Subgroups — C₂×#11729144 #13508422

Order 1792.

574 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#500	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₂₈	C₂	1	1	yes	maximal, characteristic
3	#11729144	C₂	1	8	yes	maximal, direct product factor
4	C₂×#89287	C₂	1	2	yes	maximal
5	C₂×#89284	C₂	1	2	yes	maximal
6	C₂×C₄⋊(C₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
7	C₂×C₇×C₄⋊Q₈	—	2	1	no	—
8	C₂²×Dic₂₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×Dic₁₄	—	2	1	no	—
10	C₇×#500	—	2	2	no	—
11	C₇×#500	C₂²	1	4	yes	—
12	C₂×C₇×#107	—	2	1	no	—
13	C₂×C₇×#107	C₂²	1	2	yes	—
14	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
15	C₂×#6392	C₂²	1	1	yes	characteristic
16	C₂×#6390	—	2	1	no	—
17	C₂²×C₇⋊C₁₆	C₂²	1	2	yes	—
18	C₄×Dic₂₈	—	2	2	no	—
19	C₄×Dic₂₈	C₂²	1	4	yes	—
20	#89287	C₂²	1	8	yes	—
21	#89284	C₂²	1	8	yes	—
22	C₄⋊(C₇⋊C₁₆)	C₂²	1	4	yes	—
23	C₂×#2227	—	7	1	no	maximal, Sylow
24	C₂²×C₇×Q₈	—	4	1	no	—
25	C₂²×Dic₁₄	—	2	1	no	—
26	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
27	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
28	C₇×C₄⋊Q₈	—	2	4	no	—
29	C₇×C₄⋊Q₈	—	4	2	no	—
30	C₂×C₇×C₄⋊C₄	—	2	2	no	—
31	C₂×C₇×C₄⋊C₄	—	4	1	no	—
32	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
33	C₂×C₄⋊Dic₇	—	2	1	no	—
34	C₂×Dic₇⋊C₄	—	4	1	no	—
35	C₂×C₄×Dic₇	—	4	1	no	—
36	C₂×Dic₂₈	—	2	1	no	—
37	C₂×Dic₂₈	—	2	2	no	—
38	C₂×Dic₂₈	Q₈	1	4	yes	complemented
39	C₂×Dic₂₈	C₂³	1	2	yes	—
40	C₄×Dic₁₄	—	2	4	no	—
41	C₄×Dic₁₄	—	4	2	no	—
42	C₇×#107	—	2	2	no	—
43	C₇×#107	—	2	4	no	—
44	C₇×#107	C₂³	1	4	yes	—
45	C₄×C₇×C₈	D₄	1	2	yes	—
46	C₄×C₇×C₈	C₂³	1	2	yes	—
47	#6392	—	2	1	no	—
48	#6392	C₂³	1	2	yes	—
49	#6390	—	2	4	no	—
50	C₂×C₇⋊C₁₆	—	2	4	no	—
51	C₂×C₇⋊C₁₆	C₂³	1	4	yes	—
52	C₂×#500	D₇	1	1	yes	characteristic
53	C₂×C₄×Q₁₆	—	7	1	no	—
54	#2227	—	7	8	no	—
55	C₂×#366	—	7	2	no	—
56	C₂×C₄⋊C₁₆	—	7	1	no	—
57	C₂×#358	—	7	2	no	—
58	C₂×C₇×Q₈	—	4	4	no	—
59	C₂×C₇×Q₈	—	8	4	no	—
60	C₂²×C₄×C₇	—	4	2	no	—
61	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
62	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
63	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×Dic₇	—	4	1	no	—
65	C₂²×Dic₇	—	4	1	no	—
66	C₂×Dic₁₄	—	2	4	no	—
67	C₂×Dic₁₄	—	2	2	no	—
68	C₂×Dic₁₄	—	4	1	no	—
69	C₂×Dic₁₄	—	4	2	no	—
70	C₂×C₇×C₈	—	2	2	no	—
71	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
72	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
73	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
74	C₂×C₇×C₈	#41	1	2	yes	—
75	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
76	C₇×C₄⋊C₄	—	2	4	no	—
77	C₇×C₄⋊C₄	—	4	2	no	—
78	C₇×C₄⋊C₄	—	4	4	no	—
79	C₄²×C₇	D₈	1	2	yes	—
80	C₄²×C₇	C₂×D₄	1	2	yes	—
81	C₄⋊Dic₇	—	2	2	no	—
82	C₄⋊Dic₇	—	4	1	no	—
83	Dic₇⋊C₄	—	4	4	no	—
84	C₄×Dic₇	—	4	4	no	—
85	Dic₂₈	—	2	4	no	—
86	Dic₂₈	—	2	2	no	—
87	Dic₂₈	C₂×Q₈	1	4	yes	complemented
88	C₇⋊C₁₆	—	2	4	no	—
89	C₂²×Q₁₆	—	7	1	no	—
90	C₂×C₄⋊Q₈	—	2	1	no	—
91	C₂×C₄×Q₈	—	14	1	no	—
92	C₂²×C₁₆	—	7	2	no	—
93	#500	—	2	2	no	—
94	#500	C₂×D₇	1	4	yes	—
95	C₄×Q₁₆	—	7	4	no	—
96	C₄×Q₁₆	—	14	2	no	—
97	C₂×#107	—	2	1	no	—
98	C₂×#107	—	7	1	no	—
99	C₂×#107	C₂×D₇	1	2	yes	—
100	C₂×#103	—	14	1	no	—
101	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
102	#366	—	7	8	no	—
103	C₄⋊C₁₆	—	7	4	no	—
104	#358	—	7	8	no	—
105	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
106	C₇×Q₈	—	8	8	no	—
107	C₂×C₄×C₇	—	2	2	no	—
108	C₂×C₄×C₇	—	4	4	no	—
109	C₂×C₄×C₇	—	8	4	no	—
110	C₂×C₄×C₇	Q₃₂	1	4	yes	—
111	C₂×C₄×C₇	C₂²⋊Q₈	1	2	yes	—
112	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
113	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
114	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
115	C₂×C₄×C₇	#137	1	2	yes	—
116	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
117	C₂×Dic₇	—	4	2	no	—
118	C₂×Dic₇	—	4	4	no	—
119	C₂×Dic₇	—	4	2	no	—
120	C₂×Dic₇	—	8	2	no	—
121	Dic₁₄	—	2	4	no	—
122	Dic₁₄	—	4	2	no	—
123	Dic₁₄	—	4	4	no	—
124	C₇×C₈	—	2	2	no	—
125	C₇×C₈	C₂²⋊Q₈	1	2	yes	—
126	C₇×C₈	C₂²×Q₈	1	1	yes	characteristic
127	C₇×C₈	C₂×#41	1	1	yes	characteristic
128	C₂²×Q₈	—	4	1	no	—
129	C₂²×Q₈	—	14	1	no	—
130	C₂×Q₁₆	—	7	4	no	—
131	C₂×Q₁₆	—	7	2	no	—
132	C₂×Q₁₆	—	14	2	no	—
133	C₂×Q₁₆	—	14	1	no	—
134	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
135	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
136	C₄⋊Q₈	—	2	4	no	—
137	C₄⋊Q₈	—	4	2	no	—
138	C₄×Q₈	—	14	4	no	—
139	C₄×Q₈	—	28	2	no	—
140	C₂×C₄⋊C₄	—	2	2	no	—
141	C₂×C₄⋊C₄	—	4	1	no	—
142	C₂×C₄⋊C₄	—	14	1	no	—
143	C₂×C₄⋊C₄	—	28	1	no	—
144	C₂×C₄²	—	28	1	no	—
145	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
146	C₂×C₁₆	—	7	4	no	—
147	C₂×C₁₆	—	14	4	no	—
148	#107	—	2	4	no	—
149	#107	—	2	2	no	—
150	#107	—	7	2	no	—
151	#107	—	14	1	no	—
152	#107	C₂²×D₇	1	4	yes	—
153	#103	—	14	4	no	—
154	C₄×C₈	C₂²⋊D₇	1	2	yes	—
155	C₄×C₈	C₂²×D₇	1	2	yes	—
156	C₂²×C₇	#474	1	2	yes	—
157	C₂²×C₇	C₂×Q₃₂	1	2	yes	—
158	C₂²×C₇	#509	1	2	yes	—
159	C₂²×C₇	C₂²×D₈	1	1	yes	characteristic
160	C₄×C₇	—	2	2	no	—
161	C₄×C₇	—	8	4	no	—
162	C₄×C₇	#474	1	2	yes	—
163	C₄×C₇	C₂×Q₃₂	1	4	yes	—
164	C₄×C₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
165	C₄×C₇	C₂×#137	1	1	yes	characteristic
166	Dic₇	—	4	4	no	—
167	Dic₇	—	8	2	no	—
168	C₂×Q₈	—	4	4	no	—
169	C₂×Q₈	—	8	4	no	—
170	C₂×Q₈	—	14	4	no	—
171	C₂×Q₈	—	14	2	no	—
172	C₂×Q₈	—	28	1	no	—
173	C₂×Q₈	—	28	2	no	—
174	C₂²×C₄	—	4	2	no	—
175	C₂²×C₄	—	28	1	no	—
176	C₂²×C₄	—	28	1	no	—
177	C₂²×C₄	#1141	1	1	yes	characteristic
178	C₂²×C₄	#1141	1	1	yes	characteristic
179	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
180	Q₁₆	—	7	4	no	—
181	Q₁₆	—	14	4	no	—
182	Q₁₆	—	14	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	Q₈×D₇	1	2	yes	—
185	C₂×C₈	#1161	1	2	yes	—
186	C₂×C₈	C₂×C₂²⋊D₇	1	2	yes	—
187	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
188	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
189	C₄⋊C₄	—	2	4	no	—
190	C₄⋊C₄	—	4	2	no	—
191	C₄⋊C₄	—	4	4	no	—
192	C₄⋊C₄	—	14	2	no	—
193	C₄⋊C₄	—	28	4	no	—
194	C₄⋊C₄	—	28	1	no	—
195	C₄²	—	28	4	no	—
196	C₄²	#1141	1	2	yes	—
197	C₄²	C₂×C₂²⋊D₇	1	2	yes	—
198	C₁₆	—	14	4	no	—
199	C₂×C₇	#2227	1	4	yes	complemented
200	C₂×C₇	C₂×#474	1	1	yes	characteristic
201	C₂×C₇	C₂²×Q₃₂	1	1	yes	characteristic
202	C₂×C₇	C₂×#509	1	1	yes	characteristic
203	C₂³	C₂×#1141	1	1	yes	characteristic
204	Q₈	—	8	8	no	—
205	Q₈	—	14	4	no	—
206	Q₈	—	28	4	no	—
207	Q₈	—	28	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	—	8	4	no	—
211	C₂×C₄	—	28	4	no	—
212	C₂×C₄	—	28	2	no	—
213	C₂×C₄	—	28	2	no	—
214	C₂×C₄	—	56	2	no	—
215	C₂×C₄	#6403	1	4	yes	—
216	C₂×C₄	C₂×#1141	1	2	yes	—
217	C₂×C₄	C₂×#1141	1	1	yes	characteristic
218	C₂×C₄	C₂×#1141	1	2	yes	—
219	C₂×C₄	#6505	1	2	yes	—
220	C₂×C₄	#6509	1	2	yes	—
221	C₂×C₄	C₂²×C₂²⋊D₇	1	1	yes	characteristic
222	C₈	—	2	2	no	—
223	C₈	#6509	1	2	yes	—
224	C₈	C₂×Q₈×D₇	1	1	yes	characteristic
225	C₈	C₂×#1161	1	1	yes	characteristic
226	C₇	C₂×#2227	1	1	yes	characteristic, complemented, Sylow
227	C₂²	#89859	1	2	yes	—
228	C₂²	#89919	1	2	yes	—
229	C₂²	C₂×#6403	1	2	yes	—
230	C₂²	C₂²×#1141	1	1	yes	characteristic
231	C₄	—	2	2	no	—
232	C₄	—	8	4	no	—
233	C₄	—	28	4	no	—
234	C₄	—	56	2	no	—
235	C₄	#89859	1	2	yes	—
236	C₄	C₂×#6403	1	4	yes	—
237	C₄	C₂×#6505	1	1	yes	characteristic
238	C₄	C₂×#6509	1	1	yes	characteristic
239	C₂	#11729144	1	4	yes	direct product factor
240	C₂	C₂×#89859	1	1	yes	characteristic
241	C₂	C₂×#89919	1	1	yes	characteristic
242	C₂	C₂²×#6403	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#1141
commutator, cosocle	C₂×C₇×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₇
Fitting	C₂×C₇×#500	C₂
socle, prime radical	C₂³×C₇	C₂×D₈
hypercenter	C₂×#500	D₇
hypocenter	C₇	C₂×#2227
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
3. #11729144	C₂	direct (N × H)
38. C₂×Dic₂₈	Q₈	semidirect
87. Dic₂₈	C₂×Q₈	semidirect
199. C₂×C₇	#2227	semidirect
226. C₇	C₂×#2227	semidirect
239. C₂	#11729144	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×#500	2
C₂×C₄×Dic₂₈	2
#11729144	2
C₂×#89287	2
C₂×#89284	2
C₂×C₄⋊(C₇⋊C₁₆)	2
C₂×#2227	7

Subgroups — C2×#11729144 #13508422

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11729144 #13508422