Subgroups — #13324356

Order 1792.

598 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11735920	C₂	1	1	yes	maximal, characteristic
2	#11735890	C₂	1	2	yes	maximal
3	C₂×C₇×#322	C₂	1	1	yes	maximal, characteristic
4	#11727492	C₂	1	2	yes	maximal
5	#11727484	C₂	1	1	yes	maximal, characteristic
6	#11725843	C₂	1	4	yes	maximal, complemented
7	#11725838	C₂	1	4	yes	maximal, complemented
8	#90161	C₂²	1	2	yes	—
9	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
12	#89888	C₄	1	4	yes	—
13	#89874	—	2	2	no	—
14	#89873	—	2	2	no	—
15	#89696	C₂²	1	2	yes	—
16	#89471	C₄	1	1	yes	characteristic
17	C₄×Dic₂₈	C₂²	1	4	yes	—
18	#89454	C₄	1	1	yes	characteristic
19	#89453	C₄	1	2	yes	—
20	#89451	—	2	2	no	—
21	#89451	C₂²	1	4	yes	—
22	C₈×Dic₁₄	—	2	2	no	—
23	C₈×Dic₁₄	C₂²	1	4	yes	—
24	C₇×#322	C₂²	1	2	yes	—
25	C₇×#322	C₂²	1	2	yes	—
26	#89252	C₂²	1	4	yes	—
27	#89251	C₂²	1	8	yes	—
28	#17961	—	7	1	no	maximal, Sylow
29	C₂²×C₇×C₈	—	2	1	no	—
30	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
31	C₂²×C₇×C₈	D₄	1	2	yes	—
32	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
33	#6489	—	2	2	no	—
34	#6487	—	2	2	no	—
35	#6487	C₂×C₄	1	4	yes	—
36	C₂×#1136	—	2	2	no	—
37	C₂×Dic₂₈	C₂×C₄	1	4	yes	—
38	#6433	—	2	2	no	—
39	#6432	C₂×C₄	1	2	yes	—
40	#6432	C₂×C₄	1	2	yes	—
41	C₄×Dic₁₄	—	2	2	no	—
42	C₄×Dic₁₄	C₂³	1	4	yes	—
43	C₄×C₇×C₈	—	2	1	no	—
44	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
45	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
46	C₄×C₇×C₈	D₄	1	2	yes	—
47	C₄×C₇×C₈	C₂³	1	2	yes	—
48	C₄×C₇×C₈	C₂³	1	2	yes	—
49	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
50	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
51	#6392	C₂×C₄	1	4	yes	—
52	#6390	C₂×C₄	1	8	yes	—
53	#6389	—	2	4	no	—
54	Dic₇⋊C₈	—	2	4	no	—
55	Dic₇⋊C₈	—	2	4	no	—
56	C₈×Dic₇	—	2	4	no	—
57	C₄⋊(C₇⋊C₈)	—	2	2	no	—
58	C₄⋊(C₇⋊C₈)	C₂³	1	4	yes	—
59	#3170	—	7	1	no	—
60	#2960	—	7	2	no	—
61	#1593	—	7	4	no	—
62	#1579	—	7	4	no	—
63	#1566	—	7	1	no	—
64	#1493	—	7	2	no	—
65	C₂×#322	D₇	1	1	yes	characteristic
66	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₇	#41	1	1	yes	characteristic
69	C₂×Dic₁₄	—	2	2	no	—
70	C₂×Dic₁₄	C₂²×C₄	1	4	yes	—
71	C₂×C₇×C₈	—	2	4	no	—
72	C₂×C₇×C₈	—	2	4	no	—
73	C₂×C₇×C₈	—	2	2	no	—
74	C₂×C₇×C₈	—	4	2	no	—
75	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
76	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
77	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
78	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
79	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
80	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₇	#41	1	1	yes	characteristic
82	C₄²×C₇	C₂⁴	1	1	yes	characteristic
83	C₂²⋊Dic₇	—	2	4	no	—
84	C₄⋊Dic₇	—	2	2	no	—
85	C₄⋊Dic₇	—	2	2	no	—
86	C₄⋊Dic₇	—	2	2	no	—
87	C₄⋊Dic₇	C₂²×C₄	1	4	yes	—
88	Dic₇⋊C₄	—	2	4	no	—
89	Dic₇⋊C₄	—	2	4	no	—
90	C₄×Dic₇	—	2	4	no	—
91	#1136	—	4	4	no	—
92	C₂×C₇⋊C₈	—	2	4	no	—
93	Dic₂₈	—	2	8	no	—
94	#533	—	7	2	no	—
95	#500	—	7	1	no	—
96	#494	—	7	1	no	—
97	#487	—	7	2	no	—
98	#467	—	7	4	no	—
99	#446	—	7	4	no	—
100	#446	—	14	2	no	—
101	C₈×Q₈	—	7	4	no	—
102	C₈×Q₈	—	14	2	no	—
103	C₄×Q₁₆	—	7	4	no	—
104	#424	—	14	2	no	—
105	#423	—	7	2	no	—
106	#405	—	14	2	no	—
107	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
108	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
109	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
110	#334	—	7	4	no	—
111	Q₈⋊C₈	—	7	8	no	—
112	#322	C₂×D₇	1	2	yes	—
113	#322	C₂×D₇	1	2	yes	—
114	C₂³×C₇	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₇	—	2	2	no	—
116	C₂×C₄×C₇	—	2	2	no	—
117	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₇	#137	1	4	yes	—
121	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
124	C₂×Dic₇	—	2	4	no	—
125	C₂×Dic₇	—	2	4	no	—
126	Dic₁₄	—	2	8	no	—
127	Dic₁₄	—	4	4	no	—
128	C₇×C₈	—	2	2	no	—
129	C₇×C₈	—	2	2	no	—
130	C₇×C₈	—	2	2	no	—
131	C₇×C₈	—	4	2	no	—
132	C₇×C₈	#131	1	4	yes	—
133	C₇⋊C₈	—	4	4	no	—
134	C₂×Q₁₆	—	7	4	no	—
135	C₂×M₁₆	—	14	2	no	—
136	C₂²×C₈	—	2	1	no	—
137	C₂²×C₈	C₄×D₇	1	2	yes	—
138	C₂²×C₈	D₂₈	1	2	yes	—
139	C₄⋊Q₈	—	7	2	no	—
140	C₄⋊Q₈	—	7	2	no	—
141	#125	—	14	2	no	—
142	C₂²⋊Q₈	—	7	4	no	—
143	C₂²⋊Q₈	—	14	2	no	—
144	C₄×Q₈	—	7	4	no	—
145	C₄×Q₈	—	14	2	no	—
146	#117	—	14	2	no	—
147	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
148	#107	—	7	4	no	—
149	C₄⋊C₈	—	7	4	no	—
150	C₄⋊C₈	—	14	4	no	—
151	C₄⋊C₈	—	14	4	no	—
152	C₄⋊C₈	—	14	2	no	—
153	#103	—	7	8	no	—
154	#97	—	14	4	no	—
155	C₄×C₈	—	2	1	no	—
156	C₄×C₈	—	14	4	no	—
157	C₄×C₈	C₄×D₇	1	1	yes	characteristic
158	C₄×C₈	C₄×D₇	1	1	yes	characteristic
159	C₄×C₈	D₂₈	1	2	yes	—
160	C₄×C₈	C₂²×D₇	1	2	yes	—
161	C₄×C₈	C₂²×D₇	1	2	yes	—
162	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
163	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
164	C₂²×C₇	—	2	1	no	—
165	C₂²×C₇	—	2	1	no	—
166	C₂²×C₇	#441	1	2	yes	—
167	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
168	C₄×C₇	—	2	1	no	—
169	C₄×C₇	—	2	1	no	—
170	C₄×C₇	#441	1	2	yes	—
171	C₄×C₇	#444	1	2	yes	—
172	C₄×C₇	C₂×#131	1	2	yes	—
173	C₄×C₇	C₂×#137	1	2	yes	—
174	Dic₇	—	4	4	no	—
175	Dic₇	—	4	4	no	—
176	C₂×Q₈	—	7	4	no	—
177	C₂×Q₈	—	14	2	no	—
178	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
179	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
180	C₂²×C₄	#1157	1	1	yes	characteristic
181	Q₁₆	—	14	8	no	—
182	M₁₆	—	28	4	no	—
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	—	2	4	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	4	2	no	—
187	C₂×C₈	—	14	4	no	—
188	C₂×C₈	C₂×C₄×D₇	1	2	yes	—
189	C₂×C₈	C₂×C₄×D₇	1	2	yes	—
190	C₂×C₈	C₂×D₂₈	1	2	yes	—
191	C₂×C₈	C₂×D₂₈	1	2	yes	—
192	C₄⋊C₄	—	7	4	no	—
193	C₄⋊C₄	—	14	4	no	—
194	C₄⋊C₄	—	14	2	no	—
195	C₄⋊C₄	—	14	2	no	—
196	C₄⋊C₄	—	14	2	no	—
197	C₄⋊C₄	—	14	4	no	—
198	C₂²⋊C₄	—	14	4	no	—
199	C₄²	—	14	4	no	—
200	C₄²	C₂×C₄×D₇	1	1	yes	characteristic
201	C₄²	C₂×D₂₈	1	1	yes	characteristic
202	C₄²	#1157	1	1	yes	characteristic
203	C₄²	C₂³×D₇	1	1	yes	characteristic
204	C₂×C₇	—	2	2	no	—
205	C₂×C₇	#2931	1	1	yes	characteristic
206	C₂×C₇	C₂×#441	1	1	yes	characteristic
207	C₂×C₇	C₂×#444	1	1	yes	characteristic
208	C₂³	C₄×D₂₈	1	1	yes	characteristic
209	Q₈	—	14	8	no	—
210	Q₈	—	28	4	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	14	4	no	—
214	C₂×C₄	—	14	4	no	—
215	C₂×C₄	C₄×D₂₈	1	1	yes	characteristic
216	C₂×C₄	C₄×D₂₈	1	1	yes	characteristic
217	C₂×C₄	C₄×D₂₈	1	1	yes	characteristic
218	C₂×C₄	#6472	1	4	yes	—
219	C₂×C₄	C₂²×C₄×D₇	1	1	yes	characteristic
220	C₂×C₄	C₂²×D₂₈	1	1	yes	characteristic
221	C₂×C₄	C₂×#1157	1	1	yes	characteristic
222	C₈	—	2	2	no	—
223	C₈	—	2	2	no	—
224	C₈	—	2	2	no	—
225	C₈	—	4	2	no	—
226	C₈	—	28	4	no	—
227	C₈	#6464	1	4	yes	—
228	C₇	#17961	1	1	yes	characteristic, complemented, Sylow
229	C₂²	—	2	1	no	—
230	C₂²	—	2	1	no	—
231	C₂²	#89488	1	2	yes	—
232	C₂²	C₂×C₄×D₂₈	1	1	yes	characteristic
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	28	4	no	—
236	C₄	—	28	4	no	—
237	C₄	#89488	1	2	yes	—
238	C₄	#89489	1	2	yes	—
239	C₄	C₂×#6464	1	2	yes	—
240	C₄	C₂×#6472	1	2	yes	—
241	C₂	—	2	2	no	—
242	C₂	#11735903	1	1	yes	characteristic
243	C₂	C₂×#89488	1	1	yes	characteristic
244	C₂	C₂×#89489	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₈
commutator	C₂×C₄×C₇	C₂³×C₄
Frattini	C₄²	C₂³×D₇
Fitting	C₂×C₇×#322	C₂
socle	C₂²×C₇	C₂×C₄×D₄
prime radical	C₂³×C₇	C₄×D₄
cosocle	C₄²×C₇	C₂⁴
hypercenter	C₂×#322	D₇
hypocenter	C₇	#17961
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
6. #11725843	C₂	semidirect
7. #11725838	C₂	semidirect
228. C₇	#17961	semidirect

Maximal subgroups

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

subgroup	index
#11735920	2
#11735890	2
C₂×C₇×#322	2
#11727492	2
#11727484	2
#11725843	2
#11725838	2
#17961	7