Subgroups — C₆₇×#10268 #30689519

Order 17152.

658 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#492×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#475×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#460×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1984	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#332×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#331×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₆₇×#1277	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×SD₁₆×C₆₇	—	2	1	no	—
9	C₂²×SD₁₆×C₆₇	—	2	2	no	—
10	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
11	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
13	#492×C₆₇	—	2	2	no	—
14	#492×C₆₇	—	2	2	no	—
15	#475×C₆₇	—	2	2	no	—
16	#475×C₆₇	—	2	2	no	—
17	#460×C₆₇	—	2	2	no	—
18	#460×C₆₇	—	2	2	no	—
19	C₂×#106×C₆₇	—	2	1	no	—
20	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#103×C₆₇	—	2	1	no	—
24	C₂×#102×C₆₇	—	2	1	no	—
25	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
26	#336×C₆₇	—	2	2	no	—
27	#332×C₆₇	—	2	2	no	—
28	#331×C₆₇	—	2	2	no	—
29	C₂²×Q₈×C₆₇	—	2	1	no	—
30	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
31	C₂²×D₄×C₆₇	—	2	1	no	—
32	C₂²×D₄×C₆₇	—	2	1	no	—
33	C₂²×D₄×C₆₇	—	2	1	no	—
34	C₂×SD₁₆×C₆₇	—	2	4	no	—
35	C₂×SD₁₆×C₆₇	—	2	8	no	—
36	C₂×SD₁₆×C₆₇	—	4	4	no	—
37	C₂×SD₁₆×C₆₇	—	4	2	no	—
38	C₂×SD₁₆×C₆₇	—	4	2	no	—
39	C₂×SD₁₆×C₆₇	—	4	4	no	—
40	C₂²×C₈×C₆₇	—	2	1	no	—
41	C₂²×C₈×C₆₇	—	2	1	no	—
42	C₂²×C₈×C₆₇	—	2	1	no	—
43	C₂²×C₈×C₆₇	D₄	1	2	yes	—
44	C₄⋊Q₈×C₆₇	—	2	1	no	—
45	C₄⋊Q₈×C₆₇	—	2	1	no	—
46	C₄⋊Q₈×C₆₇	—	2	1	no	—
47	C₄⋊Q₈×C₆₇	D₄	1	2	yes	—
48	#127×C₆₇	—	2	1	no	—
49	#127×C₆₇	—	2	1	no	—
50	#127×C₆₇	—	2	1	no	—
51	#127×C₆₇	D₄	1	2	yes	—
52	C₄×Q₈×C₆₇	—	2	1	no	—
53	C₄×Q₈×C₆₇	—	2	1	no	—
54	C₄×Q₈×C₆₇	—	2	1	no	—
55	C₄×Q₈×C₆₇	—	2	1	no	—
56	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
57	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
58	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
59	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
60	C₂×C₄²×C₆₇	—	2	1	no	—
61	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
62	#106×C₆₇	—	2	4	no	—
63	C₄⋊C₈×C₆₇	—	2	1	no	—
64	C₄⋊C₈×C₆₇	—	2	1	no	—
65	C₄⋊C₈×C₆₇	—	2	1	no	—
66	C₄⋊C₈×C₆₇	—	2	1	no	—
67	C₄⋊C₈×C₆₇	D₄	1	2	yes	—
68	C₄⋊C₈×C₆₇	D₄	1	2	yes	—
69	#103×C₆₇	—	2	4	no	—
70	#102×C₆₇	—	2	4	no	—
71	C₄×C₈×C₆₇	—	2	1	no	—
72	C₄×C₈×C₆₇	—	2	1	no	—
73	C₂⁴×C₆₇	—	4	1	no	—
74	C₂×Q₈×C₆₇	—	2	1	no	—
75	C₂×Q₈×C₆₇	—	2	1	no	—
76	C₂×Q₈×C₆₇	—	2	1	no	—
77	C₂×Q₈×C₆₇	—	2	2	no	—
78	C₂×Q₈×C₆₇	—	2	1	no	—
79	C₂×Q₈×C₆₇	—	2	1	no	—
80	C₂×Q₈×C₆₇	—	4	1	no	—
81	C₂×Q₈×C₆₇	—	4	2	no	—
82	C₂×Q₈×C₆₇	—	4	2	no	—
83	C₂×Q₈×C₆₇	SD₁₆	1	2	yes	complemented
84	C₂×D₄×C₆₇	—	2	2	no	—
85	C₂×D₄×C₆₇	—	2	2	no	—
86	C₂×D₄×C₆₇	—	2	2	no	—
87	C₂×D₄×C₆₇	—	4	1	no	—
88	C₂×D₄×C₆₇	—	4	1	no	—
89	C₂×D₄×C₆₇	—	4	1	no	—
90	C₂×D₄×C₆₇	—	4	1	no	—
91	C₂×D₄×C₆₇	—	4	2	no	—
92	C₂×D₄×C₆₇	—	4	1	no	—
93	C₂×D₄×C₆₇	—	4	1	no	—
94	C₂×D₄×C₆₇	—	4	2	no	—
95	C₂×D₄×C₆₇	—	4	1	no	—
96	C₂×D₄×C₆₇	—	8	2	no	—
97	C₂²×C₄×C₆₇	—	2	1	no	—
98	C₂²×C₄×C₆₇	—	2	1	no	—
99	C₂²×C₄×C₆₇	—	4	1	no	—
100	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
103	SD₁₆×C₆₇	—	4	16	no	—
104	SD₁₆×C₆₇	—	4	8	no	—
105	C₂×C₈×C₆₇	—	2	2	no	—
106	C₂×C₈×C₆₇	—	2	2	no	—
107	C₂×C₈×C₆₇	—	2	2	no	—
108	C₂×C₈×C₆₇	—	2	2	no	—
109	C₂×C₈×C₆₇	—	4	1	no	—
110	C₂×C₈×C₆₇	—	4	1	no	—
111	C₂×C₈×C₆₇	—	4	1	no	—
112	C₂×C₈×C₆₇	—	4	1	no	—
113	C₂×C₈×C₆₇	—	4	1	no	—
114	C₂×C₈×C₆₇	—	4	2	no	—
115	C₂×C₈×C₆₇	SD₁₆	1	4	yes	complemented
116	C₄⋊C₄×C₆₇	—	2	1	no	—
117	C₄⋊C₄×C₆₇	—	2	2	no	—
118	C₄⋊C₄×C₆₇	—	2	1	no	—
119	C₄⋊C₄×C₆₇	—	2	2	no	—
120	C₄⋊C₄×C₆₇	—	4	1	no	—
121	C₄⋊C₄×C₆₇	—	4	1	no	—
122	C₄⋊C₄×C₆₇	—	4	1	no	—
123	C₄⋊C₄×C₆₇	—	4	1	no	—
124	C₄²×C₆₇	—	4	1	no	—
125	C₄²×C₆₇	—	4	1	no	—
126	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
127	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
128	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
129	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
130	C₂³×C₆₇	—	4	2	no	—
131	C₂³×C₆₇	—	8	2	no	—
132	C₂³×C₆₇	—	8	1	no	—
133	C₂³×C₆₇	—	8	1	no	—
134	C₂³×C₆₇	—	8	2	no	—
135	C₂³×C₆₇	#121	1	1	yes	characteristic
136	Q₈×C₆₇	—	2	2	no	—
137	Q₈×C₆₇	—	2	2	no	—
138	Q₈×C₆₇	—	4	1	no	—
139	Q₈×C₆₇	—	4	1	no	—
140	Q₈×C₆₇	—	4	4	no	—
141	Q₈×C₆₇	—	8	1	no	—
142	Q₈×C₆₇	—	8	1	no	—
143	D₄×C₆₇	—	4	2	no	—
144	D₄×C₆₇	—	4	4	no	—
145	D₄×C₆₇	—	4	2	no	—
146	D₄×C₆₇	—	8	2	no	—
147	D₄×C₆₇	—	8	2	no	—
148	D₄×C₆₇	—	8	1	no	—
149	D₄×C₆₇	—	8	1	no	—
150	D₄×C₆₇	—	8	1	no	—
151	D₄×C₆₇	—	8	1	no	—
152	C₂×C₄×C₆₇	—	2	2	no	—
153	C₂×C₄×C₆₇	—	2	2	no	—
154	C₂×C₄×C₆₇	—	4	1	no	—
155	C₂×C₄×C₆₇	—	4	1	no	—
156	C₂×C₄×C₆₇	—	4	1	no	—
157	C₂×C₄×C₆₇	—	4	2	no	—
158	C₂×C₄×C₆₇	—	4	2	no	—
159	C₂×C₄×C₆₇	—	8	2	no	—
160	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
161	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
162	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
163	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
164	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
165	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
166	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
167	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
168	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
169	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
170	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
171	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
172	C₈×C₆₇	—	2	4	no	—
173	C₈×C₆₇	—	4	2	no	—
174	C₈×C₆₇	—	4	2	no	—
175	C₈×C₆₇	—	8	1	no	—
176	C₂²×C₆₇	—	8	2	no	—
177	C₂²×C₆₇	—	8	2	no	—
178	C₂²×C₆₇	—	8	2	no	—
179	C₂²×C₆₇	—	16	2	no	—
180	C₂²×C₆₇	—	16	2	no	—
181	C₂²×C₆₇	#392	1	1	yes	characteristic
182	C₂²×C₆₇	#460	1	2	yes	—
183	C₂²×C₆₇	#465	1	1	yes	characteristic
184	C₂²×C₆₇	#468	1	1	yes	characteristic
185	C₂²×C₆₇	#471	1	2	yes	—
186	C₄×C₆₇	—	2	1	no	—
187	C₄×C₆₇	—	2	1	no	—
188	C₄×C₆₇	—	4	2	no	—
189	C₄×C₆₇	—	8	2	no	—
190	C₄×C₆₇	—	8	1	no	—
191	C₄×C₆₇	#448	1	1	yes	characteristic
192	C₄×C₆₇	#449	1	1	yes	characteristic
193	C₄×C₆₇	#450	1	1	yes	characteristic
194	C₄×C₆₇	#450	1	2	yes	—
195	C₄×C₆₇	#452	1	1	yes	characteristic
196	C₄×C₆₇	#492	1	1	yes	characteristic
197	C₄×C₆₇	#496	1	1	yes	characteristic
198	#10268	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
199	C₂×C₆₇	—	16	2	no	—
200	C₂×C₆₇	#1666	1	2	yes	—
201	C₂×C₆₇	#1686	1	2	yes	—
202	C₂×C₆₇	#2000	1	1	yes	characteristic
203	C₂×C₆₇	#2014	1	1	yes	characteristic
204	C₂×C₆₇	#2019	1	1	yes	characteristic
205	C₂×#492	C₂×C₆₇	1	1	yes	characteristic
206	C₂×#475	C₂×C₆₇	1	1	yes	characteristic, complemented
207	C₂×#460	C₂×C₆₇	1	1	yes	characteristic
208	#1984	C₂×C₆₇	1	1	yes	characteristic, complemented
209	C₂×#332	C₂×C₆₇	1	1	yes	characteristic, complemented
210	C₂×#331	C₂×C₆₇	1	1	yes	characteristic
211	#1277	C₂×C₆₇	1	1	yes	characteristic, complemented
212	C₆₇	#10268	1	1	yes	characteristic, direct product factor, Sylow
213	C₂²×SD₁₆	—	2	1	no	—
214	C₂²×SD₁₆	—	2	2	no	—
215	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
216	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
217	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
218	#492	—	2	2	no	—
219	#492	—	2	2	no	—
220	#475	—	2	2	no	—
221	#475	—	2	2	no	—
222	#460	—	2	2	no	—
223	#460	—	2	2	no	—
224	C₂×#106	—	2	1	no	—
225	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
226	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
227	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
228	C₂×#103	—	2	1	no	—
229	C₂×#102	—	2	1	no	—
230	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
231	#336	—	2	2	no	—
232	#332	—	2	2	no	—
233	#331	—	2	2	no	—
234	C₂²×Q₈	—	2	1	no	—
235	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
236	C₂²×D₄	—	2	1	no	—
237	C₂²×D₄	—	2	1	no	—
238	C₂²×D₄	—	2	1	no	—
239	C₂×SD₁₆	—	2	4	no	—
240	C₂×SD₁₆	—	2	8	no	—
241	C₂×SD₁₆	—	4	4	no	—
242	C₂×SD₁₆	—	4	2	no	—
243	C₂×SD₁₆	—	4	2	no	—
244	C₂×SD₁₆	—	4	4	no	—
245	C₂²×C₈	—	2	1	no	—
246	C₂²×C₈	—	2	1	no	—
247	C₂²×C₈	—	2	1	no	—
248	C₂²×C₈	D₄×C₆₇	1	2	yes	—
249	C₄⋊Q₈	—	2	1	no	—
250	C₄⋊Q₈	—	2	1	no	—
251	C₄⋊Q₈	—	2	1	no	—
252	C₄⋊Q₈	D₄×C₆₇	1	2	yes	—
253	#127	—	2	1	no	—
254	#127	—	2	1	no	—
255	#127	—	2	1	no	—
256	#127	D₄×C₆₇	1	2	yes	—
257	C₄×Q₈	—	2	1	no	—
258	C₄×Q₈	—	2	1	no	—
259	C₄×Q₈	—	2	1	no	—
260	C₄×Q₈	—	2	1	no	—
261	C₂×C₄⋊C₄	—	2	1	no	—
262	C₂×C₄⋊C₄	—	2	1	no	—
263	C₂×C₄⋊C₄	—	2	1	no	—
264	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
265	C₂×C₄²	—	2	1	no	—
266	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
267	#106	—	2	4	no	—
268	C₄⋊C₈	—	2	1	no	—
269	C₄⋊C₈	—	2	1	no	—
270	C₄⋊C₈	—	2	1	no	—
271	C₄⋊C₈	—	2	1	no	—
272	C₄⋊C₈	D₄×C₆₇	1	2	yes	—
273	C₄⋊C₈	D₄×C₆₇	1	2	yes	—
274	#103	—	2	4	no	—
275	#102	—	2	4	no	—
276	C₄×C₈	—	2	1	no	—
277	C₄×C₈	—	2	1	no	—
278	C₂⁴	—	4	1	no	—
279	C₂×Q₈	—	2	1	no	—
280	C₂×Q₈	—	2	1	no	—
281	C₂×Q₈	—	2	1	no	—
282	C₂×Q₈	—	2	2	no	—
283	C₂×Q₈	—	2	1	no	—
284	C₂×Q₈	—	2	1	no	—
285	C₂×Q₈	—	4	1	no	—
286	C₂×Q₈	—	4	2	no	—
287	C₂×Q₈	—	4	2	no	—
288	C₂×Q₈	SD₁₆×C₆₇	1	2	yes	complemented
289	C₂×D₄	—	2	2	no	—
290	C₂×D₄	—	2	2	no	—
291	C₂×D₄	—	2	2	no	—
292	C₂×D₄	—	4	1	no	—
293	C₂×D₄	—	4	1	no	—
294	C₂×D₄	—	4	1	no	—
295	C₂×D₄	—	4	1	no	—
296	C₂×D₄	—	4	2	no	—
297	C₂×D₄	—	4	1	no	—
298	C₂×D₄	—	4	1	no	—
299	C₂×D₄	—	4	2	no	—
300	C₂×D₄	—	4	1	no	—
301	C₂×D₄	—	8	2	no	—
302	C₂²×C₄	—	2	1	no	—
303	C₂²×C₄	—	2	1	no	—
304	C₂²×C₄	—	4	1	no	—
305	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
306	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
307	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
308	SD₁₆	—	4	16	no	—
309	SD₁₆	—	4	8	no	—
310	C₂×C₈	—	2	2	no	—
311	C₂×C₈	—	2	2	no	—
312	C₂×C₈	—	2	2	no	—
313	C₂×C₈	—	2	2	no	—
314	C₂×C₈	—	4	1	no	—
315	C₂×C₈	—	4	1	no	—
316	C₂×C₈	—	4	1	no	—
317	C₂×C₈	—	4	1	no	—
318	C₂×C₈	—	4	1	no	—
319	C₂×C₈	—	4	2	no	—
320	C₂×C₈	SD₁₆×C₆₇	1	4	yes	complemented
321	C₄⋊C₄	—	2	1	no	—
322	C₄⋊C₄	—	2	2	no	—
323	C₄⋊C₄	—	2	1	no	—
324	C₄⋊C₄	—	2	2	no	—
325	C₄⋊C₄	—	4	1	no	—
326	C₄⋊C₄	—	4	1	no	—
327	C₄⋊C₄	—	4	1	no	—
328	C₄⋊C₄	—	4	1	no	—
329	C₄²	—	4	1	no	—
330	C₄²	—	4	1	no	—
331	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
332	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
333	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
334	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
335	C₂³	—	4	2	no	—
336	C₂³	—	8	2	no	—
337	C₂³	—	8	1	no	—
338	C₂³	—	8	1	no	—
339	C₂³	—	8	2	no	—
340	C₂³	#121×C₆₇	1	1	yes	characteristic
341	Q₈	—	2	2	no	—
342	Q₈	—	2	2	no	—
343	Q₈	—	4	1	no	—
344	Q₈	—	4	1	no	—
345	Q₈	—	4	4	no	—
346	Q₈	—	8	1	no	—
347	Q₈	—	8	1	no	—
348	D₄	—	4	2	no	—
349	D₄	—	4	4	no	—
350	D₄	—	4	2	no	—
351	D₄	—	8	2	no	—
352	D₄	—	8	2	no	—
353	D₄	—	8	1	no	—
354	D₄	—	8	1	no	—
355	D₄	—	8	1	no	—
356	D₄	—	8	1	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	4	1	no	—
360	C₂×C₄	—	4	1	no	—
361	C₂×C₄	—	4	2	no	—
362	C₂×C₄	—	4	1	no	—
363	C₂×C₄	—	4	2	no	—
364	C₂×C₄	—	8	2	no	—
365	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
366	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
367	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
368	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
369	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
370	C₂×C₄	#127×C₆₇	1	1	yes	characteristic
371	C₂×C₄	C₂×SD₁₆×C₆₇	1	1	yes	characteristic
372	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
373	C₂×C₄	#135×C₆₇	1	1	yes	characteristic
374	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
375	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
376	C₂×C₄	#137×C₆₇	1	1	yes	characteristic
377	C₈	—	2	4	no	—
378	C₈	—	4	2	no	—
379	C₈	—	4	2	no	—
380	C₈	—	8	1	no	—
381	C₂²	—	8	2	no	—
382	C₂²	—	8	2	no	—
383	C₂²	—	8	2	no	—
384	C₂²	—	16	2	no	—
385	C₂²	—	16	2	no	—
386	C₂²	#392×C₆₇	1	1	yes	characteristic
387	C₂²	#460×C₆₇	1	2	yes	—
388	C₂²	#465×C₆₇	1	1	yes	characteristic
389	C₂²	#468×C₆₇	1	1	yes	characteristic
390	C₂²	#471×C₆₇	1	2	yes	—
391	C₄	—	2	1	no	—
392	C₄	—	2	1	no	—
393	C₄	—	4	2	no	—
394	C₄	—	8	2	no	—
395	C₄	—	8	1	no	—
396	C₄	#448×C₆₇	1	1	yes	characteristic
397	C₄	#449×C₆₇	1	1	yes	characteristic
398	C₄	#450×C₆₇	1	1	yes	characteristic
399	C₄	#450×C₆₇	1	2	yes	—
400	C₄	#452×C₆₇	1	1	yes	characteristic
401	C₄	#492×C₆₇	1	1	yes	characteristic
402	C₄	#496×C₆₇	1	1	yes	characteristic
403	C₂	—	16	2	no	—
404	C₂	C₆₇×#1666	1	2	yes	—
405	C₂	C₆₇×#1686	1	2	yes	—
406	C₂	C₆₇×#2000	1	1	yes	characteristic
407	C₂	C₆₇×#2014	1	1	yes	characteristic
408	C₂	C₆₇×#2019	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	#121
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₆₇
prime radical	C₂×#127×C₆₇	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#475×C₆₇	C₂	semidirect
4. C₆₇×#1984	C₂	semidirect
5. C₂×#332×C₆₇	C₂	semidirect
7. C₆₇×#1277	C₂	semidirect
83. C₂×Q₈×C₆₇	SD₁₆	semidirect
115. C₂×C₈×C₆₇	SD₁₆	semidirect
198. #10268	C₆₇	direct (N × H)
206. C₂×#475	C₂×C₆₇	semidirect
208. #1984	C₂×C₆₇	semidirect
209. C₂×#332	C₂×C₆₇	semidirect
211. #1277	C₂×C₆₇	semidirect
212. C₆₇	#10268	direct (N × H)
288. C₂×Q₈	SD₁₆×C₆₇	semidirect
320. C₂×C₈	SD₁₆×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#492×C₆₇	2
C₂×#475×C₆₇	2
C₂×#460×C₆₇	2
C₆₇×#1984	2
C₂×#332×C₆₇	2
C₂×#331×C₆₇	2
C₆₇×#1277	2
#10268	67

Subgroups — C67×#10268 #30689519

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#10268 #30689519