Subgroups — C₆₇×#21260 #30700511

Order 17152.

2254 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#127×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#423×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#418×C₆₇	C₂	1	4	yes	maximal, complemented
4	C₄×#117×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1982	C₂	1	8	yes	maximal, complemented
6	C₂³×D₄×C₆₇	—	2	2	no	—
7	C₂³×D₄×C₆₇	C₂²	1	2	yes	—
8	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
9	C₂×#127×C₆₇	—	2	1	no	—
10	C₂×#127×C₆₇	—	2	1	no	—
11	C₂×#127×C₆₇	—	2	4	no	—
12	C₂×#127×C₆₇	C₄	1	8	yes	complemented
13	C₂×#127×C₆₇	C₂²	1	4	yes	—
14	C₂×#117×C₆₇	C₂²	1	2	yes	—
15	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
16	#423×C₆₇	—	2	4	no	—
17	C₂×C₄⋊C₈×C₆₇	C₂²	1	4	yes	complemented
18	#418×C₆₇	—	2	16	no	—
19	C₂×#102×C₆₇	C₂²	1	16	yes	complemented
20	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	complemented
21	C₄×C₂²⋊C₄×C₆₇	—	2	2	no	—
22	#376×C₆₇	—	2	1	no	—
23	C₄³×C₆₇	—	2	1	no	—
24	C₂⁵×C₆₇	—	2	2	no	—
25	C₂²×D₄×C₆₇	—	2	16	no	—
26	C₂²×D₄×C₆₇	—	2	8	no	—
27	C₂²×D₄×C₆₇	—	2	2	no	—
28	C₂²×D₄×C₆₇	—	2	4	no	—
29	C₂²×D₄×C₆₇	—	2	4	no	—
30	C₂²×D₄×C₆₇	—	2	4	no	—
31	C₂²×D₄×C₆₇	—	2	8	no	—
32	C₂²×D₄×C₆₇	—	4	16	no	—
33	C₂²×D₄×C₆₇	C₂×C₄	1	8	yes	—
34	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
35	C₂³×C₄×C₆₇	D₄	1	2	yes	—
36	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
37	C₂×M₁₆×C₆₇	—	2	8	no	—
38	C₂×M₁₆×C₆₇	—	2	4	no	—
39	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
40	#127×C₆₇	—	2	8	no	—
41	#127×C₆₇	—	2	8	no	—
42	#127×C₆₇	—	2	8	no	—
43	#127×C₆₇	C₂×C₄	1	16	yes	—
44	#117×C₆₇	—	2	4	no	—
45	#117×C₆₇	D₄	1	8	yes	complemented
46	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
47	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
48	C₂×C₄²×C₆₇	—	2	2	no	—
49	C₂×C₄²×C₆₇	—	2	4	no	—
50	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
51	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₆₇	D₄	1	4	yes	—
54	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
55	C₄⋊C₈×C₆₇	—	2	8	no	—
56	#102×C₆₇	—	2	32	no	—
57	#95×C₆₇	—	2	4	no	—
58	C₂⁴×C₆₇	—	2	8	no	—
59	C₂⁴×C₆₇	—	2	4	no	—
60	C₂⁴×C₆₇	—	4	16	no	—
61	C₂⁴×C₆₇	—	4	8	no	—
62	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
63	C₂×D₄×C₆₇	—	2	8	no	—
64	C₂×D₄×C₆₇	—	2	16	no	—
65	C₂×D₄×C₆₇	—	2	16	no	—
66	C₂×D₄×C₆₇	—	2	32	no	—
67	C₂×D₄×C₆₇	—	2	8	no	—
68	C₂×D₄×C₆₇	—	2	4	no	—
69	C₂×D₄×C₆₇	—	2	8	no	—
70	C₂×D₄×C₆₇	—	4	16	no	—
71	C₂×D₄×C₆₇	—	4	64	no	—
72	C₂×D₄×C₆₇	—	4	8	no	—
73	C₂×D₄×C₆₇	—	4	8	no	—
74	C₂×D₄×C₆₇	—	4	8	no	—
75	C₂×D₄×C₆₇	—	4	16	no	—
76	C₂×D₄×C₆₇	C₂²×C₄	1	8	yes	—
77	C₂²×C₄×C₆₇	—	2	8	no	—
78	C₂²×C₄×C₆₇	—	2	4	no	—
79	C₂²×C₄×C₆₇	C₂²⋊C₄	1	8	yes	—
80	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
86	C₂²×C₄×C₆₇	#41	1	4	yes	—
87	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
88	M₁₆×C₆₇	—	4	8	no	—
89	C₂×C₈×C₆₇	—	2	8	no	—
90	C₂×C₈×C₆₇	—	2	4	no	—
91	C₄⋊C₄×C₆₇	—	2	4	no	—
92	C₄⋊C₄×C₆₇	C₂×D₄	1	8	yes	complemented
93	C₂²⋊C₄×C₆₇	—	2	8	no	—
94	C₄²×C₆₇	—	2	8	no	—
95	C₄²×C₆₇	—	4	8	no	—
96	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
97	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
98	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
99	C₄²×C₆₇	C₂×D₄	1	4	yes	—
100	C₂³×C₆₇	—	2	4	no	—
101	C₂³×C₆₇	—	2	8	no	—
102	C₂³×C₆₇	—	4	32	no	—
103	C₂³×C₆₇	—	4	8	no	—
104	C₂³×C₆₇	—	4	16	no	—
105	C₂³×C₆₇	—	4	16	no	—
106	C₂³×C₆₇	—	8	32	no	—
107	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
108	C₂³×C₆₇	#121	1	4	yes	—
109	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
110	D₄×C₆₇	—	2	16	no	—
111	D₄×C₆₇	—	4	32	no	—
112	D₄×C₆₇	—	4	4	no	—
113	D₄×C₆₇	—	4	4	no	—
114	D₄×C₆₇	—	4	16	no	—
115	D₄×C₆₇	—	4	32	no	—
116	C₂×C₄×C₆₇	—	2	4	no	—
117	C₂×C₄×C₆₇	—	2	16	no	—
118	C₂×C₄×C₆₇	—	2	8	no	—
119	C₂×C₄×C₆₇	—	4	8	no	—
120	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
121	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
122	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
123	C₂×C₄×C₆₇	C₄×D₄	1	8	yes	—
124	C₂×C₄×C₆₇	#121	1	4	yes	—
125	C₂×C₄×C₆₇	#124	1	4	yes	—
126	C₂×C₄×C₆₇	#127	1	4	yes	—
127	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	8	yes	—
128	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
129	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
130	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
132	C₈×C₆₇	—	4	4	no	—
133	C₂²×C₆₇	—	2	4	no	—
134	C₂²×C₆₇	—	2	8	no	—
135	C₂²×C₆₇	—	4	16	no	—
136	C₂²×C₆₇	—	4	8	no	—
137	C₂²×C₆₇	—	8	32	no	—
138	C₂²×C₆₇	—	8	16	no	—
139	C₂²×C₆₇	#390	1	4	yes	—
140	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
141	C₂²×C₆₇	C₂×#121	1	2	yes	—
142	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	4	yes	—
143	C₄×C₆₇	—	2	4	no	—
144	C₄×C₆₇	—	4	8	no	—
145	C₄×C₆₇	#390	1	4	yes	—
146	C₄×C₆₇	#418	1	8	yes	complemented
147	C₄×C₆₇	C₂×C₄×D₄	1	2	yes	—
148	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
149	C₄×C₆₇	C₂×#127	1	1	yes	characteristic
150	#21260	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
151	C₂×C₆₇	—	2	4	no	—
152	C₂×C₆₇	—	8	8	no	—
153	C₂×C₆₇	C₂×#390	1	1	yes	characteristic
154	C₂×C₆₇	C₂×#418	1	2	yes	—
155	C₂×C₆₇	#3038	1	4	yes	—
156	C₂²×#127	C₂×C₆₇	1	1	yes	characteristic
157	C₂×#423	C₂×C₆₇	1	1	yes	characteristic, complemented
158	C₂×#418	C₂×C₆₇	1	4	yes	complemented
159	C₄×#117	C₂×C₆₇	1	1	yes	characteristic, complemented
160	#1982	C₂×C₆₇	1	8	yes	complemented
161	C₆₇	#21260	1	1	yes	characteristic, direct product factor, Sylow
162	C₂³×D₄	—	2	2	no	—
163	C₂³×D₄	C₂²×C₆₇	1	2	yes	—
164	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
165	C₂×#127	—	2	1	no	—
166	C₂×#127	—	2	1	no	—
167	C₂×#127	—	2	4	no	—
168	C₂×#127	C₄×C₆₇	1	8	yes	complemented
169	C₂×#127	C₂²×C₆₇	1	4	yes	—
170	C₂×#117	C₂²×C₆₇	1	2	yes	—
171	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
172	#423	—	2	4	no	—
173	C₂×C₄⋊C₈	C₂²×C₆₇	1	4	yes	complemented
174	#418	—	2	16	no	—
175	C₂×#102	C₂²×C₆₇	1	16	yes	complemented
176	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	complemented
177	C₄×C₂²⋊C₄	—	2	2	no	—
178	#376	—	2	1	no	—
179	C₄³	—	2	1	no	—
180	C₂⁵	—	2	2	no	—
181	C₂²×D₄	—	2	16	no	—
182	C₂²×D₄	—	2	8	no	—
183	C₂²×D₄	—	2	2	no	—
184	C₂²×D₄	—	2	4	no	—
185	C₂²×D₄	—	2	4	no	—
186	C₂²×D₄	—	2	4	no	—
187	C₂²×D₄	—	2	8	no	—
188	C₂²×D₄	—	4	16	no	—
189	C₂²×D₄	C₂×C₄×C₆₇	1	8	yes	—
190	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
191	C₂³×C₄	D₄×C₆₇	1	2	yes	—
192	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
193	C₂×M₁₆	—	2	8	no	—
194	C₂×M₁₆	—	2	4	no	—
195	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
196	#127	—	2	8	no	—
197	#127	—	2	8	no	—
198	#127	—	2	8	no	—
199	#127	C₂×C₄×C₆₇	1	16	yes	—
200	#117	—	2	4	no	—
201	#117	D₄×C₆₇	1	8	yes	complemented
202	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
203	C₂×C₂²⋊C₄	—	2	2	no	—
204	C₂×C₄²	—	2	2	no	—
205	C₂×C₄²	—	2	4	no	—
206	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
207	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
208	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
209	C₂×C₄²	D₄×C₆₇	1	4	yes	—
210	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
211	C₄⋊C₈	—	2	8	no	—
212	#102	—	2	32	no	—
213	#95	—	2	4	no	—
214	C₂⁴	—	2	8	no	—
215	C₂⁴	—	2	4	no	—
216	C₂⁴	—	4	16	no	—
217	C₂⁴	—	4	8	no	—
218	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
219	C₂×D₄	—	2	8	no	—
220	C₂×D₄	—	2	16	no	—
221	C₂×D₄	—	2	16	no	—
222	C₂×D₄	—	2	32	no	—
223	C₂×D₄	—	2	8	no	—
224	C₂×D₄	—	2	4	no	—
225	C₂×D₄	—	2	8	no	—
226	C₂×D₄	—	4	16	no	—
227	C₂×D₄	—	4	64	no	—
228	C₂×D₄	—	4	8	no	—
229	C₂×D₄	—	4	8	no	—
230	C₂×D₄	—	4	8	no	—
231	C₂×D₄	—	4	16	no	—
232	C₂×D₄	C₂²×C₄×C₆₇	1	8	yes	—
233	C₂²×C₄	—	2	8	no	—
234	C₂²×C₄	—	2	4	no	—
235	C₂²×C₄	C₂²⋊C₄×C₆₇	1	8	yes	—
236	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
237	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
238	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
239	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
240	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
241	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
242	C₂²×C₄	#41×C₆₇	1	4	yes	—
243	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
244	M₁₆	—	4	8	no	—
245	C₂×C₈	—	2	8	no	—
246	C₂×C₈	—	2	4	no	—
247	C₄⋊C₄	—	2	4	no	—
248	C₄⋊C₄	C₂×D₄×C₆₇	1	8	yes	complemented
249	C₂²⋊C₄	—	2	8	no	—
250	C₄²	—	2	8	no	—
251	C₄²	—	4	8	no	—
252	C₄²	C₂²⋊C₄×C₆₇	1	4	yes	—
253	C₄²	C₂²⋊C₄×C₆₇	1	4	yes	—
254	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
255	C₄²	C₂×D₄×C₆₇	1	4	yes	—
256	C₂³	—	2	4	no	—
257	C₂³	—	2	8	no	—
258	C₂³	—	4	32	no	—
259	C₂³	—	4	8	no	—
260	C₂³	—	4	16	no	—
261	C₂³	—	4	16	no	—
262	C₂³	—	8	32	no	—
263	C₂³	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
264	C₂³	#121×C₆₇	1	4	yes	—
265	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
266	D₄	—	2	16	no	—
267	D₄	—	4	32	no	—
268	D₄	—	4	4	no	—
269	D₄	—	4	4	no	—
270	D₄	—	4	16	no	—
271	D₄	—	4	32	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	16	no	—
274	C₂×C₄	—	2	8	no	—
275	C₂×C₄	—	4	8	no	—
276	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
277	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
278	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
279	C₂×C₄	C₄×D₄×C₆₇	1	8	yes	—
280	C₂×C₄	#121×C₆₇	1	4	yes	—
281	C₂×C₄	#124×C₆₇	1	4	yes	—
282	C₂×C₄	#127×C₆₇	1	4	yes	—
283	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	8	yes	—
284	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
285	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
286	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
287	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
288	C₈	—	4	4	no	—
289	C₂²	—	2	8	no	—
290	C₂²	—	2	4	no	—
291	C₂²	—	4	8	no	—
292	C₂²	—	4	16	no	—
293	C₂²	—	8	32	no	—
294	C₂²	—	8	16	no	—
295	C₂²	#390×C₆₇	1	4	yes	—
296	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
297	C₂²	C₂×#121×C₆₇	1	2	yes	—
298	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	4	yes	—
299	C₄	—	2	4	no	—
300	C₄	—	4	8	no	—
301	C₄	#390×C₆₇	1	4	yes	—
302	C₄	#418×C₆₇	1	8	yes	complemented
303	C₄	C₂×C₄×D₄×C₆₇	1	2	yes	—
304	C₄	C₂×#124×C₆₇	1	1	yes	characteristic
305	C₄	C₂×#127×C₆₇	1	1	yes	characteristic
306	C₂	—	2	4	no	—
307	C₂	—	8	8	no	—
308	C₂	C₂×#390×C₆₇	1	1	yes	characteristic
309	C₂	C₂×#418×C₆₇	1	2	yes	—
310	C₂	C₆₇×#3038	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂²×#127×C₆₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#423×C₆₇	C₂	semidirect
3. C₂×#418×C₆₇	C₂	semidirect
4. C₄×#117×C₆₇	C₂	semidirect
5. C₆₇×#1982	C₂	semidirect
12. C₂×#127×C₆₇	C₄	semidirect
17. C₂×C₄⋊C₈×C₆₇	C₂²	semidirect
19. C₂×#102×C₆₇	C₂²	semidirect
20. C₄×C₄⋊C₄×C₆₇	C₂²	semidirect
45. #117×C₆₇	D₄	semidirect
92. C₄⋊C₄×C₆₇	C₂×D₄	semidirect
146. C₄×C₆₇	#418	semidirect
150. #21260	C₆₇	direct (N × H)
157. C₂×#423	C₂×C₆₇	semidirect
158. C₂×#418	C₂×C₆₇	semidirect
159. C₄×#117	C₂×C₆₇	semidirect
160. #1982	C₂×C₆₇	semidirect
161. C₆₇	#21260	direct (N × H)
168. C₂×#127	C₄×C₆₇	semidirect
173. C₂×C₄⋊C₈	C₂²×C₆₇	semidirect
175. C₂×#102	C₂²×C₆₇	semidirect
176. C₄×C₄⋊C₄	C₂²×C₆₇	semidirect
201. #117	D₄×C₆₇	semidirect
248. C₄⋊C₄	C₂×D₄×C₆₇	semidirect
302. C₄	#418×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#127×C₆₇	2
C₂×#423×C₆₇	2
C₂×#418×C₆₇	2
C₄×#117×C₆₇	2
C₆₇×#1982	2
#21260	67

Subgroups — C67×#21260 #30700511

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21260 #30700511