Subgroups — C₂×#89326 #11729331

Order 896.

404 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#117	C₂	1	1	yes	maximal, characteristic
2	C₂×#6489	C₂	1	1	yes	maximal, characteristic
3	C₂²×#1136	C₂	1	1	yes	maximal, characteristic
4	#89326	C₂	1	4	yes	maximal, direct product factor
5	C₂³×C₄×C₇	C₂²	1	1	yes	characteristic
6	C₇×#117	—	2	1	no	—
7	C₇×#117	—	2	1	no	—
8	C₇×#117	C₄	1	2	yes	complemented
9	C₇×#117	C₂²	1	2	yes	—
10	C₂×C₇×C₄⋊C₄	—	2	1	no	—
11	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
12	C₂×C₄²×C₇	C₄	1	2	yes	complemented
13	C₂×C₂²⋊Dic₇	—	2	1	no	—
14	#6489	—	2	1	no	—
15	#6489	—	2	1	no	—
16	#6489	C₄	1	2	yes	complemented
17	#6489	C₂²	1	2	yes	—
18	C₂×C₄⋊Dic₇	—	2	1	no	—
19	C₂×C₄×Dic₇	C₄	1	2	yes	complemented
20	C₂×#1136	—	2	1	no	—
21	C₂×#1136	—	2	1	no	—
22	C₂×#1136	—	2	1	no	—
23	C₂×#1136	—	2	1	no	—
24	C₂×#1136	C₄	1	2	yes	complemented
25	C₂×#1136	C₂²	1	2	yes	—
26	C₂²×C₇⋊C₈	C₄	1	2	yes	complemented
27	C₂×H(3,ℤ/4)	—	7	1	no	maximal, Sylow
28	C₂⁴×C₇	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₇	—	2	2	no	—
30	C₂²×C₄×C₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₇	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₇	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₇	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₇	D₄	1	1	yes	characteristic
37	C₂²×C₄×C₇	D₄	1	1	yes	characteristic
38	C₂²×C₄×C₇	D₄	1	1	yes	characteristic
39	C₂²×C₄×C₇	D₄	1	1	yes	characteristic
40	C₂²×C₄×C₇	D₄	1	1	yes	characteristic
41	C₂²×C₄×C₇	Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₇	Q₈	1	1	yes	characteristic
43	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
44	C₂²×Dic₇	—	2	2	no	—
45	C₇×C₄⋊C₄	—	2	2	no	—
46	C₇×C₄⋊C₄	—	2	2	no	—
47	C₇×C₂²⋊C₄	—	2	2	no	—
48	C₇×C₂²⋊C₄	—	2	2	no	—
49	C₄²×C₇	—	2	2	no	—
50	C₄²×C₇	C₂×C₄	1	4	yes	complemented
51	C₂²⋊Dic₇	—	2	2	no	—
52	C₂²⋊Dic₇	—	2	2	no	—
53	C₄⋊Dic₇	—	2	2	no	—
54	C₄⋊Dic₇	—	2	2	no	—
55	C₄×Dic₇	—	2	2	no	—
56	C₄×Dic₇	C₂×C₄	1	4	yes	complemented
57	#1136	—	2	2	no	—
58	#1136	—	2	2	no	—
59	#1136	—	2	2	no	—
60	#1136	—	2	2	no	—
61	C₂×C₇⋊C₈	—	2	2	no	—
62	C₂×C₇⋊C₈	—	2	2	no	—
63	C₂×C₇⋊C₈	C₂×C₄	1	4	yes	complemented
64	C₂²×M₁₆	—	7	1	no	—
65	C₂×#117	—	7	1	no	—
66	C₂×#117	D₇	1	1	yes	characteristic
67	H(3,ℤ/4)	—	7	4	no	—
68	C₂³×C₇	—	4	2	no	—
69	C₂³×C₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₂³×C₇	C₂²⋊C₄	1	1	yes	characteristic
71	C₂³×C₇	C₂²⋊C₄	1	1	yes	characteristic
72	C₂³×C₇	C₂²⋊C₄	1	1	yes	characteristic
73	C₂³×C₇	C₄⋊C₄	1	1	yes	characteristic
74	C₂³×C₇	C₄⋊C₄	1	1	yes	characteristic
75	C₂³×C₇	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×C₇	—	2	4	no	—
77	C₂×C₄×C₇	—	4	4	no	—
78	C₂×C₄×C₇	C₄²	1	1	yes	characteristic
79	C₂×C₄×C₇	C₄²	1	1	yes	characteristic
80	C₂×C₄×C₇	C₄²	1	1	yes	characteristic
81	C₂×C₄×C₇	C₄²	1	1	yes	characteristic
82	C₂×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
106	C₂×Dic₇	—	2	4	no	—
107	C₂×Dic₇	—	4	4	no	—
108	C₇⋊C₈	—	2	4	no	—
109	C₂³×C₄	C₂×D₇	1	1	yes	characteristic
110	C₂×M₁₆	—	7	2	no	—
111	C₂×M₁₆	—	7	2	no	—
112	C₂×M₁₆	—	14	1	no	—
113	C₂×M₁₆	—	14	1	no	—
114	C₂×M₁₆	—	14	1	no	—
115	C₂×M₁₆	—	14	1	no	—
116	C₂²×C₈	—	7	2	no	—
117	#117	—	2	1	no	—
118	#117	—	2	1	no	—
119	#117	—	7	2	no	—
120	#117	—	7	2	no	—
121	#117	—	14	1	no	—
122	#117	—	14	1	no	—
123	#117	Dic₇	1	2	yes	complemented
124	#117	C₂×D₇	1	2	yes	—
125	C₂×C₄⋊C₄	—	2	1	no	—
126	C₂×C₄⋊C₄	—	14	1	no	—
127	C₂×C₂²⋊C₄	—	2	1	no	—
128	C₂×C₂²⋊C₄	—	14	1	no	—
129	C₂×C₄²	—	7	2	no	—
130	C₂×C₄²	Dic₇	1	2	yes	complemented
131	C₂²×C₇	—	2	2	no	—
132	C₂²×C₇	—	2	2	no	—
133	C₂²×C₇	—	2	2	no	—
134	C₂²×C₇	—	4	1	no	—
135	C₂²×C₇	—	4	1	no	—
136	C₂²×C₇	—	4	1	no	—
137	C₂²×C₇	—	4	1	no	—
138	C₂²×C₇	#95	1	1	yes	characteristic
139	C₂²×C₇	#95	1	1	yes	characteristic
140	C₂²×C₇	#95	1	1	yes	characteristic
141	C₂²×C₇	#95	1	1	yes	characteristic
142	C₂²×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
143	C₂²×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
144	C₂²×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₄×C₇	—	4	4	no	—
146	C₄×C₇	#95	1	1	yes	characteristic
147	C₄×C₇	#95	1	1	yes	characteristic
148	C₄×C₇	#95	1	1	yes	characteristic
149	C₄×C₇	#95	1	1	yes	characteristic
150	C₄×C₇	C₂×C₄²	1	1	yes	characteristic
151	C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
152	C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
153	C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
154	Dic₇	—	4	4	no	—
155	C₂⁴	C₂²⋊D₇	1	1	yes	characteristic
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	—	14	2	no	—
158	C₂²×C₄	Dic₁₄	1	1	yes	characteristic
159	C₂²×C₄	Dic₁₄	1	1	yes	characteristic
160	C₂²×C₄	C₄×D₇	1	1	yes	characteristic
161	C₂²×C₄	C₄×D₇	1	1	yes	characteristic
162	C₂²×C₄	C₄×D₇	1	1	yes	characteristic
163	C₂²×C₄	C₄×D₇	1	1	yes	characteristic
164	C₂²×C₄	D₂₈	1	1	yes	characteristic
165	C₂²×C₄	D₂₈	1	1	yes	characteristic
166	C₂²×C₄	C₂×Dic₇	1	1	yes	characteristic
167	C₂²×C₄	C₂×Dic₇	1	1	yes	characteristic
168	C₂²×C₄	C₂²⋊D₇	1	1	yes	characteristic
169	C₂²×C₄	C₂²⋊D₇	1	1	yes	characteristic
170	C₂²×C₄	C₂²⋊D₇	1	1	yes	characteristic
171	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
172	M₁₆	—	14	2	no	—
173	M₁₆	—	14	2	no	—
174	M₁₆	—	14	2	no	—
175	M₁₆	—	14	2	no	—
176	C₂×C₈	—	7	4	no	—
177	C₂×C₈	—	14	2	no	—
178	C₂×C₈	—	14	2	no	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	—	14	2	no	—
182	C₄⋊C₄	—	14	2	no	—
183	C₂²⋊C₄	—	2	2	no	—
184	C₂²⋊C₄	—	2	2	no	—
185	C₂²⋊C₄	—	14	2	no	—
186	C₂²⋊C₄	—	14	2	no	—
187	C₄²	—	2	2	no	—
188	C₄²	—	7	4	no	—
189	C₄²	—	14	2	no	—
190	C₄²	C₂×Dic₇	1	4	yes	complemented
191	C₂×C₇	—	2	1	no	—
192	C₂×C₇	—	2	1	no	—
193	C₂×C₇	—	2	1	no	—
194	C₂×C₇	—	2	1	no	—
195	C₂×C₇	—	2	1	no	—
196	C₂×C₇	—	2	1	no	—
197	C₂×C₇	H(3,ℤ/4)	1	2	yes	complemented
198	C₂×C₇	C₂×#95	1	1	yes	characteristic
199	C₂³	—	4	2	no	—
200	C₂³	Dic₇⋊C₄	1	1	yes	characteristic
201	C₂³	Dic₇⋊C₄	1	1	yes	characteristic
202	C₂³	(C₂×D₇)⋊C₄	1	1	yes	characteristic
203	C₂³	(C₂×D₇)⋊C₄	1	1	yes	characteristic
204	C₂³	C₂²⋊Dic₇	1	1	yes	characteristic
205	C₂³	C₂²⋊Dic₇	1	1	yes	characteristic
206	C₂³	C₂×C₂²⋊D₇	1	1	yes	characteristic
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	4	4	no	—
209	C₂×C₄	—	14	4	no	—
210	C₂×C₄	—	28	4	no	—
211	C₂×C₄	C₄×Dic₇	1	1	yes	characteristic
212	C₂×C₄	C₄×Dic₇	1	1	yes	characteristic
213	C₂×C₄	C₄×Dic₇	1	1	yes	characteristic
214	C₂×C₄	C₄×Dic₇	1	1	yes	characteristic
215	C₂×C₄	Dic₇⋊C₄	1	1	yes	characteristic
216	C₂×C₄	Dic₇⋊C₄	1	1	yes	characteristic
217	C₂×C₄	Dic₇⋊C₄	1	1	yes	characteristic
218	C₂×C₄	Dic₇⋊C₄	1	1	yes	characteristic
219	C₂×C₄	Dic₇⋊C₄	1	1	yes	characteristic
220	C₂×C₄	Dic₇⋊C₄	1	1	yes	characteristic
221	C₂×C₄	C₄⋊Dic₇	1	1	yes	characteristic
222	C₂×C₄	C₄⋊Dic₇	1	1	yes	characteristic
223	C₂×C₄	C₄⋊Dic₇	1	1	yes	characteristic
224	C₂×C₄	C₄⋊Dic₇	1	1	yes	characteristic
225	C₂×C₄	(C₂×D₇)⋊C₄	1	1	yes	characteristic
226	C₂×C₄	(C₂×D₇)⋊C₄	1	1	yes	characteristic
227	C₂×C₄	(C₂×D₇)⋊C₄	1	1	yes	characteristic
228	C₂×C₄	(C₂×D₇)⋊C₄	1	1	yes	characteristic
229	C₂×C₄	(C₂×D₇)⋊C₄	1	1	yes	characteristic
230	C₂×C₄	(C₂×D₇)⋊C₄	1	1	yes	characteristic
231	C₂×C₄	C₂²⋊Dic₇	1	1	yes	characteristic
232	C₂×C₄	C₂²⋊Dic₇	1	1	yes	characteristic
233	C₂×C₄	C₂×Dic₁₄	1	1	yes	characteristic
234	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
235	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
236	C₂×C₄	C₂×D₂₈	1	1	yes	characteristic
237	C₂×C₄	C₂²×Dic₇	1	1	yes	characteristic
238	C₂×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
239	C₈	—	14	4	no	—
240	C₇	C₂×H(3,ℤ/4)	1	1	yes	characteristic, complemented, Sylow
241	C₂²	—	2	2	no	—
242	C₂²	—	2	2	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	4	1	no	—
245	C₂²	—	4	1	no	—
246	C₂²	—	4	1	no	—
247	C₂²	—	4	1	no	—
248	C₂²	C₇⋊#95	1	1	yes	characteristic
249	C₂²	C₇⋊#95	1	1	yes	characteristic
250	C₂²	C₇⋊#95	1	1	yes	characteristic
251	C₂²	C₇⋊#95	1	1	yes	characteristic
252	C₂²	C₂×Dic₇⋊C₄	1	1	yes	characteristic
253	C₂²	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
254	C₂²	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
255	C₄	—	4	4	no	—
256	C₄	—	28	4	no	—
257	C₄	C₇⋊#95	1	1	yes	characteristic
258	C₄	C₇⋊#95	1	1	yes	characteristic
259	C₄	C₇⋊#95	1	1	yes	characteristic
260	C₄	C₇⋊#95	1	1	yes	characteristic
261	C₄	C₂×C₄×Dic₇	1	1	yes	characteristic
262	C₄	C₂×Dic₇⋊C₄	1	1	yes	characteristic
263	C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
264	C₄	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
265	C₂	—	2	1	no	—
266	C₂	—	2	1	no	—
267	C₂	—	2	1	no	—
268	C₂	—	2	1	no	—
269	C₂	—	2	1	no	—
270	C₂	—	2	1	no	—
271	C₂	#89326	1	2	yes	direct product factor
272	C₂	C₂×C₇⋊#95	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₄×Dic₇
commutator	C₄×C₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₇
Fitting	C₂×C₇×#117	C₂
socle	C₂²×C₇	#95
prime radical	C₂⁴×C₇	D₄
cosocle	C₂²×C₄×C₇	C₂³
hypercenter	C₂×#117	D₇
hypocenter	C₇	C₂×H(3,ℤ/4)
LCS	G ▷ C₄×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
4. #89326	C₂	direct (N × H)
8. C₇×#117	C₄	semidirect
12. C₂×C₄²×C₇	C₄	semidirect
16. #6489	C₄	semidirect
19. C₂×C₄×Dic₇	C₄	semidirect
24. C₂×#1136	C₄	semidirect
26. C₂²×C₇⋊C₈	C₄	semidirect
50. C₄²×C₇	C₂×C₄	semidirect
56. C₄×Dic₇	C₂×C₄	semidirect
63. C₂×C₇⋊C₈	C₂×C₄	semidirect
123. #117	Dic₇	semidirect
130. C₂×C₄²	Dic₇	semidirect
190. C₄²	C₂×Dic₇	semidirect
197. C₂×C₇	H(3,ℤ/4)	semidirect
240. C₇	C₂×H(3,ℤ/4)	semidirect
271. C₂	#89326	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₇×#117	2
C₂×#6489	2
C₂²×#1136	2
#89326	2
C₂×H(3,ℤ/4)	7

Subgroups — C2×#89326 #11729331

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#89326 #11729331