Subgroups — C₆₇×#21958 #30701209

Order 17152.

790 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	C₂×#433×C₆₇	C₂	1	2	yes	maximal
2	C₆₇×#2296	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#1940	C₂	1	4	yes	maximal, complemented
4	C₆₇×#1939	C₂	1	4	yes	maximal, complemented
5	C₆₇×#1798	C₂	1	2	yes	maximal
6	C₆₇×#1797	C₂	1	2	yes	maximal
7	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
8	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
9	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
10	#433×C₆₇	—	2	8	no	—
11	C₂×C₄⋊C₈×C₆₇	C₂²	1	4	yes	—
12	C₂×C₄⋊C₈×C₆₇	C₂²	1	4	yes	—
13	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
14	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
15	C₂×C₂²⋊C₈×C₆₇	C₂²	1	4	yes	—
16	C₂×#97×C₆₇	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
18	#389×C₆₇	C₄	1	2	yes	—
19	#389×C₆₇	C₄	1	2	yes	—
20	#385×C₆₇	C₄	1	2	yes	—
21	#385×C₆₇	C₄	1	2	yes	—
22	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
23	#336×C₆₇	C₂²	1	2	yes	—
24	#336×C₆₇	C₂²	1	2	yes	—
25	#336×C₆₇	C₂²	1	4	yes	—
26	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
29	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
30	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
31	#117×C₆₇	—	4	2	no	—
32	#117×C₆₇	D₄	1	4	yes	—
33	#117×C₆₇	Q₈	1	4	yes	—
34	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
35	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
38	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
39	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
40	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₆₇	—	4	2	no	—
42	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
44	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
45	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
46	C₄⋊C₈×C₆₇	—	2	8	no	—
47	C₄⋊C₈×C₆₇	—	4	4	no	—
48	C₄⋊C₈×C₆₇	—	4	2	no	—
49	C₄⋊C₈×C₆₇	—	4	2	no	—
50	C₂²⋊C₈×C₆₇	—	2	8	no	—
51	#97×C₆₇	—	2	4	no	—
52	C₄×C₈×C₆₇	—	2	4	no	—
53	#95×C₆₇	C₂×C₄	1	4	yes	—
54	#95×C₆₇	C₂×C₄	1	4	yes	—
55	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₆₇	—	2	2	no	—
57	C₂²×C₄×C₆₇	—	2	2	no	—
58	C₂²×C₄×C₆₇	—	2	4	no	—
59	C₂²×C₄×C₆₇	—	4	1	no	—
60	C₂²×C₄×C₆₇	—	4	1	no	—
61	C₂²×C₄×C₆₇	—	4	2	no	—
62	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
67	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
70	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
71	C₂×C₈×C₆₇	—	2	4	no	—
72	C₂×C₈×C₆₇	—	2	4	no	—
73	C₂×C₈×C₆₇	—	2	4	no	—
74	C₂×C₈×C₆₇	—	2	4	no	—
75	C₂×C₈×C₆₇	—	2	4	no	—
76	C₂×C₈×C₆₇	—	2	4	no	—
77	C₄⋊C₄×C₆₇	—	2	8	no	—
78	C₄⋊C₄×C₆₇	—	4	4	no	—
79	C₄⋊C₄×C₆₇	—	4	1	no	—
80	C₄⋊C₄×C₆₇	—	4	1	no	—
81	C₄⋊C₄×C₆₇	C₄⋊C₄	1	8	yes	—
82	C₂²⋊C₄×C₆₇	—	4	2	no	—
83	C₂²⋊C₄×C₆₇	C₄⋊C₄	1	8	yes	—
84	C₄²×C₆₇	—	2	4	no	—
85	C₄²×C₆₇	—	2	2	no	—
86	C₄²×C₆₇	—	2	2	no	—
87	C₄²×C₆₇	—	4	2	no	—
88	C₄²×C₆₇	C₂×D₄	1	4	yes	—
89	C₄²×C₆₇	C₂×Q₈	1	4	yes	—
90	C₂³×C₆₇	—	2	1	no	—
91	C₂³×C₆₇	—	2	1	no	—
92	C₂³×C₆₇	—	2	2	no	—
93	C₂³×C₆₇	—	2	2	no	—
94	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
95	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
96	C₂×C₄×C₆₇	—	2	2	no	—
97	C₂×C₄×C₆₇	—	2	1	no	—
98	C₂×C₄×C₆₇	—	2	1	no	—
99	C₂×C₄×C₆₇	—	2	4	no	—
100	C₂×C₄×C₆₇	—	2	2	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	2	no	—
105	C₂×C₄×C₆₇	—	2	2	no	—
106	C₂×C₄×C₆₇	—	2	2	no	—
107	C₂×C₄×C₆₇	—	2	4	no	—
108	C₂×C₄×C₆₇	—	2	2	no	—
109	C₂×C₄×C₆₇	—	2	4	no	—
110	C₂×C₄×C₆₇	—	2	2	no	—
111	C₂×C₄×C₆₇	—	4	4	no	—
112	C₂×C₄×C₆₇	—	4	4	no	—
113	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	8	yes	—
114	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
115	C₂×C₄×C₆₇	#131	1	4	yes	—
116	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
118	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
119	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
120	C₈×C₆₇	—	4	4	no	—
121	C₈×C₆₇	—	4	4	no	—
122	C₂²×C₆₇	—	2	2	no	—
123	C₂²×C₆₇	—	2	4	no	—
124	C₂²×C₆₇	—	4	1	no	—
125	C₂²×C₆₇	—	4	1	no	—
126	C₂²×C₆₇	—	4	2	no	—
127	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
128	C₂²×C₆₇	#518	1	1	yes	characteristic
129	C₂²×C₆₇	#519	1	1	yes	characteristic
130	C₂²×C₆₇	C₂×#131	1	2	yes	—
131	C₂²×C₆₇	#568	1	2	yes	—
132	C₄×C₆₇	—	2	2	no	—
133	C₄×C₆₇	—	2	8	no	—
134	C₄×C₆₇	—	2	4	no	—
135	C₄×C₆₇	—	4	2	no	—
136	C₄×C₆₇	—	4	2	no	—
137	C₄×C₆₇	—	4	2	no	—
138	C₄×C₆₇	#537	1	2	yes	—
139	C₄×C₆₇	#544	1	2	yes	—
140	#21958	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
141	C₂×C₆₇	—	4	2	no	—
142	C₂×C₆₇	#2316	1	1	yes	characteristic
143	C₂×C₆₇	#2906	1	2	yes	—
144	C₂×C₆₇	#2977	1	1	yes	characteristic
145	C₂×C₆₇	#2978	1	1	yes	characteristic
146	C₂×C₆₇	#2985	1	2	yes	—
147	C₂×#433	C₂×C₆₇	1	2	yes	—
148	#2296	C₂×C₆₇	1	1	yes	characteristic
149	#1940	C₂×C₆₇	1	4	yes	complemented
150	#1939	C₂×C₆₇	1	4	yes	complemented
151	#1798	C₂×C₆₇	1	2	yes	—
152	#1797	C₂×C₆₇	1	2	yes	—
153	C₆₇	#21958	1	1	yes	characteristic, direct product factor, Sylow
154	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
155	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
156	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
157	#433	—	2	8	no	—
158	C₂×C₄⋊C₈	C₂²×C₆₇	1	4	yes	—
159	C₂×C₄⋊C₈	C₂²×C₆₇	1	4	yes	—
160	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
161	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
162	C₂×C₂²⋊C₈	C₂²×C₆₇	1	4	yes	—
163	C₂×#97	C₂²×C₆₇	1	2	yes	—
164	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
165	#389	C₄×C₆₇	1	2	yes	—
166	#389	C₄×C₆₇	1	2	yes	—
167	#385	C₄×C₆₇	1	2	yes	—
168	#385	C₄×C₆₇	1	2	yes	—
169	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
170	#336	C₂²×C₆₇	1	2	yes	—
171	#336	C₂²×C₆₇	1	2	yes	—
172	#336	C₂²×C₆₇	1	4	yes	—
173	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
174	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
175	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
176	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
177	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
178	#117	—	4	2	no	—
179	#117	D₄×C₆₇	1	4	yes	—
180	#117	Q₈×C₆₇	1	4	yes	—
181	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
182	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
183	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
184	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
185	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
186	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
187	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
188	C₂×C₄²	—	4	2	no	—
189	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
190	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
191	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
192	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
193	C₄⋊C₈	—	2	8	no	—
194	C₄⋊C₈	—	4	4	no	—
195	C₄⋊C₈	—	4	2	no	—
196	C₄⋊C₈	—	4	2	no	—
197	C₂²⋊C₈	—	2	8	no	—
198	#97	—	2	4	no	—
199	C₄×C₈	—	2	4	no	—
200	#95	C₂×C₄×C₆₇	1	4	yes	—
201	#95	C₂×C₄×C₆₇	1	4	yes	—
202	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	4	1	no	—
207	C₂²×C₄	—	4	1	no	—
208	C₂²×C₄	—	4	2	no	—
209	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
210	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
211	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
212	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
213	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
214	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
215	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
216	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
217	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	2	4	no	—
224	C₄⋊C₄	—	2	8	no	—
225	C₄⋊C₄	—	4	1	no	—
226	C₄⋊C₄	—	4	1	no	—
227	C₄⋊C₄	—	4	4	no	—
228	C₄⋊C₄	C₄⋊C₄×C₆₇	1	8	yes	—
229	C₂²⋊C₄	—	4	2	no	—
230	C₂²⋊C₄	C₄⋊C₄×C₆₇	1	8	yes	—
231	C₄²	—	2	4	no	—
232	C₄²	—	2	2	no	—
233	C₄²	—	2	2	no	—
234	C₄²	—	4	2	no	—
235	C₄²	C₂×D₄×C₆₇	1	4	yes	—
236	C₄²	C₂×Q₈×C₆₇	1	4	yes	—
237	C₂³	—	2	1	no	—
238	C₂³	—	2	1	no	—
239	C₂³	—	2	2	no	—
240	C₂³	—	2	2	no	—
241	C₂³	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
242	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	8	yes	—
261	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
262	C₂×C₄	#131×C₆₇	1	4	yes	—
263	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
264	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
265	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
266	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
267	C₈	—	4	4	no	—
268	C₈	—	4	4	no	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	4	no	—
271	C₂²	—	4	1	no	—
272	C₂²	—	4	1	no	—
273	C₂²	—	4	2	no	—
274	C₂²	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
275	C₂²	#518×C₆₇	1	1	yes	characteristic
276	C₂²	#519×C₆₇	1	1	yes	characteristic
277	C₂²	C₂×#131×C₆₇	1	2	yes	—
278	C₂²	#568×C₆₇	1	2	yes	—
279	C₄	—	2	2	no	—
280	C₄	—	2	8	no	—
281	C₄	—	2	4	no	—
282	C₄	—	4	2	no	—
283	C₄	—	4	2	no	—
284	C₄	—	4	2	no	—
285	C₄	#537×C₆₇	1	2	yes	—
286	C₄	#544×C₆₇	1	2	yes	—
287	C₂	—	4	2	no	—
288	C₂	C₆₇×#2316	1	1	yes	characteristic
289	C₂	C₆₇×#2906	1	2	yes	—
290	C₂	C₆₇×#2977	1	1	yes	characteristic
291	C₂	C₆₇×#2978	1	1	yes	characteristic
292	C₂	C₆₇×#2985	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂⁴
commutator	C₂³	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂³×C₆₇	C₂³×C₄
prime radical	C₂⁴×C₆₇	C₂²×C₄
LCS	G ▷ C₂³ ▷ 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×#1940	C₂	semidirect
4. C₆₇×#1939	C₂	semidirect
140. #21958	C₆₇	direct (N × H)
149. #1940	C₂×C₆₇	semidirect
150. #1939	C₂×C₆₇	semidirect
153. C₆₇	#21958	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#433×C₆₇	2
C₆₇×#2296	2
C₆₇×#1940	2
C₆₇×#1939	2
C₆₇×#1798	2
C₆₇×#1797	2
#21958	67

Subgroups — C67×#21958 #30701209

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21958 #30701209