Subgroups — #13337043

Order 1792.

606 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11736758	C₂	1	2	yes	maximal
2	#11735890	C₂	1	1	yes	maximal, characteristic
3	C₂×#89454	C₂	1	1	yes	maximal, characteristic
4	C₇×#1564	C₂	1	1	yes	maximal, characteristic
5	#11728534	C₂	1	2	yes	maximal
6	#11726342	C₂	1	4	yes	maximal, complemented
7	#11726306	C₂	1	4	yes	maximal, complemented
8	#90161	C₂²	1	1	yes	characteristic
9	C₂×#6432	C₂²	1	2	yes	—
10	C₇×#431	C₂²	1	2	yes	—
11	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
12	C₂×#6392	—	2	1	no	—
13	C₂×#6392	—	2	1	no	—
14	#89874	—	2	1	no	—
15	#89873	—	2	1	no	—
16	#89696	C₂²	1	1	yes	characteristic
17	#89621	C₂²	1	8	yes	—
18	#89511	—	2	4	no	—
19	#89471	—	2	2	no	—
20	#89454	—	2	1	no	—
21	#89454	—	2	1	no	—
22	#89454	C₂²	1	2	yes	—
23	#89454	C₂²	1	2	yes	—
24	#89453	—	2	2	no	—
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	C₇×#335	C₂²	1	2	yes	—
30	C₇×#334	C₂²	1	2	yes	—
31	#89277	C₂²	1	8	yes	—
32	#19208	—	7	1	no	maximal, Sylow
33	C₂²×Dic₁₄	—	2	2	no	—
34	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
35	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
37	#6489	—	2	1	no	—
38	C₂×C₄⋊Dic₇	—	2	2	no	—
39	C₂×C₄⋊Dic₇	—	2	2	no	—
40	#6487	—	2	1	no	—
41	#6487	D₄	1	2	yes	—
42	C₂×#1136	—	2	1	no	—
43	C₂×Dic₂₈	—	2	8	no	—
44	#6433	—	2	1	no	—
45	#6432	—	2	2	no	—
46	#6432	—	2	2	no	—
47	#6432	—	2	2	no	—
48	#6432	D₄	1	1	yes	characteristic
49	#6432	D₄	1	1	yes	characteristic
50	#6432	C₂³	1	4	yes	—
51	C₄×Dic₁₄	—	2	1	no	—
52	C₄×Dic₁₄	C₂³	1	2	yes	—
53	C₇×C₄⋊C₈	C₂³	1	4	yes	—
54	C₇×C₂²⋊C₈	—	2	2	no	—
55	C₄×C₇×C₈	—	2	2	no	—
56	C₄×C₇×C₈	D₄	1	1	yes	characteristic
57	C₄×C₇×C₈	D₄	1	1	yes	characteristic
58	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
59	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
60	#6392	—	2	2	no	—
61	#6392	—	2	2	no	—
62	#6392	—	2	2	no	—
63	#6392	—	2	2	no	—
64	#6390	—	2	8	no	—
65	#6389	—	2	2	no	—
66	Dic₇⋊C₈	—	2	2	no	—
67	Dic₇⋊C₈	—	2	2	no	—
68	C₈×Dic₇	—	2	2	no	—
69	C₄⋊(C₇⋊C₈)	—	2	1	no	—
70	C₄⋊(C₇⋊C₈)	C₂³	1	2	yes	—
71	#3209	—	7	2	no	—
72	C₂×#500	—	7	1	no	—
73	#2960	—	7	1	no	—
74	#1694	—	7	4	no	—
75	#1674	—	7	4	no	—
76	#1564	D₇	1	1	yes	characteristic
77	#1549	—	7	2	no	—
78	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₇	#41	1	1	yes	characteristic
81	C₂²×Dic₇	—	4	2	no	—
82	C₂×Dic₁₄	—	2	1	no	—
83	C₂×Dic₁₄	—	2	4	no	—
84	C₂×Dic₁₄	—	4	2	no	—
85	C₂×Dic₁₄	—	8	4	no	—
86	C₂×Dic₁₄	C₂×D₄	1	2	yes	—
87	C₂×C₇×C₈	—	2	2	no	—
88	C₂×C₇×C₈	—	2	4	no	—
89	C₂×C₇×C₈	Q₁₆	1	4	yes	complemented
90	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
91	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
92	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
93	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₇	#41	1	1	yes	characteristic
97	C₄²×C₇	C₂⁴	1	1	yes	characteristic
98	C₂²⋊Dic₇	—	2	2	no	—
99	C₄⋊Dic₇	—	2	1	no	—
100	C₄⋊Dic₇	—	2	1	no	—
101	C₄⋊Dic₇	—	2	1	no	—
102	C₄⋊Dic₇	—	2	4	no	—
103	C₄⋊Dic₇	—	2	4	no	—
104	C₄⋊Dic₇	—	4	2	no	—
105	C₄⋊Dic₇	—	4	2	no	—
106	C₄⋊Dic₇	C₂×D₄	1	2	yes	—
107	Dic₇⋊C₄	—	2	2	no	—
108	Dic₇⋊C₄	—	2	2	no	—
109	C₄×Dic₇	—	2	2	no	—
110	#1136	—	4	2	no	—
111	C₂×C₇⋊C₈	—	2	2	no	—
112	Dic₂₈	—	8	8	no	—
113	#533	—	7	1	no	—
114	C₂×C₄⋊Q₈	—	7	2	no	—
115	#500	—	7	2	no	—
116	#500	—	7	2	no	—
117	#500	—	14	1	no	—
118	#500	—	14	1	no	—
119	#494	—	14	2	no	—
120	#487	—	14	2	no	—
121	#462	—	7	8	no	—
122	#451	—	14	4	no	—
123	#446	—	7	2	no	—
124	#446	—	14	1	no	—
125	C₈×Q₈	—	7	2	no	—
126	C₈×Q₈	—	14	1	no	—
127	#431	C₂×D₇	1	2	yes	—
128	C₂×#107	—	14	1	no	—
129	C₂×#107	—	14	1	no	—
130	#424	—	14	1	no	—
131	#423	—	7	1	no	—
132	#405	—	14	1	no	—
133	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
134	#335	C₂×D₇	1	2	yes	—
135	#334	C₂×D₇	1	2	yes	—
136	#332	—	7	8	no	—
137	C₂³×C₇	#121	1	1	yes	characteristic
138	C₂×C₄×C₇	—	2	2	no	—
139	C₂×C₄×C₇	—	2	2	no	—
140	C₂×C₄×C₇	#121	1	1	yes	characteristic
141	C₂×C₄×C₇	#121	1	1	yes	characteristic
142	C₂×C₄×C₇	#121	1	1	yes	characteristic
143	C₂×C₄×C₇	C₂×Q₁₆	1	2	yes	—
144	C₂×C₄×C₇	#137	1	2	yes	—
145	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
148	C₂×Dic₇	—	2	2	no	—
149	C₂×Dic₇	—	2	2	no	—
150	C₂×Dic₇	—	4	4	no	—
151	C₂×Dic₇	—	8	4	no	—
152	Dic₁₄	—	2	4	no	—
153	Dic₁₄	—	4	2	no	—
154	Dic₁₄	—	4	8	no	—
155	Dic₁₄	—	8	4	no	—
156	C₇×C₈	—	2	1	no	—
157	C₇×C₈	—	2	1	no	—
158	C₇×C₈	—	4	4	no	—
159	C₇×C₈	C₂×Q₁₆	1	2	yes	—
160	C₇×C₈	C₂×Q₁₆	1	2	yes	—
161	C₇⋊C₈	—	4	2	no	—
162	C₂²×Q₈	—	14	2	no	—
163	C₂×Q₁₆	—	14	8	no	—
164	C₂×M₁₆	—	14	1	no	—
165	C₂²×C₈	D₂₈	1	1	yes	characteristic
166	C₂²×C₈	D₂₈	1	1	yes	characteristic
167	C₄⋊Q₈	—	7	1	no	—
168	C₄⋊Q₈	—	7	1	no	—
169	C₄⋊Q₈	—	7	4	no	—
170	C₄⋊Q₈	—	14	2	no	—
171	C₄⋊Q₈	—	14	2	no	—
172	C₄⋊Q₈	—	14	2	no	—
173	#125	—	14	1	no	—
174	C₂²⋊Q₈	—	7	2	no	—
175	C₂²⋊Q₈	—	14	1	no	—
176	C₄×Q₈	—	7	2	no	—
177	C₄×Q₈	—	14	1	no	—
178	#117	—	14	1	no	—
179	C₂×C₄⋊C₄	—	14	2	no	—
180	C₂×C₄⋊C₄	—	14	2	no	—
181	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
182	#107	—	14	2	no	—
183	#107	—	14	2	no	—
184	#107	—	14	2	no	—
185	#107	—	14	2	no	—
186	C₄⋊C₈	—	7	2	no	—
187	C₄⋊C₈	—	14	2	no	—
188	C₄⋊C₈	—	14	2	no	—
189	C₄⋊C₈	—	14	1	no	—
190	C₄⋊C₈	C₂²×D₇	1	4	yes	—
191	#103	—	14	8	no	—
192	C₂²⋊C₈	—	2	2	no	—
193	#97	—	14	2	no	—
194	C₄×C₈	—	2	2	no	—
195	C₄×C₈	—	14	2	no	—
196	C₄×C₈	D₂₈	1	1	yes	characteristic
197	C₄×C₈	D₂₈	1	1	yes	characteristic
198	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
199	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
200	C₂²×C₇	—	2	1	no	—
201	C₂²×C₇	—	2	1	no	—
202	C₂²×C₇	#462	1	2	yes	—
203	C₂²×C₇	C₂×#121	1	1	yes	characteristic
204	C₄×C₇	—	2	1	no	—
205	C₄×C₇	—	2	1	no	—
206	C₄×C₇	#462	1	2	yes	—
207	C₄×C₇	#473	1	2	yes	—
208	C₄×C₇	C₂²×Q₁₆	1	1	yes	characteristic
209	C₄×C₇	C₂×#137	1	1	yes	characteristic
210	C₄×C₇	#576	1	2	yes	—
211	Dic₇	—	4	2	no	—
212	Dic₇	—	4	2	no	—
213	Dic₇	—	8	4	no	—
214	C₂×Q₈	—	7	2	no	—
215	C₂×Q₈	—	14	1	no	—
216	C₂×Q₈	—	14	4	no	—
217	C₂×Q₈	—	28	2	no	—
218	C₂×Q₈	—	56	4	no	—
219	C₂²×C₄	—	28	2	no	—
220	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
221	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
222	C₂²×C₄	#1161	1	1	yes	characteristic
223	Q₁₆	—	56	8	no	—
224	M₁₆	—	28	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	14	2	no	—
228	C₂×C₈	Dic₂₈	1	4	yes	complemented
229	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
230	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
231	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
232	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
233	C₄⋊C₄	—	7	2	no	—
234	C₄⋊C₄	—	14	2	no	—
235	C₄⋊C₄	—	14	1	no	—
236	C₄⋊C₄	—	14	1	no	—
237	C₄⋊C₄	—	14	1	no	—
238	C₄⋊C₄	—	14	2	no	—
239	C₄⋊C₄	—	14	4	no	—
240	C₄⋊C₄	—	14	4	no	—
241	C₄⋊C₄	—	28	2	no	—
242	C₄⋊C₄	—	28	2	no	—
243	C₂²⋊C₄	—	14	2	no	—
244	C₄²	—	14	2	no	—
245	C₄²	C₂×D₂₈	1	1	yes	characteristic
246	C₄²	D₄×D₇	1	1	yes	characteristic
247	C₄²	#1161	1	1	yes	characteristic
248	C₄²	C₂³×D₇	1	1	yes	characteristic
249	C₂×C₇	—	2	2	no	—
250	C₂×C₇	C₂×#462	1	1	yes	characteristic
251	C₂×C₇	#3058	1	1	yes	characteristic
252	C₂×C₇	C₂×#473	1	1	yes	characteristic
253	C₂³	#6458	1	1	yes	characteristic
254	Q₈	—	14	4	no	—
255	Q₈	—	28	2	no	—
256	Q₈	—	28	8	no	—
257	Q₈	—	56	4	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	14	2	no	—
261	C₂×C₄	—	14	2	no	—
262	C₂×C₄	—	28	4	no	—
263	C₂×C₄	—	56	4	no	—
264	C₂×C₄	#6458	1	1	yes	characteristic
265	C₂×C₄	#6458	1	1	yes	characteristic
266	C₂×C₄	#6458	1	1	yes	characteristic
267	C₂×C₄	C₂×Dic₂₈	1	2	yes	—
268	C₂×C₄	#6472	1	2	yes	—
269	C₂×C₄	C₂²×D₂₈	1	1	yes	characteristic
270	C₂×C₄	C₂×D₄×D₇	1	1	yes	characteristic
271	C₂×C₄	C₂×#1161	1	1	yes	characteristic
272	C₈	—	2	1	no	—
273	C₈	—	2	1	no	—
274	C₈	—	4	4	no	—
275	C₈	—	28	2	no	—
276	C₈	C₂×Dic₂₈	1	2	yes	—
277	C₈	C₂×Dic₂₈	1	2	yes	—
278	C₇	#19208	1	1	yes	characteristic, complemented, Sylow
279	C₂²	—	2	1	no	—
280	C₂²	—	2	1	no	—
281	C₂²	#89621	1	2	yes	—
282	C₂²	C₂×#6458	1	1	yes	characteristic
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	28	2	no	—
286	C₄	—	28	2	no	—
287	C₄	—	56	4	no	—
288	C₄	#89621	1	2	yes	—
289	C₄	#89668	1	2	yes	—
290	C₄	C₂²×Dic₂₈	1	1	yes	characteristic
291	C₄	C₂×#6472	1	1	yes	characteristic
292	C₄	#90460	1	2	yes	—
293	C₂	—	2	2	no	—
294	C₂	C₂×#89621	1	1	yes	characteristic
295	C₂	#11736628	1	1	yes	characteristic
296	C₂	C₂×#89668	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#6458
commutator, cosocle	C₄²×C₇	C₂⁴
Frattini	C₄²	C₂³×D₇
Fitting	C₇×#1564	C₂
socle	C₂²×C₇	C₂×#121
prime radical	C₂³×C₇	#121
hypercenter	#1564	D₇
hypocenter	C₇	#19208
LCS	G ▷ C₄²×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
6. #11726342	C₂	semidirect
7. #11726306	C₂	semidirect
89. C₂×C₇×C₈	Q₁₆	semidirect
228. C₂×C₈	Dic₂₈	semidirect
278. C₇	#19208	semidirect

Maximal subgroups

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

subgroup	index
#11736758	2
#11735890	2
C₂×#89454	2
C₇×#1564	2
#11728534	2
#11726342	2
#11726306	2
#19208	7