Subgroups — C₂²×#89969 #14218093

Order 1792.

2710 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×#135	C₂	1	1	yes	maximal, characteristic
2	C₂²×#6511	C₂	1	2	yes	maximal, complemented
3	C₂²×#6411	C₂	1	2	yes	maximal, complemented
4	C₂×#89969	C₂	1	24	yes	maximal, direct product factor
5	C₂²×#6393	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#6389	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₇×#41	C₂²	1	2	yes	—
8	C₂×C₇×#135	C₄	1	12	yes	complemented
9	C₂×C₇×#135	C₂²	1	12	yes	—
10	C₂²×C₇×Q₁₆	C₄	1	1	yes	characteristic, complemented
11	C₂²×C₇×SD₁₆	C₄	1	2	yes	complemented
12	C₂²×C₇×D₈	C₄	1	1	yes	characteristic, complemented
13	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
14	C₂×#6511	—	2	12	no	—
15	C₂×#6511	C₂²	1	24	yes	complemented
16	C₂²×C₄×Dic₇	C₂²	1	1	yes	characteristic
17	C₂²×#1136	—	2	2	no	—
18	C₂²×#1136	C₂²	1	2	yes	—
19	C₂³×C₇⋊C₈	—	2	2	no	—
20	C₂³×C₇⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×#6411	C₂²	1	24	yes	complemented
22	#89969	C₂²	1	64	yes	direct product factor
23	C₂×#6393	C₂²	1	12	yes	complemented
24	C₂×#6389	C₂²	1	12	yes	complemented
25	C₂²×#444	—	7	1	no	maximal, Sylow
26	C₂×C₇×#41	—	2	6	no	—
27	C₂×C₇×#41	—	2	6	no	—
28	C₂×C₇×#41	C₂×C₄	1	12	yes	—
29	C₂×C₇×#41	C₂³	1	12	yes	—
30	C₂²×C₇×Q₈	C₂×C₄	1	2	yes	—
31	C₂²×C₇×D₄	—	2	2	no	—
32	C₂²×C₇×D₄	C₂×C₄	1	2	yes	—
33	C₂³×C₄×C₇	—	2	2	no	—
34	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
35	C₂³×Dic₇	—	2	1	no	—
36	C₇×#135	C₂×C₄	1	48	yes	complemented
37	C₇×#135	C₂³	1	16	yes	—
38	C₂×C₇×Q₁₆	C₂×C₄	1	12	yes	complemented
39	C₂×C₇×SD₁₆	C₂×C₄	1	24	yes	complemented
40	C₂×C₇×D₈	C₂×C₄	1	12	yes	complemented
41	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₇×C₈	C₂×C₄	1	6	yes	—
44	C₂²×C₇×C₈	C₂³	1	6	yes	—
45	#6511	—	2	48	no	—
46	#6511	C₂³	1	32	yes	complemented
47	C₂×C₄×Dic₇	—	2	3	no	—
48	C₂×C₄×Dic₇	C₂³	1	6	yes	—
49	C₂×#1136	—	2	6	no	—
50	C₂×#1136	—	2	24	no	—
51	C₂×#1136	C₂³	1	12	yes	—
52	C₂²×C₇⋊C₈	—	2	2	no	—
53	C₂²×C₇⋊C₈	—	2	24	no	—
54	C₂²×C₇⋊C₈	D₄	1	2	yes	—
55	C₂²×C₇⋊C₈	D₄	1	6	yes	—
56	C₂²×C₇⋊C₈	C₂³	1	6	yes	—
57	#6411	C₂³	1	32	yes	complemented
58	#6393	C₂³	1	16	yes	complemented
59	#6389	C₂³	1	16	yes	complemented
60	C₂²×#135	D₇	1	1	yes	characteristic
61	C₂²×#131	—	7	2	no	—
62	C₂×#444	—	7	24	no	—
63	C₂²×#108	—	7	1	no	—
64	C₂²×#104	—	7	2	no	—
65	C₂²×#97	—	7	1	no	—
66	C₂⁴×C₇	—	2	2	no	—
67	C₂⁴×C₇	#41	1	1	yes	characteristic
68	C₇×#41	—	2	24	no	—
69	C₇×#41	—	2	12	no	—
70	C₇×#41	—	2	12	no	—
71	C₇×#41	C₂²×C₄	1	24	yes	—
72	C₇×#41	C₂⁴	1	8	yes	—
73	C₂×C₇×Q₈	—	2	6	no	—
74	C₂×C₇×Q₈	C₂²×C₄	1	12	yes	—
75	C₂×C₇×D₄	—	2	12	no	—
76	C₂×C₇×D₄	—	2	12	no	—
77	C₂×C₇×D₄	—	2	6	no	—
78	C₂×C₇×D₄	C₂²×C₄	1	12	yes	—
79	C₂²×C₄×C₇	—	2	2	no	—
80	C₂²×C₄×C₇	—	2	12	no	—
81	C₂²×C₄×C₇	—	2	12	no	—
82	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₇	C₂²×C₄	1	3	yes	—
84	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
85	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₇	#41	1	3	yes	—
87	C₂²×C₄×C₇	C₂⁴	1	3	yes	—
88	C₂²×Dic₇	—	2	6	no	—
89	C₂²×Dic₇	—	4	4	no	—
90	C₇×Q₁₆	C₂²×C₄	1	16	yes	complemented
91	C₇×SD₁₆	C₂²×C₄	1	32	yes	complemented
92	C₇×D₈	C₂²×C₄	1	16	yes	complemented
93	C₂×C₇×C₈	C₂²×C₄	1	6	yes	—
94	C₂×C₇×C₈	C₂²×C₄	1	6	yes	—
95	C₂×C₇×C₈	C₂²×C₄	1	12	yes	—
96	C₂×C₇×C₈	C₂⁴	1	4	yes	—
97	C₄×Dic₇	—	2	6	no	—
98	C₄×Dic₇	C₂⁴	1	4	yes	—
99	#1136	—	2	12	no	—
100	#1136	—	2	32	no	—
101	#1136	C₂⁴	1	8	yes	—
102	C₂×C₇⋊C₈	—	2	12	no	—
103	C₂×C₇⋊C₈	—	2	32	no	—
104	C₂×C₇⋊C₈	C₂×D₄	1	12	yes	—
105	C₂×C₇⋊C₈	C₂×D₄	1	12	yes	—
106	C₂×C₇⋊C₈	C₂⁴	1	4	yes	—
107	C₂²×#41	C₂×D₇	1	2	yes	—
108	C₂×#135	Dic₇	1	12	yes	complemented
109	C₂×#135	C₂×D₇	1	12	yes	—
110	C₂²×Q₁₆	Dic₇	1	1	yes	characteristic, complemented
111	C₂²×SD₁₆	Dic₇	1	2	yes	complemented
112	C₂²×D₈	Dic₇	1	1	yes	characteristic, complemented
113	C₂×#131	—	7	24	no	—
114	C₂×#131	—	14	12	no	—
115	C₂²×M₁₆	—	7	2	no	—
116	C₂²×M₁₆	—	14	2	no	—
117	C₂³×C₈	—	7	1	no	—
118	C₂³×C₈	—	14	2	no	—
119	C₂³×C₈	C₂×D₇	1	1	yes	characteristic
120	C₂²×C₄²	—	7	1	no	—
121	#444	—	7	64	no	—
122	C₂×#108	—	7	12	no	—
123	C₂×#104	—	7	24	no	—
124	C₂×#97	—	7	12	no	—
125	C₂³×C₇	—	2	4	no	—
126	C₂³×C₇	—	2	12	no	—
127	C₂³×C₇	—	4	8	no	—
128	C₂³×C₇	C₄×D₄	1	3	yes	—
129	C₂³×C₇	C₄×D₄	1	1	yes	characteristic
130	C₂³×C₇	C₂×#41	1	3	yes	—
131	C₇×Q₈	—	2	12	no	—
132	C₇×Q₈	C₂³×C₄	1	8	yes	—
133	C₇×D₄	—	2	24	no	—
134	C₇×D₄	—	2	8	no	—
135	C₇×D₄	—	2	12	no	—
136	C₇×D₄	C₂³×C₄	1	8	yes	—
137	C₂×C₄×C₇	—	2	12	no	—
138	C₂×C₄×C₇	—	2	24	no	—
139	C₂×C₄×C₇	—	2	8	no	—
140	C₂×C₄×C₇	C₄×D₄	1	3	yes	—
141	C₂×C₄×C₇	C₄×D₄	1	3	yes	—
142	C₂×C₄×C₇	C₄×D₄	1	6	yes	—
143	C₂×C₄×C₇	C₂³×C₄	1	3	yes	—
144	C₂×C₄×C₇	C₂³×C₄	1	3	yes	—
145	C₂×C₄×C₇	C₂²×D₄	1	3	yes	—
146	C₂×C₄×C₇	C₂²×D₄	1	3	yes	—
147	C₂×C₄×C₇	C₂×#41	1	3	yes	—
148	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
149	C₂×Dic₇	—	2	4	no	—
150	C₂×Dic₇	—	4	12	no	—
151	C₇×C₈	C₂³×C₄	1	4	yes	—
152	C₇×C₈	C₂³×C₄	1	4	yes	—
153	C₇⋊C₈	—	2	8	no	—
154	C₇⋊C₈	C₂²×D₄	1	8	yes	—
155	C₂×#41	—	2	6	no	—
156	C₂×#41	—	2	6	no	—
157	C₂×#41	C₂×Dic₇	1	12	yes	—
158	C₂×#41	C₂²×D₇	1	12	yes	—
159	C₂²×Q₈	C₂×Dic₇	1	2	yes	—
160	C₂²×D₄	—	2	2	no	—
161	C₂²×D₄	C₂×Dic₇	1	2	yes	—
162	C₂³×C₄	—	2	2	no	—
163	C₂³×C₄	—	14	1	no	—
164	C₂³×C₄	C₂²×D₇	1	1	yes	characteristic
165	#135	C₂×Dic₇	1	48	yes	complemented
166	#135	C₂²×D₇	1	16	yes	—
167	C₂×Q₁₆	C₂×Dic₇	1	12	yes	complemented
168	C₂×SD₁₆	C₂×Dic₇	1	24	yes	complemented
169	C₂×D₈	C₂×Dic₇	1	12	yes	complemented
170	#131	—	7	32	no	—
171	#131	—	14	48	no	—
172	C₂×M₁₆	—	7	12	no	—
173	C₂×M₁₆	—	14	24	no	—
174	C₂×M₁₆	—	14	6	no	—
175	C₂²×C₈	—	7	6	no	—
176	C₂²×C₈	—	7	2	no	—
177	C₂²×C₈	—	7	6	no	—
178	C₂²×C₈	—	14	24	no	—
179	C₂²×C₈	—	14	2	no	—
180	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
181	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
182	C₂²×C₈	C₂×Dic₇	1	6	yes	—
183	C₂²×C₈	C₂²×D₇	1	6	yes	—
184	C₂×C₄²	—	7	6	no	—
185	C₂×C₄²	—	14	3	no	—
186	#108	—	7	16	no	—
187	#104	—	7	32	no	—
188	#97	—	7	16	no	—
189	C₂²×C₇	—	2	6	no	—
190	C₂²×C₇	—	2	6	no	—
191	C₂²×C₇	—	2	8	no	—
192	C₂²×C₇	—	4	24	no	—
193	C₂²×C₇	#444	1	4	yes	complemented
194	C₂²×C₇	C₂×C₄×D₄	1	3	yes	—
195	C₂²×C₇	C₂×C₄×D₄	1	3	yes	—
196	C₂²×C₇	C₂²×#41	1	1	yes	characteristic
197	C₄×C₇	—	2	8	no	—
198	C₄×C₇	C₂×C₄×D₄	1	3	yes	—
199	C₄×C₇	C₂×C₄×D₄	1	3	yes	—
200	C₄×C₇	C₂⁴×C₄	1	1	yes	characteristic
201	C₄×C₇	C₂³×D₄	1	1	yes	characteristic
202	Dic₇	—	4	4	no	—
203	C₂⁴	—	2	2	no	—
204	C₂⁴	#1159	1	1	yes	characteristic
205	#41	—	2	24	no	—
206	#41	—	2	12	no	—
207	#41	—	2	12	no	—
208	#41	C₂²×Dic₇	1	24	yes	—
209	#41	C₂³×D₇	1	8	yes	—
210	C₂×Q₈	—	2	6	no	—
211	C₂×Q₈	C₂²×Dic₇	1	12	yes	—
212	C₂×D₄	—	2	12	no	—
213	C₂×D₄	—	2	12	no	—
214	C₂×D₄	—	2	6	no	—
215	C₂×D₄	C₂²×Dic₇	1	12	yes	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	2	12	no	—
218	C₂²×C₄	—	2	12	no	—
219	C₂²×C₄	—	14	6	no	—
220	C₂²×C₄	—	28	4	no	—
221	C₂²×C₄	D₄×D₇	1	3	yes	—
222	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
223	C₂²×C₄	#1159	1	3	yes	—
224	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
225	C₂²×C₄	C₂²×Dic₇	1	3	yes	—
226	C₂²×C₄	C₂³×D₇	1	3	yes	—
227	Q₁₆	C₂²×Dic₇	1	16	yes	complemented
228	SD₁₆	C₂²×Dic₇	1	32	yes	complemented
229	D₈	C₂²×Dic₇	1	16	yes	complemented
230	M₁₆	—	7	8	no	—
231	M₁₆	—	14	32	no	—
232	M₁₆	—	14	12	no	—
233	C₂×C₈	—	7	12	no	—
234	C₂×C₈	—	7	12	no	—
235	C₂×C₈	—	7	4	no	—
236	C₂×C₈	—	14	32	no	—
237	C₂×C₈	—	14	12	no	—
238	C₂×C₈	C₂²×Dic₇	1	6	yes	—
239	C₂×C₈	C₂²×Dic₇	1	6	yes	—
240	C₂×C₈	C₂²×Dic₇	1	12	yes	—
241	C₂×C₈	C₂³×D₇	1	4	yes	—
242	C₄²	—	7	4	no	—
243	C₄²	—	14	6	no	—
244	C₂×C₇	—	2	1	no	—
245	C₂×C₇	—	2	3	no	—
246	C₂×C₇	—	4	8	no	—
247	C₂×C₇	C₂×#444	1	6	yes	complemented
248	C₂×C₇	C₂²×C₄×D₄	1	1	yes	characteristic
249	C₂³	—	2	4	no	—
250	C₂³	—	2	12	no	—
251	C₂³	—	4	8	no	—
252	C₂³	D₄×Dic₇	1	3	yes	—
253	C₂³	D₄×Dic₇	1	1	yes	characteristic
254	C₂³	C₂×#1159	1	3	yes	—
255	Q₈	—	2	12	no	—
256	Q₈	C₂³×Dic₇	1	8	yes	—
257	D₄	—	2	24	no	—
258	D₄	—	2	8	no	—
259	D₄	—	2	12	no	—
260	D₄	C₂³×Dic₇	1	8	yes	—
261	C₂×C₄	—	2	12	no	—
262	C₂×C₄	—	2	24	no	—
263	C₂×C₄	—	2	8	no	—
264	C₂×C₄	—	14	4	no	—
265	C₂×C₄	—	28	12	no	—
266	C₂×C₄	D₄×Dic₇	1	3	yes	—
267	C₂×C₄	D₄×Dic₇	1	3	yes	—
268	C₂×C₄	D₄×Dic₇	1	6	yes	—
269	C₂×C₄	C₂×D₄×D₇	1	3	yes	—
270	C₂×C₄	C₂×D₄×D₇	1	3	yes	—
271	C₂×C₄	C₂×#1159	1	3	yes	—
272	C₂×C₄	C₂³×Dic₇	1	3	yes	—
273	C₂×C₄	C₂³×Dic₇	1	3	yes	—
274	C₂×C₄	C₂⁴×D₇	1	1	yes	characteristic
275	C₈	—	7	8	no	—
276	C₈	—	14	8	no	—
277	C₈	C₂³×Dic₇	1	4	yes	—
278	C₈	C₂³×Dic₇	1	4	yes	—
279	C₇	C₂²×#444	1	1	yes	characteristic, complemented, Sylow
280	C₂²	—	2	6	no	—
281	C₂²	—	2	6	no	—
282	C₂²	—	2	8	no	—
283	C₂²	—	4	24	no	—
284	C₂²	#89969	1	4	yes	direct product factor
285	C₂²	C₂×D₄×Dic₇	1	3	yes	—
286	C₂²	C₂×D₄×Dic₇	1	3	yes	—
287	C₂²	C₂²×#1159	1	1	yes	characteristic
288	C₄	—	2	8	no	—
289	C₄	—	28	4	no	—
290	C₄	C₂×D₄×Dic₇	1	3	yes	—
291	C₄	C₂×D₄×Dic₇	1	3	yes	—
292	C₄	C₂²×D₄×D₇	1	1	yes	characteristic
293	C₄	C₂⁴×Dic₇	1	1	yes	characteristic
294	C₂	—	2	1	no	—
295	C₂	—	2	3	no	—
296	C₂	—	4	8	no	—
297	C₂	C₂×#89969	1	6	yes	direct product factor
298	C₂	C₂²×D₄×Dic₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₇
commutator	C₄×C₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₇
Fitting, prime radical	C₂²×C₇×#135	C₂
socle	C₂³×C₇	C₄×D₄
cosocle	C₂×C₄×C₇	C₂⁵
hypercenter	C₂²×#135	D₇
hypocenter	C₇	C₂²×#444
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂²×#135

Complements

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

N	H ≅ G/N	type
2. C₂²×#6511	C₂	semidirect
3. C₂²×#6411	C₂	semidirect
4. C₂×#89969	C₂	direct (N × H)
5. C₂²×#6393	C₂	semidirect
6. C₂²×#6389	C₂	semidirect
8. C₂×C₇×#135	C₄	semidirect
10. C₂²×C₇×Q₁₆	C₄	semidirect
11. C₂²×C₇×SD₁₆	C₄	semidirect
12. C₂²×C₇×D₈	C₄	semidirect
15. C₂×#6511	C₂²	semidirect
21. C₂×#6411	C₂²	semidirect
22. #89969	C₂²	direct (N × H)
23. C₂×#6393	C₂²	semidirect
24. C₂×#6389	C₂²	semidirect
36. C₇×#135	C₂×C₄	semidirect
38. C₂×C₇×Q₁₆	C₂×C₄	semidirect
39. C₂×C₇×SD₁₆	C₂×C₄	semidirect
40. C₂×C₇×D₈	C₂×C₄	semidirect
46. #6511	C₂³	semidirect
57. #6411	C₂³	semidirect
58. #6393	C₂³	semidirect
59. #6389	C₂³	semidirect
90. C₇×Q₁₆	C₂²×C₄	semidirect
91. C₇×SD₁₆	C₂²×C₄	semidirect
92. C₇×D₈	C₂²×C₄	semidirect
108. C₂×#135	Dic₇	semidirect
110. C₂²×Q₁₆	Dic₇	semidirect
111. C₂²×SD₁₆	Dic₇	semidirect
112. C₂²×D₈	Dic₇	semidirect
165. #135	C₂×Dic₇	semidirect
167. C₂×Q₁₆	C₂×Dic₇	semidirect
168. C₂×SD₁₆	C₂×Dic₇	semidirect
169. C₂×D₈	C₂×Dic₇	semidirect
193. C₂²×C₇	#444	semidirect
227. Q₁₆	C₂²×Dic₇	semidirect
228. SD₁₆	C₂²×Dic₇	semidirect
229. D₈	C₂²×Dic₇	semidirect
247. C₂×C₇	C₂×#444	semidirect
279. C₇	C₂²×#444	semidirect
284. C₂²	#89969	direct (N × H)
297. C₂	C₂×#89969	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₇×#135	2
C₂²×#6511	2
C₂²×#6411	2
C₂×#89969	2
C₂²×#6393	2
C₂²×#6389	2
C₂²×#444	7

Subgroups — C22×#89969 #14218093

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#89969 #14218093