Subgroups — C₆₇×#21258 #30700509

Order 17152.

1706 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#532×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#423×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#417×C₆₇	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1983	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1982	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1981	C₂	1	2	yes	maximal, complemented
8	C₂²×#41×C₆₇	C₂²	1	2	yes	—
9	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
10	#532×C₆₇	—	2	4	no	—
11	#532×C₆₇	C₄	1	8	yes	complemented
12	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#124×C₆₇	C₂²	1	2	yes	—
15	C₂×#121×C₆₇	—	2	1	no	—
16	C₂×#121×C₆₇	—	2	1	no	—
17	C₂×C₄×D₄×C₆₇	—	2	1	no	—
18	C₂×C₄×D₄×C₆₇	—	2	1	no	—
19	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
20	C₂²×C₄²×C₆₇	—	2	1	no	—
21	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
22	#423×C₆₇	—	2	4	no	—
23	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
24	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
25	#417×C₆₇	—	2	16	no	—
26	C₂×#103×C₆₇	C₂²	1	8	yes	—
27	C₂×#102×C₆₇	C₂²	1	8	yes	—
28	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
29	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
30	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
31	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
32	C₂×#95×C₆₇	—	2	1	no	—
33	C₂×#41×C₆₇	—	2	4	no	—
34	C₂×#41×C₆₇	—	2	8	no	—
35	C₂×#41×C₆₇	—	2	8	no	—
36	C₂×#41×C₆₇	C₂×C₄	1	8	yes	—
37	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
38	C₂²×D₄×C₆₇	—	2	2	no	—
39	C₂²×D₄×C₆₇	—	2	1	no	—
40	C₂²×D₄×C₆₇	—	2	1	no	—
41	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
42	C₂³×C₄×C₆₇	—	2	2	no	—
43	C₂³×C₄×C₆₇	—	2	2	no	—
44	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
45	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
46	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
47	C₂×M₁₆×C₆₇	—	2	8	no	—
48	C₂×M₁₆×C₆₇	—	2	4	no	—
49	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
50	C₄⋊Q₈×C₆₇	—	2	2	no	—
51	C₄⋊Q₈×C₆₇	C₂×C₄	1	4	yes	complemented
52	#127×C₆₇	—	2	2	no	—
53	#127×C₆₇	C₂×C₄	1	4	yes	complemented
54	#124×C₆₇	—	2	4	no	—
55	#124×C₆₇	C₂×C₄	1	8	yes	complemented
56	#121×C₆₇	—	2	8	no	—
57	#121×C₆₇	—	2	8	no	—
58	C₄×D₄×C₆₇	—	2	8	no	—
59	C₄×D₄×C₆₇	—	2	8	no	—
60	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
61	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
62	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
63	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
64	C₂×C₄⋊C₄×C₆₇	D₄	1	8	yes	complemented
65	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
66	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
67	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
68	C₂×C₄²×C₆₇	—	2	2	no	—
69	C₂×C₄²×C₆₇	—	2	2	no	—
70	C₂×C₄²×C₆₇	—	4	4	no	—
71	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
72	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
73	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
74	C₂×C₄²×C₆₇	D₄	1	4	yes	—
75	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
76	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
77	C₄⋊C₈×C₆₇	—	2	4	no	—
78	C₄⋊C₈×C₆₇	—	2	4	no	—
79	#103×C₆₇	—	2	16	no	—
80	#102×C₆₇	—	2	16	no	—
81	#95×C₆₇	—	2	2	no	—
82	#95×C₆₇	—	2	2	no	—
83	C₂⁴×C₆₇	—	2	2	no	—
84	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
85	#41×C₆₇	—	2	32	no	—
86	#41×C₆₇	—	4	8	no	—
87	#41×C₆₇	—	4	8	no	—
88	C₂×Q₈×C₆₇	—	2	2	no	—
89	C₂×Q₈×C₆₇	—	2	4	no	—
90	C₂×Q₈×C₆₇	—	2	4	no	—
91	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
92	C₂×D₄×C₆₇	—	2	8	no	—
93	C₂×D₄×C₆₇	—	2	4	no	—
94	C₂×D₄×C₆₇	—	2	4	no	—
95	C₂×D₄×C₆₇	—	2	4	no	—
96	C₂×D₄×C₆₇	—	2	2	no	—
97	C₂×D₄×C₆₇	—	2	4	no	—
98	C₂×D₄×C₆₇	—	4	4	no	—
99	C₂×D₄×C₆₇	—	4	8	no	—
100	C₂×D₄×C₆₇	—	4	2	no	—
101	C₂×D₄×C₆₇	—	4	2	no	—
102	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
103	C₂²×C₄×C₆₇	—	2	2	no	—
104	C₂²×C₄×C₆₇	—	2	2	no	—
105	C₂²×C₄×C₆₇	—	2	8	no	—
106	C₂²×C₄×C₆₇	—	2	4	no	—
107	C₂²×C₄×C₆₇	—	2	2	no	—
108	C₂²×C₄×C₆₇	—	2	8	no	—
109	C₂²×C₄×C₆₇	—	4	2	no	—
110	C₂²×C₄×C₆₇	—	4	4	no	—
111	C₂²×C₄×C₆₇	—	4	4	no	—
112	C₂²×C₄×C₆₇	—	4	8	no	—
113	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
114	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
115	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
116	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
122	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
123	C₂²×C₄×C₆₇	#41	1	4	yes	—
124	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
125	M₁₆×C₆₇	—	4	8	no	—
126	C₂×C₈×C₆₇	—	2	8	no	—
127	C₂×C₈×C₆₇	—	2	4	no	—
128	C₄⋊C₄×C₆₇	—	2	4	no	—
129	C₄⋊C₄×C₆₇	—	2	4	no	—
130	C₄⋊C₄×C₆₇	—	2	8	no	—
131	C₄⋊C₄×C₆₇	—	2	4	no	—
132	C₄⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
133	C₂²⋊C₄×C₆₇	—	2	8	no	—
134	C₂²⋊C₄×C₆₇	—	2	8	no	—
135	C₄²×C₆₇	—	4	4	no	—
136	C₄²×C₆₇	—	4	4	no	—
137	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
138	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
139	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
140	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
141	C₄²×C₆₇	C₂×D₄	1	2	yes	—
142	C₄²×C₆₇	C₂×D₄	1	2	yes	—
143	C₂³×C₆₇	—	2	1	no	—
144	C₂³×C₆₇	—	2	1	no	—
145	C₂³×C₆₇	—	2	2	no	—
146	C₂³×C₆₇	—	2	4	no	—
147	C₂³×C₆₇	—	4	4	no	—
148	C₂³×C₆₇	—	4	2	no	—
149	C₂³×C₆₇	—	4	2	no	—
150	C₂³×C₆₇	—	4	4	no	—
151	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
152	C₂³×C₆₇	#121	1	4	yes	—
153	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
154	Q₈×C₆₇	—	2	8	no	—
155	Q₈×C₆₇	—	4	2	no	—
156	Q₈×C₆₇	—	4	2	no	—
157	D₄×C₆₇	—	2	8	no	—
158	D₄×C₆₇	—	4	8	no	—
159	D₄×C₆₇	—	4	16	no	—
160	D₄×C₆₇	—	4	2	no	—
161	D₄×C₆₇	—	4	2	no	—
162	D₄×C₆₇	—	4	8	no	—
163	C₂×C₄×C₆₇	—	2	1	no	—
164	C₂×C₄×C₆₇	—	2	1	no	—
165	C₂×C₄×C₆₇	—	2	2	no	—
166	C₂×C₄×C₆₇	—	2	2	no	—
167	C₂×C₄×C₆₇	—	2	8	no	—
168	C₂×C₄×C₆₇	—	2	4	no	—
169	C₂×C₄×C₆₇	—	2	8	no	—
170	C₂×C₄×C₆₇	—	4	4	no	—
171	C₂×C₄×C₆₇	—	4	16	no	—
172	C₂×C₄×C₆₇	—	4	4	no	—
173	C₂×C₄×C₆₇	—	4	4	no	—
174	C₂×C₄×C₆₇	—	4	4	no	—
175	C₂×C₄×C₆₇	—	4	16	no	—
176	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
177	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
178	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
179	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
180	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
181	C₂×C₄×C₆₇	C₄×D₄	1	8	yes	—
182	C₂×C₄×C₆₇	#121	1	4	yes	—
183	C₂×C₄×C₆₇	#124	1	4	yes	—
184	C₂×C₄×C₆₇	#127	1	4	yes	—
185	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
186	C₂×C₄×C₆₇	#137	1	2	yes	—
187	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
188	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
189	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
190	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
191	C₈×C₆₇	—	4	4	no	—
192	C₂²×C₆₇	—	2	2	no	—
193	C₂²×C₆₇	—	2	2	no	—
194	C₂²×C₆₇	—	2	4	no	—
195	C₂²×C₆₇	—	2	4	no	—
196	C₂²×C₆₇	—	4	4	no	—
197	C₂²×C₆₇	—	4	4	no	—
198	C₂²×C₆₇	—	4	4	no	—
199	C₂²×C₆₇	—	8	4	no	—
200	C₂²×C₆₇	—	8	4	no	—
201	C₂²×C₆₇	#390	1	4	yes	—
202	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
203	C₂²×C₆₇	C₂×#121	1	2	yes	—
204	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
205	C₂²×C₆₇	C₂×#137	1	1	yes	characteristic
206	C₂²×C₆₇	#575	1	2	yes	—
207	C₄×C₆₇	—	2	2	no	—
208	C₄×C₆₇	—	2	2	no	—
209	C₄×C₆₇	—	2	2	no	—
210	C₄×C₆₇	—	4	8	no	—
211	C₄×C₆₇	—	4	4	no	—
212	C₄×C₆₇	#390	1	4	yes	—
213	C₄×C₆₇	#417	1	4	yes	complemented
214	C₄×C₆₇	C₂×C₄×D₄	1	2	yes	—
215	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
216	C₄×C₆₇	C₂×#127	1	1	yes	characteristic
217	#21258	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
218	C₂×C₆₇	—	2	4	no	—
219	C₂×C₆₇	—	8	4	no	—
220	C₂×C₆₇	C₂×#390	1	1	yes	characteristic
221	C₂×C₆₇	C₂×#417	1	1	yes	characteristic
222	C₂×C₆₇	#2896	1	1	yes	characteristic
223	C₂×C₆₇	#3036	1	2	yes	—
224	C₂×C₆₇	#3037	1	2	yes	—
225	C₂×#532	C₂×C₆₇	1	1	yes	characteristic
226	C₂×#423	C₂×C₆₇	1	1	yes	characteristic, complemented
227	C₂×#417	C₂×C₆₇	1	4	yes	complemented
228	C₂×C₄×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic, complemented
229	#1983	C₂×C₆₇	1	4	yes	complemented
230	#1982	C₂×C₆₇	1	2	yes	complemented
231	#1981	C₂×C₆₇	1	2	yes	complemented
232	C₆₇	#21258	1	1	yes	characteristic, direct product factor, Sylow
233	C₂²×#41	C₂²×C₆₇	1	2	yes	—
234	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
235	#532	—	2	4	no	—
236	#532	C₄×C₆₇	1	8	yes	complemented
237	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
238	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
239	C₂×#124	C₂²×C₆₇	1	2	yes	—
240	C₂×#121	—	2	1	no	—
241	C₂×#121	—	2	1	no	—
242	C₂×C₄×D₄	—	2	1	no	—
243	C₂×C₄×D₄	—	2	1	no	—
244	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
245	C₂²×C₄²	—	2	1	no	—
246	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
247	#423	—	2	4	no	—
248	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
249	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
250	#417	—	2	16	no	—
251	C₂×#103	C₂²×C₆₇	1	8	yes	—
252	C₂×#102	C₂²×C₆₇	1	8	yes	—
253	C₄×C₄⋊C₄	—	2	1	no	—
254	C₄×C₄⋊C₄	—	2	1	no	—
255	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
256	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
257	C₂×#95	—	2	1	no	—
258	C₂×#41	—	2	4	no	—
259	C₂×#41	—	2	8	no	—
260	C₂×#41	—	2	8	no	—
261	C₂×#41	C₂×C₄×C₆₇	1	8	yes	—
262	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
263	C₂²×D₄	—	2	2	no	—
264	C₂²×D₄	—	2	1	no	—
265	C₂²×D₄	—	2	1	no	—
266	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
267	C₂³×C₄	—	2	2	no	—
268	C₂³×C₄	—	2	2	no	—
269	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
270	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
271	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
272	C₂×M₁₆	—	2	8	no	—
273	C₂×M₁₆	—	2	4	no	—
274	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
275	C₄⋊Q₈	—	2	2	no	—
276	C₄⋊Q₈	C₂×C₄×C₆₇	1	4	yes	complemented
277	#127	—	2	2	no	—
278	#127	C₂×C₄×C₆₇	1	4	yes	complemented
279	#124	—	2	4	no	—
280	#124	C₂×C₄×C₆₇	1	8	yes	complemented
281	#121	—	2	8	no	—
282	#121	—	2	8	no	—
283	C₄×D₄	—	2	8	no	—
284	C₄×D₄	—	2	8	no	—
285	C₂×C₄⋊C₄	—	2	2	no	—
286	C₂×C₄⋊C₄	—	2	1	no	—
287	C₂×C₄⋊C₄	—	2	1	no	—
288	C₂×C₄⋊C₄	—	2	4	no	—
289	C₂×C₄⋊C₄	D₄×C₆₇	1	8	yes	complemented
290	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
291	C₂×C₂²⋊C₄	—	2	2	no	—
292	C₂×C₂²⋊C₄	—	2	2	no	—
293	C₂×C₄²	—	2	2	no	—
294	C₂×C₄²	—	2	2	no	—
295	C₂×C₄²	—	4	4	no	—
296	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
297	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
298	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
299	C₂×C₄²	D₄×C₆₇	1	4	yes	—
300	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
301	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
302	C₄⋊C₈	—	2	4	no	—
303	C₄⋊C₈	—	2	4	no	—
304	#103	—	2	16	no	—
305	#102	—	2	16	no	—
306	#95	—	2	2	no	—
307	#95	—	2	2	no	—
308	C₂⁴	—	2	2	no	—
309	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
310	#41	—	2	32	no	—
311	#41	—	4	8	no	—
312	#41	—	4	8	no	—
313	C₂×Q₈	—	2	2	no	—
314	C₂×Q₈	—	2	4	no	—
315	C₂×Q₈	—	2	4	no	—
316	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
317	C₂×D₄	—	2	8	no	—
318	C₂×D₄	—	2	4	no	—
319	C₂×D₄	—	2	4	no	—
320	C₂×D₄	—	2	4	no	—
321	C₂×D₄	—	2	2	no	—
322	C₂×D₄	—	2	4	no	—
323	C₂×D₄	—	4	4	no	—
324	C₂×D₄	—	4	8	no	—
325	C₂×D₄	—	4	2	no	—
326	C₂×D₄	—	4	2	no	—
327	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
328	C₂²×C₄	—	2	2	no	—
329	C₂²×C₄	—	2	2	no	—
330	C₂²×C₄	—	2	8	no	—
331	C₂²×C₄	—	2	4	no	—
332	C₂²×C₄	—	2	2	no	—
333	C₂²×C₄	—	2	8	no	—
334	C₂²×C₄	—	4	2	no	—
335	C₂²×C₄	—	4	4	no	—
336	C₂²×C₄	—	4	4	no	—
337	C₂²×C₄	—	4	8	no	—
338	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
339	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
340	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
341	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
342	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
343	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
344	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
345	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
346	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
347	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
348	C₂²×C₄	#41×C₆₇	1	4	yes	—
349	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
350	M₁₆	—	4	8	no	—
351	C₂×C₈	—	2	8	no	—
352	C₂×C₈	—	2	4	no	—
353	C₄⋊C₄	—	2	4	no	—
354	C₄⋊C₄	—	2	4	no	—
355	C₄⋊C₄	—	2	8	no	—
356	C₄⋊C₄	—	2	4	no	—
357	C₄⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
358	C₂²⋊C₄	—	2	8	no	—
359	C₂²⋊C₄	—	2	8	no	—
360	C₄²	—	4	4	no	—
361	C₄²	—	4	4	no	—
362	C₄²	C₂²⋊C₄×C₆₇	1	4	yes	—
363	C₄²	C₂²⋊C₄×C₆₇	1	4	yes	—
364	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
365	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
366	C₄²	C₂×D₄×C₆₇	1	2	yes	—
367	C₄²	C₂×D₄×C₆₇	1	2	yes	—
368	C₂³	—	2	1	no	—
369	C₂³	—	2	1	no	—
370	C₂³	—	2	2	no	—
371	C₂³	—	2	4	no	—
372	C₂³	—	4	4	no	—
373	C₂³	—	4	2	no	—
374	C₂³	—	4	2	no	—
375	C₂³	—	4	4	no	—
376	C₂³	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
377	C₂³	#121×C₆₇	1	4	yes	—
378	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
379	Q₈	—	2	8	no	—
380	Q₈	—	4	2	no	—
381	Q₈	—	4	2	no	—
382	D₄	—	2	8	no	—
383	D₄	—	4	8	no	—
384	D₄	—	4	16	no	—
385	D₄	—	4	2	no	—
386	D₄	—	4	2	no	—
387	D₄	—	4	8	no	—
388	C₂×C₄	—	2	1	no	—
389	C₂×C₄	—	2	1	no	—
390	C₂×C₄	—	2	2	no	—
391	C₂×C₄	—	2	2	no	—
392	C₂×C₄	—	2	8	no	—
393	C₂×C₄	—	2	4	no	—
394	C₂×C₄	—	2	8	no	—
395	C₂×C₄	—	4	4	no	—
396	C₂×C₄	—	4	16	no	—
397	C₂×C₄	—	4	4	no	—
398	C₂×C₄	—	4	4	no	—
399	C₂×C₄	—	4	4	no	—
400	C₂×C₄	—	4	16	no	—
401	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
402	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
403	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
404	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
405	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
406	C₂×C₄	C₄×D₄×C₆₇	1	8	yes	—
407	C₂×C₄	#121×C₆₇	1	4	yes	—
408	C₂×C₄	#124×C₆₇	1	4	yes	—
409	C₂×C₄	#127×C₆₇	1	4	yes	—
410	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
411	C₂×C₄	#137×C₆₇	1	2	yes	—
412	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
413	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
414	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
415	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
416	C₈	—	4	4	no	—
417	C₂²	—	2	4	no	—
418	C₂²	—	2	4	no	—
419	C₂²	—	2	2	no	—
420	C₂²	—	2	2	no	—
421	C₂²	—	4	4	no	—
422	C₂²	—	4	4	no	—
423	C₂²	—	4	4	no	—
424	C₂²	—	8	4	no	—
425	C₂²	—	8	4	no	—
426	C₂²	#390×C₆₇	1	4	yes	—
427	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
428	C₂²	C₂×#121×C₆₇	1	2	yes	—
429	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
430	C₂²	C₂×#137×C₆₇	1	1	yes	characteristic
431	C₂²	#575×C₆₇	1	2	yes	—
432	C₄	—	2	2	no	—
433	C₄	—	2	2	no	—
434	C₄	—	2	2	no	—
435	C₄	—	4	8	no	—
436	C₄	—	4	4	no	—
437	C₄	#390×C₆₇	1	4	yes	—
438	C₄	#417×C₆₇	1	4	yes	complemented
439	C₄	C₂×C₄×D₄×C₆₇	1	2	yes	—
440	C₄	C₂×#124×C₆₇	1	1	yes	characteristic
441	C₄	C₂×#127×C₆₇	1	1	yes	characteristic
442	C₂	—	2	4	no	—
443	C₂	—	8	4	no	—
444	C₂	C₂×#390×C₆₇	1	1	yes	characteristic
445	C₂	C₂×#417×C₆₇	1	1	yes	characteristic
446	C₂	C₆₇×#2896	1	1	yes	characteristic
447	C₂	C₆₇×#3036	1	2	yes	—
448	C₂	C₆₇×#3037	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#532×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×#417×C₆₇	C₂	semidirect
4. C₂×C₄×C₄⋊C₄×C₆₇	C₂	semidirect
5. C₆₇×#1983	C₂	semidirect
6. C₆₇×#1982	C₂	semidirect
7. C₆₇×#1981	C₂	semidirect
11. #532×C₆₇	C₄	semidirect
51. C₄⋊Q₈×C₆₇	C₂×C₄	semidirect
53. #127×C₆₇	C₂×C₄	semidirect
55. #124×C₆₇	C₂×C₄	semidirect
64. C₂×C₄⋊C₄×C₆₇	D₄	semidirect
213. C₄×C₆₇	#417	semidirect
217. #21258	C₆₇	direct (N × H)
226. C₂×#423	C₂×C₆₇	semidirect
227. C₂×#417	C₂×C₆₇	semidirect
228. C₂×C₄×C₄⋊C₄	C₂×C₆₇	semidirect
229. #1983	C₂×C₆₇	semidirect
230. #1982	C₂×C₆₇	semidirect
231. #1981	C₂×C₆₇	semidirect
232. C₆₇	#21258	direct (N × H)
236. #532	C₄×C₆₇	semidirect
276. C₄⋊Q₈	C₂×C₄×C₆₇	semidirect
278. #127	C₂×C₄×C₆₇	semidirect
280. #124	C₂×C₄×C₆₇	semidirect
289. C₂×C₄⋊C₄	D₄×C₆₇	semidirect
438. C₄	#417×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#532×C₆₇	2
C₂×#423×C₆₇	2
C₂×#417×C₆₇	2
C₂×C₄×C₄⋊C₄×C₆₇	2
C₆₇×#1983	2
C₆₇×#1982	2
C₆₇×#1981	2
#21258	67

Subgroups — C67×#21258 #30700509

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21258 #30700509