Subgroups — C₂×#22558091 #30745879

Order 17152.

830 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523954	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863262	C₂	1	1	yes	maximal, characteristic
3	#22558091	C₂	1	4	yes	maximal, direct product factor
4	C₂²×#97×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₆₇⋊C₈	—	2	1	no	—
6	C₂²×C₄⋊Dic₆₇	C₄	1	2	yes	—
7	C₂×#14523954	—	2	1	no	—
8	C₂×#14523954	—	2	2	no	—
9	C₂×#14523954	C₄	1	4	yes	complemented
10	C₂×#14523954	C₂²	1	2	yes	—
11	#16863262	—	2	1	no	—
12	#16863262	—	2	1	no	—
13	#16863262	C₄	1	2	yes	—
14	#16863262	C₂²	1	2	yes	—
15	C₂×C₆₇⋊#95	—	2	1	no	—
16	C₂×C₆₇⋊#95	—	2	1	no	—
17	C₂³×C₈×C₆₇	—	2	1	no	—
18	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×#97×C₆₇	—	2	1	no	—
20	C₂×#97×C₆₇	—	2	2	no	—
21	C₂×#97×C₆₇	C₄	1	4	yes	complemented
22	C₂×#97×C₆₇	C₂²	1	2	yes	—
23	C₂³×Dic₆₇	—	2	2	no	—
24	C₂²×C₆₇⋊C₈	—	2	8	no	—
25	C₂²×C₆₇⋊C₈	—	2	4	no	—
26	C₂²×C₆₇⋊C₈	—	2	2	no	—
27	C₂×C₄⋊Dic₆₇	—	2	2	no	—
28	C₂×C₄⋊Dic₆₇	—	4	4	no	—
29	C₂×C₄⋊Dic₆₇	C₂×C₄	1	4	yes	—
30	#14523954	—	2	4	no	—
31	#14523954	—	2	2	no	—
32	#14523954	C₂×C₄	1	4	yes	complemented
33	C₆₇⋊#95	—	2	2	no	—
34	C₆₇⋊#95	—	2	2	no	—
35	C₆₇⋊#95	—	2	2	no	—
36	C₆₇⋊#95	—	2	2	no	—
37	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
38	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
39	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₆₇	—	2	8	no	—
41	C₂²×C₈×C₆₇	—	2	4	no	—
42	C₂²×C₈×C₆₇	—	2	2	no	—
43	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₆₇	D₄	1	2	yes	—
49	C₂×C₄²×C₆₇	Q₈	1	2	yes	—
50	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
51	#97×C₆₇	—	2	4	no	—
52	#97×C₆₇	—	2	2	no	—
53	#97×C₆₇	C₂×C₄	1	4	yes	complemented
54	C₂²×Dic₆₇	—	2	4	no	—
55	C₂²×Dic₆₇	—	4	4	no	—
56	C₂²×Dic₆₇	—	4	4	no	—
57	C₂²×Dic₆₇	—	4	4	no	—
58	C₂×C₆₇⋊C₈	—	2	16	no	—
59	C₂×C₆₇⋊C₈	—	2	8	no	—
60	C₂×C₆₇⋊C₈	—	2	4	no	—
61	C₄⋊Dic₆₇	—	4	4	no	—
62	C₄⋊Dic₆₇	—	4	4	no	—
63	C₂⁴×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₆₇	—	2	2	no	—
65	C₂²×C₄×C₆₇	—	2	2	no	—
66	C₂²×C₄×C₆₇	—	2	4	no	—
67	C₂²×C₄×C₆₇	—	2	4	no	—
68	C₂²×C₄×C₆₇	C₄²	1	2	yes	—
69	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
73	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
74	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
75	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
76	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
77	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₆₇	—	2	16	no	—
81	C₂×C₈×C₆₇	—	2	8	no	—
82	C₂×C₈×C₆₇	—	2	4	no	—
83	C₄²×C₆₇	C₄²	1	2	yes	—
84	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
85	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
86	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
87	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
88	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
89	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
91	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
93	C₂³×C₆₇	#95	1	2	yes	—
94	C₂³×C₆₇	#100	1	2	yes	—
95	C₂³×C₆₇	#101	1	2	yes	—
96	C₂³×C₆₇	#104	1	4	yes	—
97	C₂³×C₆₇	#104	1	4	yes	—
98	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₆₇	—	2	4	no	—
100	C₂×C₄×C₆₇	—	2	4	no	—
101	C₂×C₄×C₆₇	—	2	4	no	—
102	C₂×C₄×C₆₇	—	2	4	no	—
103	C₂×C₄×C₆₇	—	2	4	no	—
104	C₂×C₄×C₆₇	—	2	4	no	—
105	C₂×C₄×C₆₇	—	2	1	no	—
106	C₂×C₄×C₆₇	—	2	1	no	—
107	C₂×C₄×C₆₇	—	2	1	no	—
108	C₂×C₄×C₆₇	—	2	1	no	—
109	C₂×C₄×C₆₇	—	2	2	no	—
110	C₂×C₄×C₆₇	—	2	2	no	—
111	C₂×C₄×C₆₇	—	2	2	no	—
112	C₂×C₄×C₆₇	—	2	2	no	—
113	C₂×C₄×C₆₇	#95	1	2	yes	—
114	C₂×C₄×C₆₇	#95	1	2	yes	—
115	C₂×C₄×C₆₇	#95	1	2	yes	—
116	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
117	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×Dic₆₇	—	4	4	no	—
123	C₂×Dic₆₇	—	4	4	no	—
124	C₂×Dic₆₇	—	4	4	no	—
125	C₂×Dic₆₇	—	8	8	no	—
126	C₈×C₆₇	—	2	8	no	—
127	C₆₇⋊C₈	—	2	8	no	—
128	C₂²×C₆₇	#329	1	8	yes	—
129	C₂²×C₆₇	#330	1	8	yes	—
130	C₂²×C₆₇	#339	1	4	yes	—
131	C₂²×C₆₇	#339	1	4	yes	—
132	C₂²×C₆₇	#343	1	4	yes	—
133	C₂²×C₆₇	C₂×#95	1	1	yes	characteristic
134	C₂²×C₆₇	C₂×#100	1	1	yes	characteristic
135	C₂²×C₆₇	C₂×#101	1	1	yes	characteristic
136	C₂²×C₆₇	C₂×#104	1	2	yes	—
137	C₂²×C₆₇	C₂×#104	1	2	yes	—
138	C₄×C₆₇	—	2	1	no	—
139	C₄×C₆₇	—	2	1	no	—
140	C₄×C₆₇	—	2	2	no	—
141	C₄×C₆₇	—	2	2	no	—
142	C₄×C₆₇	—	2	1	no	—
143	C₄×C₆₇	—	2	1	no	—
144	C₄×C₆₇	—	2	2	no	—
145	C₄×C₆₇	—	2	1	no	—
146	C₄×C₆₇	—	2	1	no	—
147	Dic₆₇	—	8	4	no	—
148	C₂×#1295	—	67	1	no	maximal, Sylow
149	C₂×C₆₇	#1295	1	8	yes	complemented
150	C₂×C₆₇	C₂×#329	1	2	yes	—
151	C₂×C₆₇	C₂×#330	1	2	yes	—
152	C₂×C₆₇	C₂×#339	1	1	yes	characteristic
153	C₂×C₆₇	C₂×#339	1	1	yes	characteristic
154	C₂×C₆₇	C₂×#343	1	1	yes	characteristic
155	C₂²×#97	—	67	1	no	—
156	C₂²×#97	D₆₇	1	1	yes	characteristic
157	C₂×#382	—	67	1	no	—
158	#1295	—	67	4	no	—
159	C₆₇	C₂×#1295	1	1	yes	characteristic, complemented, Sylow
160	C₂³×C₈	—	2	1	no	—
161	C₂³×C₈	—	134	1	no	—
162	C₂²×C₄⋊C₄	—	67	2	no	—
163	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
164	C₂×#97	—	2	1	no	—
165	C₂×#97	—	2	2	no	—
166	C₂×#97	—	67	4	no	—
167	C₂×#97	—	67	2	no	—
168	C₂×#97	—	134	1	no	—
169	C₂×#97	—	134	2	no	—
170	C₂×#97	Dic₆₇	1	4	yes	complemented
171	C₂×#97	C₂×D₆₇	1	2	yes	—
172	#382	—	67	2	no	—
173	#382	—	67	2	no	—
174	#382	—	134	1	no	—
175	#382	—	134	1	no	—
176	C₂×#95	—	134	1	no	—
177	C₂×#95	—	134	1	no	—
178	C₂³×C₄	—	134	2	no	—
179	C₂³×C₄	C₄×D₆₇	1	1	yes	characteristic
180	C₂³×C₄	D₂₆₈	1	1	yes	characteristic
181	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
182	C₂²×C₈	—	2	8	no	—
183	C₂²×C₈	—	2	4	no	—
184	C₂²×C₈	—	2	2	no	—
185	C₂²×C₈	—	134	8	no	—
186	C₂²×C₈	—	134	4	no	—
187	C₂²×C₈	—	134	2	no	—
188	C₂×C₄⋊C₄	—	268	4	no	—
189	C₂×C₄⋊C₄	—	67	4	no	—
190	C₂×C₄⋊C₄	—	134	2	no	—
191	C₂×C₄²	Dic₁₃₄	1	2	yes	—
192	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
193	C₂×C₄²	C₄×D₆₇	1	2	yes	—
194	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
195	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
196	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
197	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
198	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
199	#97	—	2	4	no	—
200	#97	—	2	2	no	—
201	#97	—	67	4	no	—
202	#97	—	134	4	no	—
203	#97	—	134	2	no	—
204	#97	C₂×Dic₆₇	1	4	yes	complemented
205	#95	—	134	2	no	—
206	#95	—	134	2	no	—
207	#95	—	134	2	no	—
208	#95	—	134	2	no	—
209	C₂⁴	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	—	2	4	no	—
214	C₂²×C₄	—	268	4	no	—
215	C₂²×C₄	—	268	4	no	—
216	C₂²×C₄	—	268	4	no	—
217	C₂²×C₄	—	134	4	no	—
218	C₂²×C₄	C₄×Dic₆₇	1	2	yes	—
219	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
220	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
221	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
222	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
223	C₂²×C₄	C₂²⋊Dic₆₇	1	2	yes	—
224	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
225	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
226	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
227	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
229	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
230	C₂×C₈	—	2	8	no	—
231	C₂×C₈	—	2	16	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	134	8	no	—
234	C₂×C₈	—	134	16	no	—
235	C₂×C₈	—	134	4	no	—
236	C₄⋊C₄	—	268	4	no	—
237	C₄⋊C₄	—	268	4	no	—
238	C₄²	C₄×Dic₆₇	1	2	yes	—
239	C₄²	(C₂×D₆₇)⋊C₄	1	2	yes	—
240	C₄²	C₂²⋊Dic₆₇	1	2	yes	—
241	C₄²	Dic₆₇⋊C₄	1	2	yes	—
242	C₄²	Dic₆₇⋊C₄	1	2	yes	—
243	C₄²	C₄⋊Dic₆₇	1	2	yes	—
244	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
245	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
246	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
247	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
248	C₂³	C₆₇⋊#95	1	2	yes	—
249	C₂³	#14523959	1	2	yes	—
250	C₂³	#14523961	1	2	yes	—
251	C₂³	#14523969	1	4	yes	—
252	C₂³	#14523971	1	4	yes	—
253	C₂³	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	268	4	no	—
269	C₂×C₄	—	268	4	no	—
270	C₂×C₄	—	268	4	no	—
271	C₂×C₄	—	536	8	no	—
272	C₂×C₄	C₆₇⋊#95	1	2	yes	—
273	C₂×C₄	C₆₇⋊#95	1	2	yes	—
274	C₂×C₄	C₆₇⋊#95	1	2	yes	—
275	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
276	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
277	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
278	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
279	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
280	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
281	C₈	—	2	8	no	—
282	C₈	—	134	8	no	—
283	C₂²	#16863135	1	8	yes	—
284	C₂²	#16863137	1	8	yes	—
285	C₂²	#16863156	1	4	yes	—
286	C₂²	#16863157	1	4	yes	—
287	C₂²	#16863164	1	4	yes	—
288	C₂²	C₂×C₆₇⋊#95	1	1	yes	characteristic
289	C₂²	C₂×#14523959	1	1	yes	characteristic
290	C₂²	C₂×#14523961	1	1	yes	characteristic
291	C₂²	C₂×#14523969	1	2	yes	—
292	C₂²	C₂×#14523971	1	2	yes	—
293	C₄	—	2	1	no	—
294	C₄	—	2	2	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	2	no	—
298	C₄	—	2	1	no	—
299	C₄	—	2	2	no	—
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	536	4	no	—
303	C₂	#22558091	1	8	yes	direct product factor
304	C₂	C₂×#16863135	1	2	yes	—
305	C₂	C₂×#16863137	1	2	yes	—
306	C₂	C₂×#16863156	1	1	yes	characteristic
307	C₂	C₂×#16863157	1	1	yes	characteristic
308	C₂	C₂×#16863164	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	(C₂×D₆₇)⋊C₄
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂²×#97×C₆₇	C₂
socle, prime radical	C₂⁴×C₆₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂²×#97	D₆₇
hypocenter	C₆₇	C₂×#1295
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
3. #22558091	C₂	direct (N × H)
9. C₂×#14523954	C₄	semidirect
21. C₂×#97×C₆₇	C₄	semidirect
32. #14523954	C₂×C₄	semidirect
53. #97×C₆₇	C₂×C₄	semidirect
149. C₂×C₆₇	#1295	semidirect
159. C₆₇	C₂×#1295	semidirect
170. C₂×#97	Dic₆₇	semidirect
204. #97	C₂×Dic₆₇	semidirect
303. C₂	#22558091	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14523954	2
C₂×#16863262	2
#22558091	2
C₂²×#97×C₆₇	2
C₂×#1295	67

Subgroups — C2×#22558091 #30745879

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22558091 #30745879