Subgroups — C₂×#11729109 #13509294

Order 1792.

518 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#499	C₂	1	1	yes	maximal, characteristic
2	C₂×#89454	C₂	1	1	yes	maximal, characteristic
3	#11729109	C₂	1	8	yes	maximal, direct product factor
4	C₂×#89285	C₂	1	2	yes	maximal
5	C₂×#89284	C₂	1	2	yes	maximal
6	C₂×#89256	C₂	1	1	yes	maximal, characteristic
7	C₂×C₇×#125	—	2	1	no	—
8	C₂×#6432	—	2	1	no	—
9	C₇×#499	—	2	2	no	—
10	C₇×#499	C₂²	1	4	yes	—
11	C₂×C₇×#107	C₂²	1	1	yes	characteristic
12	C₂×C₇×#107	C₂²	1	1	yes	characteristic
13	C₂×C₇×#106	—	2	1	no	—
14	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
15	C₂×#6392	—	2	1	no	—
16	C₂×#6392	C₂²	1	2	yes	—
17	C₂²×C₇⋊C₁₆	C₂²	1	2	yes	—
18	#89454	—	2	2	no	—
19	#89454	C₂²	1	4	yes	—
20	#89285	C₂²	1	8	yes	—
21	#89284	C₂²	1	8	yes	—
22	#89256	C₂²	1	4	yes	—
23	C₂×#2255	—	7	1	no	maximal, Sylow
24	C₂²×Dic₁₄	—	4	1	no	—
25	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
26	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
27	C₇×#125	—	2	4	no	—
28	C₇×#125	—	4	2	no	—
29	C₂×C₇×C₄⋊C₄	—	2	1	no	—
30	C₂×C₇×C₄⋊C₄	—	2	1	no	—
31	C₂×C₇×C₄⋊C₄	—	4	1	no	—
32	C₂×C₇×C₄⋊C₄	—	4	1	no	—
33	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
34	C₂×C₄⋊Dic₇	—	2	2	no	—
35	C₂×C₄⋊Dic₇	—	4	1	no	—
36	#6432	—	2	4	no	—
37	#6432	—	4	2	no	—
38	C₇×#107	—	2	1	no	—
39	C₇×#107	—	2	1	no	—
40	C₇×#107	C₂³	1	2	yes	—
41	C₇×#107	C₂³	1	2	yes	—
42	C₇×#106	—	2	4	no	—
43	C₄×C₇×C₈	D₄	1	2	yes	—
44	C₄×C₇×C₈	C₂³	1	2	yes	—
45	#6392	—	2	4	no	—
46	#6392	—	2	2	no	—
47	#6392	C₂³	1	4	yes	—
48	C₂×C₇⋊C₁₆	Q₈	1	8	yes	complemented
49	C₂×C₇⋊C₁₆	C₂³	1	4	yes	—
50	C₂×#500	—	7	1	no	—
51	C₂×#499	D₇	1	1	yes	characteristic
52	#2255	—	7	8	no	—
53	C₂×#367	—	7	2	no	—
54	C₂×#366	—	7	2	no	—
55	C₂×#346	—	7	1	no	—
56	C₂²×C₄×C₇	—	4	1	no	—
57	C₂²×C₄×C₇	—	4	1	no	—
58	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
61	C₂²×Dic₇	—	4	2	no	—
62	C₂×Dic₁₄	—	4	4	no	—
63	C₂×Dic₁₄	—	8	4	no	—
64	C₂×C₇×C₈	Q₁₆	1	4	yes	—
65	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
66	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
67	C₂×C₇×C₈	C₂×Q₈	1	4	yes	—
68	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
69	C₇×C₄⋊C₄	—	2	2	no	—
70	C₇×C₄⋊C₄	—	2	2	no	—
71	C₇×C₄⋊C₄	—	4	1	no	—
72	C₇×C₄⋊C₄	—	4	1	no	—
73	C₇×C₄⋊C₄	—	4	4	no	—
74	C₇×C₄⋊C₄	—	4	4	no	—
75	C₄²×C₇	D₈	1	2	yes	—
76	C₄²×C₇	C₂×D₄	1	2	yes	—
77	C₄⋊Dic₇	—	2	4	no	—
78	C₄⋊Dic₇	—	4	4	no	—
79	C₄⋊Dic₇	—	4	2	no	—
80	C₇⋊C₁₆	C₂×Q₈	1	8	yes	complemented
81	C₂×C₄⋊Q₈	—	14	1	no	—
82	C₂×#125	—	2	1	no	—
83	C₂²×C₁₆	—	7	2	no	—
84	#500	—	7	4	no	—
85	#500	—	14	2	no	—
86	#499	—	2	2	no	—
87	#499	C₂×D₇	1	4	yes	—
88	C₂×#107	—	7	2	no	—
89	C₂×#107	—	14	1	no	—
90	C₂×#107	C₂×D₇	1	1	yes	characteristic
91	C₂×#107	C₂×D₇	1	1	yes	characteristic
92	C₂×#106	—	2	1	no	—
93	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
94	#367	—	7	8	no	—
95	#366	—	7	8	no	—
96	#346	—	7	4	no	—
97	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
98	C₂×C₄×C₇	—	2	2	no	—
99	C₂×C₄×C₇	—	2	2	no	—
100	C₂×C₄×C₇	—	4	2	no	—
101	C₂×C₄×C₇	—	4	2	no	—
102	C₂×C₄×C₇	—	8	2	no	—
103	C₂×C₄×C₇	—	8	2	no	—
104	C₂×C₄×C₇	C₄⋊Q₈	1	2	yes	—
105	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
106	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
107	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
108	C₂×C₄×C₇	C₂×Q₁₆	1	2	yes	—
109	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
110	C₂×Dic₇	—	4	4	no	—
111	C₂×Dic₇	—	8	4	no	—
112	Dic₁₄	—	8	8	no	—
113	C₇×C₈	C₄⋊Q₈	1	2	yes	—
114	C₇×C₈	C₂×Q₁₆	1	4	yes	—
115	C₇×C₈	C₂²×Q₈	1	2	yes	—
116	C₂²×Q₈	—	28	1	no	—
117	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
118	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
119	C₄⋊Q₈	—	14	4	no	—
120	C₄⋊Q₈	—	28	2	no	—
121	#125	—	2	4	no	—
122	#125	—	4	2	no	—
123	C₂×C₄⋊C₄	—	2	1	no	—
124	C₂×C₄⋊C₄	—	2	1	no	—
125	C₂×C₄⋊C₄	—	4	1	no	—
126	C₂×C₄⋊C₄	—	4	1	no	—
127	C₂×C₄⋊C₄	—	14	2	no	—
128	C₂×C₄⋊C₄	—	28	1	no	—
129	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
130	C₂×C₁₆	—	7	8	no	—
131	C₂×C₁₆	—	7	4	no	—
132	#107	—	2	1	no	—
133	#107	—	2	1	no	—
134	#107	—	7	4	no	—
135	#107	—	14	2	no	—
136	#107	—	14	4	no	—
137	#107	C₂²×D₇	1	2	yes	—
138	#107	C₂²×D₇	1	2	yes	—
139	#106	—	2	4	no	—
140	C₄×C₈	C₂²⋊D₇	1	2	yes	—
141	C₄×C₈	C₂²×D₇	1	2	yes	—
142	C₂²×C₇	#500	1	2	yes	—
143	C₂²×C₇	#509	1	2	yes	—
144	C₂²×C₇	#510	1	2	yes	—
145	C₂²×C₇	C₂²×D₈	1	1	yes	characteristic
146	C₄×C₇	—	2	2	no	—
147	C₄×C₇	—	2	2	no	—
148	C₄×C₇	—	8	2	no	—
149	C₄×C₇	—	8	2	no	—
150	C₄×C₇	#500	1	2	yes	—
151	C₄×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
152	C₄×C₇	C₂²×Q₁₆	1	1	yes	characteristic
153	Dic₇	—	8	4	no	—
154	C₂×Q₈	—	28	4	no	—
155	C₂×Q₈	—	56	4	no	—
156	C₂²×C₄	—	4	1	no	—
157	C₂²×C₄	—	4	1	no	—
158	C₂²×C₄	—	28	2	no	—
159	C₂²×C₄	#1141	1	1	yes	characteristic
160	C₂²×C₄	#1141	1	1	yes	characteristic
161	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
162	C₂×C₈	#1144	1	4	yes	—
163	C₂×C₈	Q₈×D₇	1	4	yes	—
164	C₂×C₈	C₂×C₂²⋊D₇	1	2	yes	—
165	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
166	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
167	C₄⋊C₄	—	2	2	no	—
168	C₄⋊C₄	—	2	2	no	—
169	C₄⋊C₄	—	4	1	no	—
170	C₄⋊C₄	—	4	1	no	—
171	C₄⋊C₄	—	4	4	no	—
172	C₄⋊C₄	—	4	4	no	—
173	C₄⋊C₄	—	14	4	no	—
174	C₄⋊C₄	—	28	2	no	—
175	C₄⋊C₄	—	28	4	no	—
176	C₄²	#1141	1	2	yes	—
177	C₄²	C₂×C₂²⋊D₇	1	2	yes	—
178	C₁₆	—	7	8	no	—
179	C₂×C₇	#2255	1	4	yes	complemented
180	C₂×C₇	C₂×#500	1	1	yes	characteristic
181	C₂×C₇	C₂×#509	1	1	yes	characteristic
182	C₂×C₇	C₂×#510	1	1	yes	characteristic
183	C₂³	C₂×#1141	1	1	yes	characteristic
184	Q₈	—	56	8	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	8	2	no	—
190	C₂×C₄	—	8	2	no	—
191	C₂×C₄	—	28	4	no	—
192	C₂×C₄	—	56	4	no	—
193	C₂×C₄	C₂×#1141	1	2	yes	—
194	C₂×C₄	C₂×#1141	1	2	yes	—
195	C₂×C₄	C₂×#1141	1	1	yes	characteristic
196	C₂×C₄	C₂×#1144	1	2	yes	—
197	C₂×C₄	#6507	1	2	yes	—
198	C₂×C₄	C₂²×C₂²⋊D₇	1	1	yes	characteristic
199	C₈	C₂×#1144	1	4	yes	—
200	C₈	#6507	1	2	yes	—
201	C₈	C₂×Q₈×D₇	1	2	yes	—
202	C₇	C₂×#2255	1	1	yes	characteristic, complemented, Sylow
203	C₂²	#89850	1	2	yes	—
204	C₂²	#89965	1	2	yes	—
205	C₂²	#89967	1	2	yes	—
206	C₂²	C₂²×#1141	1	1	yes	characteristic
207	C₄	—	2	2	no	—
208	C₄	—	2	2	no	—
209	C₄	—	8	2	no	—
210	C₄	—	8	2	no	—
211	C₄	—	56	4	no	—
212	C₄	#89850	1	2	yes	—
213	C₄	C₂²×#1144	1	1	yes	characteristic
214	C₄	C₂×#6507	1	1	yes	characteristic
215	C₂	#11729109	1	4	yes	direct product factor
216	C₂	C₂×#89850	1	1	yes	characteristic
217	C₂	C₂×#89965	1	1	yes	characteristic
218	C₂	C₂×#89967	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×#499	C₂
socle, prime radical	C₂³×C₇	C₂×D₈
hypercenter	C₂×#499	D₇
hypocenter	C₇	C₂×#2255
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#499

Complements

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

N	H ≅ G/N	type
3. #11729109	C₂	direct (N × H)
48. C₂×C₇⋊C₁₆	Q₈	semidirect
80. C₇⋊C₁₆	C₂×Q₈	semidirect
179. C₂×C₇	#2255	semidirect
202. C₇	C₂×#2255	semidirect
215. C₂	#11729109	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×#499	2
C₂×#89454	2
#11729109	2
C₂×#89285	2
C₂×#89284	2
C₂×#89256	2
C₂×#2255	7

Subgroups — C2×#11729109 #13509294

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11729109 #13509294