Subgroups — C₆₇×#21930 #30701181

Order 17152.

1306 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂²⋊C₈×C₆₇	C₂	1	2	yes	maximal
2	C₂×C₄×C₂²⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#1850	C₂	1	8	yes	maximal, complemented
4	C₆₇×#1797	C₂	1	4	yes	maximal, complemented
5	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
6	C₂³×C₈×C₆₇	C₂²	1	4	yes	—
7	C₂²×C₂²⋊C₄×C₆₇	C₄	1	1	yes	characteristic
8	C₂²×C₂²⋊C₄×C₆₇	C₄	1	1	yes	characteristic
9	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
10	C₂×C₄⋊C₈×C₆₇	C₂²	1	8	yes	complemented
11	C₂×C₂²⋊C₈×C₆₇	—	2	8	no	—
12	C₂×C₂²⋊C₈×C₆₇	C₂²	1	4	yes	—
13	C₂×C₂²⋊C₈×C₆₇	C₂²	1	4	yes	—
14	C₄×C₂²⋊C₄×C₆₇	C₄	1	4	yes	—
15	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
16	C₂×#95×C₆₇	C₄	1	2	yes	—
17	#336×C₆₇	C₂²	1	8	yes	complemented
18	C₂⁵×C₆₇	C₂×C₄	1	1	yes	characteristic
19	C₂³×C₄×C₆₇	—	4	4	no	—
20	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
21	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
23	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
24	C₂³×C₄×C₆₇	D₄	1	4	yes	—
25	C₂³×C₄×C₆₇	Q₈	1	4	yes	—
26	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
27	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
28	C₂²×C₈×C₆₇	—	2	8	no	—
29	C₂²×C₈×C₆₇	—	4	8	no	—
30	C₂²×C₈×C₆₇	C₂³	1	8	yes	—
31	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
32	C₂×C₂²⋊C₄×C₆₇	C₈	1	8	yes	complemented
33	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
34	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
35	C₂×C₄²×C₆₇	—	4	4	no	—
36	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
37	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
38	C₄⋊C₈×C₆₇	—	4	8	no	—
39	C₂²⋊C₈×C₆₇	—	2	8	no	—
40	C₂²⋊C₈×C₆₇	—	2	8	no	—
41	#95×C₆₇	C₂×C₄	1	8	yes	—
42	C₂⁴×C₆₇	—	4	2	no	—
43	C₂⁴×C₆₇	—	4	2	no	—
44	C₂⁴×C₆₇	C₄⋊C₄	1	4	yes	—
45	C₂⁴×C₆₇	C₂×C₈	1	4	yes	—
46	C₂⁴×C₆₇	M₁₆	1	4	yes	—
47	C₂⁴×C₆₇	C₂²×C₄	1	2	yes	—
48	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₆₇	—	2	4	no	—
50	C₂²×C₄×C₆₇	—	2	8	no	—
51	C₂²×C₄×C₆₇	—	2	4	no	—
52	C₂²×C₄×C₆₇	—	4	8	no	—
53	C₂²×C₄×C₆₇	—	4	8	no	—
54	C₂²×C₄×C₆₇	—	4	2	no	—
55	C₂²×C₄×C₆₇	—	4	2	no	—
56	C₂²×C₄×C₆₇	—	4	4	no	—
57	C₂²×C₄×C₆₇	—	4	2	no	—
58	C₂²×C₄×C₆₇	—	4	2	no	—
59	C₂²×C₄×C₆₇	—	4	2	no	—
60	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
61	C₂²×C₄×C₆₇	C₄⋊C₄	1	8	yes	—
62	C₂²×C₄×C₆₇	C₂×C₈	1	8	yes	complemented
63	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
64	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
67	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
68	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
70	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
71	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
72	C₂×C₈×C₆₇	—	2	8	no	—
73	C₂×C₈×C₆₇	—	2	16	no	—
74	C₂×C₈×C₆₇	—	4	16	no	—
75	C₂²⋊C₄×C₆₇	—	2	8	no	—
76	C₂²⋊C₄×C₆₇	C₂×C₈	1	16	yes	complemented
77	C₄²×C₆₇	—	2	8	no	—
78	C₄²×C₆₇	—	4	4	no	—
79	C₂³×C₆₇	—	2	2	no	—
80	C₂³×C₆₇	—	2	4	no	—
81	C₂³×C₆₇	—	2	2	no	—
82	C₂³×C₆₇	—	2	4	no	—
83	C₂³×C₆₇	—	4	4	no	—
84	C₂³×C₆₇	—	4	4	no	—
85	C₂³×C₆₇	—	4	8	no	—
86	C₂³×C₆₇	—	4	8	no	—
87	C₂³×C₆₇	C₄⋊C₈	1	16	yes	complemented
88	C₂³×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
89	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
90	C₂³×C₆₇	C₂²×C₈	1	4	yes	—
91	C₂³×C₆₇	C₂²×C₈	1	2	yes	—
92	C₂³×C₆₇	C₂×M₁₆	1	4	yes	—
93	C₂³×C₆₇	C₂×M₁₆	1	2	yes	—
94	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
95	C₂×C₄×C₆₇	—	2	4	no	—
96	C₂×C₄×C₆₇	—	2	2	no	—
97	C₂×C₄×C₆₇	—	2	4	no	—
98	C₂×C₄×C₆₇	—	2	4	no	—
99	C₂×C₄×C₆₇	—	2	8	no	—
100	C₂×C₄×C₆₇	—	2	4	no	—
101	C₂×C₄×C₆₇	—	2	4	no	—
102	C₂×C₄×C₆₇	—	2	8	no	—
103	C₂×C₄×C₆₇	—	2	4	no	—
104	C₂×C₄×C₆₇	—	2	4	no	—
105	C₂×C₄×C₆₇	—	2	4	no	—
106	C₂×C₄×C₆₇	—	4	8	no	—
107	C₂×C₄×C₆₇	—	4	4	no	—
108	C₂×C₄×C₆₇	—	4	8	no	—
109	C₂×C₄×C₆₇	—	4	8	no	—
110	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
112	C₂×C₄×C₆₇	C₂²×C₈	1	8	yes	—
113	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
115	C₂×C₄×C₆₇	H(5,𝔽₂)	1	4	yes	—
116	C₈×C₆₇	—	4	8	no	—
117	C₂²×C₆₇	—	2	4	no	—
118	C₂²×C₆₇	—	2	2	no	—
119	C₂²×C₆₇	—	2	2	no	—
120	C₂²×C₆₇	—	2	4	no	—
121	C₂²×C₆₇	—	2	4	no	—
122	C₂²×C₆₇	—	2	8	no	—
123	C₂²×C₆₇	—	2	4	no	—
124	C₂²×C₆₇	—	2	8	no	—
125	C₂²×C₆₇	—	4	16	no	—
126	C₂²×C₆₇	C₂×C₄⋊C₈	1	8	yes	—
127	C₂²×C₆₇	C₂×C₄⋊C₈	1	4	yes	—
128	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
129	C₂²×C₆₇	#518	1	2	yes	—
130	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
131	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
132	C₂²×C₆₇	#568	1	2	yes	—
133	C₄×C₆₇	—	2	2	no	—
134	C₄×C₆₇	—	2	4	no	—
135	C₄×C₆₇	—	2	8	no	—
136	C₄×C₆₇	—	4	4	no	—
137	C₄×C₆₇	#534	1	2	yes	—
138	C₄×C₆₇	#543	1	2	yes	—
139	#21930	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
140	C₂×C₆₇	—	2	4	no	—
141	C₂×C₆₇	—	2	8	no	—
142	C₂×C₆₇	#2315	1	1	yes	characteristic
143	C₂×C₆₇	C₂²×C₄⋊C₈	1	1	yes	characteristic
144	C₂×C₆₇	#2905	1	1	yes	characteristic
145	C₂×C₆₇	#2972	1	2	yes	—
146	C₂×C₆₇	#2973	1	2	yes	—
147	C₂²×C₂²⋊C₈	C₂×C₆₇	1	2	yes	—
148	C₂×C₄×C₂²⋊C₄	C₂×C₆₇	1	1	yes	characteristic
149	#1850	C₂×C₆₇	1	8	yes	complemented
150	#1797	C₂×C₆₇	1	4	yes	complemented
151	C₆₇	#21930	1	1	yes	characteristic, direct product factor, Sylow
152	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
153	C₂³×C₈	C₂²×C₆₇	1	4	yes	—
154	C₂²×C₂²⋊C₄	C₄×C₆₇	1	1	yes	characteristic
155	C₂²×C₂²⋊C₄	C₄×C₆₇	1	1	yes	characteristic
156	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
157	C₂×C₄⋊C₈	C₂²×C₆₇	1	8	yes	complemented
158	C₂×C₂²⋊C₈	—	2	8	no	—
159	C₂×C₂²⋊C₈	C₂²×C₆₇	1	4	yes	—
160	C₂×C₂²⋊C₈	C₂²×C₆₇	1	4	yes	—
161	C₄×C₂²⋊C₄	C₄×C₆₇	1	4	yes	—
162	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
163	C₂×#95	C₄×C₆₇	1	2	yes	—
164	#336	C₂²×C₆₇	1	8	yes	complemented
165	C₂⁵	C₂×C₄×C₆₇	1	1	yes	characteristic
166	C₂³×C₄	—	4	4	no	—
167	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
168	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
169	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
170	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
171	C₂³×C₄	D₄×C₆₇	1	4	yes	—
172	C₂³×C₄	Q₈×C₆₇	1	4	yes	—
173	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
174	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
175	C₂²×C₈	—	2	8	no	—
176	C₂²×C₈	—	4	8	no	—
177	C₂²×C₈	C₂³×C₆₇	1	8	yes	—
178	C₂×C₂²⋊C₄	—	2	4	no	—
179	C₂×C₂²⋊C₄	C₈×C₆₇	1	8	yes	complemented
180	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
181	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
182	C₂×C₄²	—	4	4	no	—
183	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
184	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
185	C₄⋊C₈	—	4	8	no	—
186	C₂²⋊C₈	—	2	8	no	—
187	C₂²⋊C₈	—	2	8	no	—
188	#95	C₂×C₄×C₆₇	1	8	yes	—
189	C₂⁴	—	4	2	no	—
190	C₂⁴	—	4	2	no	—
191	C₂⁴	C₄⋊C₄×C₆₇	1	4	yes	—
192	C₂⁴	C₂×C₈×C₆₇	1	4	yes	—
193	C₂⁴	M₁₆×C₆₇	1	4	yes	—
194	C₂⁴	C₂²×C₄×C₆₇	1	2	yes	—
195	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	—	2	8	no	—
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	—	4	8	no	—
200	C₂²×C₄	—	4	8	no	—
201	C₂²×C₄	—	4	2	no	—
202	C₂²×C₄	—	4	2	no	—
203	C₂²×C₄	—	4	4	no	—
204	C₂²×C₄	—	4	2	no	—
205	C₂²×C₄	—	4	2	no	—
206	C₂²×C₄	—	4	2	no	—
207	C₂²×C₄	C₄⋊C₄×C₆₇	1	4	yes	—
208	C₂²×C₄	C₄⋊C₄×C₆₇	1	8	yes	—
209	C₂²×C₄	C₂×C₈×C₆₇	1	8	yes	complemented
210	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
211	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
212	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
213	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
214	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
215	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
216	C₂²×C₄	C₂×Q₈×C₆₇	1	4	yes	—
217	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
218	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
219	C₂×C₈	—	2	8	no	—
220	C₂×C₈	—	2	16	no	—
221	C₂×C₈	—	4	16	no	—
222	C₂²⋊C₄	—	2	8	no	—
223	C₂²⋊C₄	C₂×C₈×C₆₇	1	16	yes	complemented
224	C₄²	—	2	8	no	—
225	C₄²	—	4	4	no	—
226	C₂³	—	2	2	no	—
227	C₂³	—	2	4	no	—
228	C₂³	—	2	2	no	—
229	C₂³	—	2	4	no	—
230	C₂³	—	4	4	no	—
231	C₂³	—	4	4	no	—
232	C₂³	—	4	8	no	—
233	C₂³	—	4	8	no	—
234	C₂³	C₄⋊C₈×C₆₇	1	16	yes	complemented
235	C₂³	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
236	C₂³	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
237	C₂³	C₂²×C₈×C₆₇	1	4	yes	—
238	C₂³	C₂²×C₈×C₆₇	1	2	yes	—
239	C₂³	C₂×M₁₆×C₆₇	1	4	yes	—
240	C₂³	C₂×M₁₆×C₆₇	1	2	yes	—
241	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	8	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	8	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	4	8	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	4	8	no	—
256	C₂×C₄	—	4	8	no	—
257	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
258	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
259	C₂×C₄	C₂²×C₈×C₆₇	1	8	yes	—
260	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
261	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
262	C₂×C₄	H(5,𝔽₂)×C₆₇	1	4	yes	—
263	C₈	—	4	8	no	—
264	C₂²	—	2	4	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	4	no	—
268	C₂²	—	2	4	no	—
269	C₂²	—	2	8	no	—
270	C₂²	—	2	4	no	—
271	C₂²	—	2	8	no	—
272	C₂²	—	4	16	no	—
273	C₂²	C₂×C₄⋊C₈×C₆₇	1	8	yes	—
274	C₂²	C₂×C₄⋊C₈×C₆₇	1	4	yes	—
275	C₂²	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
276	C₂²	#518×C₆₇	1	2	yes	—
277	C₂²	C₂³×C₈×C₆₇	1	1	yes	characteristic
278	C₂²	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
279	C₂²	#568×C₆₇	1	2	yes	—
280	C₄	—	2	2	no	—
281	C₄	—	2	4	no	—
282	C₄	—	2	8	no	—
283	C₄	—	4	4	no	—
284	C₄	#534×C₆₇	1	2	yes	—
285	C₄	#543×C₆₇	1	2	yes	—
286	C₂	—	2	4	no	—
287	C₂	—	2	8	no	—
288	C₂	C₆₇×#2315	1	1	yes	characteristic
289	C₂	C₂²×C₄⋊C₈×C₆₇	1	1	yes	characteristic
290	C₂	C₆₇×#2905	1	1	yes	characteristic
291	C₂	C₆₇×#2972	1	2	yes	—
292	C₂	C₆₇×#2973	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₆₇×#1850	C₂	semidirect
4. C₆₇×#1797	C₂	semidirect
10. C₂×C₄⋊C₈×C₆₇	C₂²	semidirect
17. #336×C₆₇	C₂²	semidirect
32. C₂×C₂²⋊C₄×C₆₇	C₈	semidirect
62. C₂²×C₄×C₆₇	C₂×C₈	semidirect
76. C₂²⋊C₄×C₆₇	C₂×C₈	semidirect
87. C₂³×C₆₇	C₄⋊C₈	semidirect
139. #21930	C₆₇	direct (N × H)
149. #1850	C₂×C₆₇	semidirect
150. #1797	C₂×C₆₇	semidirect
151. C₆₇	#21930	direct (N × H)
157. C₂×C₄⋊C₈	C₂²×C₆₇	semidirect
164. #336	C₂²×C₆₇	semidirect
179. C₂×C₂²⋊C₄	C₈×C₆₇	semidirect
209. C₂²×C₄	C₂×C₈×C₆₇	semidirect
223. C₂²⋊C₄	C₂×C₈×C₆₇	semidirect
234. C₂³	C₄⋊C₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₂²⋊C₈×C₆₇	2
C₂×C₄×C₂²⋊C₄×C₆₇	2
C₆₇×#1850	2
C₆₇×#1797	2
#21930	67

Subgroups — C67×#21930 #30701181

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21930 #30701181