Subgroups — #13061646

Order 1344.

584 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#66730	C₂	1	2	yes	maximal, complemented
2	C₃×C₄×C₇×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#10621413	C₂	1	2	yes	maximal, complemented
4	C₂×C₄⋊Dic₂₁	C₂	1	2	yes	maximal, complemented
5	#10621380	C₂	1	2	yes	maximal, complemented
6	#10621369	C₂	1	4	yes	maximal, complemented
7	C₄×D₈₄	C₂	1	1	yes	maximal, characteristic, complemented
8	#10621352	C₂	1	1	yes	maximal, characteristic, complemented
9	#90246	—	3	1	no	maximal, Hall
10	C₂×C₃×C₇×D₄	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₄×C₇	C₂²	1	2	yes	—
12	C₂×#4268	C₂²	1	4	yes	—
13	C₂²×Dic₂₁	C₂²	1	4	yes	—
14	#66730	—	2	8	no	—
15	C₂×D₈₄	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₂₁	C₂²	1	2	yes	—
17	C₂×Dic₄₂	C₂²	1	2	yes	—
18	C₃×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₃×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
20	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic
21	#66631	C₂²	1	2	yes	—
22	#66631	C₂²	1	4	yes	—
23	C₄⋊Dic₂₁	C₂²	1	2	yes	—
24	C₄⋊Dic₂₁	C₂²	1	2	yes	—
25	C₄⋊Dic₂₁	C₂²	1	1	yes	characteristic
26	C₄⋊Dic₂₁	C₂²	1	4	yes	—
27	C₂×#1159	—	3	2	no	—
28	C₄×C₇×D₄	D₃	1	1	yes	characteristic, complemented
29	#6493	—	3	2	no	—
30	C₂×C₄⋊Dic₇	—	3	2	no	—
31	#6460	—	3	2	no	—
32	#6449	—	3	4	no	—
33	C₄×D₂₈	—	3	1	no	—
34	#6432	—	3	1	no	—
35	#5555	—	7	1	no	maximal, Hall
36	C₂³×C₃×C₇	C₂³	1	2	yes	—
37	C₂²×D₂₁	C₂³	1	2	yes	—
38	C₃×C₇×D₄	D₄	1	4	yes	complemented
39	C₂×C₃×C₄×C₇	—	2	4	no	—
40	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
41	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
42	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
43	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
44	#4268	—	2	8	no	—
45	C₂×Dic₂₁	—	2	8	no	—
46	C₂×Dic₂₁	C₂³	1	4	yes	—
47	C₂×Dic₂₁	C₂³	1	2	yes	—
48	D₈₄	—	2	2	no	—
49	C₄×D₂₁	—	2	4	no	—
50	Dic₄₂	—	2	4	no	—
51	C₂×C₇×D₄	C₂×D₃	1	1	yes	characteristic
52	C₂²×C₄×C₇	C₂×D₃	1	2	yes	—
53	C₂×C₂²⋊D₇	—	3	4	no	—
54	C₂²×Dic₇	—	3	4	no	—
55	#1159	—	6	8	no	—
56	C₂×D₂₈	—	3	1	no	—
57	C₂×C₄×D₇	—	3	2	no	—
58	C₂×Dic₁₄	—	3	2	no	—
59	C₇×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
60	C₇×C₂²⋊C₄	C₂×D₃	1	2	yes	—
61	C₄²×C₇	C₂×D₃	1	1	yes	characteristic
62	(C₂×D₇)⋊C₄	—	3	2	no	—
63	(C₂×D₇)⋊C₄	—	3	4	no	—
64	C₄⋊Dic₇	—	3	2	no	—
65	C₄⋊Dic₇	—	3	2	no	—
66	C₄⋊Dic₇	—	3	1	no	—
67	C₄⋊Dic₇	—	3	4	no	—
68	C₂×#237	—	7	2	no	—
69	C₃×C₄×D₄	D₇	1	1	yes	characteristic, complemented
70	#930	—	7	2	no	—
71	C₂×C₄⋊Dic₃	—	7	2	no	—
72	#897	—	7	2	no	—
73	#886	—	7	4	no	—
74	C₄×D₁₂	—	7	1	no	—
75	#869	—	7	1	no	—
76	C₂²×C₃×C₇	—	2	2	no	—
77	C₂²×C₃×C₇	—	2	2	no	—
78	C₂²×C₃×C₇	C₂×D₄	1	4	yes	—
79	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
80	C₂×D₂₁	—	2	2	no	—
81	C₂×D₂₁	—	2	2	no	—
82	C₂×D₂₁	—	2	2	no	—
83	C₃×C₄×C₇	—	2	2	no	—
84	C₃×C₄×C₇	—	2	1	no	—
85	C₃×C₄×C₇	C₂×D₄	1	2	yes	—
86	C₃×C₄×C₇	#41	1	2	yes	complemented
87	Dic₂₁	—	2	4	no	—
88	Dic₂₁	—	2	2	no	—
89	#547	—	21	1	no	Sylow
90	C₂³×C₇	C₂²×D₃	1	2	yes	—
91	C₂²×D₇	—	3	2	no	—
92	C₇×D₄	D₁₂	1	4	yes	complemented
93	C₂×C₄×C₇	—	2	4	no	—
94	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
95	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
96	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
97	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
98	C₂²⋊D₇	—	6	8	no	—
99	C₂×Dic₇	—	3	4	no	—
100	C₂×Dic₇	—	3	2	no	—
101	C₂×Dic₇	—	6	8	no	—
102	D₂₈	—	6	2	no	—
103	C₄×D₇	—	6	4	no	—
104	Dic₁₄	—	6	4	no	—
105	C₂×C₃×D₄	C₂×D₇	1	1	yes	characteristic
106	C₂²×C₃×C₄	C₂×D₇	1	2	yes	—
107	C₂×#67	—	7	4	no	—
108	C₂²×Dic₃	—	7	4	no	—
109	#237	—	14	8	no	—
110	C₂×D₁₂	—	7	1	no	—
111	C₂×C₄×D₃	—	7	2	no	—
112	C₂×Dic₆	—	7	2	no	—
113	C₃×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
114	C₃×C₂²⋊C₄	C₂×D₇	1	2	yes	—
115	C₃×C₄²	C₂×D₇	1	1	yes	characteristic
116	(C₂×D₃)⋊C₄	—	7	2	no	—
117	(C₂×D₃)⋊C₄	—	7	4	no	—
118	C₄⋊Dic₃	—	7	2	no	—
119	C₄⋊Dic₃	—	7	2	no	—
120	C₄⋊Dic₃	—	7	1	no	—
121	C₄⋊Dic₃	—	7	4	no	—
122	C₂×C₃×C₇	—	2	4	no	—
123	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
125	C₂×C₃×C₇	2₋¹⁺⁴	1	1	yes	characteristic
126	D₂₁	—	4	2	no	—
127	C₂×#41	—	21	2	no	—
128	C₄⋊Q₈	—	21	1	no	—
129	#123	—	21	4	no	—
130	C₂²⋊Q₈	—	21	2	no	—
131	#121	—	21	2	no	—
132	C₄×D₄	—	21	1	no	—
133	C₄×D₄	D₂₁	1	1	yes	characteristic, complemented
134	C₂×C₄⋊C₄	—	21	2	no	—
135	C₂²×C₇	—	2	2	no	—
136	C₂²×C₇	—	2	2	no	—
137	C₂²×C₇	C₂×D₁₂	1	4	yes	—
138	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
139	C₂×D₇	—	6	2	no	—
140	C₂×D₇	—	6	2	no	—
141	C₂×D₇	—	6	2	no	—
142	C₄×C₇	—	2	2	no	—
143	C₄×C₇	—	2	1	no	—
144	C₄×C₇	C₂×D₁₂	1	2	yes	—
145	C₄×C₇	#237	1	2	yes	complemented
146	Dic₇	—	6	4	no	—
147	Dic₇	—	6	2	no	—
148	C₂³×C₃	C₂²×D₇	1	2	yes	—
149	C₂²×D₃	—	7	2	no	—
150	C₃×D₄	D₂₈	1	4	yes	complemented
151	C₂×C₃×C₄	—	2	4	no	—
152	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
153	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
154	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
155	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
156	#67	—	14	8	no	—
157	C₂×Dic₃	—	7	4	no	—
158	C₂×Dic₃	—	7	2	no	—
159	C₂×Dic₃	—	14	8	no	—
160	D₁₂	—	14	2	no	—
161	C₄×D₃	—	14	4	no	—
162	Dic₆	—	14	4	no	—
163	C₃×C₇	#547	1	1	yes	characteristic, complemented, Hall
164	#41	—	42	8	no	—
165	C₂×Q₈	—	21	2	no	—
166	C₂×D₄	—	21	4	no	—
167	C₂×D₄	—	21	1	no	—
168	C₂×D₄	C₂×D₂₁	1	1	yes	characteristic
169	C₂²×C₄	—	21	4	no	—
170	C₂²×C₄	—	21	2	no	—
171	C₂²×C₄	C₂×D₂₁	1	2	yes	—
172	C₄⋊C₄	—	21	1	no	—
173	C₄⋊C₄	—	21	2	no	—
174	C₄⋊C₄	—	21	4	no	—
175	C₄⋊C₄	—	21	2	no	—
176	C₄⋊C₄	C₂×D₂₁	1	1	yes	characteristic
177	C₂²⋊C₄	—	21	2	no	—
178	C₂²⋊C₄	—	21	4	no	—
179	C₂²⋊C₄	C₂×D₂₁	1	2	yes	—
180	C₄²	C₂×D₂₁	1	1	yes	characteristic
181	C₂×C₇	—	2	4	no	—
182	C₂×C₇	C₂²×D₁₂	1	1	yes	characteristic
183	C₂×C₇	C₂×#237	1	1	yes	characteristic
184	C₂×C₇	#1010	1	1	yes	characteristic
185	D₇	—	12	2	no	—
186	C₂²×C₃	—	2	2	no	—
187	C₂²×C₃	—	2	2	no	—
188	C₂²×C₃	C₂×D₂₈	1	4	yes	—
189	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
190	C₂×D₃	—	14	2	no	—
191	C₂×D₃	—	14	2	no	—
192	C₂×D₃	—	14	2	no	—
193	C₃×C₄	—	2	2	no	—
194	C₃×C₄	—	2	1	no	—
195	C₃×C₄	C₂×D₂₈	1	2	yes	—
196	C₃×C₄	#1159	1	2	yes	complemented
197	Dic₃	—	14	4	no	—
198	Dic₃	—	14	2	no	—
199	C₂³	—	21	2	no	—
200	C₂³	C₂²×D₂₁	1	2	yes	—
201	Q₈	—	42	4	no	—
202	D₄	—	42	8	no	—
203	D₄	—	42	2	no	—
204	D₄	D₈₄	1	4	yes	complemented
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	21	2	no	—
207	C₂×C₄	—	21	4	no	—
208	C₂×C₄	—	42	8	no	—
209	C₂×C₄	—	42	4	no	—
210	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
211	C₂×C₄	C₂²×D₂₁	1	2	yes	—
212	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
213	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
214	C₇	#5555	1	1	yes	characteristic, complemented, Sylow
215	C₂×C₃	—	2	4	no	—
216	C₂×C₃	C₂²×D₂₈	1	1	yes	characteristic
217	C₂×C₃	C₂×#1159	1	1	yes	characteristic
218	C₂×C₃	#6554	1	1	yes	characteristic
219	D₃	—	28	2	no	—
220	C₂²	—	2	2	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	42	2	no	—
223	C₂²	—	42	2	no	—
224	C₂²	—	42	2	no	—
225	C₂²	C₂×D₈₄	1	4	yes	—
226	C₂²	C₂³×D₂₁	1	1	yes	characteristic
227	C₄	—	2	2	no	—
228	C₄	—	2	1	no	—
229	C₄	—	42	2	no	—
230	C₄	—	42	4	no	—
231	C₄	C₂×D₈₄	1	2	yes	—
232	C₄	#66730	1	2	yes	complemented
233	C₃	#90246	1	1	yes	characteristic, complemented, Sylow
234	C₂	—	2	4	no	—
235	C₂	—	84	2	no	—
236	C₂	C₂²×D₈₄	1	1	yes	characteristic
237	C₂	C₂×#66730	1	1	yes	characteristic
238	C₂	#10621663	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₁
commutator, socle, cosocle	C₂²×C₃×C₇	C₂⁴
Fitting	C₃×C₄×C₇×D₄	C₂
hypercenter	C₄×D₄	D₂₁
hypocenter	C₃×C₇	#547
LCS	G ▷ C₂²×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#66730	C₂	semidirect
2. C₃×C₄×C₇×D₄	C₂	semidirect
3. #10621413	C₂	semidirect
4. C₂×C₄⋊Dic₂₁	C₂	semidirect
5. #10621380	C₂	semidirect
6. #10621369	C₂	semidirect
7. C₄×D₈₄	C₂	semidirect
8. #10621352	C₂	semidirect
28. C₄×C₇×D₄	D₃	semidirect
38. C₃×C₇×D₄	D₄	semidirect
69. C₃×C₄×D₄	D₇	semidirect
86. C₃×C₄×C₇	#41	semidirect
92. C₇×D₄	D₁₂	semidirect
133. C₄×D₄	D₂₁	semidirect
145. C₄×C₇	#237	semidirect
150. C₃×D₄	D₂₈	semidirect
163. C₃×C₇	#547	semidirect
196. C₃×C₄	#1159	semidirect
204. D₄	D₈₄	semidirect
214. C₇	#5555	semidirect
232. C₄	#66730	semidirect
233. C₃	#90246	semidirect

Maximal subgroups

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

subgroup	index
C₂×#66730	2
C₃×C₄×C₇×D₄	2
#10621413	2
C₂×C₄⋊Dic₂₁	2
#10621380	2
#10621369	2
C₄×D₈₄	2
#10621352	2
#90246	3
#5555	7