
80	C₂×C₇×D₄	—	2	1	no	—
81	C₂×C₇×D₄	—	4	1	no	—
82	C₂²×C₄×C₇	—	2	1	no	—
83	C₂²×C₄×C₇	—	4	1	no	—
84	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
85	C₂×Dic₁₄	—	4	2	no	—
86	C₇×Q₁₆	—	2	1	no	—
87	C₇×Q₁₆	C₂×Q₈	1	2	yes	—
88	C₇×SD₁₆	—	2	2	no	—
89	C₇×SD₁₆	—	2	2	no	—
90	C₇×D₈	—	2	1	no	—
91	C₇×D₈	C₂×Q₈	1	2	yes	—
92	C₇×M₁₆	—	2	2	no	—
93	C₇×M₁₆	—	4	1	no	—
94	C₇×M₁₆	—	4	2	no	—
95	C₇×M₁₆	—	4	2	no	—
96	C₂×C₇×C₈	—	4	1	no	—
97	C₂×C₇×C₈	—	4	2	no	—
98	C₂×C₇×C₈	—	4	2	no	—
99	C₂×C₇×C₈	SD₁₆	1	1	yes	characteristic
100	C₂×C₇×C₈	SD₁₆	1	1	yes	characteristic
101	C₂×C₇×C₈	SD₁₆	1	1	yes	characteristic
102	C₂×C₇×C₈	SD₁₆	1	1	yes	characteristic
103	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
104	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
105	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
106	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
107	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
108	C₂×C₇×C₈	#41	1	1	yes	characteristic
109	C₂×C₇×C₈	#41	1	1	yes	characteristic
110	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
111	C₄²×C₇	—	2	1	no	—
112	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
114	C₂²⋊Dic₇	—	4	2	no	—
115	C₄⋊Dic₇	—	2	4	no	—
116	C₄⋊Dic₇	—	4	2	no	—
117	Dic₇⋊C₄	—	4	4	no	—
118	C₄×Dic₇	—	4	2	no	—
119	C₇⋊C₁₆	—	2	4	no	—
120	C₂×#135	C₂×D₇	1	1	yes	characteristic
121	C₂×#131	—	2	1	no	—
122	#533	—	14	1	no	—
123	C₂×M₃₂	—	7	2	no	—
124	#500	—	14	1	no	—
125	#499	—	7	2	no	—
126	#499	—	14	1	no	—
127	#498	—	7	2	no	—
128	#443	—	2	1	no	—
129	#443	—	2	1	no	—
130	#443	C₂×D₇	1	2	yes	—
131	#443	C₂×D₇	1	2	yes	—
132	C₂×#108	C₂×D₇	1	1	yes	characteristic
133	#427	—	7	2	no	—
134	#427	—	14	1	no	—
135	C₂×#104	—	2	1	no	—
136	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
137	#365	—	7	8	no	—
138	#364	—	7	2	no	—
139	#364	—	7	2	no	—
140	#360	—	7	8	no	—
141	C₂³×C₇	—	4	1	no	—
142	C₂³×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
143	C₇×Q₈	—	2	2	no	—
144	C₇×Q₈	—	4	1	no	—
145	C₇×D₄	—	2	2	no	—
146	C₇×D₄	—	4	2	no	—
147	C₇×D₄	—	4	2	no	—
148	C₇×D₄	—	4	1	no	—
149	C₂×C₄×C₇	—	2	1	no	—
150	C₂×C₄×C₇	—	2	1	no	—
151	C₂×C₄×C₇	—	4	2	no	—
152	C₂×C₄×C₇	—	4	1	no	—
153	C₂×C₄×C₇	—	4	2	no	—
154	C₂×C₄×C₇	—	4	2	no	—
155	C₂×C₄×C₇	C₂²⋊Q₈	1	1	yes	characteristic
156	C₂×C₄×C₇	C₂²⋊Q₈	1	1	yes	characteristic
157	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
158	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
159	C₂×C₄×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
160	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
161	C₂×Dic₇	—	4	4	no	—
162	Dic₁₄	—	8	4	no	—
163	C₇×C₈	—	4	2	no	—
164	C₇×C₈	C₂²⋊Q₈	1	1	yes	characteristic
165	C₇×C₈	C₂²⋊Q₈	1	1	yes	characteristic
166	C₇×C₈	C₂×SD₁₆	1	1	yes	characteristic
167	C₇×C₈	C₂×SD₁₆	1	1	yes	characteristic
168	C₇×C₈	C₂×SD₁₆	1	1	yes	characteristic
169	C₇×C₈	C₂×SD₁₆	1	1	yes	characteristic
170	C₇×C₈	C₂²×Q₈	1	1	yes	characteristic
171	C₇×C₈	C₂×#41	1	1	yes	characteristic
172	C₂×#41	—	2	1	no	—
173	#135	—	2	1	no	—
174	#135	—	2	1	no	—
175	#135	Dic₁₄	1	2	yes	complemented
176	#135	C₂²×D₇	1	2	yes	—
177	C₂×Q₁₆	Dic₁₄	1	1	yes	characteristic, complemented
178	C₂×SD₁₆	—	2	1	no	—
179	C₂×D₈	Dic₁₄	1	1	yes	characteristic, complemented
180	#131	—	2	2	no	—
181	#131	—	2	2	no	—
182	#131	—	4	1	no	—
183	#131	—	4	1	no	—
184	C₂×M₁₆	—	2	1	no	—
185	C₂×M₁₆	—	4	1	no	—
186	C₂²×C₈	—	4	1	no	—
187	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
188	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
189	C₄⋊Q₈	—	14	2	no	—
190	#125	—	14	2	no	—
191	C₂²⋊Q₈	—	28	2	no	—
192	C₄×Q₈	—	28	2	no	—
193	#117	—	14	2	no	—
194	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
195	M₃₂	—	7	4	no	—
196	M₃₂	—	14	2	no	—
197	C₂×C₁₆	—	14	2	no	—
198	#108	—	2	1	no	—
199	#108	C₂²×D₇	1	2	yes	—
200	#107	—	14	2	no	—
201	#107	—	14	2	no	—
202	#106	—	7	4	no	—
203	#106	—	14	2	no	—
204	#104	—	2	2	no	—
205	#104	—	2	2	no	—
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	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
210	C₂²×C₇	—	4	1	no	—
211	C₂²×C₇	—	4	2	no	—
212	C₂²×C₇	—	8	2	no	—
213	C₂²×C₇	#476	1	1	yes	characteristic
214	C₂²×C₇	#476	1	1	yes	characteristic
215	C₂²×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
216	C₄×C₇	—	2	2	no	—
217	C₄×C₇	—	4	1	no	—
218	C₄×C₇	—	4	2	no	—
219	C₄×C₇	#476	1	1	yes	characteristic
220	C₄×C₇	#476	1	1	yes	characteristic
221	C₄×C₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
222	C₄×C₇	C₂²×SD₁₆	1	1	yes	characteristic
223	Dic₇	—	8	4	no	—
224	#41	—	2	2	no	—
225	#41	—	2	2	no	—
226	#41	—	4	1	no	—
227	#41	—	4	1	no	—
228	C₂×Q₈	—	2	1	no	—
229	C₂×Q₈	—	28	2	no	—
230	C₂×D₄	—	2	1	no	—
231	C₂×D₄	—	4	1	no	—
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	4	1	no	—
234	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
235	Q₁₆	—	2	1	no	—
236	Q₁₆	C₂×Dic₁₄	1	2	yes	—
237	SD₁₆	—	2	2	no	—
238	SD₁₆	—	2	2	no	—
239	D₈	—	2	1	no	—
240	D₈	C₂×Dic₁₄	1	2	yes	—
241	M₁₆	—	2	2	no	—
242	M₁₆	—	4	1	no	—
243	M₁₆	—	4	2	no	—
244	M₁₆	—	4	2	no	—
245	C₂×C₈	—	4	1	no	—
246	C₂×C₈	—	4	2	no	—
247	C₂×C₈	—	4	2	no	—
248	C₂×C₈	#1142	1	1	yes	characteristic
249	C₂×C₈	#1142	1	1	yes	characteristic
250	C₂×C₈	#1142	1	1	yes	characteristic
251	C₂×C₈	#1142	1	1	yes	characteristic
252	C₂×C₈	C₂×Dic₁₄	1	1	yes	characteristic
253	C₂×C₈	C₂×Dic₁₄	1	1	yes	characteristic
254	C₂×C₈	#1157	1	1	yes	characteristic
255	C₂×C₈	#1157	1	1	yes	characteristic
256	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
257	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
258	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
259	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
260	C₄⋊C₄	—	14	4	no	—
261	C₄⋊C₄	—	28	4	no	—
262	C₄⋊C₄	—	28	2	no	—
263	C₂²⋊C₄	—	28	2	no	—
264	C₄²	—	2	1	no	—
265	C₄²	—	28	2	no	—
266	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
267	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
268	C₁₆	—	14	4	no	—
269	C₂×C₇	—	2	1	no	—
270	C₂×C₇	—	2	1	no	—
271	C₂×C₇	—	2	1	no	—
272	C₂×C₇	—	8	2	no	—
273	C₂×C₇	C₂×#476	1	1	yes	characteristic
274	C₂³	—	4	1	no	—
275	C₂³	#6512	1	1	yes	characteristic
276	Q₈	—	2	2	no	—
277	Q₈	—	4	1	no	—
278	Q₈	—	56	4	no	—
279	D₄	—	2	2	no	—
280	D₄	—	4	2	no	—
281	D₄	—	4	2	no	—
282	D₄	—	4	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	1	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	28	4	no	—
290	C₂×C₄	#6487	1	1	yes	characteristic
291	C₂×C₄	#6487	1	1	yes	characteristic
292	C₂×C₄	C₂×#1142	1	1	yes	characteristic
293	C₂×C₄	C₂×#1142	1	1	yes	characteristic
294	C₂×C₄	#6512	1	1	yes	characteristic
295	C₂×C₄	C₂²×C₂²⋊D₇	1	1	yes	characteristic
296	C₈	—	4	2	no	—
297	C₈	#6487	1	1	yes	characteristic
298	C₈	#6487	1	1	yes	characteristic
299	C₈	C₂×#1142	1	1	yes	characteristic
300	C₈	C₂×#1142	1	1	yes	characteristic
301	C₈	C₂×#1142	1	1	yes	characteristic
302	C₈	C₂×#1142	1	1	yes	characteristic
303	C₈	C₂²×Dic₁₄	1	1	yes	characteristic
304	C₈	C₂×#1157	1	1	yes	characteristic
305	C₇	#33637	1	1	yes	characteristic, complemented, Sylow
306	C₂²	—	4	1	no	—
307	C₂²	—	4	2	no	—
308	C₂²	—	8	2	no	—
309	C₂²	#89780	1	1	yes	characteristic
310	C₂²	#89780	1	1	yes	characteristic
311	C₂²	C₂×#6512	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	—	4	1	no	—
314	C₄	—	4	2	no	—
315	C₄	—	56	4	no	—
316	C₄	#89780	1	1	yes	characteristic
317	C₄	#89780	1	1	yes	characteristic
318	C₄	C₂×#6487	1	1	yes	characteristic
319	C₄	C₂²×#1142	1	1	yes	characteristic
320	C₂	—	2	1	no	—
321	C₂	—	2	1	no	—
322	C₂	—	2	1	no	—
323	C₂	—	8	2	no	—
324	C₂	C₂×#89780	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#89780
commutator, cosocle	C₂×C₇×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₇
Fitting	C₂×C₇×#443	C₂
socle	C₂×C₇	C₂×#476
prime radical	C₂×C₇×#135	C₂²
hypercenter	C₂×#443	D₇
hypocenter	C₇	#33637
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#443

Complements

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

N	H ≅ G/N	type
2. #11735887	C₂	semidirect
4. #11728674	C₂	semidirect
5. #11728673	C₂	semidirect
6. #11728439	C₂	semidirect
7. C₂×#89258	C₂	semidirect
22. #89455	C₂²	semidirect
24. #89452	C₂²	semidirect
26. #89289	C₂²	semidirect
27. #89258	C₂²	semidirect
28. #89258	C₂²	semidirect
33. C₇×#135	Q₈	semidirect
35. C₂×C₇×Q₁₆	Q₈	semidirect
37. C₂×C₇×D₈	Q₈	semidirect
175. #135	Dic₁₄	semidirect
177. C₂×Q₁₆	Dic₁₄	semidirect
179. C₂×D₈	Dic₁₄	semidirect
305. 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×C₇×#443	2
#11735887	2
#11729830	2
#11728674	2
#11728673	2
#11728439	2
C₂×#89258	2
#33637	7