Subgroups — C₂×#22558300 #30770707

Order 17152.

694 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14523977	C₂	1	1	yes	maximal, characteristic
3	#22558300	C₂	1	4	yes	maximal, direct product factor
4	C₂²×#107×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Dic₆₇	—	2	1	no	—
6	C₂²×C₆₇⋊C₁₆	—	2	2	no	—
7	C₂²×C₆₇⋊C₁₆	—	2	2	no	—
8	C₂²×C₆₇⋊C₁₆	C₄	1	4	yes	complemented
9	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
10	C₂×#14523977	—	2	2	no	—
11	C₂×#14523977	—	2	1	no	—
12	C₂×#14523977	C₄	1	4	yes	complemented
13	C₂×#14523977	C₂²	1	2	yes	—
14	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
16	C₂×#107×C₆₇	—	2	1	no	—
17	C₂×#107×C₆₇	—	2	2	no	—
18	C₂×#107×C₆₇	C₄	1	4	yes	complemented
19	C₂×#107×C₆₇	C₂²	1	2	yes	—
20	C₂³×Dic₆₇	—	4	1	no	—
21	C₂×C₄⋊Dic₆₇	—	2	4	no	—
22	C₂×C₄⋊Dic₆₇	—	2	2	no	—
23	C₂×C₄⋊Dic₆₇	—	4	1	no	—
24	C₂×C₄⋊Dic₆₇	—	4	2	no	—
25	C₂×C₆₇⋊C₁₆	—	2	8	no	—
26	C₂×C₆₇⋊C₁₆	—	2	2	no	—
27	C₂×C₆₇⋊C₁₆	—	2	2	no	—
28	C₂×C₆₇⋊C₁₆	C₂×C₄	1	4	yes	complemented
29	#14523977	—	2	4	no	—
30	#14523977	—	2	2	no	—
31	#14523977	C₂×C₄	1	4	yes	complemented
32	C₂³×C₄×C₆₇	—	4	1	no	—
33	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
37	C₂²×C₈×C₆₇	D₄	1	2	yes	—
38	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₆₇	D₄	1	2	yes	—
40	C₂²×C₈×C₆₇	Q₈	1	2	yes	—
41	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
42	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
43	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
44	C₂×C₄⋊C₄×C₆₇	—	4	1	no	—
45	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
46	#107×C₆₇	—	2	4	no	—
47	#107×C₆₇	—	2	2	no	—
48	#107×C₆₇	C₂×C₄	1	4	yes	complemented
49	C₂²×Dic₆₇	—	4	4	no	—
50	C₂²×Dic₆₇	—	4	2	no	—
51	C₂²×Dic₆₇	—	8	2	no	—
52	C₂²×Dic₆₇	—	8	2	no	—
53	C₄⋊Dic₆₇	—	2	4	no	—
54	C₄⋊Dic₆₇	—	4	4	no	—
55	C₄⋊Dic₆₇	—	4	2	no	—
56	C₆₇⋊C₁₆	—	2	4	no	—
57	C₂⁴×C₆₇	D₈	1	1	yes	characteristic
58	C₂²×C₄×C₆₇	—	4	4	no	—
59	C₂²×C₄×C₆₇	—	4	2	no	—
60	C₂²×C₄×C₆₇	—	8	2	no	—
61	C₂²×C₄×C₆₇	—	8	2	no	—
62	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
66	C₂²×C₄×C₆₇	SD₁₆	1	2	yes	—
67	C₂²×C₄×C₆₇	SD₁₆	1	2	yes	—
68	C₂²×C₄×C₆₇	Q₁₆	1	2	yes	—
69	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₆₇	C₄²	1	4	yes	—
71	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
72	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
73	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
74	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
75	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
76	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
77	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
83	C₄⋊C₄×C₆₇	—	2	4	no	—
84	C₄⋊C₄×C₆₇	—	4	4	no	—
85	C₄⋊C₄×C₆₇	—	4	2	no	—
86	C₂³×C₆₇	#102	1	2	yes	—
87	C₂³×C₆₇	#102	1	2	yes	—
88	C₂³×C₆₇	#107	1	2	yes	—
89	C₂³×C₆₇	D₁₆	1	2	yes	—
90	C₂³×C₆₇	SD₃₂	1	2	yes	—
91	C₂³×C₆₇	SD₃₂	1	2	yes	—
92	C₂³×C₆₇	Q₃₂	1	2	yes	—
93	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
94	C₂×C₄×C₆₇	—	4	4	no	—
95	C₂×C₄×C₆₇	—	8	8	no	—
96	C₂×C₄×C₆₇	—	8	2	no	—
97	C₂×C₄×C₆₇	—	8	2	no	—
98	C₂×C₄×C₆₇	#95	1	4	yes	—
99	C₂×C₄×C₆₇	#102	1	2	yes	—
100	C₂×C₄×C₆₇	#102	1	2	yes	—
101	C₂×C₄×C₆₇	#103	1	4	yes	—
102	C₂×C₄×C₆₇	#103	1	4	yes	—
103	C₂×C₄×C₆₇	#106	1	4	yes	—
104	C₂×C₄×C₆₇	#107	1	2	yes	—
105	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
109	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
110	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
111	C₂×Dic₆₇	—	4	4	no	—
112	C₂×Dic₆₇	—	8	8	no	—
113	C₂×Dic₆₇	—	8	2	no	—
114	C₂×Dic₆₇	—	8	2	no	—
115	C₈×C₆₇	#95	1	4	yes	—
116	C₈×C₆₇	C₂×C₄²	1	1	yes	characteristic
117	C₈×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂²×C₆₇	#340	1	4	yes	—
121	C₂²×C₆₇	#357	1	4	yes	—
122	C₂²×C₆₇	#357	1	4	yes	—
123	C₂²×C₆₇	#358	1	4	yes	—
124	C₂²×C₆₇	#358	1	4	yes	—
125	C₂²×C₆₇	#366	1	4	yes	—
126	C₂²×C₆₇	#367	1	4	yes	—
127	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
128	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
129	C₂²×C₆₇	C₂×#107	1	1	yes	characteristic
130	C₂²×C₆₇	C₂×D₁₆	1	1	yes	characteristic
131	C₂²×C₆₇	C₂×SD₃₂	1	1	yes	characteristic
132	C₂²×C₆₇	C₂×SD₃₂	1	1	yes	characteristic
133	C₂²×C₆₇	C₂×Q₃₂	1	1	yes	characteristic
134	C₄×C₆₇	—	8	4	no	—
135	C₄×C₆₇	#340	1	4	yes	—
136	C₄×C₆₇	C₂×#95	1	1	yes	characteristic
137	C₄×C₆₇	C₂×#103	1	1	yes	characteristic
138	C₄×C₆₇	C₂×#103	1	1	yes	characteristic
139	C₄×C₆₇	C₂×#106	1	1	yes	characteristic
140	Dic₆₇	—	8	4	no	—
141	C₂×#1380	—	67	1	no	maximal, Sylow
142	C₂×C₆₇	#1380	1	8	yes	complemented
143	C₂×C₆₇	C₂×#340	1	1	yes	characteristic
144	C₂×C₆₇	C₂×#357	1	1	yes	characteristic
145	C₂×C₆₇	C₂×#357	1	1	yes	characteristic
146	C₂×C₆₇	C₂×#358	1	1	yes	characteristic
147	C₂×C₆₇	C₂×#358	1	1	yes	characteristic
148	C₂×C₆₇	C₂×#366	1	1	yes	characteristic
149	C₂×C₆₇	C₂×#367	1	1	yes	characteristic
150	C₂³×C₁₆	—	67	1	no	—
151	C₂²×#107	—	67	1	no	—
152	C₂²×#107	D₆₇	1	1	yes	characteristic
153	#1380	—	67	4	no	—
154	C₆₇	C₂×#1380	1	1	yes	characteristic, complemented, Sylow
155	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
156	C₂²×C₄⋊C₄	—	2	1	no	—
157	C₂²×C₄⋊C₄	—	134	1	no	—
158	C₂²×C₁₆	—	67	4	no	—
159	C₂²×C₁₆	—	67	2	no	—
160	C₂²×C₁₆	—	134	2	no	—
161	C₂²×C₁₆	—	134	2	no	—
162	C₂×#107	—	2	1	no	—
163	C₂×#107	—	2	2	no	—
164	C₂×#107	—	67	4	no	—
165	C₂×#107	—	67	2	no	—
166	C₂×#107	—	134	2	no	—
167	C₂×#107	—	134	1	no	—
168	C₂×#107	Dic₆₇	1	4	yes	complemented
169	C₂×#107	C₂×D₆₇	1	2	yes	—
170	C₂³×C₄	—	4	1	no	—
171	C₂³×C₄	—	268	1	no	—
172	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
173	C₂²×C₈	Dic₁₃₄	1	2	yes	—
174	C₂²×C₈	C₄×D₆₇	1	2	yes	—
175	C₂²×C₈	C₄×D₆₇	1	2	yes	—
176	C₂²×C₈	D₂₆₈	1	2	yes	—
177	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
178	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
179	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
180	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
181	C₂×C₄⋊C₄	—	2	4	no	—
182	C₂×C₄⋊C₄	—	2	2	no	—
183	C₂×C₄⋊C₄	—	4	1	no	—
184	C₂×C₄⋊C₄	—	4	2	no	—
185	C₂×C₄⋊C₄	—	268	1	no	—
186	C₂×C₄⋊C₄	—	268	2	no	—
187	C₂×C₄⋊C₄	—	134	4	no	—
188	C₂×C₄⋊C₄	—	134	2	no	—
189	C₂×C₁₆	—	67	4	no	—
190	C₂×C₁₆	—	134	8	no	—
191	C₂×C₁₆	—	134	2	no	—
192	C₂×C₁₆	—	134	2	no	—
193	#107	—	2	4	no	—
194	#107	—	2	2	no	—
195	#107	—	67	4	no	—
196	#107	—	134	4	no	—
197	#107	—	134	2	no	—
198	#107	C₂×Dic₆₇	1	4	yes	complemented
199	C₂⁴	#11761613	1	1	yes	characteristic
200	C₂²×C₄	—	4	4	no	—
201	C₂²×C₄	—	4	2	no	—
202	C₂²×C₄	—	8	2	no	—
203	C₂²×C₄	—	8	2	no	—
204	C₂²×C₄	—	268	4	no	—
205	C₂²×C₄	—	268	2	no	—
206	C₂²×C₄	—	536	2	no	—
207	C₂²×C₄	—	536	2	no	—
208	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
209	C₂²×C₄	C₂²⋊Dic₆₇	1	2	yes	—
210	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
211	C₂²×C₄	#11761613	1	1	yes	characteristic
212	C₂²×C₄	#11761615	1	2	yes	—
213	C₂²×C₄	#11761616	1	2	yes	—
214	C₂²×C₄	#11761618	1	2	yes	—
215	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
216	C₂×C₈	C₄×Dic₆₇	1	4	yes	—
217	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
218	C₂×C₈	(C₂×D₆₇)⋊C₄	1	4	yes	—
219	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
220	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
221	C₂×C₈	Dic₆₇⋊C₄	1	4	yes	—
222	C₂×C₈	C₄⋊Dic₆₇	1	4	yes	—
223	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
224	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
225	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
226	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
227	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
228	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
229	C₄⋊C₄	—	2	4	no	—
230	C₄⋊C₄	—	4	4	no	—
231	C₄⋊C₄	—	4	2	no	—
232	C₄⋊C₄	—	268	4	no	—
233	C₄⋊C₄	—	268	2	no	—
234	C₄⋊C₄	—	134	4	no	—
235	C₁₆	—	134	4	no	—
236	C₂³	#14523963	1	2	yes	—
237	C₂³	#14523964	1	2	yes	—
238	C₂³	#14523976	1	2	yes	—
239	C₂³	#14523984	1	2	yes	—
240	C₂³	#14523986	1	2	yes	—
241	C₂³	#14523987	1	2	yes	—
242	C₂³	#14523989	1	2	yes	—
243	C₂³	C₂×#11761613	1	1	yes	characteristic
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	—	8	2	no	—
246	C₂×C₄	—	8	2	no	—
247	C₂×C₄	—	8	8	no	—
248	C₂×C₄	—	268	4	no	—
249	C₂×C₄	—	536	2	no	—
250	C₂×C₄	—	536	2	no	—
251	C₂×C₄	—	536	8	no	—
252	C₂×C₄	C₆₇⋊#95	1	4	yes	—
253	C₂×C₄	#14523963	1	2	yes	—
254	C₂×C₄	#14523964	1	2	yes	—
255	C₂×C₄	#14523966	1	4	yes	—
256	C₂×C₄	#14523967	1	4	yes	—
257	C₂×C₄	#14523974	1	4	yes	—
258	C₂×C₄	#14523976	1	2	yes	—
259	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
260	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
261	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
262	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
263	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
264	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
265	C₈	C₆₇⋊#95	1	4	yes	—
266	C₈	C₂×C₄×Dic₆₇	1	1	yes	characteristic
267	C₈	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
268	C₈	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
269	C₈	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
270	C₂²	#16863158	1	4	yes	—
271	C₂²	#16863196	1	4	yes	—
272	C₂²	#16863197	1	4	yes	—
273	C₂²	#16863199	1	4	yes	—
274	C₂²	#16863200	1	4	yes	—
275	C₂²	#16863221	1	4	yes	—
276	C₂²	#16863223	1	4	yes	—
277	C₂²	C₂×#14523963	1	1	yes	characteristic
278	C₂²	C₂×#14523964	1	1	yes	characteristic
279	C₂²	C₂×#14523976	1	1	yes	characteristic
280	C₂²	C₂×#14523984	1	1	yes	characteristic
281	C₂²	C₂×#14523986	1	1	yes	characteristic
282	C₂²	C₂×#14523987	1	1	yes	characteristic
283	C₂²	C₂×#14523989	1	1	yes	characteristic
284	C₄	—	8	4	no	—
285	C₄	—	536	4	no	—
286	C₄	#16863158	1	4	yes	—
287	C₄	C₂×C₆₇⋊#95	1	1	yes	characteristic
288	C₄	C₂×#14523966	1	1	yes	characteristic
289	C₄	C₂×#14523967	1	1	yes	characteristic
290	C₄	C₂×#14523974	1	1	yes	characteristic
291	C₂	#22558300	1	8	yes	direct product factor
292	C₂	C₂×#16863158	1	1	yes	characteristic
293	C₂	C₂×#16863196	1	1	yes	characteristic
294	C₂	C₂×#16863197	1	1	yes	characteristic
295	C₂	C₂×#16863199	1	1	yes	characteristic
296	C₂	C₂×#16863200	1	1	yes	characteristic
297	C₂	C₂×#16863221	1	1	yes	characteristic
298	C₂	C₂×#16863223	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	#11761613
commutator	C₈×C₆₇	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂²×#107×C₆₇	C₂
socle, prime radical	C₂⁴×C₆₇	D₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂²×#107	D₆₇
hypocenter	C₆₇	C₂×#1380
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×#107

Complements

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

N	H ≅ G/N	type
3. #22558300	C₂	direct (N × H)
8. C₂²×C₆₇⋊C₁₆	C₄	semidirect
12. C₂×#14523977	C₄	semidirect
18. C₂×#107×C₆₇	C₄	semidirect
28. C₂×C₆₇⋊C₁₆	C₂×C₄	semidirect
31. #14523977	C₂×C₄	semidirect
48. #107×C₆₇	C₂×C₄	semidirect
142. C₂×C₆₇	#1380	semidirect
154. C₆₇	C₂×#1380	semidirect
168. C₂×#107	Dic₆₇	semidirect
198. #107	C₂×Dic₆₇	semidirect
291. C₂	#22558300	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₆₇⋊C₁₆	2
C₂²×#14523977	2
#22558300	2
C₂²×#107×C₆₇	2
C₂×#1380	67

Subgroups — C2×#22558300 #30770707

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22558300 #30770707