Subgroups — C₂²×#89968 #14218087

Order 1792.

2750 conjugacy classes of subgroups.

The subgroup lattice has 306 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×#89968	C₂	1	24	yes	maximal, direct product factor
5	C₂²×#6393	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×Dic₇	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	#89968	C₂²	1	64	yes	direct product factor
23	C₂×#6393	C₂²	1	12	yes	complemented
24	C₂×C₈×Dic₇	C₂²	1	12	yes	complemented
25	C₂²×#443	—	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	1	yes	characteristic
55	C₂²×C₇⋊C₈	D₄	1	1	yes	characteristic
56	C₂²×C₇⋊C₈	D₄	1	6	yes	—
57	C₂²×C₇⋊C₈	C₂³	1	6	yes	—
58	#6411	C₂³	1	32	yes	complemented
59	#6393	C₂³	1	16	yes	complemented
60	C₈×Dic₇	C₂³	1	16	yes	complemented
61	C₂²×#135	D₇	1	1	yes	characteristic
62	C₂²×#131	—	7	2	no	—
63	C₂×#443	—	7	24	no	—
64	C₂²×#108	—	7	1	no	—
65	C₂²×#104	—	7	2	no	—
66	C₂²×C₄×C₈	—	7	1	no	—
67	C₂⁴×C₇	—	2	2	no	—
68	C₂⁴×C₇	#41	1	1	yes	characteristic
69	C₇×#41	—	2	24	no	—
70	C₇×#41	—	2	12	no	—
71	C₇×#41	—	2	12	no	—
72	C₇×#41	C₂²×C₄	1	24	yes	—
73	C₇×#41	C₂⁴	1	8	yes	—
74	C₂×C₇×Q₈	—	2	6	no	—
75	C₂×C₇×Q₈	C₂²×C₄	1	12	yes	—
76	C₂×C₇×D₄	—	2	12	no	—
77	C₂×C₇×D₄	—	2	12	no	—
78	C₂×C₇×D₄	—	2	6	no	—
79	C₂×C₇×D₄	C₂²×C₄	1	12	yes	—
80	C₂²×C₄×C₇	—	2	2	no	—
81	C₂²×C₄×C₇	—	2	12	no	—
82	C₂²×C₄×C₇	—	2	12	no	—
83	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₇	C₂²×C₄	1	3	yes	—
85	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
86	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₇	#41	1	3	yes	—
88	C₂²×C₄×C₇	C₂⁴	1	3	yes	—
89	C₂²×Dic₇	—	2	8	no	—
90	C₂²×Dic₇	—	2	6	no	—
91	C₇×Q₁₆	C₂²×C₄	1	16	yes	complemented
92	C₇×SD₁₆	C₂²×C₄	1	32	yes	complemented
93	C₇×D₈	C₂²×C₄	1	16	yes	complemented
94	C₂×C₇×C₈	C₂²×C₄	1	6	yes	—
95	C₂×C₇×C₈	C₂²×C₄	1	6	yes	—
96	C₂×C₇×C₈	C₂²×C₄	1	12	yes	—
97	C₂×C₇×C₈	C₂⁴	1	4	yes	—
98	C₄×Dic₇	—	2	6	no	—
99	C₄×Dic₇	C₂⁴	1	4	yes	—
100	#1136	—	2	12	no	—
101	#1136	—	2	32	no	—
102	#1136	C₂⁴	1	8	yes	—
103	C₂×C₇⋊C₈	—	2	12	no	—
104	C₂×C₇⋊C₈	—	2	32	no	—
105	C₂×C₇⋊C₈	C₂×D₄	1	6	yes	—
106	C₂×C₇⋊C₈	C₂×D₄	1	6	yes	—
107	C₂×C₇⋊C₈	C₂×D₄	1	12	yes	—
108	C₂×C₇⋊C₈	C₂⁴	1	4	yes	—
109	C₂²×#41	C₂×D₇	1	2	yes	—
110	C₂×#135	Dic₇	1	12	yes	complemented
111	C₂×#135	C₂×D₇	1	12	yes	—
112	C₂²×Q₁₆	Dic₇	1	1	yes	characteristic, complemented
113	C₂²×SD₁₆	Dic₇	1	2	yes	complemented
114	C₂²×D₈	Dic₇	1	1	yes	characteristic, complemented
115	C₂×#131	—	7	24	no	—
116	C₂×#131	—	14	12	no	—
117	C₂²×M₁₆	—	7	2	no	—
118	C₂²×M₁₆	—	14	2	no	—
119	C₂³×C₈	—	7	1	no	—
120	C₂³×C₈	—	14	2	no	—
121	C₂³×C₈	C₂×D₇	1	1	yes	characteristic
122	C₂²×C₄²	—	7	1	no	—
123	#443	—	7	64	no	—
124	C₂×#108	—	7	12	no	—
125	C₂×#104	—	7	24	no	—
126	C₂×C₄×C₈	—	7	12	no	—
127	C₂³×C₇	—	2	4	no	—
128	C₂³×C₇	—	2	12	no	—
129	C₂³×C₇	—	4	8	no	—
130	C₂³×C₇	C₄×D₄	1	3	yes	—
131	C₂³×C₇	C₄×D₄	1	1	yes	characteristic
132	C₂³×C₇	C₂×#41	1	3	yes	—
133	C₇×Q₈	—	2	12	no	—
134	C₇×Q₈	C₂³×C₄	1	8	yes	—
135	C₇×D₄	—	2	24	no	—
136	C₇×D₄	—	2	8	no	—
137	C₇×D₄	—	2	12	no	—
138	C₇×D₄	C₂³×C₄	1	8	yes	—
139	C₂×C₄×C₇	—	2	12	no	—
140	C₂×C₄×C₇	—	2	24	no	—
141	C₂×C₄×C₇	—	2	8	no	—
142	C₂×C₄×C₇	C₄×D₄	1	3	yes	—
143	C₂×C₄×C₇	C₄×D₄	1	3	yes	—
144	C₂×C₄×C₇	C₄×D₄	1	6	yes	—
145	C₂×C₄×C₇	C₂³×C₄	1	3	yes	—
146	C₂×C₄×C₇	C₂³×C₄	1	3	yes	—
147	C₂×C₄×C₇	C₂²×D₄	1	3	yes	—
148	C₂×C₄×C₇	C₂²×D₄	1	3	yes	—
149	C₂×C₄×C₇	C₂×#41	1	3	yes	—
150	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
151	C₂×Dic₇	—	2	24	no	—
152	C₂×Dic₇	—	2	4	no	—
153	C₇×C₈	C₂³×C₄	1	4	yes	—
154	C₇×C₈	C₂³×C₄	1	4	yes	—
155	C₇⋊C₈	—	2	8	no	—
156	C₇⋊C₈	C₂²×D₄	1	4	yes	—
157	C₇⋊C₈	C₂²×D₄	1	4	yes	—
158	C₂×#41	—	2	6	no	—
159	C₂×#41	—	2	6	no	—
160	C₂×#41	C₂×Dic₇	1	12	yes	—
161	C₂×#41	C₂²×D₇	1	12	yes	—
162	C₂²×Q₈	C₂×Dic₇	1	2	yes	—
163	C₂²×D₄	—	2	2	no	—
164	C₂²×D₄	C₂×Dic₇	1	2	yes	—
165	C₂³×C₄	—	2	2	no	—
166	C₂³×C₄	—	14	1	no	—
167	C₂³×C₄	C₂²×D₇	1	1	yes	characteristic
168	#135	C₂×Dic₇	1	48	yes	complemented
169	#135	C₂²×D₇	1	16	yes	—
170	C₂×Q₁₆	C₂×Dic₇	1	12	yes	complemented
171	C₂×SD₁₆	C₂×Dic₇	1	24	yes	complemented
172	C₂×D₈	C₂×Dic₇	1	12	yes	complemented
173	#131	—	7	32	no	—
174	#131	—	14	48	no	—
175	C₂×M₁₆	—	7	12	no	—
176	C₂×M₁₆	—	14	24	no	—
177	C₂×M₁₆	—	14	6	no	—
178	C₂²×C₈	—	7	6	no	—
179	C₂²×C₈	—	7	1	no	—
180	C₂²×C₈	—	7	1	no	—
181	C₂²×C₈	—	7	6	no	—
182	C₂²×C₈	—	14	24	no	—
183	C₂²×C₈	—	14	2	no	—
184	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
185	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
186	C₂²×C₈	C₂×Dic₇	1	6	yes	—
187	C₂²×C₈	C₂²×D₇	1	6	yes	—
188	C₂×C₄²	—	7	6	no	—
189	C₂×C₄²	—	14	3	no	—
190	#108	—	7	16	no	—
191	#104	—	7	32	no	—
192	C₄×C₈	—	7	16	no	—
193	C₂²×C₇	—	2	6	no	—
194	C₂²×C₇	—	2	6	no	—
195	C₂²×C₇	—	2	8	no	—
196	C₂²×C₇	—	4	24	no	—
197	C₂²×C₇	#443	1	4	yes	complemented
198	C₂²×C₇	C₂×C₄×D₄	1	3	yes	—
199	C₂²×C₇	C₂×C₄×D₄	1	3	yes	—
200	C₂²×C₇	C₂²×#41	1	1	yes	characteristic
201	C₄×C₇	—	2	8	no	—
202	C₄×C₇	C₂×C₄×D₄	1	3	yes	—
203	C₄×C₇	C₂×C₄×D₄	1	3	yes	—
204	C₄×C₇	C₂⁴×C₄	1	1	yes	characteristic
205	C₄×C₇	C₂³×D₄	1	1	yes	characteristic
206	Dic₇	—	2	8	no	—
207	C₂⁴	—	2	2	no	—
208	C₂⁴	#1159	1	1	yes	characteristic
209	#41	—	2	24	no	—
210	#41	—	2	12	no	—
211	#41	—	2	12	no	—
212	#41	C₂²×Dic₇	1	24	yes	—
213	#41	C₂³×D₇	1	8	yes	—
214	C₂×Q₈	—	2	6	no	—
215	C₂×Q₈	C₂²×Dic₇	1	12	yes	—
216	C₂×D₄	—	2	12	no	—
217	C₂×D₄	—	2	12	no	—
218	C₂×D₄	—	2	6	no	—
219	C₂×D₄	C₂²×Dic₇	1	12	yes	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	12	no	—
222	C₂²×C₄	—	2	12	no	—
223	C₂²×C₄	—	14	8	no	—
224	C₂²×C₄	—	14	6	no	—
225	C₂²×C₄	D₄×D₇	1	3	yes	—
226	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
227	C₂²×C₄	#1159	1	3	yes	—
228	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
229	C₂²×C₄	C₂²×Dic₇	1	3	yes	—
230	C₂²×C₄	C₂³×D₇	1	3	yes	—
231	Q₁₆	C₂²×Dic₇	1	16	yes	complemented
232	SD₁₆	C₂²×Dic₇	1	32	yes	complemented
233	D₈	C₂²×Dic₇	1	16	yes	complemented
234	M₁₆	—	7	8	no	—
235	M₁₆	—	14	32	no	—
236	M₁₆	—	14	12	no	—
237	C₂×C₈	—	7	6	no	—
238	C₂×C₈	—	7	6	no	—
239	C₂×C₈	—	7	12	no	—
240	C₂×C₈	—	7	4	no	—
241	C₂×C₈	—	14	32	no	—
242	C₂×C₈	—	14	12	no	—
243	C₂×C₈	C₂²×Dic₇	1	6	yes	—
244	C₂×C₈	C₂²×Dic₇	1	6	yes	—
245	C₂×C₈	C₂²×Dic₇	1	12	yes	—
246	C₂×C₈	C₂³×D₇	1	4	yes	—
247	C₄²	—	7	4	no	—
248	C₄²	—	14	6	no	—
249	C₂×C₇	—	2	1	no	—
250	C₂×C₇	—	2	3	no	—
251	C₂×C₇	—	4	8	no	—
252	C₂×C₇	C₂×#443	1	6	yes	complemented
253	C₂×C₇	C₂²×C₄×D₄	1	1	yes	characteristic
254	C₂³	—	2	4	no	—
255	C₂³	—	2	12	no	—
256	C₂³	—	4	8	no	—
257	C₂³	D₄×Dic₇	1	3	yes	—
258	C₂³	D₄×Dic₇	1	1	yes	characteristic
259	C₂³	C₂×#1159	1	3	yes	—
260	Q₈	—	2	12	no	—
261	Q₈	C₂³×Dic₇	1	8	yes	—
262	D₄	—	2	24	no	—
263	D₄	—	2	8	no	—
264	D₄	—	2	12	no	—
265	D₄	C₂³×Dic₇	1	8	yes	—
266	C₂×C₄	—	2	12	no	—
267	C₂×C₄	—	2	24	no	—
268	C₂×C₄	—	2	8	no	—
269	C₂×C₄	—	14	24	no	—
270	C₂×C₄	—	14	4	no	—
271	C₂×C₄	D₄×Dic₇	1	3	yes	—
272	C₂×C₄	D₄×Dic₇	1	3	yes	—
273	C₂×C₄	D₄×Dic₇	1	6	yes	—
274	C₂×C₄	C₂×D₄×D₇	1	3	yes	—
275	C₂×C₄	C₂×D₄×D₇	1	3	yes	—
276	C₂×C₄	C₂×#1159	1	3	yes	—
277	C₂×C₄	C₂³×Dic₇	1	3	yes	—
278	C₂×C₄	C₂³×Dic₇	1	3	yes	—
279	C₂×C₄	C₂⁴×D₇	1	1	yes	characteristic
280	C₈	—	7	4	no	—
281	C₈	—	7	4	no	—
282	C₈	—	14	8	no	—
283	C₈	C₂³×Dic₇	1	4	yes	—
284	C₈	C₂³×Dic₇	1	4	yes	—
285	C₇	C₂²×#443	1	1	yes	characteristic, complemented, Sylow
286	C₂²	—	2	6	no	—
287	C₂²	—	2	6	no	—
288	C₂²	—	2	8	no	—
289	C₂²	—	4	24	no	—
290	C₂²	#89968	1	4	yes	direct product factor
291	C₂²	C₂×D₄×Dic₇	1	3	yes	—
292	C₂²	C₂×D₄×Dic₇	1	3	yes	—
293	C₂²	C₂²×#1159	1	1	yes	characteristic
294	C₄	—	2	8	no	—
295	C₄	—	14	8	no	—
296	C₄	C₂×D₄×Dic₇	1	3	yes	—
297	C₄	C₂×D₄×Dic₇	1	3	yes	—
298	C₄	C₂²×D₄×D₇	1	1	yes	characteristic
299	C₄	C₂⁴×Dic₇	1	1	yes	characteristic
300	C₂	—	2	1	no	—
301	C₂	—	2	3	no	—
302	C₂	—	4	8	no	—
303	C₂	C₂×#89968	1	6	yes	direct product factor
304	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×#443
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×#89968	C₂	direct (N × H)
5. C₂²×#6393	C₂	semidirect
6. C₂²×C₈×Dic₇	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. #89968	C₂²	direct (N × H)
23. C₂×#6393	C₂²	semidirect
24. C₂×C₈×Dic₇	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
58. #6411	C₂³	semidirect
59. #6393	C₂³	semidirect
60. C₈×Dic₇	C₂³	semidirect
91. C₇×Q₁₆	C₂²×C₄	semidirect
92. C₇×SD₁₆	C₂²×C₄	semidirect
93. C₇×D₈	C₂²×C₄	semidirect
110. C₂×#135	Dic₇	semidirect
112. C₂²×Q₁₆	Dic₇	semidirect
113. C₂²×SD₁₆	Dic₇	semidirect
114. C₂²×D₈	Dic₇	semidirect
168. #135	C₂×Dic₇	semidirect
170. C₂×Q₁₆	C₂×Dic₇	semidirect
171. C₂×SD₁₆	C₂×Dic₇	semidirect
172. C₂×D₈	C₂×Dic₇	semidirect
197. C₂²×C₇	#443	semidirect
231. Q₁₆	C₂²×Dic₇	semidirect
232. SD₁₆	C₂²×Dic₇	semidirect
233. D₈	C₂²×Dic₇	semidirect
252. C₂×C₇	C₂×#443	semidirect
285. C₇	C₂²×#443	semidirect
290. C₂²	#89968	direct (N × H)
303. C₂	C₂×#89968	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×#89968	2
C₂²×#6393	2
C₂²×C₈×Dic₇	2
C₂²×#443	7

Subgroups — C22×#89968 #14218087

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#89968 #14218087