Subgroups — C₂×#11726271 #13323188

Order 1792.

718 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#89602	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₂₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₇×#334	C₂	1	1	yes	maximal, characteristic
4	C₂×#89279	C₂	1	2	yes	maximal
5	C₂×#89250	C₂	1	1	yes	maximal, characteristic
6	#11726271	C₂	1	8	yes	maximal, direct product factor
7	C₂²×Dic₂₈	C₄	1	1	yes	characteristic
8	C₂×C₄×Dic₁₄	C₂²	1	2	yes	—
9	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
11	C₂×#6392	C₄	1	1	yes	characteristic
12	C₂×#6390	C₄	1	2	yes	—
13	C₂×Dic₇⋊C₈	—	2	2	no	—
14	C₂×C₈×Dic₇	—	2	2	no	—
15	C₂×C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
17	#89602	—	2	4	no	—
18	#89602	C₂²	1	8	yes	—
19	C₄×Dic₂₈	C₄	1	4	yes	—
20	C₄×Dic₂₈	C₂²	1	4	yes	—
21	C₇×#334	C₂²	1	4	yes	—
22	#89279	C₂²	1	8	yes	—
23	#89250	C₂²	1	4	yes	—
24	C₂×#1579	—	7	1	no	maximal, Sylow
25	C₂²×Dic₁₄	C₂×C₄	1	2	yes	—
26	C₂²×C₇×C₈	—	2	2	no	—
27	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
30	C₂×C₄⋊Dic₇	C₂×C₄	1	2	yes	—
31	C₂×Dic₇⋊C₄	—	2	2	no	—
32	C₂×C₄×Dic₇	—	2	2	no	—
33	C₂²×C₇⋊C₈	—	2	1	no	—
34	C₂²×C₇⋊C₈	D₄	1	2	yes	—
35	C₂×Dic₂₈	C₈	1	8	yes	complemented
36	C₂×Dic₂₈	C₂×C₄	1	4	yes	—
37	C₄×Dic₁₄	—	2	2	no	—
38	C₄×Dic₁₄	—	2	2	no	—
39	C₄×Dic₁₄	C₂×C₄	1	4	yes	—
40	C₄×Dic₁₄	C₂³	1	4	yes	—
41	C₇×C₄⋊C₈	—	2	2	no	—
42	C₇×C₄⋊C₈	C₂³	1	4	yes	—
43	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
44	C₄×C₇×C₈	C₂³	1	2	yes	—
45	#6392	C₂×C₄	1	4	yes	—
46	#6390	C₂×C₄	1	8	yes	—
47	Dic₇⋊C₈	—	2	8	no	—
48	C₈×Dic₇	—	2	8	no	—
49	C₄×C₇⋊C₈	—	2	1	no	—
50	C₄×C₇⋊C₈	D₄	1	2	yes	—
51	C₄×C₇⋊C₈	C₂³	1	2	yes	—
52	C₄×C₇⋊C₈	C₂³	1	2	yes	—
53	C₂×C₈×Q₈	—	7	2	no	—
54	C₂×C₄×Q₁₆	—	7	1	no	—
55	#1579	—	7	8	no	—
56	C₂×#334	D₇	1	1	yes	characteristic
57	C₂×Q₈⋊C₈	—	7	2	no	—
58	C₂×#322	—	7	1	no	—
59	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₇	#41	1	1	yes	characteristic
62	C₂²×Dic₇	—	2	2	no	—
63	C₂²×Dic₇	—	2	2	no	—
64	C₂×Dic₁₄	—	2	2	no	—
65	C₂×Dic₁₄	—	2	4	no	—
66	C₂×Dic₁₄	C₂×C₈	1	8	yes	—
67	C₂×Dic₁₄	C₂²×C₄	1	4	yes	—
68	C₂×C₇×C₈	—	2	2	no	—
69	C₂×C₇×C₈	—	2	4	no	—
70	C₂×C₇×C₈	—	4	4	no	—
71	C₂×C₇×C₈	C₂×C₈	1	4	yes	—
72	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
73	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
74	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
75	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₇	#41	1	1	yes	characteristic
77	C₄²×C₇	C₂⁴	1	1	yes	characteristic
78	C₄⋊Dic₇	—	2	2	no	—
79	C₄⋊Dic₇	C₂²×C₄	1	4	yes	—
80	Dic₇⋊C₄	—	2	4	no	—
81	Dic₇⋊C₄	—	2	4	no	—
82	C₄×Dic₇	—	2	4	no	—
83	C₄×Dic₇	—	2	4	no	—
84	C₂×C₇⋊C₈	—	2	4	no	—
85	C₂×C₇⋊C₈	—	2	2	no	—
86	C₂×C₇⋊C₈	—	4	2	no	—
87	C₂×C₇⋊C₈	C₂×D₄	1	4	yes	—
88	Dic₂₈	C₂×C₈	1	16	yes	complemented
89	C₂²×Q₁₆	—	7	1	no	—
90	C₂×C₄×Q₈	—	7	2	no	—
91	C₈×Q₈	—	7	8	no	—
92	C₈×Q₈	—	14	4	no	—
93	C₄×Q₁₆	—	7	4	no	—
94	C₄×Q₁₆	—	7	4	no	—
95	C₂×#107	—	7	1	no	—
96	C₂×C₄⋊C₈	—	14	2	no	—
97	C₂×C₄⋊C₈	C₂×D₇	1	2	yes	—
98	C₂×#103	—	7	2	no	—
99	C₂×C₄×C₈	—	7	1	no	—
100	C₂×C₄×C₈	—	7	1	no	—
101	C₂×C₄×C₈	—	14	2	no	—
102	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
103	#334	C₂×D₇	1	4	yes	—
104	Q₈⋊C₈	—	7	8	no	—
105	#322	—	7	4	no	—
106	C₂³×C₇	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₇	—	2	2	no	—
108	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₇	C₂²×C₈	1	2	yes	—
112	C₂×C₄×C₇	#131	1	2	yes	—
113	C₂×C₄×C₇	#137	1	4	yes	—
114	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
115	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
117	C₂×Dic₇	—	2	4	no	—
118	C₂×Dic₇	—	2	8	no	—
119	C₂×Dic₇	—	2	4	no	—
120	C₂×Dic₇	—	4	4	no	—
121	Dic₁₄	—	2	4	no	—
122	Dic₁₄	—	2	8	no	—
123	Dic₁₄	C₂²×C₈	1	8	yes	—
124	C₇×C₈	—	2	2	no	—
125	C₇×C₈	—	4	4	no	—
126	C₇×C₈	C₂²×C₈	1	4	yes	—
127	C₇⋊C₈	—	2	4	no	—
128	C₇⋊C₈	—	4	2	no	—
129	C₂²×Q₈	—	7	2	no	—
130	C₂×Q₁₆	—	7	8	no	—
131	C₂×Q₁₆	—	7	4	no	—
132	C₂²×C₈	—	2	2	no	—
133	C₂²×C₈	—	7	2	no	—
134	C₂²×C₈	—	14	1	no	—
135	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
136	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
137	C₄×Q₈	—	7	4	no	—
138	C₄×Q₈	—	7	4	no	—
139	C₄×Q₈	—	14	2	no	—
140	C₄×Q₈	—	14	2	no	—
141	C₂×C₄⋊C₄	—	7	2	no	—
142	C₂×C₄⋊C₄	—	14	2	no	—
143	C₂×C₄²	—	14	2	no	—
144	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
145	#107	—	7	4	no	—
146	C₄⋊C₈	—	2	2	no	—
147	C₄⋊C₈	—	14	8	no	—
148	C₄⋊C₈	C₂²×D₇	1	4	yes	—
149	#103	—	7	8	no	—
150	C₄×C₈	—	7	2	no	—
151	C₄×C₈	—	7	2	no	—
152	C₄×C₈	—	7	2	no	—
153	C₄×C₈	—	14	8	no	—
154	C₄×C₈	—	14	1	no	—
155	C₄×C₈	C₄×D₇	1	2	yes	—
156	C₄×C₈	C₂²×D₇	1	2	yes	—
157	C₂²×C₇	C₈×D₄	1	2	yes	—
158	C₂²×C₇	#441	1	2	yes	—
159	C₂²×C₇	#444	1	2	yes	—
160	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
161	C₄×C₇	—	2	1	no	—
162	C₄×C₇	—	2	1	no	—
163	C₄×C₇	C₈×D₄	1	2	yes	—
164	C₄×C₇	#441	1	2	yes	—
165	C₄×C₇	C₂³×C₈	1	1	yes	characteristic
166	C₄×C₇	C₂×#131	1	1	yes	characteristic
167	C₄×C₇	C₂×#137	1	2	yes	—
168	Dic₇	—	2	8	no	—
169	Dic₇	—	4	4	no	—
170	C₂×Q₈	—	7	8	no	—
171	C₂×Q₈	—	7	4	no	—
172	C₂×Q₈	—	14	2	no	—
173	C₂×Q₈	—	14	4	no	—
174	C₂²×C₄	—	14	2	no	—
175	C₂²×C₄	—	14	2	no	—
176	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
177	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
178	C₂²×C₄	#1161	1	1	yes	characteristic
179	Q₁₆	—	7	16	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	2	4	no	—
182	C₂×C₈	—	4	4	no	—
183	C₂×C₈	—	7	4	no	—
184	C₂×C₈	—	14	4	no	—
185	C₂×C₈	—	14	2	no	—
186	C₂×C₈	—	28	2	no	—
187	C₂×C₈	C₈×D₇	1	4	yes	—
188	C₂×C₈	C₂×C₄×D₇	1	2	yes	—
189	C₂×C₈	C₂×C₄×D₇	1	2	yes	—
190	C₄⋊C₄	—	7	4	no	—
191	C₄⋊C₄	—	14	4	no	—
192	C₄⋊C₄	—	14	2	no	—
193	C₄⋊C₄	—	14	4	no	—
194	C₄²	—	14	4	no	—
195	C₄²	—	14	4	no	—
196	C₄²	C₂×C₄×D₇	1	1	yes	characteristic
197	C₄²	D₄×D₇	1	1	yes	characteristic
198	C₄²	#1161	1	1	yes	characteristic
199	C₄²	C₂³×D₇	1	1	yes	characteristic
200	C₂×C₇	#1579	1	4	yes	complemented
201	C₂×C₇	C₂×C₈×D₄	1	1	yes	characteristic
202	C₂×C₇	C₂×#441	1	1	yes	characteristic
203	C₂×C₇	C₂×#444	1	1	yes	characteristic
204	C₂³	#6456	1	1	yes	characteristic
205	Q₈	—	7	8	no	—
206	Q₈	—	14	8	no	—
207	Q₈	—	14	4	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	14	8	no	—
210	C₂×C₄	—	14	4	no	—
211	C₂×C₄	—	14	4	no	—
212	C₂×C₄	—	28	4	no	—
213	C₂×C₄	#6456	1	1	yes	characteristic
214	C₂×C₄	#6456	1	1	yes	characteristic
215	C₂×C₄	#6456	1	1	yes	characteristic
216	C₂×C₄	C₂×C₈×D₇	1	2	yes	—
217	C₂×C₄	#6470	1	2	yes	—
218	C₂×C₄	#6478	1	4	yes	—
219	C₂×C₄	C₂²×C₄×D₇	1	1	yes	characteristic
220	C₂×C₄	C₂×D₄×D₇	1	1	yes	characteristic
221	C₂×C₄	C₂×#1161	1	1	yes	characteristic
222	C₈	—	2	2	no	—
223	C₈	—	4	4	no	—
224	C₈	—	14	4	no	—
225	C₈	—	28	2	no	—
226	C₈	C₂×C₈×D₇	1	4	yes	—
227	C₇	C₂×#1579	1	1	yes	characteristic, complemented, Sylow
228	C₂²	#89606	1	2	yes	—
229	C₂²	#89625	1	2	yes	—
230	C₂²	#89666	1	2	yes	—
231	C₂²	C₂×#6456	1	1	yes	characteristic
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	14	8	no	—
235	C₄	—	28	4	no	—
236	C₄	#89606	1	2	yes	—
237	C₄	#89625	1	2	yes	—
238	C₄	C₂²×C₈×D₇	1	1	yes	characteristic
239	C₄	C₂×#6470	1	1	yes	characteristic
240	C₄	C₂×#6478	1	2	yes	—
241	C₂	#11726271	1	4	yes	direct product factor
242	C₂	C₂×#89606	1	1	yes	characteristic
243	C₂	C₂×#89625	1	1	yes	characteristic
244	C₂	C₂×#89666	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₇
commutator	C₄×C₇	C₂³×C₈
Frattini	C₄²	C₂³×D₇
Fitting	C₂×C₇×#334	C₂
socle, prime radical	C₂³×C₇	C₄×D₄
cosocle	C₄²×C₇	C₂⁴
hypercenter	C₂×#334	D₇
hypocenter	C₇	C₂×#1579
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
6. #11726271	C₂	direct (N × H)
35. C₂×Dic₂₈	C₈	semidirect
88. Dic₂₈	C₂×C₈	semidirect
200. C₂×C₇	#1579	semidirect
227. C₇	C₂×#1579	semidirect
241. C₂	#11726271	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×#89602	2
C₂×C₄×Dic₂₈	2
C₂×C₇×#334	2
C₂×#89279	2
C₂×#89250	2
#11726271	2
C₂×#1579	7

Subgroups — C2×#11726271 #13323188

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11726271 #13323188