Subgroups — #13337042

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	C₇×#2960	C₂	1	1	yes	maximal, characteristic
2	#11740815	C₂	1	2	yes	maximal
3	C₂×#89454	C₂	1	1	yes	maximal, characteristic
4	#11728718	C₂	1	4	yes	maximal, complemented
5	#11728698	C₂	1	4	yes	maximal, complemented
6	#11728262	C₂	1	2	yes	maximal
7	#11727483	C₂	1	1	yes	maximal, characteristic
8	C₇×#533	C₂²	1	1	yes	characteristic
9	C₂×#6432	C₂²	1	2	yes	—
10	C₇×#446	—	2	1	no	—
11	C₇×#446	C₂²	1	2	yes	—
12	C₇×C₈×Q₈	—	2	1	no	—
13	C₇×C₈×Q₈	C₂²	1	2	yes	—
14	C₇×#424	—	2	1	no	—
15	C₇×#423	C₂²	1	1	yes	characteristic
16	C₇×#405	—	2	1	no	—
17	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
18	C₂×#6392	—	2	1	no	—
19	C₂×#6392	—	2	1	no	—
20	#89865	—	2	2	no	—
21	#89864	—	2	2	no	—
22	#89822	—	2	4	no	—
23	#89803	C₂²	1	8	yes	—
24	#89697	C₂²	1	2	yes	—
25	#89454	—	2	1	no	—
26	#89454	—	2	1	no	—
27	#89454	C₂²	1	2	yes	—
28	#89454	C₂²	1	2	yes	—
29	#89281	C₂²	1	8	yes	—
30	#89253	C₂²	1	2	yes	—
31	#89252	C₂²	1	2	yes	—
32	#19208	—	7	1	no	maximal, Sylow
33	C₂²×Dic₁₄	—	2	2	no	—
34	C₂×C₇×M₁₆	—	2	1	no	—
35	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
36	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
37	C₇×C₄⋊Q₈	D₄	1	1	yes	characteristic
38	C₇×C₄⋊Q₈	D₄	1	1	yes	characteristic
39	C₇×#125	—	2	1	no	—
40	C₇×C₂²⋊Q₈	—	2	1	no	—
41	C₇×C₂²⋊Q₈	D₄	1	2	yes	—
42	C₄×C₇×Q₈	—	2	1	no	—
43	C₄×C₇×Q₈	C₂³	1	2	yes	—
44	C₇×#117	—	2	1	no	—
45	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
46	C₂×#1144	—	2	8	no	—
47	C₂×C₄⋊Dic₇	—	2	2	no	—
48	C₂×C₄⋊Dic₇	—	2	2	no	—
49	#6432	—	2	2	no	—
50	#6432	—	2	2	no	—
51	#6432	—	2	2	no	—
52	#6432	C₂³	1	4	yes	—
53	C₇×C₄⋊C₈	—	2	2	no	—
54	C₇×C₄⋊C₈	—	2	2	no	—
55	C₇×C₄⋊C₈	—	2	1	no	—
56	C₇×C₄⋊C₈	C₂³	1	2	yes	—
57	C₇×#97	—	2	2	no	—
58	C₄×C₇×C₈	—	2	2	no	—
59	C₄×C₇×C₈	D₄	1	1	yes	characteristic
60	C₄×C₇×C₈	D₄	1	1	yes	characteristic
61	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
62	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
63	C₂²⋊(C₇⋊C₈)	—	2	2	no	—
64	#6392	—	2	2	no	—
65	#6392	—	2	2	no	—
66	#6392	—	2	2	no	—
67	#6392	—	2	2	no	—
68	#6384	—	2	8	no	—
69	C₄⋊(C₇⋊C₈)	C₂³	1	4	yes	—
70	C₄×C₇⋊C₈	—	2	2	no	—
71	#3209	—	7	2	no	—
72	C₂×#500	—	7	1	no	—
73	#2960	D₇	1	1	yes	characteristic
74	#1694	—	7	4	no	—
75	#1674	—	7	4	no	—
76	#1564	—	7	1	no	—
77	#1549	—	7	2	no	—
78	C₂×C₇×Q₈	—	2	1	no	—
79	C₂×C₇×Q₈	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₇	#41	1	1	yes	characteristic
83	C₂²×Dic₇	—	4	2	no	—
84	C₂×Dic₁₄	—	2	4	no	—
85	C₂×Dic₁₄	—	4	2	no	—
86	C₂×Dic₁₄	—	8	4	no	—
87	C₇×M₁₆	—	4	2	no	—
88	C₂×C₇×C₈	—	2	2	no	—
89	C₂×C₇×C₈	—	2	2	no	—
90	C₂×C₇×C₈	Q₁₆	1	4	yes	complemented
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	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
95	C₇×C₄⋊C₄	—	2	1	no	—
96	C₇×C₄⋊C₄	—	2	1	no	—
97	C₇×C₄⋊C₄	—	2	1	no	—
98	C₇×C₄⋊C₄	—	2	2	no	—
99	C₇×C₄⋊C₄	—	2	2	no	—
100	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
101	C₇×C₂²⋊C₄	—	2	2	no	—
102	C₄²×C₇	—	2	2	no	—
103	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₇	#41	1	1	yes	characteristic
106	C₄²×C₇	C₂⁴	1	1	yes	characteristic
107	#1144	—	8	8	no	—
108	C₄⋊Dic₇	—	2	4	no	—
109	C₄⋊Dic₇	—	2	4	no	—
110	C₄⋊Dic₇	—	4	2	no	—
111	C₄⋊Dic₇	—	4	2	no	—
112	C₂×C₇⋊C₈	—	2	4	no	—
113	#533	C₂×D₇	1	1	yes	characteristic
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	—	2	1	no	—
124	#446	C₂×D₇	1	2	yes	—
125	C₈×Q₈	—	2	1	no	—
126	C₈×Q₈	C₂×D₇	1	2	yes	—
127	#431	—	7	2	no	—
128	C₂×#107	—	14	1	no	—
129	C₂×#107	—	14	1	no	—
130	#424	—	2	1	no	—
131	#423	C₂×D₇	1	1	yes	characteristic
132	#405	—	2	1	no	—
133	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
134	#335	—	7	2	no	—
135	#334	—	7	2	no	—
136	#332	—	7	8	no	—
137	C₂³×C₇	#121	1	1	yes	characteristic
138	C₇×Q₈	—	2	4	no	—
139	C₇×Q₈	—	4	2	no	—
140	C₂×C₄×C₇	—	2	2	no	—
141	C₂×C₄×C₇	—	2	2	no	—
142	C₂×C₄×C₇	—	2	2	no	—
143	C₂×C₄×C₇	—	2	2	no	—
144	C₂×C₄×C₇	#121	1	1	yes	characteristic
145	C₂×C₄×C₇	#121	1	1	yes	characteristic
146	C₂×C₄×C₇	#121	1	1	yes	characteristic
147	C₂×C₄×C₇	C₂×Q₁₆	1	2	yes	—
148	C₂×C₄×C₇	#137	1	2	yes	—
149	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
152	C₂×Dic₇	—	4	4	no	—
153	C₂×Dic₇	—	8	4	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	2	no	—
159	C₇×C₈	C₂×Q₁₆	1	2	yes	—
160	C₇×C₈	C₂×Q₁₆	1	2	yes	—
161	C₇⋊C₈	—	4	4	no	—
162	C₂²×Q₈	—	14	2	no	—
163	C₂×Q₁₆	—	14	8	no	—
164	C₂×M₁₆	—	2	1	no	—
165	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
166	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
167	C₄⋊Q₈	—	7	4	no	—
168	C₄⋊Q₈	—	14	2	no	—
169	C₄⋊Q₈	—	14	2	no	—
170	C₄⋊Q₈	—	14	2	no	—
171	C₄⋊Q₈	D₂₈	1	1	yes	characteristic
172	C₄⋊Q₈	D₂₈	1	1	yes	characteristic
173	#125	—	2	1	no	—
174	C₂²⋊Q₈	—	2	1	no	—
175	C₂²⋊Q₈	D₂₈	1	2	yes	—
176	C₄×Q₈	—	2	1	no	—
177	C₄×Q₈	C₂²×D₇	1	2	yes	—
178	#117	—	2	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	—	2	2	no	—
187	C₄⋊C₈	—	2	2	no	—
188	C₄⋊C₈	—	2	1	no	—
189	C₄⋊C₈	—	7	4	no	—
190	C₄⋊C₈	C₂²×D₇	1	2	yes	—
191	#103	—	14	8	no	—
192	C₂²⋊C₈	—	14	2	no	—
193	#97	—	2	2	no	—
194	C₄×C₈	—	2	2	no	—
195	C₄×C₈	—	14	2	no	—
196	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
197	C₄×C₈	C₂²⋊D₇	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	—	4	2	no	—
207	C₄×C₇	—	4	2	no	—
208	C₄×C₇	#462	1	2	yes	—
209	C₄×C₇	#473	1	2	yes	—
210	C₄×C₇	C₂²×Q₁₆	1	1	yes	characteristic
211	C₄×C₇	C₂×#137	1	1	yes	characteristic
212	C₄×C₇	#576	1	2	yes	—
213	Dic₇	—	8	4	no	—
214	C₂×Q₈	—	2	1	no	—
215	C₂×Q₈	—	14	4	no	—
216	C₂×Q₈	—	28	2	no	—
217	C₂×Q₈	—	56	4	no	—
218	C₂×Q₈	C₂×D₂₈	1	2	yes	—
219	C₂²×C₄	—	28	2	no	—
220	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
221	C₂²×C₄	#1157	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
223	Q₁₆	—	56	8	no	—
224	M₁₆	—	4	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	14	4	no	—
228	C₂×C₈	#1144	1	4	yes	complemented
229	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
230	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
231	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
232	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	2	1	no	—
235	C₄⋊C₄	—	2	1	no	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	14	4	no	—
239	C₄⋊C₄	—	14	4	no	—
240	C₄⋊C₄	—	28	2	no	—
241	C₄⋊C₄	—	28	2	no	—
242	C₄⋊C₄	C₂×D₂₈	1	2	yes	—
243	C₂²⋊C₄	—	2	2	no	—
244	C₄²	—	2	2	no	—
245	C₄²	C₂×D₂₈	1	1	yes	characteristic
246	C₄²	#1157	1	1	yes	characteristic
247	C₄²	C₂×C₂²⋊D₇	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	#6493	1	1	yes	characteristic
254	Q₈	—	2	4	no	—
255	Q₈	—	4	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	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	28	4	no	—
263	C₂×C₄	—	56	4	no	—
264	C₂×C₄	#6493	1	1	yes	characteristic
265	C₂×C₄	#6493	1	1	yes	characteristic
266	C₂×C₄	#6493	1	1	yes	characteristic
267	C₂×C₄	C₂×#1144	1	2	yes	—
268	C₂×C₄	#6514	1	2	yes	—
269	C₂×C₄	C₂²×D₂₈	1	1	yes	characteristic
270	C₂×C₄	C₂×#1157	1	1	yes	characteristic
271	C₂×C₄	C₂²×C₂²⋊D₇	1	1	yes	characteristic
272	C₈	—	2	1	no	—
273	C₈	—	2	1	no	—
274	C₈	—	4	2	no	—
275	C₈	—	28	4	no	—
276	C₈	C₂×#1144	1	2	yes	—
277	C₈	C₂×#1144	1	2	yes	—
278	C₇	#19208	1	1	yes	characteristic, complemented, Sylow
279	C₂²	—	2	1	no	—
280	C₂²	—	2	1	no	—
281	C₂²	#89803	1	2	yes	—
282	C₂²	C₂×#6493	1	1	yes	characteristic
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	4	2	no	—
286	C₄	—	4	2	no	—
287	C₄	—	56	4	no	—
288	C₄	#89803	1	2	yes	—
289	C₄	#89915	1	2	yes	—
290	C₄	#90443	1	2	yes	—
291	C₄	C₂²×#1144	1	1	yes	characteristic
292	C₄	C₂×#6514	1	1	yes	characteristic
293	C₂	—	2	2	no	—
294	C₂	C₂×#89803	1	1	yes	characteristic
295	C₂	#11741244	1	1	yes	characteristic
296	C₂	C₂×#89915	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#6493
commutator, cosocle	C₄²×C₇	C₂⁴
Frattini	C₄²	C₂³×D₇
Fitting	C₇×#2960	C₂
socle	C₂²×C₇	C₂×#121
prime radical	C₂³×C₇	#121
hypercenter	#2960	D₇
hypocenter	C₇	#19208
LCS	G ▷ C₄²×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960

Complements

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

N	H ≅ G/N	type
4. #11728718	C₂	semidirect
5. #11728698	C₂	semidirect
90. C₂×C₇×C₈	Q₁₆	semidirect
228. C₂×C₈	#1144	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
C₇×#2960	2
#11740815	2
C₂×#89454	2
#11728718	2
#11728698	2
#11728262	2
#11727483	2
#19208	7