Subgroups — #13253886

Order 1792.

484 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11730753	C₂	1	1	yes	maximal, characteristic
2	C₇×#1767	C₂	1	1	yes	maximal, characteristic
3	C₄×#6377	C₂	1	1	yes	maximal, characteristic
4	#11725465	C₂	1	4	yes	maximal
5	C₂×#6432	C₄	1	1	yes	characteristic
6	C₂×#6432	C₄	1	1	yes	characteristic
7	C₂×C₇×C₄⋊C₈	—	2	1	no	—
8	C₂×C₇×#97	C₂²	1	2	yes	—
9	C₄³×C₇	C₂²	1	1	yes	characteristic
10	#89708	C₄	1	2	yes	—
11	#89708	C₂²	1	2	yes	—
12	#89701	—	2	1	no	—
13	#89701	—	2	1	no	—
14	#89701	—	2	1	no	—
15	#89701	—	2	1	no	—
16	C₂×#6377	—	2	1	no	—
17	C₂×#6377	C₂²	1	2	yes	—
18	C₂×C₄×C₇⋊C₈	—	2	1	no	—
19	C₇×#336	—	2	1	no	—
20	#9890	—	7	1	no	maximal, Sylow
21	C₂²×Dic₁₄	—	2	1	no	—
22	C₂²×Dic₁₄	—	2	1	no	—
23	C₂²×C₇×C₈	—	2	2	no	—
24	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
31	C₂×C₄⋊Dic₇	—	2	1	no	—
32	C₂×C₄⋊Dic₇	—	2	1	no	—
33	C₂×C₄⋊Dic₇	C₂×C₄	1	2	yes	—
34	C₂×C₄⋊Dic₇	C₂×C₄	1	2	yes	—
35	C₂²×C₇⋊C₈	—	2	2	no	—
36	#6432	—	4	2	no	—
37	#6432	—	4	2	no	—
38	C₇×C₄⋊C₈	—	2	4	no	—
39	C₇×#97	—	2	4	no	—
40	C₇⋊#95	—	2	2	no	—
41	C₇⋊#95	—	2	2	no	—
42	C₇⋊#95	—	2	2	no	—
43	C₇⋊#95	—	2	2	no	—
44	#6377	—	2	4	no	—
45	#6377	—	2	2	no	—
46	#6377	Q₈	1	4	yes	complemented
47	C₄×C₇⋊C₈	—	2	4	no	—
48	#2862	—	7	1	no	—
49	#1767	D₇	1	1	yes	characteristic
50	C₄×#97	—	7	1	no	—
51	#1295	—	7	4	no	—
52	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×Dic₇	—	2	2	no	—
60	C₂²×Dic₇	—	2	2	no	—
61	C₂×Dic₁₄	—	4	2	no	—
62	C₂×Dic₁₄	—	4	2	no	—
63	C₂×Dic₁₄	—	4	2	no	—
64	C₂×Dic₁₄	—	4	2	no	—
65	C₂×Dic₁₄	—	4	2	no	—
66	C₂×Dic₁₄	—	4	2	no	—
67	C₂×C₇×C₈	—	2	4	no	—
68	C₂×C₇×C₈	—	4	2	no	—
69	C₂×C₇×C₈	—	4	2	no	—
70	C₄²×C₇	—	2	2	no	—
71	C₄²×C₇	—	2	2	no	—
72	C₄²×C₇	—	2	2	no	—
73	C₄²×C₇	—	2	2	no	—
74	C₄²×C₇	—	2	2	no	—
75	C₄²×C₇	—	4	1	no	—
76	C₄²×C₇	C₂×Q₈	1	2	yes	—
77	C₄²×C₇	#41	1	2	yes	—
78	C₄⋊Dic₇	—	4	2	no	—
79	C₄⋊Dic₇	—	4	2	no	—
80	C₄⋊Dic₇	—	4	2	no	—
81	C₄⋊Dic₇	—	4	2	no	—
82	C₂×C₇⋊C₈	—	2	4	no	—
83	C₂×C₇⋊C₈	—	2	4	no	—
84	C₂×C₇⋊C₈	—	2	4	no	—
85	C₂×C₄⋊Q₈	—	7	1	no	—
86	C₂×C₄⋊Q₈	—	7	1	no	—
87	C₂×C₄⋊C₈	—	2	1	no	—
88	C₂×#97	—	7	2	no	—
89	C₂×#97	—	14	1	no	—
90	C₂×#97	C₂×D₇	1	2	yes	—
91	C₂×C₄×C₈	—	14	1	no	—
92	#391	—	14	1	no	—
93	#391	—	14	1	no	—
94	#391	—	14	1	no	—
95	#391	—	14	1	no	—
96	#382	—	7	2	no	—
97	#382	—	7	2	no	—
98	C₄³	C₂×D₇	1	1	yes	characteristic
99	#336	—	2	1	no	—
100	C₂³×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₇	—	2	1	no	—
102	C₂×C₄×C₇	—	2	1	no	—
103	C₂×C₄×C₇	—	2	1	no	—
104	C₂×C₄×C₇	—	2	1	no	—
105	C₂×C₄×C₇	—	2	1	no	—
106	C₂×C₄×C₇	—	2	1	no	—
107	C₂×C₄×C₇	—	2	1	no	—
108	C₂×C₄×C₇	—	2	1	no	—
109	C₂×C₄×C₇	—	2	1	no	—
110	C₂×C₄×C₇	—	2	1	no	—
111	C₂×C₄×C₇	—	2	1	no	—
112	C₂×C₄×C₇	—	2	1	no	—
113	C₂×C₄×C₇	—	2	1	no	—
114	C₂×C₄×C₇	#100	1	2	yes	—
115	C₂×C₄×C₇	#101	1	2	yes	—
116	C₂×C₄×C₇	#104	1	2	yes	—
117	C₂×C₄×C₇	#104	1	2	yes	—
118	C₂×C₄×C₇	#104	1	2	yes	—
119	C₂×C₄×C₇	C₄×Q₈	1	2	yes	—
120	C₂×C₄×C₇	C₂²⋊Q₈	1	2	yes	—
121	C₂×C₄×C₇	#124	1	1	yes	characteristic
122	C₂×C₄×C₇	C₄⋊Q₈	1	1	yes	characteristic
123	C₂×Dic₇	—	4	2	no	—
124	C₂×Dic₇	—	4	2	no	—
125	C₂×Dic₇	—	4	2	no	—
126	C₂×Dic₇	—	4	2	no	—
127	C₂×Dic₇	—	4	2	no	—
128	C₂×Dic₇	—	4	2	no	—
129	Dic₁₄	—	8	4	no	—
130	Dic₁₄	—	8	4	no	—
131	C₇×C₈	—	4	4	no	—
132	C₇⋊C₈	—	2	8	no	—
133	C₂²×Q₈	—	14	1	no	—
134	C₂²×Q₈	—	14	1	no	—
135	C₂²×C₈	—	2	2	no	—
136	C₂²×C₈	—	14	2	no	—
137	C₄⋊Q₈	—	28	2	no	—
138	C₄⋊Q₈	—	28	2	no	—
139	C₂×C₄⋊C₄	—	7	2	no	—
140	C₂×C₄⋊C₄	—	7	2	no	—
141	C₂×C₄⋊C₄	—	14	1	no	—
142	C₂×C₄⋊C₄	—	14	1	no	—
143	C₂×C₄²	C₄×D₇	1	1	yes	characteristic
144	C₂×C₄²	C₄×D₇	1	1	yes	characteristic
145	C₂×C₄²	D₂₈	1	1	yes	characteristic
146	C₂×C₄²	D₂₈	1	1	yes	characteristic
147	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
148	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
149	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
150	C₄⋊C₈	—	2	4	no	—
151	#97	—	2	4	no	—
152	#97	—	7	4	no	—
153	#97	—	14	4	no	—
154	#97	—	14	2	no	—
155	C₄×C₈	—	14	4	no	—
156	#95	—	14	2	no	—
157	#95	—	14	2	no	—
158	#95	—	14	2	no	—
159	#95	—	14	2	no	—
160	C₂²×C₇	#391	1	1	yes	characteristic
161	C₂²×C₇	C₂×#100	1	1	yes	characteristic
162	C₂²×C₇	C₂×#101	1	1	yes	characteristic
163	C₂²×C₇	C₂×#104	1	1	yes	characteristic
164	C₂²×C₇	C₂×#104	1	1	yes	characteristic
165	C₂²×C₇	C₂×#104	1	1	yes	characteristic
166	C₂²×C₇	#421	1	1	yes	characteristic
167	C₄×C₇	—	2	1	no	—
168	C₄×C₇	—	2	1	no	—
169	C₄×C₇	—	2	2	no	—
170	C₄×C₇	—	2	2	no	—
171	C₄×C₇	—	2	2	no	—
172	C₄×C₇	—	2	2	no	—
173	C₄×C₇	—	4	1	no	—
174	C₄×C₇	#329	1	4	yes	—
175	Dic₇	—	8	2	no	—
176	Dic₇	—	8	2	no	—
177	C₂×Q₈	—	28	2	no	—
178	C₂×Q₈	—	28	2	no	—
179	C₂×Q₈	—	28	2	no	—
180	C₂×Q₈	—	28	2	no	—
181	C₂×Q₈	—	28	2	no	—
182	C₂×Q₈	—	28	2	no	—
183	C₂²×C₄	—	14	2	no	—
184	C₂²×C₄	—	14	2	no	—
185	C₂²×C₄	(C₂×D₇)⋊C₄	1	1	yes	characteristic
186	C₂²×C₄	(C₂×D₇)⋊C₄	1	1	yes	characteristic
187	C₂²×C₄	(C₂×D₇)⋊C₄	1	1	yes	characteristic
188	C₂²×C₄	(C₂×D₇)⋊C₄	1	1	yes	characteristic
189	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
190	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
191	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
192	C₂×C₈	—	2	4	no	—
193	C₂×C₈	—	4	2	no	—
194	C₂×C₈	—	4	2	no	—
195	C₂×C₈	—	14	4	no	—
196	C₂×C₈	—	14	4	no	—
197	C₂×C₈	—	14	4	no	—
198	C₄⋊C₄	—	28	2	no	—
199	C₄⋊C₄	—	28	2	no	—
200	C₄⋊C₄	—	28	2	no	—
201	C₄⋊C₄	—	28	2	no	—
202	C₄²	—	2	2	no	—
203	C₄²	—	2	2	no	—
204	C₄²	—	2	2	no	—
205	C₄²	—	2	2	no	—
206	C₄²	—	2	2	no	—
207	C₄²	—	4	1	no	—
208	C₄²	#1159	1	2	yes	—
209	C₄²	Q₈×D₇	1	2	yes	—
210	C₂×C₇	C₂×#329	1	1	yes	characteristic
211	C₂×C₇	#1526	1	1	yes	characteristic
212	C₂×C₇	#1527	1	1	yes	characteristic
213	C₂×C₇	#1531	1	1	yes	characteristic
214	C₂×C₇	#1986	1	1	yes	characteristic
215	C₂×C₇	#1994	1	1	yes	characteristic
216	C₂×C₇	#1997	1	1	yes	characteristic
217	C₂³	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
218	Q₈	—	56	4	no	—
219	Q₈	—	56	4	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	28	2	no	—
234	C₂×C₄	—	28	2	no	—
235	C₂×C₄	—	28	2	no	—
236	C₂×C₄	—	28	2	no	—
237	C₂×C₄	—	28	2	no	—
238	C₂×C₄	—	28	2	no	—
239	C₂×C₄	#6379	1	2	yes	—
240	C₂×C₄	#6397	1	2	yes	—
241	C₂×C₄	#6398	1	2	yes	—
242	C₂×C₄	#6399	1	2	yes	—
243	C₂×C₄	#6399	1	2	yes	—
244	C₂×C₄	#6450	1	2	yes	—
245	C₂×C₄	#6460	1	2	yes	—
246	C₂×C₄	#6499	1	1	yes	characteristic
247	C₂×C₄	#6507	1	1	yes	characteristic
248	C₈	—	4	4	no	—
249	C₈	—	14	8	no	—
250	C₇	#9890	1	1	yes	characteristic, complemented, Sylow
251	C₂²	C₂×#6379	1	1	yes	characteristic
252	C₂²	#89746	1	1	yes	characteristic
253	C₂²	#89777	1	1	yes	characteristic
254	C₂²	C₂×#6397	1	1	yes	characteristic
255	C₂²	C₂×#6398	1	1	yes	characteristic
256	C₂²	C₂×#6399	1	1	yes	characteristic
257	C₂²	C₂×#6399	1	1	yes	characteristic
258	C₄	—	2	2	no	—
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	2	2	no	—
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	4	1	no	—
265	C₄	—	56	2	no	—
266	C₄	—	56	2	no	—
267	C₄	#89259	1	4	yes	—
268	C₂	#11727247	1	1	yes	characteristic
269	C₂	C₂×#89259	1	1	yes	characteristic
270	C₂	#11727616	1	1	yes	characteristic
271	C₂	#11728268	1	1	yes	characteristic
272	C₂	#11728278	1	1	yes	characteristic
273	C₂	#11728558	1	1	yes	characteristic
274	C₂	#11728567	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₇)⋊C₄
commutator	C₂²×C₄×C₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₇×#1767	C₂
socle, prime radical	C₂³×C₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	#1767	D₇
hypocenter	C₇	#9890
LCS	G ▷ C₂²×C₄×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1767

Complements

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

N	H ≅ G/N	type
46. #6377	Q₈	semidirect
250. C₇	#9890	semidirect

Maximal subgroups

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

subgroup	index
#11730753	2
C₇×#1767	2
C₄×#6377	2
#11725465	2
#9890	7