Subgroups — C₂²×#16863150 #30853181

Order 17152.

3086 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄×Dic₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863150	C₂	1	12	yes	maximal, direct product factor
4	C₂⁴×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂⁴×Dic₆₇	C₄	1	2	yes	—
6	C₂³×C₆₇⋊C₈	—	2	4	no	—
7	C₂³×C₆₇⋊C₈	—	2	4	no	—
8	C₂³×C₆₇⋊C₈	C₄	1	8	yes	complemented
9	C₂³×C₆₇⋊C₈	C₂²	1	6	yes	—
10	C₂²×C₄×Dic₆₇	—	2	4	no	—
11	C₂²×C₄×Dic₆₇	—	2	4	no	—
12	C₂²×C₄×Dic₆₇	C₄	1	6	yes	—
13	C₂²×C₄×Dic₆₇	C₂²	1	6	yes	—
14	#16863150	C₂²	1	16	yes	direct product factor
15	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
16	C₂³×C₈×C₆₇	—	2	4	no	—
17	C₂³×C₈×C₆₇	—	2	4	no	—
18	C₂³×C₈×C₆₇	C₄	1	8	yes	complemented
19	C₂³×C₈×C₆₇	C₂²	1	6	yes	—
20	C₂³×Dic₆₇	—	2	8	no	—
21	C₂³×Dic₆₇	—	2	8	no	—
22	C₂³×Dic₆₇	C₈	1	16	yes	complemented
23	C₂³×Dic₆₇	C₂×C₄	1	12	yes	—
24	C₂²×C₆₇⋊C₈	—	2	32	no	—
25	C₂²×C₆₇⋊C₈	—	2	12	no	—
26	C₂²×C₆₇⋊C₈	—	2	12	no	—
27	C₂²×C₆₇⋊C₈	C₂×C₄	1	24	yes	complemented
28	C₂²×C₆₇⋊C₈	C₂³	1	4	yes	—
29	C₂×C₄×Dic₆₇	—	2	24	no	—
30	C₂×C₄×Dic₆₇	—	2	12	no	—
31	C₂×C₄×Dic₆₇	—	2	12	no	—
32	C₂×C₄×Dic₆₇	C₂×C₄	1	12	yes	—
33	C₂×C₄×Dic₆₇	C₂³	1	4	yes	—
34	C₂⁵×C₆₇	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₄×C₆₇	C₂×C₄	1	3	yes	—
36	C₂³×C₄×C₆₇	C₂×C₄	1	4	yes	—
37	C₂³×C₄×C₆₇	C₂×C₄	1	4	yes	—
38	C₂³×C₄×C₆₇	D₄	1	4	yes	—
39	C₂³×C₄×C₆₇	D₄	1	4	yes	—
40	C₂³×C₄×C₆₇	D₄	1	4	yes	—
41	C₂³×C₄×C₆₇	Q₈	1	4	yes	—
42	C₂³×C₄×C₆₇	C₂³	1	3	yes	—
43	C₂²×C₈×C₆₇	—	2	32	no	—
44	C₂²×C₈×C₆₇	—	2	12	no	—
45	C₂²×C₈×C₆₇	—	2	12	no	—
46	C₂²×C₈×C₆₇	C₂×C₄	1	24	yes	complemented
47	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
48	C₂²×Dic₆₇	—	2	64	no	—
49	C₂²×Dic₆₇	—	2	24	no	—
50	C₂²×Dic₆₇	—	2	24	no	—
51	C₂²×Dic₆₇	C₂×C₈	1	48	yes	complemented
52	C₂²×Dic₆₇	C₂²×C₄	1	8	yes	—
53	C₂×C₆₇⋊C₈	—	2	48	no	—
54	C₂×C₆₇⋊C₈	—	2	4	no	—
55	C₂×C₆₇⋊C₈	—	2	4	no	—
56	C₂×C₆₇⋊C₈	C₂²×C₄	1	8	yes	complemented
57	C₄×Dic₆₇	—	2	24	no	—
58	C₄×Dic₆₇	—	2	4	no	—
59	C₄×Dic₆₇	—	2	4	no	—
60	C₂⁴×C₆₇	C₄²	1	4	yes	—
61	C₂⁴×C₆₇	C₂²⋊C₄	1	4	yes	—
62	C₂⁴×C₆₇	C₄⋊C₄	1	4	yes	—
63	C₂⁴×C₆₇	C₂×C₈	1	8	yes	—
64	C₂⁴×C₆₇	M₁₆	1	8	yes	—
65	C₂⁴×C₆₇	C₂²×C₄	1	3	yes	—
66	C₂²×C₄×C₆₇	C₄²	1	12	yes	—
67	C₂²×C₄×C₆₇	C₂²⋊C₄	1	12	yes	—
68	C₂²×C₄×C₆₇	C₂²⋊C₄	1	16	yes	—
69	C₂²×C₄×C₆₇	C₂²⋊C₄	1	16	yes	—
70	C₂²×C₄×C₆₇	C₄⋊C₄	1	12	yes	—
71	C₂²×C₄×C₆₇	C₄⋊C₄	1	16	yes	—
72	C₂²×C₄×C₆₇	C₄⋊C₄	1	16	yes	—
73	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
74	C₂²×C₄×C₆₇	C₂²×C₄	1	6	yes	—
75	C₂²×C₄×C₆₇	C₂²×C₄	1	6	yes	—
76	C₂²×C₄×C₆₇	C₂×D₄	1	6	yes	—
77	C₂²×C₄×C₆₇	C₂×D₄	1	6	yes	—
78	C₂²×C₄×C₆₇	C₂×D₄	1	6	yes	—
79	C₂²×C₄×C₆₇	C₂×Q₈	1	6	yes	—
80	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
81	C₂×C₈×C₆₇	—	2	48	no	—
82	C₂×C₈×C₆₇	—	2	4	no	—
83	C₂×C₈×C₆₇	—	2	4	no	—
84	C₂×C₈×C₆₇	C₂²×C₄	1	8	yes	complemented
85	C₂³×C₆₇	#95	1	16	yes	—
86	C₂³×C₆₇	C₄×C₈	1	16	yes	—
87	C₂³×C₆₇	#97	1	16	yes	—
88	C₂³×C₆₇	C₂²⋊C₈	1	32	yes	—
89	C₂³×C₆₇	C₄⋊C₈	1	32	yes	—
90	C₂³×C₆₇	C₂×C₄²	1	6	yes	—
91	C₂³×C₆₇	C₂×C₂²⋊C₄	1	6	yes	—
92	C₂³×C₆₇	C₂×C₄⋊C₄	1	6	yes	—
93	C₂³×C₆₇	C₂²×C₈	1	12	yes	—
94	C₂³×C₆₇	C₂×M₁₆	1	12	yes	—
95	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
96	C₂×C₄×C₆₇	#95	1	48	yes	—
97	C₂×C₄×C₆₇	C₂×C₄²	1	6	yes	—
98	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	12	yes	—
99	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	6	yes	—
100	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	12	yes	—
101	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	12	yes	—
102	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	6	yes	—
103	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	12	yes	—
104	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
105	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
110	C₂×Dic₆₇	—	2	8	no	—
111	C₂×Dic₆₇	—	2	8	no	—
112	C₂×Dic₆₇	—	2	96	no	—
113	C₂×Dic₆₇	C₂²×C₈	1	16	yes	complemented
114	C₈×C₆₇	—	2	8	no	—
115	C₆₇⋊C₈	—	2	8	no	—
116	C₂²×C₆₇	#336	1	64	yes	complemented
117	C₂²×C₆₇	C₂×#95	1	12	yes	—
118	C₂²×C₆₇	C₂×C₄×C₈	1	12	yes	—
119	C₂²×C₆₇	C₂×#97	1	12	yes	—
120	C₂²×C₆₇	C₂×C₂²⋊C₈	1	24	yes	—
121	C₂²×C₆₇	C₂×C₄⋊C₈	1	24	yes	—
122	C₂²×C₆₇	C₂²×C₄²	1	1	yes	characteristic
123	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
124	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
125	C₂²×C₆₇	C₂³×C₈	1	2	yes	—
126	C₂²×C₆₇	C₂²×M₁₆	1	2	yes	—
127	C₄×C₆₇	C₂×#95	1	12	yes	—
128	C₄×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
129	C₄×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
130	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
131	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
132	Dic₆₇	—	2	16	no	—
133	C₂²×#336	—	67	1	no	maximal, Sylow
134	C₂×C₆₇	C₂×#336	1	24	yes	complemented
135	C₂×C₆₇	C₂²×#95	1	1	yes	characteristic
136	C₂×C₆₇	C₂²×C₄×C₈	1	1	yes	characteristic
137	C₂×C₆₇	C₂²×#97	1	1	yes	characteristic
138	C₂×C₆₇	C₂²×C₂²⋊C₈	1	2	yes	—
139	C₂×C₆₇	C₂²×C₄⋊C₈	1	2	yes	—
140	C₂⁴×C₈	—	67	1	no	—
141	C₂⁴×C₈	D₆₇	1	1	yes	characteristic
142	C₂³×C₄²	—	67	1	no	—
143	C₂×#336	—	67	12	no	—
144	C₆₇	C₂²×#336	1	1	yes	characteristic, complemented, Sylow
145	C₂⁴×C₄	—	67	2	no	—
146	C₂⁴×C₄	C₂×D₆₇	1	1	yes	characteristic
147	C₂³×C₈	—	2	4	no	—
148	C₂³×C₈	—	2	4	no	—
149	C₂³×C₈	—	67	8	no	—
150	C₂³×C₈	—	67	6	no	—
151	C₂³×C₈	—	134	4	no	—
152	C₂³×C₈	—	134	4	no	—
153	C₂³×C₈	Dic₆₇	1	8	yes	complemented
154	C₂³×C₈	C₂×D₆₇	1	6	yes	—
155	C₂²×C₄²	—	67	6	no	—
156	C₂²×C₄²	—	67	6	no	—
157	C₂²×C₄²	—	134	4	no	—
158	C₂²×C₄²	—	134	4	no	—
159	#336	—	67	16	no	—
160	C₂⁵	C₄×D₆₇	1	1	yes	characteristic
161	C₂³×C₄	—	67	16	no	—
162	C₂³×C₄	—	67	12	no	—
163	C₂³×C₄	—	134	8	no	—
164	C₂³×C₄	—	134	8	no	—
165	C₂³×C₄	Dic₁₃₄	1	4	yes	—
166	C₂³×C₄	C₄×D₆₇	1	3	yes	—
167	C₂³×C₄	C₄×D₆₇	1	4	yes	—
168	C₂³×C₄	D₂₆₈	1	4	yes	—
169	C₂³×C₄	C₂×Dic₆₇	1	4	yes	—
170	C₂³×C₄	C₂²⋊D₆₇	1	4	yes	—
171	C₂³×C₄	C₂²⋊D₆₇	1	4	yes	—
172	C₂³×C₄	C₂²×D₆₇	1	3	yes	—
173	C₂²×C₈	—	2	32	no	—
174	C₂²×C₈	—	2	12	no	—
175	C₂²×C₈	—	2	12	no	—
176	C₂²×C₈	—	67	24	no	—
177	C₂²×C₈	—	67	4	no	—
178	C₂²×C₈	—	134	32	no	—
179	C₂²×C₈	—	134	12	no	—
180	C₂²×C₈	—	134	12	no	—
181	C₂²×C₈	C₂×Dic₆₇	1	24	yes	complemented
182	C₂²×C₈	C₂²×D₆₇	1	4	yes	—
183	C₂×C₄²	—	67	12	no	—
184	C₂×C₄²	—	67	4	no	—
185	C₂×C₄²	—	134	24	no	—
186	C₂×C₄²	—	134	12	no	—
187	C₂×C₄²	—	134	12	no	—
188	C₂⁴	C₄×Dic₆₇	1	4	yes	—
189	C₂⁴	(C₂×D₆₇)⋊C₄	1	4	yes	—
190	C₂⁴	Dic₆₇⋊C₄	1	4	yes	—
191	C₂⁴	C₈×D₆₇	1	8	yes	—
192	C₂⁴	#11761611	1	8	yes	—
193	C₂⁴	C₂×C₄×D₆₇	1	3	yes	—
194	C₂²×C₄	—	67	48	no	—
195	C₂²×C₄	—	67	8	no	—
196	C₂²×C₄	—	134	64	no	—
197	C₂²×C₄	—	134	24	no	—
198	C₂²×C₄	—	134	24	no	—
199	C₂²×C₄	C₄×Dic₆₇	1	12	yes	—
200	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	12	yes	—
201	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	16	yes	—
202	C₂²×C₄	C₂²⋊Dic₆₇	1	16	yes	—
203	C₂²×C₄	Dic₆₇⋊C₄	1	12	yes	—
204	C₂²×C₄	Dic₆₇⋊C₄	1	16	yes	—
205	C₂²×C₄	C₄⋊Dic₆₇	1	16	yes	—
206	C₂²×C₄	C₂×C₄×D₆₇	1	3	yes	—
207	C₂²×C₄	C₂×C₄×D₆₇	1	6	yes	—
208	C₂²×C₄	C₂²×Dic₆₇	1	6	yes	—
209	C₂²×C₄	C₂×C₂²⋊D₆₇	1	6	yes	—
210	C₂²×C₄	C₂×C₂²⋊D₆₇	1	6	yes	—
211	C₂²×C₄	C₂×D₂₆₈	1	6	yes	—
212	C₂²×C₄	C₂×Dic₁₃₄	1	6	yes	—
213	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
214	C₂×C₈	—	2	48	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	67	8	no	—
218	C₂×C₈	—	134	48	no	—
219	C₂×C₈	—	134	4	no	—
220	C₂×C₈	—	134	4	no	—
221	C₂×C₈	C₂²×Dic₆₇	1	8	yes	complemented
222	C₄²	—	134	24	no	—
223	C₄²	—	134	4	no	—
224	C₄²	—	134	4	no	—
225	C₂³	C₆₇⋊#95	1	16	yes	—
226	C₂³	C₈×Dic₆₇	1	16	yes	—
227	C₂³	#14523953	1	16	yes	—
228	C₂³	(C₂×D₆₇)⋊C₈	1	32	yes	—
229	C₂³	Dic₆₇⋊C₈	1	32	yes	—
230	C₂³	C₂×C₄×Dic₆₇	1	6	yes	—
231	C₂³	C₂×(C₂×D₆₇)⋊C₄	1	6	yes	—
232	C₂³	C₂×Dic₆₇⋊C₄	1	6	yes	—
233	C₂³	C₂×C₈×D₆₇	1	12	yes	—
234	C₂³	C₂×#11761611	1	12	yes	—
235	C₂³	C₂²×C₄×D₆₇	1	1	yes	characteristic
236	C₂×C₄	—	67	16	no	—
237	C₂×C₄	—	134	8	no	—
238	C₂×C₄	—	134	96	no	—
239	C₂×C₄	—	134	8	no	—
240	C₂×C₄	C₆₇⋊#95	1	48	yes	—
241	C₂×C₄	C₂×C₄×Dic₆₇	1	6	yes	—
242	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	6	yes	—
243	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	12	yes	—
244	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	12	yes	—
245	C₂×C₄	C₂×Dic₆₇⋊C₄	1	6	yes	—
246	C₂×C₄	C₂×Dic₆₇⋊C₄	1	12	yes	—
247	C₂×C₄	C₂×C₄⋊Dic₆₇	1	12	yes	—
248	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
249	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
250	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
251	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
252	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
253	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
254	C₈	—	2	8	no	—
255	C₈	—	134	8	no	—
256	C₂²	#16863150	1	64	yes	direct product factor
257	C₂²	C₂×C₆₇⋊#95	1	12	yes	—
258	C₂²	C₂×C₈×Dic₆₇	1	12	yes	—
259	C₂²	C₂×#14523953	1	12	yes	—
260	C₂²	C₂×(C₂×D₆₇)⋊C₈	1	24	yes	—
261	C₂²	C₂×Dic₆₇⋊C₈	1	24	yes	—
262	C₂²	C₂²×C₄×Dic₆₇	1	1	yes	characteristic
263	C₂²	C₂²×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
264	C₂²	C₂²×Dic₆₇⋊C₄	1	1	yes	characteristic
265	C₂²	C₂²×C₈×D₆₇	1	2	yes	—
266	C₂²	C₂²×#11761611	1	2	yes	—
267	C₄	—	134	16	no	—
268	C₄	C₂×C₆₇⋊#95	1	12	yes	—
269	C₄	C₂²×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
270	C₄	C₂²×C₂²⋊Dic₆₇	1	1	yes	characteristic
271	C₄	C₂²×Dic₆₇⋊C₄	1	1	yes	characteristic
272	C₄	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
273	C₂	C₂×#16863150	1	24	yes	direct product factor
274	C₂	C₂²×C₆₇⋊#95	1	1	yes	characteristic
275	C₂	C₂²×C₈×Dic₆₇	1	1	yes	characteristic
276	C₂	C₂²×#14523953	1	1	yes	characteristic
277	C₂	C₂²×(C₂×D₆₇)⋊C₈	1	2	yes	—
278	C₂	C₂²×Dic₆₇⋊C₈	1	2	yes	—

	subgroup	G/H
center	C₂⁴×C₄	C₂×D₆₇
commutator	C₂×C₆₇	C₂²×C₄×C₈
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂⁴×C₈×C₆₇	C₂
socle, prime radical	C₂⁵×C₆₇	C₂×C₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂⁴×C₈	D₆₇
hypocenter	C₆₇	C₂²×#336
LCS	G ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈

Complements

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

N	H ≅ G/N	type
3. C₂×#16863150	C₂	direct (N × H)
8. C₂³×C₆₇⋊C₈	C₄	semidirect
14. #16863150	C₂²	direct (N × H)
18. C₂³×C₈×C₆₇	C₄	semidirect
22. C₂³×Dic₆₇	C₈	semidirect
27. C₂²×C₆₇⋊C₈	C₂×C₄	semidirect
46. C₂²×C₈×C₆₇	C₂×C₄	semidirect
51. C₂²×Dic₆₇	C₂×C₈	semidirect
56. C₂×C₆₇⋊C₈	C₂²×C₄	semidirect
84. C₂×C₈×C₆₇	C₂²×C₄	semidirect
113. C₂×Dic₆₇	C₂²×C₈	semidirect
116. C₂²×C₆₇	#336	semidirect
134. C₂×C₆₇	C₂×#336	semidirect
144. C₆₇	C₂²×#336	semidirect
153. C₂³×C₈	Dic₆₇	semidirect
181. C₂²×C₈	C₂×Dic₆₇	semidirect
221. C₂×C₈	C₂²×Dic₆₇	semidirect
256. C₂²	#16863150	direct (N × H)
273. C₂	C₂×#16863150	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₆₇⋊C₈	2
C₂³×C₄×Dic₆₇	2
C₂×#16863150	2
C₂⁴×C₈×C₆₇	2
C₂²×#336	67

Subgroups — C22×#16863150 #30853181

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16863150 #30853181