Subgroups — C₂×#11763684 #14544474

Order 2176.

392 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₇⋊M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×#63242	C₂	1	1	yes	maximal, characteristic
3	#11763684	C₂	1	4	yes	maximal, direct product factor
4	C₂×M₁₆×D₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄×D₁₇	C₂²	1	1	yes	characteristic
6	C₂×#63247	—	2	1	no	—
7	C₂²×#3624	—	2	1	no	—
8	D₁₇⋊M₁₆	—	2	1	no	—
9	D₁₇⋊M₁₆	—	2	1	no	—
10	D₁₇⋊M₁₆	C₄	1	2	yes	—
11	D₁₇⋊M₁₆	C₂²	1	2	yes	—
12	C₂×#63243	C₄	1	2	yes	—
13	C₂×#63242	—	2	1	no	—
14	C₂×#63242	—	2	1	no	—
15	C₂×#63242	—	2	1	no	—
16	C₂×#63242	—	2	1	no	—
17	C₂×#63242	C₄	1	2	yes	—
18	C₂×#63242	C₄	1	2	yes	—
19	C₂×#63242	C₂²	1	2	yes	—
20	C₂×M₁₆×C₁₇	C₄	1	1	yes	characteristic
21	C₂×#63223	C₄	1	1	yes	characteristic
22	M₁₆×D₁₇	—	2	1	no	—
23	M₁₆×D₁₇	—	2	1	no	—
24	M₁₆×D₁₇	C₄	1	2	yes	—
25	M₁₆×D₁₇	C₂²	1	2	yes	—
26	C₂×#63218	—	2	1	no	—
27	C₂×C₈×D₁₇	—	2	1	no	—
28	C₂³×D₁₇	Q₈	1	1	yes	characteristic
29	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×Dic₁₇	D₄	1	1	yes	characteristic
31	C₂×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×D₁₇	D₄	1	1	yes	characteristic
37	C₂×C₄×D₁₇	D₄	1	1	yes	characteristic
38	C₂×C₄×D₁₇	D₄	1	1	yes	characteristic
39	C₂×C₄×D₁₇	D₄	1	1	yes	characteristic
40	C₂×C₄×D₁₇	D₄	1	1	yes	characteristic
41	C₂×C₄×D₁₇	Q₈	1	1	yes	characteristic
42	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
43	#63247	—	2	2	no	—
44	#63247	—	2	2	no	—
45	C₂×#3624	—	2	2	no	—
46	C₂×#3624	—	2	2	no	—
47	C₂×#3624	—	2	2	no	—
48	C₂×#3624	—	2	2	no	—
49	#63243	—	2	2	no	—
50	#63243	C₂×C₄	1	4	yes	—
51	#63242	—	2	2	no	—
52	#63242	—	2	2	no	—
53	#63242	—	2	2	no	—
54	#63242	—	2	2	no	—
55	#63242	—	2	2	no	—
56	#63242	C₂×C₄	1	4	yes	—
57	M₁₆×C₁₇	—	2	1	no	—
58	M₁₆×C₁₇	C₂×C₄	1	2	yes	—
59	C₂×C₈×C₁₇	—	2	1	no	—
60	#63223	—	2	1	no	—
61	#63223	C₂×C₄	1	2	yes	—
62	C₂×#3622	—	2	1	no	—
63	#63218	—	2	2	no	—
64	#63218	—	2	2	no	—
65	C₈×D₁₇	—	2	2	no	—
66	C₈×D₁₇	—	2	2	no	—
67	C₂³×C₁₇	C₄⋊C₄	1	1	yes	characteristic
68	C₂²×D₁₇	—	2	4	no	—
69	C₂²×D₁₇	C₄⋊C₄	1	1	yes	characteristic
70	C₂²×D₁₇	C₄⋊C₄	1	1	yes	characteristic
71	C₂²×D₁₇	C₄⋊C₄	1	1	yes	characteristic
72	C₂²×D₁₇	C₄⋊C₄	1	1	yes	characteristic
73	C₂²×D₁₇	C₄⋊C₄	1	1	yes	characteristic
74	C₂²×D₁₇	C₂×Q₈	1	1	yes	characteristic
75	C₂×C₄×C₁₇	C₄²	1	1	yes	characteristic
76	C₂×C₄×C₁₇	C₄²	1	1	yes	characteristic
77	C₂×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
81	C₂×Dic₁₇	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×Dic₁₇	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×Dic₁₇	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×Dic₁₇	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
86	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
87	C₄×D₁₇	C₄²	1	1	yes	characteristic
88	C₄×D₁₇	C₄²	1	1	yes	characteristic
89	C₄×D₁₇	C₂²⋊C₄	1	1	yes	characteristic
90	C₄×D₁₇	C₂²⋊C₄	1	1	yes	characteristic
91	C₄×D₁₇	C₂²⋊C₄	1	1	yes	characteristic
92	C₄×D₁₇	C₂²⋊C₄	1	1	yes	characteristic
93	C₄×D₁₇	C₂²⋊C₄	1	1	yes	characteristic
94	C₄×D₁₇	C₂²⋊C₄	1	1	yes	characteristic
95	C₄×D₁₇	C₄⋊C₄	1	1	yes	characteristic
96	C₄×D₁₇	C₄⋊C₄	1	1	yes	characteristic
97	C₄×D₁₇	C₄⋊C₄	1	1	yes	characteristic
98	C₄×D₁₇	C₄⋊C₄	1	1	yes	characteristic
99	C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
100	C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
101	C₄×D₁₇	C₂×D₄	1	1	yes	characteristic
102	C₄×D₁₇	C₂×D₄	1	1	yes	characteristic
103	#3624	—	2	4	no	—
104	#3624	—	2	4	no	—
105	C₈×C₁₇	—	2	2	no	—
106	#3622	—	2	2	no	—
107	C₂×#341	—	17	1	no	maximal, Sylow
108	C₂²×C₁₇	—	2	2	no	—
109	C₂²×C₁₇	#95	1	1	yes	characteristic
110	C₂²×C₁₇	#95	1	1	yes	characteristic
111	C₂²×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×D₁₇	—	2	2	no	—
113	C₂×D₁₇	—	2	2	no	—
114	C₂×D₁₇	—	2	2	no	—
115	C₂×D₁₇	—	2	2	no	—
116	C₂×D₁₇	—	2	2	no	—
117	C₂×D₁₇	#95	1	1	yes	characteristic
118	C₂×D₁₇	#95	1	1	yes	characteristic
119	C₂×D₁₇	#108	1	4	yes	—
120	C₂×D₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×D₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₄×C₁₇	#95	1	1	yes	characteristic
123	C₄×C₁₇	#95	1	1	yes	characteristic
124	C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
125	C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	Dic₁₇	#95	1	1	yes	characteristic
127	Dic₁₇	#95	1	1	yes	characteristic
128	Dic₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	Dic₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂²×M₁₆	—	17	1	no	—
131	C₂²×M₁₆	—	17	1	no	—
132	C₂³×C₈	—	17	1	no	—
133	#341	—	17	4	no	—
134	C₂×C₁₇	—	2	1	no	—
135	C₂×C₁₇	—	2	1	no	—
136	C₂×C₁₇	#341	1	2	yes	complemented
137	C₂×C₁₇	C₂×#95	1	1	yes	characteristic
138	D₁₇	—	2	1	no	—
139	D₁₇	—	2	1	no	—
140	D₁₇	#341	1	2	yes	complemented
141	D₁₇	C₂×#108	1	2	yes	—
142	C₂³×C₄	—	17	1	no	—
143	C₂×M₁₆	—	17	2	no	—
144	C₂×M₁₆	—	17	2	no	—
145	C₂×M₁₆	—	17	2	no	—
146	C₂×M₁₆	—	17	1	no	—
147	C₂×M₁₆	—	17	2	no	—
148	C₂×M₁₆	—	17	2	no	—
149	C₂×M₁₆	—	34	1	no	—
150	C₂×M₁₆	—	34	1	no	—
151	C₂×M₁₆	—	34	1	no	—
152	C₂×M₁₆	—	34	1	no	—
153	C₂×M₁₆	—	34	1	no	—
154	C₂×M₁₆	—	34	1	no	—
155	C₂×M₁₆	#595	1	1	yes	characteristic
156	C₂²×C₈	—	17	2	no	—
157	C₂²×C₈	—	17	2	no	—
158	C₂²×C₈	—	17	2	no	—
159	C₂²×C₈	—	34	1	no	—
160	C₂²×C₈	—	34	1	no	—
161	C₂²×C₈	—	34	1	no	—
162	C₂²×C₈	—	34	1	no	—
163	C₂²×C₈	—	34	1	no	—
164	C₂²×C₈	—	34	1	no	—
165	C₁₇	C₂×#341	1	1	yes	characteristic, complemented, Sylow
166	C₂⁴	—	17	1	no	—
167	C₂²×C₄	—	17	1	no	—
168	C₂²×C₄	—	17	1	no	—
169	C₂²×C₄	—	17	1	no	—
170	C₂²×C₄	—	17	1	no	—
171	C₂²×C₄	—	17	1	no	—
172	C₂²×C₄	—	17	1	no	—
173	C₂²×C₄	—	17	1	no	—
174	C₂²×C₄	—	17	1	no	—
175	C₂²×C₄	—	17	1	no	—
176	C₂²×C₄	—	17	1	no	—
177	C₂²×C₄	—	17	1	no	—
178	C₂²×C₄	—	17	1	no	—
179	C₂²×C₄	—	17	1	no	—
180	C₂²×C₄	C₂×#595	1	1	yes	characteristic
181	M₁₆	—	2	1	no	—
182	M₁₆	—	17	4	no	—
183	M₁₆	—	17	2	no	—
184	M₁₆	—	34	2	no	—
185	M₁₆	—	34	2	no	—
186	M₁₆	—	34	2	no	—
187	M₁₆	—	34	2	no	—
188	M₁₆	—	34	2	no	—
189	M₁₆	—	34	1	no	—
190	M₁₆	C₂×#595	1	2	yes	—
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	17	4	no	—
193	C₂×C₈	—	34	2	no	—
194	C₂×C₈	—	34	2	no	—
195	C₂×C₈	—	34	2	no	—
196	C₂×C₈	—	34	2	no	—
197	C₂×C₈	—	34	2	no	—
198	C₂×C₈	—	34	2	no	—
199	C₂×C₈	—	34	2	no	—
200	C₂×C₈	—	34	2	no	—
201	C₂×C₈	—	34	2	no	—
202	C₂×C₈	—	34	2	no	—
203	C₂×C₈	—	34	2	no	—
204	C₂×C₈	—	34	1	no	—
205	C₂³	—	17	1	no	—
206	C₂³	—	17	1	no	—
207	C₂³	—	17	1	no	—
208	C₂³	—	17	1	no	—
209	C₂³	—	17	1	no	—
210	C₂³	—	17	1	no	—
211	C₂³	—	34	4	no	—
212	C₂³	C₄⋊#595	1	1	yes	characteristic
213	C₂×C₄	—	17	1	no	—
214	C₂×C₄	—	17	1	no	—
215	C₂×C₄	—	17	1	no	—
216	C₂×C₄	—	17	1	no	—
217	C₂×C₄	—	17	1	no	—
218	C₂×C₄	—	17	1	no	—
219	C₂×C₄	—	17	1	no	—
220	C₂×C₄	—	17	1	no	—
221	C₂×C₄	—	17	1	no	—
222	C₂×C₄	—	17	1	no	—
223	C₂×C₄	—	17	1	no	—
224	C₂×C₄	—	17	1	no	—
225	C₂×C₄	—	17	1	no	—
226	C₂×C₄	—	17	1	no	—
227	C₂×C₄	—	17	1	no	—
228	C₂×C₄	—	17	1	no	—
229	C₂×C₄	—	17	1	no	—
230	C₂×C₄	—	17	1	no	—
231	C₂×C₄	—	17	1	no	—
232	C₂×C₄	—	17	1	no	—
233	C₂×C₄	—	17	1	no	—
234	C₂×C₄	—	17	1	no	—
235	C₂×C₄	C₄×#595	1	1	yes	characteristic
236	C₂×C₄	C₄×#595	1	1	yes	characteristic
237	C₂×C₄	C₄⋊#595	1	1	yes	characteristic
238	C₂×C₄	C₂²⋊#595	1	1	yes	characteristic
239	C₂×C₄	C₂²⋊#595	1	1	yes	characteristic
240	C₂×C₄	C₂²×#595	1	1	yes	characteristic
241	C₈	—	2	2	no	—
242	C₈	—	34	4	no	—
243	C₈	—	34	4	no	—
244	C₈	—	34	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	17	1	no	—
247	C₂²	—	17	1	no	—
248	C₂²	—	17	4	no	—
249	C₂²	—	17	1	no	—
250	C₂²	—	17	1	no	—
251	C₂²	—	34	2	no	—
252	C₂²	—	34	2	no	—
253	C₂²	—	34	2	no	—
254	C₂²	—	34	2	no	—
255	C₂²	—	34	2	no	—
256	C₂²	(C₂×#595)⋊C₄	1	1	yes	characteristic
257	C₂²	(C₂×#595)⋊C₄	1	1	yes	characteristic
258	C₂²	C₂×C₄⋊#595	1	1	yes	characteristic
259	C₄	—	17	1	no	—
260	C₄	—	17	1	no	—
261	C₄	—	17	1	no	—
262	C₄	—	17	1	no	—
263	C₄	(C₂×#595)⋊C₄	1	1	yes	characteristic
264	C₄	(C₂×#595)⋊C₄	1	1	yes	characteristic
265	C₄	C₂×C₄×#595	1	1	yes	characteristic
266	C₄	C₂×C₂²⋊#595	1	1	yes	characteristic
267	C₂	—	2	1	no	—
268	C₂	—	2	1	no	—
269	C₂	—	17	2	no	—
270	C₂	—	17	2	no	—
271	C₂	—	34	1	no	—
272	C₂	—	34	1	no	—
273	C₂	#11763684	1	2	yes	direct product factor
274	C₂	C₂×(C₂×#595)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊#595
commutator	C₄×C₁₇	C₂×C₄²
Frattini	C₂×C₄	C₂²×#595
Fitting	C₂×M₁₆×C₁₇	C₄
socle	C₂²×C₁₇	#95
prime radical	C₂³×D₁₇	Q₈
cosocle	C₂×C₄×D₁₇	C₂³
hypercenter	C₂×M₁₆	#595
hypocenter	C₁₇	C₂×#341
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
3. #11763684	C₂	direct (N × H)
136. C₂×C₁₇	#341	semidirect
140. D₁₇	#341	semidirect
165. C₁₇	C₂×#341	semidirect
273. C₂	#11763684	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×D₁₇⋊M₁₆	2
C₂²×#63242	2
#11763684	2
C₂×M₁₆×D₁₇	2
C₂×#341	17

Subgroups — C2×#11763684 #14544474

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11763684 #14544474