Subgroups — #13372807

Order 1792.

622 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#446	C₂	1	1	yes	maximal, characteristic
2	Q₈×#1136	C₂	1	2	yes	maximal
3	#11728228	C₂	1	4	yes	maximal
4	#11728228	C₂	1	2	yes	maximal
5	#11727595	C₂	1	2	yes	maximal
6	#11727594	C₂	1	2	yes	maximal
7	#11727545	C₂	1	1	yes	maximal, characteristic
8	C₄×#6393	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
10	C₇×#446	C₄	1	4	yes	—
11	C₇×#446	C₄	1	4	yes	—
12	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₇×#97	C₂²	1	2	yes	—
15	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
16	C₂×#6393	C₂²	1	2	yes	—
17	C₂×#6393	C₂²	1	4	yes	—
18	#89796	—	2	2	no	—
19	#89796	—	2	4	no	—
20	Q₈×C₇⋊C₈	—	2	2	no	—
21	#89696	C₂²	1	2	yes	—
22	#89696	C₂²	1	4	yes	—
23	C₄×#1136	C₂²	1	2	yes	—
24	C₄×#1136	C₂²	1	4	yes	—
25	#89347	C₂²	1	2	yes	—
26	#89347	C₂²	1	4	yes	—
27	#89347	C₂²	1	1	yes	characteristic
28	#89347	C₂²	1	1	yes	characteristic
29	#89347	C₂²	1	2	yes	—
30	#22747	—	7	1	no	maximal, Sylow
31	C₂²×C₇×Q₈	Q₈	1	1	yes	characteristic
32	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
33	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
34	C₂²×C₇×C₈	C₂³	1	2	yes	—
35	C₄×C₇×Q₈	C₂×C₄	1	2	yes	—
36	C₄×C₇×Q₈	C₂×C₄	1	2	yes	—
37	C₄×C₇×Q₈	D₄	1	2	yes	—
38	C₄×C₇×Q₈	D₄	1	2	yes	—
39	C₂×C₇×C₄⋊C₄	Q₈	1	1	yes	characteristic
40	C₂×C₇×C₄⋊C₄	Q₈	1	2	yes	—
41	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₇	C₂³	1	2	yes	—
43	C₂×#1136	C₂³	1	4	yes	—
44	C₂×#1136	C₂³	1	2	yes	—
45	C₂×#1136	C₂³	1	2	yes	—
46	C₇×C₄⋊C₈	C₂×C₄	1	2	yes	—
47	C₇×C₄⋊C₈	C₂×C₄	1	2	yes	—
48	C₇×C₄⋊C₈	C₂×C₄	1	8	yes	—
49	C₇×#97	C₂×C₄	1	8	yes	—
50	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
51	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
52	#6393	—	2	4	no	—
53	#6393	—	2	8	no	—
54	C₄⋊(C₇⋊C₈)	—	2	2	no	—
55	C₄⋊(C₇⋊C₈)	—	2	4	no	—
56	C₄⋊(C₇⋊C₈)	—	2	4	no	—
57	C₄⋊(C₇⋊C₈)	—	2	2	no	—
58	#6377	—	2	4	no	—
59	#6377	—	2	2	no	—
60	C₄×C₇⋊C₈	—	2	2	no	—
61	C₄×C₇⋊C₈	—	2	4	no	—
62	Q₈×M₁₆	—	7	2	no	—
63	C₂×#446	D₇	1	1	yes	characteristic
64	#1950	—	7	4	no	—
65	#1950	—	7	2	no	—
66	#1838	—	7	2	no	—
67	#1837	—	7	1	no	—
68	#1778	—	7	2	no	—
69	C₄×#108	—	7	1	no	—
70	C₂×C₇×Q₈	—	4	2	no	—
71	C₂×C₇×Q₈	C₄⋊C₄	1	2	yes	—
72	C₂×C₇×Q₈	C₄⋊C₄	1	2	yes	—
73	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
74	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
75	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
76	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
77	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
78	C₂×C₇×C₈	—	2	2	no	—
79	C₂×C₇×C₈	—	2	4	no	—
80	C₂×C₇×C₈	—	2	2	no	—
81	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
82	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
83	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
84	C₇×C₄⋊C₄	C₄⋊C₄	1	4	yes	—
85	C₇×C₄⋊C₄	C₄⋊C₄	1	4	yes	—
86	C₇×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
87	C₇×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
88	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₇	C₂²×C₄	1	4	yes	—
91	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₇	C₂×D₄	1	4	yes	—
93	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
94	#1136	—	2	2	no	—
95	#1136	—	2	4	no	—
96	#1136	—	2	2	no	—
97	#1136	—	2	4	no	—
98	#1136	—	2	2	no	—
99	#1136	#41	1	4	yes	—
100	C₂×C₇⋊C₈	—	2	2	no	—
101	C₂×C₇⋊C₈	—	2	4	no	—
102	C₂×C₇⋊C₈	—	2	2	no	—
103	C₂×C₄×Q₈	C₂×D₇	1	1	yes	characteristic
104	#446	—	14	2	no	—
105	#446	—	14	4	no	—
106	#446	Dic₇	1	4	yes	—
107	#446	Dic₇	1	4	yes	—
108	C₈×Q₈	—	14	2	no	—
109	C₂×#108	—	7	2	no	—
110	C₂×#108	—	7	4	no	—
111	#423	—	7	2	no	—
112	#423	—	7	4	no	—
113	C₂×C₄⋊C₈	C₂×D₇	1	1	yes	characteristic
114	C₂×C₄⋊C₈	C₂×D₇	1	2	yes	—
115	C₄×M₁₆	—	7	2	no	—
116	C₄×M₁₆	—	7	4	no	—
117	C₂×#97	C₂×D₇	1	2	yes	—
118	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
119	#341	—	7	2	no	—
120	#341	—	7	2	no	—
121	#341	—	7	1	no	—
122	#341	—	7	4	no	—
123	#341	—	7	1	no	—
124	C₂³×C₇	C₂²×Q₈	1	1	yes	characteristic
125	C₇×Q₈	—	2	4	no	—
126	C₇×Q₈	—	4	2	no	—
127	C₂×C₄×C₇	—	2	2	no	—
128	C₂×C₄×C₇	—	4	1	no	—
129	C₂×C₄×C₇	—	4	2	no	—
130	C₂×C₄×C₇	—	4	2	no	—
131	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
134	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
136	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
138	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₇	C₂×#41	1	2	yes	—
140	C₂×C₄×C₇	2₋¹⁺⁴	1	2	yes	—
141	C₇×C₈	—	2	2	no	—
142	C₇×C₈	—	2	4	no	—
143	C₇×C₈	—	2	2	no	—
144	C₇⋊C₈	—	2	4	no	—
145	C₇⋊C₈	—	4	4	no	—
146	C₇⋊C₈	—	4	2	no	—
147	C₂²×Q₈	Dic₁₄	1	1	yes	characteristic
148	C₂×M₁₆	—	7	2	no	—
149	C₂×M₁₆	—	7	4	no	—
150	C₂×M₁₆	—	7	2	no	—
151	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
152	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
153	C₂²×C₈	C₂²×D₇	1	2	yes	—
154	C₄×Q₈	D₂₈	1	2	yes	—
155	C₄×Q₈	D₂₈	1	2	yes	—
156	C₄×Q₈	C₂×Dic₇	1	2	yes	—
157	C₄×Q₈	C₂×Dic₇	1	2	yes	—
158	C₂×C₄⋊C₄	Dic₁₄	1	1	yes	characteristic
159	C₂×C₄⋊C₄	Dic₁₄	1	2	yes	—
160	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
161	C₂×C₄²	C₂²×D₇	1	2	yes	—
162	#108	—	14	4	no	—
163	#108	—	14	8	no	—
164	C₄⋊C₈	—	14	2	no	—
165	C₄⋊C₈	—	14	4	no	—
166	C₄⋊C₈	—	14	4	no	—
167	C₄⋊C₈	—	14	2	no	—
168	C₄⋊C₈	C₂×Dic₇	1	2	yes	—
169	C₄⋊C₈	C₂×Dic₇	1	2	yes	—
170	C₄⋊C₈	C₂×Dic₇	1	8	yes	—
171	#97	—	14	4	no	—
172	#97	—	14	2	no	—
173	#97	C₂×Dic₇	1	8	yes	—
174	C₄×C₈	—	14	2	no	—
175	C₄×C₈	—	14	4	no	—
176	C₄×C₈	C₂×Dic₇	1	2	yes	—
177	C₄×C₈	C₂×Dic₇	1	2	yes	—
178	C₂²×C₇	—	2	1	no	—
179	C₂²×C₇	—	2	1	no	—
180	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
181	C₂²×C₇	#557	1	2	yes	—
182	C₄×C₇	—	2	1	no	—
183	C₄×C₇	—	2	4	no	—
184	C₄×C₇	—	2	1	no	—
185	C₄×C₇	—	2	1	no	—
186	C₄×C₇	—	2	4	no	—
187	C₄×C₇	#430	1	2	yes	—
188	C₄×C₇	C₄×#41	1	1	yes	characteristic
189	C₄×C₇	#519	1	1	yes	characteristic
190	C₄×C₇	#548	1	2	yes	—
191	C₂×Q₈	—	4	2	no	—
192	C₂×Q₈	C₄⋊Dic₇	1	2	yes	—
193	C₂×Q₈	C₄⋊Dic₇	1	2	yes	—
194	C₂²×C₄	C₂×Dic₁₄	1	1	yes	characteristic
195	C₂²×C₄	C₂×Dic₁₄	1	1	yes	characteristic
196	C₂²×C₄	C₂×Dic₁₄	1	2	yes	—
197	C₂²×C₄	C₂×Dic₁₄	1	2	yes	—
198	C₂²×C₄	C₂³×D₇	1	1	yes	characteristic
199	M₁₆	—	7	4	no	—
200	M₁₆	—	14	2	no	—
201	M₁₆	—	14	4	no	—
202	M₁₆	—	14	2	no	—
203	M₁₆	—	14	2	no	—
204	M₁₆	—	14	4	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	4	no	—
208	C₂×C₈	—	14	2	no	—
209	C₂×C₈	—	14	2	no	—
210	C₂×C₈	—	14	4	no	—
211	C₂×C₈	C₂²×Dic₇	1	2	yes	—
212	C₂×C₈	C₂²×Dic₇	1	2	yes	—
213	C₂×C₈	C₂²×Dic₇	1	4	yes	—
214	C₄⋊C₄	C₄⋊Dic₇	1	4	yes	—
215	C₄⋊C₄	C₄⋊Dic₇	1	2	yes	—
216	C₄⋊C₄	C₄⋊Dic₇	1	4	yes	—
217	C₄⋊C₄	C₄⋊Dic₇	1	2	yes	—
218	C₄²	C₂×D₂₈	1	1	yes	characteristic
219	C₄²	C₂×D₂₈	1	4	yes	—
220	C₄²	C₂×D₂₈	1	1	yes	characteristic
221	C₄²	C₂²×Dic₇	1	1	yes	characteristic
222	C₄²	C₂²×Dic₇	1	1	yes	characteristic
223	C₄²	C₂²×Dic₇	1	4	yes	—
224	C₂×C₇	—	2	2	no	—
225	C₂×C₇	#2351	1	1	yes	characteristic
226	C₂×C₇	C₂×#430	1	1	yes	characteristic
227	C₂×C₇	#2916	1	1	yes	characteristic
228	C₂³	C₂²×Dic₁₄	1	1	yes	characteristic
229	Q₈	—	2	4	no	—
230	Q₈	—	4	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	4	1	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
236	C₂×C₄	C₂×C₄⋊Dic₇	1	4	yes	—
237	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
238	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
239	C₂×C₄	C₂×C₄⋊Dic₇	1	4	yes	—
240	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
241	C₂×C₄	C₂²×D₂₈	1	1	yes	characteristic
242	C₂×C₄	#6551	1	2	yes	—
243	C₂×C₄	D₇×#41	1	2	yes	—
244	C₂×C₄	C₂³×Dic₇	1	1	yes	characteristic
245	C₈	—	2	2	no	—
246	C₈	—	2	2	no	—
247	C₈	—	2	4	no	—
248	C₈	—	14	4	no	—
249	C₈	—	28	2	no	—
250	C₈	—	28	4	no	—
251	C₇	#22747	1	1	yes	characteristic, complemented, Sylow
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	#90260	1	2	yes	—
255	C₂²	C₂²×C₄⋊Dic₇	1	1	yes	characteristic
256	C₄	—	2	1	no	—
257	C₄	—	2	4	no	—
258	C₄	—	2	1	no	—
259	C₄	—	2	4	no	—
260	C₄	—	2	1	no	—
261	C₄	#89897	1	2	yes	—
262	C₄	#90267	1	2	yes	—
263	C₄	#90503	1	1	yes	characteristic
264	C₄	#41×Dic₇	1	1	yes	characteristic
265	C₂	—	2	2	no	—
266	C₂	#11739905	1	1	yes	characteristic
267	C₂	C₂×#89897	1	1	yes	characteristic
268	C₂	#11741234	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₈
commutator	C₂×C₄×C₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇
Fitting	C₂×C₇×#446	C₂
socle	C₂²×C₇	C₂²×C₄⋊C₄
prime radical	C₂³×C₇	C₂²×Q₈
cosocle	C₂²×C₄×C₇	C₂⁴
hypercenter	C₂×#446	D₇
hypocenter	C₇	#22747
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈ ◁ C₂×#446

Complements

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

N	H ≅ G/N	type
251. C₇	#22747	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×#446	2
Q₈×#1136	2
#11728228	2
#11727595	2
#11727594	2
#11727545	2
C₄×#6393	2
#22747	7