Subgroups — C₂²×#454×C₆₇ #30739631

Order 17152.

4894 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×2₋¹⁺⁴×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×AGL(1,ℤ/8)×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₂²×#124×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#454×C₆₇	C₂	1	24	yes	maximal, direct product factor
5	C₂²×#104×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂²×#101×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×2₋¹⁺⁴×C₆₇	—	2	12	no	—
8	C₂×2₋¹⁺⁴×C₆₇	—	2	6	no	—
9	C₂×2₋¹⁺⁴×C₆₇	C₂²	1	12	yes	complemented
10	C₂²×#41×C₆₇	—	2	2	no	—
11	C₂²×#41×C₆₇	—	2	2	no	—
12	C₂²×#41×C₆₇	C₂²	1	2	yes	—
13	C₂³×Q₈×C₆₇	—	2	2	no	—
14	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×AGL(1,ℤ/8)×C₆₇	—	2	12	no	—
17	C₂×AGL(1,ℤ/8)×C₆₇	C₂²	1	24	yes	complemented
18	C₂²×SD₁₆×C₆₇	—	2	2	no	—
19	C₂²×D₈×C₆₇	—	2	2	no	—
20	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
21	C₂×#124×C₆₇	—	2	6	no	—
22	C₂×#124×C₆₇	C₂²	1	12	yes	complemented
23	C₂²×C₂²⋊C₄×C₆₇	—	2	2	no	—
24	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
25	#454×C₆₇	C₂²	1	64	yes	direct product factor
26	C₂×#104×C₆₇	C₂²	1	24	yes	complemented
27	C₂×#101×C₆₇	C₂²	1	12	yes	complemented
28	C₂⁵×C₆₇	—	2	1	no	—
29	2₋¹⁺⁴×C₆₇	—	2	48	no	—
30	2₋¹⁺⁴×C₆₇	—	2	48	no	—
31	2₋¹⁺⁴×C₆₇	—	2	24	no	—
32	2₋¹⁺⁴×C₆₇	C₂³	1	16	yes	complemented
33	C₂×#41×C₆₇	—	2	24	no	—
34	C₂×#41×C₆₇	—	2	24	no	—
35	C₂×#41×C₆₇	—	2	24	no	—
36	C₂×#41×C₆₇	—	2	24	no	—
37	C₂×#41×C₆₇	—	2	6	no	—
38	C₂×#41×C₆₇	—	2	6	no	—
39	C₂×#41×C₆₇	D₄	1	12	yes	—
40	C₂×#41×C₆₇	C₂³	1	12	yes	—
41	C₂²×Q₈×C₆₇	—	2	2	no	—
42	C₂²×Q₈×C₆₇	—	2	24	no	—
43	C₂²×Q₈×C₆₇	—	2	24	no	—
44	C₂²×Q₈×C₆₇	—	2	2	no	—
45	C₂²×Q₈×C₆₇	—	2	6	no	—
46	C₂²×Q₈×C₆₇	D₄	1	2	yes	—
47	C₂²×Q₈×C₆₇	D₄	1	6	yes	—
48	C₂²×Q₈×C₆₇	C₂³	1	6	yes	—
49	C₂²×D₄×C₆₇	—	2	2	no	—
50	C₂²×D₄×C₆₇	—	2	2	no	—
51	C₂²×D₄×C₆₇	—	2	2	no	—
52	C₂²×D₄×C₆₇	—	2	6	no	—
53	C₂²×D₄×C₆₇	—	2	3	no	—
54	C₂²×D₄×C₆₇	D₄	1	2	yes	—
55	C₂²×D₄×C₆₇	C₂³	1	6	yes	—
56	C₂³×C₄×C₆₇	—	2	4	no	—
57	C₂³×C₄×C₆₇	—	2	2	no	—
58	C₂³×C₄×C₆₇	—	2	1	no	—
59	C₂³×C₄×C₆₇	D₄	1	2	yes	—
60	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
61	AGL(1,ℤ/8)×C₆₇	—	2	48	no	—
62	AGL(1,ℤ/8)×C₆₇	C₂³	1	32	yes	complemented
63	C₂×SD₁₆×C₆₇	—	2	24	no	—
64	C₂×D₈×C₆₇	—	2	24	no	—
65	C₂×M₁₆×C₆₇	—	2	6	no	—
66	C₂×M₁₆×C₆₇	C₂³	1	12	yes	—
67	C₂²×C₈×C₆₇	—	2	2	no	—
68	#124×C₆₇	—	2	24	no	—
69	#124×C₆₇	C₂³	1	16	yes	complemented
70	C₂×C₂²⋊C₄×C₆₇	—	2	24	no	—
71	C₂×C₄²×C₆₇	—	2	3	no	—
72	C₂×C₄²×C₆₇	C₂³	1	6	yes	—
73	#104×C₆₇	C₂³	1	32	yes	complemented
74	#101×C₆₇	C₂³	1	16	yes	complemented
75	C₂⁴×C₆₇	—	2	2	no	—
76	C₂⁴×C₆₇	—	2	6	no	—
77	C₂⁴×C₆₇	—	4	8	no	—
78	C₂⁴×C₆₇	—	4	3	no	—
79	C₂⁴×C₆₇	—	4	1	no	—
80	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
81	#41×C₆₇	—	2	96	no	—
82	#41×C₆₇	—	2	96	no	—
83	#41×C₆₇	—	2	32	no	—
84	#41×C₆₇	—	2	24	no	—
85	#41×C₆₇	—	2	32	no	—
86	#41×C₆₇	—	2	12	no	—
87	#41×C₆₇	—	2	12	no	—
88	#41×C₆₇	C₂×D₄	1	24	yes	—
89	#41×C₆₇	C₂⁴	1	8	yes	—
90	C₂×Q₈×C₆₇	—	2	24	no	—
91	C₂×Q₈×C₆₇	—	2	96	no	—
92	C₂×Q₈×C₆₇	—	2	32	no	—
93	C₂×Q₈×C₆₇	—	2	6	no	—
94	C₂×Q₈×C₆₇	—	2	12	no	—
95	C₂×Q₈×C₆₇	—	2	12	no	—
96	C₂×Q₈×C₆₇	—	2	12	no	—
97	C₂×Q₈×C₆₇	—	2	12	no	—
98	C₂×Q₈×C₆₇	C₂×D₄	1	12	yes	—
99	C₂×Q₈×C₆₇	C₂×D₄	1	12	yes	—
100	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
101	C₂×D₄×C₆₇	—	2	24	no	—
102	C₂×D₄×C₆₇	—	2	12	no	—
103	C₂×D₄×C₆₇	—	2	12	no	—
104	C₂×D₄×C₆₇	—	2	6	no	—
105	C₂×D₄×C₆₇	—	2	12	no	—
106	C₂×D₄×C₆₇	—	2	12	no	—
107	C₂×D₄×C₆₇	—	2	6	no	—
108	C₂×D₄×C₆₇	—	4	6	no	—
109	C₂×D₄×C₆₇	—	4	6	no	—
110	C₂×D₄×C₆₇	C₂×D₄	1	12	yes	—
111	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
112	C₂²×C₄×C₆₇	—	2	48	no	—
113	C₂²×C₄×C₆₇	—	2	6	no	—
114	C₂²×C₄×C₆₇	—	2	2	no	—
115	C₂²×C₄×C₆₇	—	2	2	no	—
116	C₂²×C₄×C₆₇	—	2	12	no	—
117	C₂²×C₄×C₆₇	—	2	12	no	—
118	C₂²×C₄×C₆₇	—	2	6	no	—
119	C₂²×C₄×C₆₇	—	4	4	no	—
120	C₂²×C₄×C₆₇	C₂×D₄	1	12	yes	—
121	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
122	C₂²×C₄×C₆₇	C₂×D₄	1	6	yes	—
123	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
124	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
125	SD₁₆×C₆₇	—	2	32	no	—
126	D₈×C₆₇	—	2	32	no	—
127	M₁₆×C₆₇	—	2	12	no	—
128	M₁₆×C₆₇	C₂⁴	1	8	yes	—
129	C₂×C₈×C₆₇	—	2	12	no	—
130	C₂²⋊C₄×C₆₇	—	2	32	no	—
131	C₄²×C₆₇	—	2	6	no	—
132	C₄²×C₆₇	C₂⁴	1	4	yes	—
133	C₂³×C₆₇	—	2	4	no	—
134	C₂³×C₆₇	—	2	12	no	—
135	C₂³×C₆₇	—	2	4	no	—
136	C₂³×C₆₇	—	4	24	no	—
137	C₂³×C₆₇	—	4	8	no	—
138	C₂³×C₆₇	—	4	6	no	—
139	C₂³×C₆₇	—	4	6	no	—
140	C₂³×C₆₇	—	8	12	no	—
141	C₂³×C₆₇	—	8	4	no	—
142	C₂³×C₆₇	#120	1	3	yes	—
143	C₂³×C₆₇	#120	1	1	yes	characteristic
144	C₂³×C₆₇	C₂²×D₄	1	3	yes	—
145	Q₈×C₆₇	—	2	32	no	—
146	Q₈×C₆₇	—	2	12	no	—
147	Q₈×C₆₇	—	2	24	no	—
148	Q₈×C₆₇	—	2	8	no	—
149	Q₈×C₆₇	C₂²×D₄	1	8	yes	—
150	D₄×C₆₇	—	2	32	no	—
151	D₄×C₆₇	—	2	24	no	—
152	D₄×C₆₇	—	2	8	no	—
153	D₄×C₆₇	—	2	12	no	—
154	D₄×C₆₇	—	2	8	no	—
155	D₄×C₆₇	—	4	12	no	—
156	D₄×C₆₇	C₂²×D₄	1	8	yes	—
157	C₂×C₄×C₆₇	—	2	64	no	—
158	C₂×C₄×C₆₇	—	2	12	no	—
159	C₂×C₄×C₆₇	—	2	12	no	—
160	C₂×C₄×C₆₇	—	2	12	no	—
161	C₂×C₄×C₆₇	—	2	24	no	—
162	C₂×C₄×C₆₇	—	2	8	no	—
163	C₂×C₄×C₆₇	—	2	4	no	—
164	C₂×C₄×C₆₇	—	4	12	no	—
165	C₂×C₄×C₆₇	#120	1	6	yes	—
166	C₂×C₄×C₆₇	#120	1	6	yes	—
167	C₂×C₄×C₆₇	C₂²×D₄	1	8	yes	—
168	C₂×C₄×C₆₇	C₂²×D₄	1	3	yes	—
169	C₂×C₄×C₆₇	C₂²×D₄	1	6	yes	—
170	C₂×C₄×C₆₇	C₂²×D₄	1	6	yes	—
171	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
172	C₈×C₆₇	—	2	8	no	—
173	C₂²×C₆₇	—	2	6	no	—
174	C₂²×C₆₇	—	2	6	no	—
175	C₂²×C₆₇	—	2	8	no	—
176	C₂²×C₆₇	—	4	8	no	—
177	C₂²×C₆₇	—	4	24	no	—
178	C₂²×C₆₇	—	4	4	no	—
179	C₂²×C₆₇	—	8	12	no	—
180	C₂²×C₆₇	—	8	12	no	—
181	C₂²×C₆₇	#454	1	4	yes	direct product factor
182	C₂²×C₆₇	C₂×#120	1	3	yes	—
183	C₂²×C₆₇	C₂×#120	1	3	yes	—
184	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
185	C₄×C₆₇	—	2	8	no	—
186	C₄×C₆₇	—	2	8	no	—
187	C₄×C₆₇	—	4	4	no	—
188	C₄×C₆₇	C₂×#120	1	6	yes	—
189	C₄×C₆₇	C₂³×D₄	1	2	yes	—
190	C₂²×#454	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
191	C₂×C₆₇	—	2	1	no	—
192	C₂×C₆₇	—	2	3	no	—
193	C₂×C₆₇	—	4	8	no	—
194	C₂×C₆₇	—	8	4	no	—
195	C₂×C₆₇	C₂×#454	1	6	yes	direct product factor
196	C₂×C₆₇	C₂²×#120	1	1	yes	characteristic
197	C₂²×2₋¹⁺⁴	C₂×C₆₇	1	1	yes	characteristic, complemented
198	C₂²×AGL(1,ℤ/8)	C₂×C₆₇	1	2	yes	complemented
199	C₂²×#124	C₂×C₆₇	1	1	yes	characteristic, complemented
200	C₂×#454	C₂×C₆₇	1	24	yes	direct product factor
201	C₂²×#104	C₂×C₆₇	1	2	yes	complemented
202	C₂²×#101	C₂×C₆₇	1	1	yes	characteristic, complemented
203	C₆₇	C₂²×#454	1	1	yes	characteristic, direct product factor, Sylow
204	C₂×2₋¹⁺⁴	—	2	12	no	—
205	C₂×2₋¹⁺⁴	—	2	6	no	—
206	C₂×2₋¹⁺⁴	C₂²×C₆₇	1	12	yes	complemented
207	C₂²×#41	—	2	2	no	—
208	C₂²×#41	—	2	2	no	—
209	C₂²×#41	C₂²×C₆₇	1	2	yes	—
210	C₂³×Q₈	—	2	2	no	—
211	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
212	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
213	C₂×AGL(1,ℤ/8)	—	2	12	no	—
214	C₂×AGL(1,ℤ/8)	C₂²×C₆₇	1	24	yes	complemented
215	C₂²×SD₁₆	—	2	2	no	—
216	C₂²×D₈	—	2	2	no	—
217	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
218	C₂×#124	—	2	6	no	—
219	C₂×#124	C₂²×C₆₇	1	12	yes	complemented
220	C₂²×C₂²⋊C₄	—	2	2	no	—
221	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
222	#454	C₂²×C₆₇	1	64	yes	direct product factor
223	C₂×#104	C₂²×C₆₇	1	24	yes	complemented
224	C₂×#101	C₂²×C₆₇	1	12	yes	complemented
225	C₂⁵	—	2	1	no	—
226	2₋¹⁺⁴	—	2	48	no	—
227	2₋¹⁺⁴	—	2	48	no	—
228	2₋¹⁺⁴	—	2	24	no	—
229	2₋¹⁺⁴	C₂³×C₆₇	1	16	yes	complemented
230	C₂×#41	—	2	24	no	—
231	C₂×#41	—	2	24	no	—
232	C₂×#41	—	2	24	no	—
233	C₂×#41	—	2	24	no	—
234	C₂×#41	—	2	6	no	—
235	C₂×#41	—	2	6	no	—
236	C₂×#41	D₄×C₆₇	1	12	yes	—
237	C₂×#41	C₂³×C₆₇	1	12	yes	—
238	C₂²×Q₈	—	2	2	no	—
239	C₂²×Q₈	—	2	24	no	—
240	C₂²×Q₈	—	2	24	no	—
241	C₂²×Q₈	—	2	2	no	—
242	C₂²×Q₈	—	2	6	no	—
243	C₂²×Q₈	D₄×C₆₇	1	2	yes	—
244	C₂²×Q₈	D₄×C₆₇	1	6	yes	—
245	C₂²×Q₈	C₂³×C₆₇	1	6	yes	—
246	C₂²×D₄	—	2	2	no	—
247	C₂²×D₄	—	2	2	no	—
248	C₂²×D₄	—	2	2	no	—
249	C₂²×D₄	—	2	6	no	—
250	C₂²×D₄	—	2	3	no	—
251	C₂²×D₄	D₄×C₆₇	1	2	yes	—
252	C₂²×D₄	C₂³×C₆₇	1	6	yes	—
253	C₂³×C₄	—	2	4	no	—
254	C₂³×C₄	—	2	2	no	—
255	C₂³×C₄	—	2	1	no	—
256	C₂³×C₄	D₄×C₆₇	1	2	yes	—
257	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
258	AGL(1,ℤ/8)	—	2	48	no	—
259	AGL(1,ℤ/8)	C₂³×C₆₇	1	32	yes	complemented
260	C₂×SD₁₆	—	2	24	no	—
261	C₂×D₈	—	2	24	no	—
262	C₂×M₁₆	—	2	6	no	—
263	C₂×M₁₆	C₂³×C₆₇	1	12	yes	—
264	C₂²×C₈	—	2	2	no	—
265	#124	—	2	24	no	—
266	#124	C₂³×C₆₇	1	16	yes	complemented
267	C₂×C₂²⋊C₄	—	2	24	no	—
268	C₂×C₄²	—	2	3	no	—
269	C₂×C₄²	C₂³×C₆₇	1	6	yes	—
270	#104	C₂³×C₆₇	1	32	yes	complemented
271	#101	C₂³×C₆₇	1	16	yes	complemented
272	C₂⁴	—	2	2	no	—
273	C₂⁴	—	2	6	no	—
274	C₂⁴	—	4	8	no	—
275	C₂⁴	—	4	3	no	—
276	C₂⁴	—	4	1	no	—
277	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
278	#41	—	2	96	no	—
279	#41	—	2	96	no	—
280	#41	—	2	32	no	—
281	#41	—	2	24	no	—
282	#41	—	2	32	no	—
283	#41	—	2	12	no	—
284	#41	—	2	12	no	—
285	#41	C₂×D₄×C₆₇	1	24	yes	—
286	#41	C₂⁴×C₆₇	1	8	yes	—
287	C₂×Q₈	—	2	24	no	—
288	C₂×Q₈	—	2	96	no	—
289	C₂×Q₈	—	2	32	no	—
290	C₂×Q₈	—	2	6	no	—
291	C₂×Q₈	—	2	12	no	—
292	C₂×Q₈	—	2	12	no	—
293	C₂×Q₈	—	2	12	no	—
294	C₂×Q₈	—	2	12	no	—
295	C₂×Q₈	C₂×D₄×C₆₇	1	12	yes	—
296	C₂×Q₈	C₂×D₄×C₆₇	1	12	yes	—
297	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
298	C₂×D₄	—	2	24	no	—
299	C₂×D₄	—	2	12	no	—
300	C₂×D₄	—	2	12	no	—
301	C₂×D₄	—	2	6	no	—
302	C₂×D₄	—	2	12	no	—
303	C₂×D₄	—	2	12	no	—
304	C₂×D₄	—	2	6	no	—
305	C₂×D₄	—	4	6	no	—
306	C₂×D₄	—	4	6	no	—
307	C₂×D₄	C₂×D₄×C₆₇	1	12	yes	—
308	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
309	C₂²×C₄	—	2	48	no	—
310	C₂²×C₄	—	2	2	no	—
311	C₂²×C₄	—	2	2	no	—
312	C₂²×C₄	—	2	6	no	—
313	C₂²×C₄	—	2	12	no	—
314	C₂²×C₄	—	2	12	no	—
315	C₂²×C₄	—	2	6	no	—
316	C₂²×C₄	—	4	4	no	—
317	C₂²×C₄	C₂×D₄×C₆₇	1	12	yes	—
318	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
319	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
320	C₂²×C₄	C₂×D₄×C₆₇	1	6	yes	—
321	C₂²×C₄	C₂⁴×C₆₇	1	3	yes	—
322	SD₁₆	—	2	32	no	—
323	D₈	—	2	32	no	—
324	M₁₆	—	2	12	no	—
325	M₁₆	C₂⁴×C₆₇	1	8	yes	—
326	C₂×C₈	—	2	12	no	—
327	C₂²⋊C₄	—	2	32	no	—
328	C₄²	—	2	6	no	—
329	C₄²	C₂⁴×C₆₇	1	4	yes	—
330	C₂³	—	2	4	no	—
331	C₂³	—	2	12	no	—
332	C₂³	—	2	4	no	—
333	C₂³	—	4	24	no	—
334	C₂³	—	4	8	no	—
335	C₂³	—	4	6	no	—
336	C₂³	—	4	6	no	—
337	C₂³	—	8	12	no	—
338	C₂³	—	8	4	no	—
339	C₂³	#120×C₆₇	1	3	yes	—
340	C₂³	#120×C₆₇	1	1	yes	characteristic
341	C₂³	C₂²×D₄×C₆₇	1	3	yes	—
342	Q₈	—	2	32	no	—
343	Q₈	—	2	12	no	—
344	Q₈	—	2	24	no	—
345	Q₈	—	2	8	no	—
346	Q₈	C₂²×D₄×C₆₇	1	8	yes	—
347	D₄	—	2	32	no	—
348	D₄	—	2	24	no	—
349	D₄	—	2	8	no	—
350	D₄	—	2	12	no	—
351	D₄	—	2	8	no	—
352	D₄	—	4	12	no	—
353	D₄	C₂²×D₄×C₆₇	1	8	yes	—
354	C₂×C₄	—	2	64	no	—
355	C₂×C₄	—	2	12	no	—
356	C₂×C₄	—	2	12	no	—
357	C₂×C₄	—	2	12	no	—
358	C₂×C₄	—	2	24	no	—
359	C₂×C₄	—	2	8	no	—
360	C₂×C₄	—	2	4	no	—
361	C₂×C₄	—	4	12	no	—
362	C₂×C₄	#120×C₆₇	1	6	yes	—
363	C₂×C₄	#120×C₆₇	1	6	yes	—
364	C₂×C₄	C₂²×D₄×C₆₇	1	8	yes	—
365	C₂×C₄	C₂²×D₄×C₆₇	1	3	yes	—
366	C₂×C₄	C₂²×D₄×C₆₇	1	6	yes	—
367	C₂×C₄	C₂²×D₄×C₆₇	1	6	yes	—
368	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
369	C₈	—	2	8	no	—
370	C₂²	—	2	6	no	—
371	C₂²	—	2	6	no	—
372	C₂²	—	2	8	no	—
373	C₂²	—	4	8	no	—
374	C₂²	—	4	24	no	—
375	C₂²	—	4	4	no	—
376	C₂²	—	8	12	no	—
377	C₂²	—	8	12	no	—
378	C₂²	#454×C₆₇	1	4	yes	direct product factor
379	C₂²	C₂×#120×C₆₇	1	3	yes	—
380	C₂²	C₂×#120×C₆₇	1	3	yes	—
381	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
382	C₄	—	2	8	no	—
383	C₄	—	2	8	no	—
384	C₄	—	4	4	no	—
385	C₄	C₂×#120×C₆₇	1	6	yes	—
386	C₄	C₂³×D₄×C₆₇	1	2	yes	—
387	C₂	—	2	1	no	—
388	C₂	—	2	3	no	—
389	C₂	—	4	8	no	—
390	C₂	—	8	4	no	—
391	C₂	C₂×#454×C₆₇	1	6	yes	direct product factor
392	C₂	C₂²×#120×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	#120
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂³×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×2₋¹⁺⁴×C₆₇	C₂	semidirect
2. C₂²×AGL(1,ℤ/8)×C₆₇	C₂	semidirect
3. C₂²×#124×C₆₇	C₂	semidirect
4. C₂×#454×C₆₇	C₂	direct (N × H)
5. C₂²×#104×C₆₇	C₂	semidirect
6. C₂²×#101×C₆₇	C₂	semidirect
9. C₂×2₋¹⁺⁴×C₆₇	C₂²	semidirect
17. C₂×AGL(1,ℤ/8)×C₆₇	C₂²	semidirect
22. C₂×#124×C₆₇	C₂²	semidirect
25. #454×C₆₇	C₂²	direct (N × H)
26. C₂×#104×C₆₇	C₂²	semidirect
27. C₂×#101×C₆₇	C₂²	semidirect
32. 2₋¹⁺⁴×C₆₇	C₂³	semidirect
62. AGL(1,ℤ/8)×C₆₇	C₂³	semidirect
69. #124×C₆₇	C₂³	semidirect
73. #104×C₆₇	C₂³	semidirect
74. #101×C₆₇	C₂³	semidirect
181. C₂²×C₆₇	#454	direct (N × H)
190. C₂²×#454	C₆₇	direct (N × H)
195. C₂×C₆₇	C₂×#454	direct (N × H)
197. C₂²×2₋¹⁺⁴	C₂×C₆₇	semidirect
198. C₂²×AGL(1,ℤ/8)	C₂×C₆₇	semidirect
199. C₂²×#124	C₂×C₆₇	semidirect
200. C₂×#454	C₂×C₆₇	direct (N × H)
201. C₂²×#104	C₂×C₆₇	semidirect
202. C₂²×#101	C₂×C₆₇	semidirect
203. C₆₇	C₂²×#454	direct (N × H)
206. C₂×2₋¹⁺⁴	C₂²×C₆₇	semidirect
214. C₂×AGL(1,ℤ/8)	C₂²×C₆₇	semidirect
219. C₂×#124	C₂²×C₆₇	semidirect
222. #454	C₂²×C₆₇	direct (N × H)
223. C₂×#104	C₂²×C₆₇	semidirect
224. C₂×#101	C₂²×C₆₇	semidirect
229. 2₋¹⁺⁴	C₂³×C₆₇	semidirect
259. AGL(1,ℤ/8)	C₂³×C₆₇	semidirect
266. #124	C₂³×C₆₇	semidirect
270. #104	C₂³×C₆₇	semidirect
271. #101	C₂³×C₆₇	semidirect
378. C₂²	#454×C₆₇	direct (N × H)
391. C₂	C₂×#454×C₆₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×2₋¹⁺⁴×C₆₇	2
C₂²×AGL(1,ℤ/8)×C₆₇	2
C₂²×#124×C₆₇	2
C₂×#454×C₆₇	2
C₂²×#104×C₆₇	2
C₂²×#101×C₆₇	2
C₂²×#454	67

Subgroups — C22×#454×C67 #30739631

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#454×C₆₇ #30739631