Subgroups — C₆₇×#19900 #30699151

Order 17152.

710 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3393	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₇×#3324	C₂	1	2	yes	maximal, complemented
3	D₄×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1709	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1701	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1669	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1668	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1639	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1628	C₂	1	2	yes	maximal, complemented
10	C₆₇×#1589	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×SD₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₈×C₆₇	—	2	1	no	—
13	#556×C₆₇	C₂²	1	2	yes	complemented
14	D₄²×C₆₇	C₂²	1	2	yes	—
15	#495×C₆₇	—	2	1	no	—
16	#494×C₆₇	C₂²	1	1	yes	characteristic, complemented
17	#493×C₆₇	C₂²	1	1	yes	characteristic
18	#486×C₆₇	C₂²	1	2	yes	complemented
19	#486×C₆₇	C₂²	1	2	yes	complemented
20	#482×C₆₇	—	2	2	no	—
21	#474×C₆₇	C₂²	1	2	yes	—
22	#468×C₆₇	—	2	2	no	—
23	#466×C₆₇	—	2	1	no	—
24	#464×C₆₇	—	2	2	no	—
25	#461×C₆₇	C₂²	1	2	yes	complemented
26	#460×C₆₇	C₂²	1	2	yes	complemented
27	#459×C₆₇	C₂²	1	2	yes	—
28	#450×C₆₇	—	2	2	no	—
29	#447×C₆₇	—	2	2	no	—
30	C₈×Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
31	C₄×Q₁₆×C₆₇	—	2	1	no	—
32	C₄×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic
33	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
34	#436×C₆₇	C₂²	1	2	yes	complemented
35	C₈×D₄×C₆₇	C₂²	1	1	yes	characteristic
36	#418×C₆₇	—	2	2	no	—
37	#335×C₆₇	C₂²	1	1	yes	characteristic, complemented
38	#334×C₆₇	C₂²	1	1	yes	characteristic, complemented
39	#332×C₆₇	C₂²	1	2	yes	complemented
40	#331×C₆₇	C₂²	1	2	yes	complemented
41	Q₈⋊C₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
42	D₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
43	D₄⋊C₈×C₆₇	C₂²	1	2	yes	complemented
44	C₈²×C₆₇	C₂²	1	1	yes	characteristic, complemented
45	C₂²×D₄×C₆₇	—	2	2	no	—
46	C₂²×D₄×C₆₇	—	2	2	no	—
47	C₂×Q₁₆×C₆₇	—	2	1	no	—
48	C₂×Q₁₆×C₆₇	—	2	1	no	—
49	C₂×SD₁₆×C₆₇	—	2	2	no	—
50	C₂×SD₁₆×C₆₇	—	2	2	no	—
51	C₂×SD₁₆×C₆₇	D₄	1	1	yes	characteristic
52	C₂×D₈×C₆₇	—	2	2	no	—
53	C₂×D₈×C₆₇	—	2	1	no	—
54	C₂×D₈×C₆₇	—	2	1	no	—
55	C₂×D₈×C₆₇	—	4	4	no	—
56	C₂×D₈×C₆₇	D₄	1	1	yes	characteristic
57	C₂×M₁₆×C₆₇	—	2	2	no	—
58	C₂²×C₈×C₆₇	—	2	1	no	—
59	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
60	#127×C₆₇	C₂³	1	2	yes	—
61	#126×C₆₇	—	4	2	no	—
62	#124×C₆₇	—	2	2	no	—
63	C₂²⋊Q₈×C₆₇	—	2	2	no	—
64	#121×C₆₇	—	2	2	no	—
65	#121×C₆₇	—	2	2	no	—
66	#120×C₆₇	—	4	2	no	—
67	C₄×Q₈×C₆₇	—	2	2	no	—
68	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
69	C₄×D₄×C₆₇	C₂³	1	2	yes	—
70	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
71	#117×C₆₇	—	2	2	no	—
72	#107×C₆₇	—	2	2	no	—
73	#107×C₆₇	D₄	1	1	yes	characteristic
74	#106×C₆₇	D₄	1	1	yes	characteristic
75	C₄⋊C₈×C₆₇	—	2	1	no	—
76	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
77	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
78	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
79	#103×C₆₇	—	2	2	no	—
80	#103×C₆₇	D₄	1	1	yes	characteristic
81	#102×C₆₇	—	2	2	no	—
82	#102×C₆₇	—	2	2	no	—
83	#102×C₆₇	—	2	2	no	—
84	#102×C₆₇	—	2	2	no	—
85	#102×C₆₇	—	2	2	no	—
86	#102×C₆₇	—	4	2	no	—
87	#102×C₆₇	D₄	1	2	yes	—
88	#102×C₆₇	D₄	1	1	yes	characteristic
89	C₂²⋊C₈×C₆₇	—	2	2	no	—
90	C₂²⋊C₈×C₆₇	—	2	1	no	—
91	C₄×C₈×C₆₇	—	2	1	no	—
92	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
94	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
95	C₂⁴×C₆₇	—	4	2	no	—
96	C₂×Q₈×C₆₇	—	2	2	no	—
97	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
98	C₂×D₄×C₆₇	—	2	2	no	—
99	C₂×D₄×C₆₇	—	2	2	no	—
100	C₂×D₄×C₆₇	—	2	2	no	—
101	C₂×D₄×C₆₇	—	4	2	no	—
102	C₂×D₄×C₆₇	—	4	2	no	—
103	C₂×D₄×C₆₇	—	4	4	no	—
104	C₂×D₄×C₆₇	—	4	4	no	—
105	C₂×D₄×C₆₇	—	8	2	no	—
106	C₂×D₄×C₆₇	—	8	2	no	—
107	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
108	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₆₇	—	2	2	no	—
110	C₂²×C₄×C₆₇	—	2	1	no	—
111	Q₁₆×C₆₇	—	4	2	no	—
112	Q₁₆×C₆₇	—	8	1	no	—
113	SD₁₆×C₆₇	—	2	2	no	—
114	SD₁₆×C₆₇	—	8	2	no	—
115	SD₁₆×C₆₇	—	8	2	no	—
116	D₈×C₆₇	—	2	2	no	—
117	D₈×C₆₇	—	4	4	no	—
118	D₈×C₆₇	—	4	2	no	—
119	D₈×C₆₇	—	8	1	no	—
120	M₁₆×C₆₇	—	8	2	no	—
121	C₂×C₈×C₆₇	—	2	2	no	—
122	C₂×C₈×C₆₇	—	2	1	no	—
123	C₂×C₈×C₆₇	—	2	1	no	—
124	C₂×C₈×C₆₇	—	2	1	no	—
125	C₂×C₈×C₆₇	—	4	2	no	—
126	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
127	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
128	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
129	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
130	C₄⋊C₄×C₆₇	—	2	1	no	—
131	C₄⋊C₄×C₆₇	—	2	2	no	—
132	C₄⋊C₄×C₆₇	—	2	2	no	—
133	C₄⋊C₄×C₆₇	—	4	2	no	—
134	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
135	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
136	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
137	C₂²⋊C₄×C₆₇	—	2	1	no	—
138	C₂²⋊C₄×C₆₇	—	2	2	no	—
139	C₂²⋊C₄×C₆₇	—	4	2	no	—
140	C₂²⋊C₄×C₆₇	—	4	2	no	—
141	C₂²⋊C₄×C₆₇	—	4	2	no	—
142	C₄²×C₆₇	—	2	1	no	—
143	C₄²×C₆₇	—	4	2	no	—
144	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
145	C₂³×C₆₇	—	2	1	no	—
146	C₂³×C₆₇	—	2	2	no	—
147	C₂³×C₆₇	—	4	2	no	—
148	C₂³×C₆₇	—	8	4	no	—
149	C₂³×C₆₇	—	8	4	no	—
150	C₂³×C₆₇	—	16	2	no	—
151	Q₈×C₆₇	—	2	1	no	—
152	Q₈×C₆₇	—	2	1	no	—
153	Q₈×C₆₇	—	4	2	no	—
154	Q₈×C₆₇	—	4	2	no	—
155	D₄×C₆₇	—	2	2	no	—
156	D₄×C₆₇	—	2	1	no	—
157	D₄×C₆₇	—	2	2	no	—
158	D₄×C₆₇	—	4	2	no	—
159	D₄×C₆₇	—	4	2	no	—
160	D₄×C₆₇	—	4	2	no	—
161	D₄×C₆₇	—	4	2	no	—
162	D₄×C₆₇	—	8	2	no	—
163	D₄×C₆₇	—	8	4	no	—
164	D₄×C₆₇	—	8	4	no	—
165	D₄×C₆₇	#135	1	2	yes	—
166	C₂×C₄×C₆₇	—	2	1	no	—
167	C₂×C₄×C₆₇	—	2	1	no	—
168	C₂×C₄×C₆₇	—	2	2	no	—
169	C₂×C₄×C₆₇	—	2	1	no	—
170	C₂×C₄×C₆₇	—	4	2	no	—
171	C₂×C₄×C₆₇	—	4	1	no	—
172	C₂×C₄×C₆₇	—	4	2	no	—
173	C₂×C₄×C₆₇	—	4	1	no	—
174	C₂×C₄×C₆₇	—	4	2	no	—
175	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
176	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
177	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
178	C₈×C₆₇	—	2	1	no	—
179	C₈×C₆₇	—	2	1	no	—
180	C₈×C₆₇	—	2	1	no	—
181	C₈×C₆₇	—	4	1	no	—
182	C₈×C₆₇	—	4	1	no	—
183	C₈×C₆₇	—	4	2	no	—
184	C₈×C₆₇	—	4	1	no	—
185	C₈×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
186	C₂²×C₆₇	—	2	2	no	—
187	C₂²×C₆₇	—	4	1	no	—
188	C₂²×C₆₇	—	4	1	no	—
189	C₂²×C₆₇	—	4	2	no	—
190	C₂²×C₆₇	—	4	2	no	—
191	C₂²×C₆₇	—	4	2	no	—
192	C₂²×C₆₇	—	8	2	no	—
193	C₂²×C₆₇	—	8	2	no	—
194	C₂²×C₆₇	—	8	2	no	—
195	C₂²×C₆₇	—	16	8	no	—
196	C₂²×C₆₇	D₄²	1	1	yes	characteristic
197	C₄×C₆₇	—	2	1	no	—
198	C₄×C₆₇	—	4	1	no	—
199	C₄×C₆₇	—	4	2	no	—
200	C₄×C₆₇	—	4	1	no	—
201	C₄×C₆₇	—	4	1	no	—
202	C₄×C₆₇	—	8	2	no	—
203	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
204	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
205	C₄×C₆₇	#576	1	1	yes	characteristic
206	C₄×C₆₇	#576	1	1	yes	characteristic
207	#19900	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
208	C₂×C₆₇	—	4	2	no	—
209	C₂×C₆₇	—	8	2	no	—
210	C₂×C₆₇	—	16	2	no	—
211	C₂×C₆₇	#3274	1	1	yes	characteristic
212	C₂×C₆₇	#3280	1	1	yes	characteristic
213	C₂×C₆₇	#3290	1	1	yes	characteristic
214	#3393	C₂×C₆₇	1	1	yes	characteristic, complemented
215	#3324	C₂×C₆₇	1	2	yes	complemented
216	D₄×D₈	C₂×C₆₇	1	1	yes	characteristic
217	#1709	C₂×C₆₇	1	1	yes	characteristic, complemented
218	#1701	C₂×C₆₇	1	2	yes	complemented
219	#1669	C₂×C₆₇	1	1	yes	characteristic, complemented
220	#1668	C₂×C₆₇	1	1	yes	characteristic, complemented
221	#1639	C₂×C₆₇	1	2	yes	complemented
222	#1628	C₂×C₆₇	1	2	yes	complemented
223	#1589	C₂×C₆₇	1	1	yes	characteristic, complemented
224	C₈×SD₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
225	C₆₇	#19900	1	1	yes	characteristic, direct product factor, Sylow
226	C₂²×D₈	—	2	1	no	—
227	#556	C₂²×C₆₇	1	2	yes	complemented
228	D₄²	C₂²×C₆₇	1	2	yes	—
229	#495	—	2	1	no	—
230	#494	C₂²×C₆₇	1	1	yes	characteristic, complemented
231	#493	C₂²×C₆₇	1	1	yes	characteristic
232	#486	C₂²×C₆₇	1	2	yes	complemented
233	#486	C₂²×C₆₇	1	2	yes	complemented
234	#482	—	2	2	no	—
235	#474	C₂²×C₆₇	1	2	yes	—
236	#468	—	2	2	no	—
237	#466	—	2	1	no	—
238	#464	—	2	2	no	—
239	#461	C₂²×C₆₇	1	2	yes	complemented
240	#460	C₂²×C₆₇	1	2	yes	complemented
241	#459	C₂²×C₆₇	1	2	yes	—
242	#450	—	2	2	no	—
243	#447	—	2	2	no	—
244	C₈×Q₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
245	C₄×Q₁₆	—	2	1	no	—
246	C₄×SD₁₆	C₂²×C₆₇	1	1	yes	characteristic
247	C₄×D₈	C₂²×C₆₇	1	1	yes	characteristic
248	#436	C₂²×C₆₇	1	2	yes	complemented
249	C₈×D₄	C₂²×C₆₇	1	1	yes	characteristic
250	#418	—	2	2	no	—
251	#335	C₂²×C₆₇	1	1	yes	characteristic, complemented
252	#334	C₂²×C₆₇	1	1	yes	characteristic, complemented
253	#332	C₂²×C₆₇	1	2	yes	complemented
254	#331	C₂²×C₆₇	1	2	yes	complemented
255	Q₈⋊C₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
256	D₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
257	D₄⋊C₈	C₂²×C₆₇	1	2	yes	complemented
258	C₈²	C₂²×C₆₇	1	1	yes	characteristic, complemented
259	C₂²×D₄	—	2	2	no	—
260	C₂²×D₄	—	2	2	no	—
261	C₂×Q₁₆	—	2	1	no	—
262	C₂×Q₁₆	—	2	1	no	—
263	C₂×SD₁₆	—	2	2	no	—
264	C₂×SD₁₆	—	2	2	no	—
265	C₂×SD₁₆	D₄×C₆₇	1	1	yes	characteristic
266	C₂×D₈	—	2	2	no	—
267	C₂×D₈	—	2	1	no	—
268	C₂×D₈	—	2	1	no	—
269	C₂×D₈	—	4	4	no	—
270	C₂×D₈	D₄×C₆₇	1	1	yes	characteristic
271	C₂×M₁₆	—	2	2	no	—
272	C₂²×C₈	—	2	1	no	—
273	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
274	#127	C₂³×C₆₇	1	2	yes	—
275	#126	—	4	2	no	—
276	#124	—	2	2	no	—
277	C₂²⋊Q₈	—	2	2	no	—
278	#121	—	2	2	no	—
279	#121	—	2	2	no	—
280	#120	—	4	2	no	—
281	C₄×Q₈	—	2	2	no	—
282	C₄×Q₈	C₂³×C₆₇	1	1	yes	characteristic
283	C₄×D₄	C₂³×C₆₇	1	1	yes	characteristic
284	C₄×D₄	C₂³×C₆₇	1	2	yes	—
285	#117	—	2	2	no	—
286	#107	—	2	2	no	—
287	#107	D₄×C₆₇	1	1	yes	characteristic
288	#106	D₄×C₆₇	1	1	yes	characteristic
289	C₄⋊C₈	—	2	1	no	—
290	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
291	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
292	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
293	#103	—	2	2	no	—
294	#103	D₄×C₆₇	1	1	yes	characteristic
295	#102	—	2	2	no	—
296	#102	—	2	2	no	—
297	#102	—	2	2	no	—
298	#102	—	2	2	no	—
299	#102	—	2	2	no	—
300	#102	—	4	2	no	—
301	#102	D₄×C₆₇	1	1	yes	characteristic
302	#102	D₄×C₆₇	1	2	yes	—
303	C₂²⋊C₈	—	2	1	no	—
304	C₂²⋊C₈	—	2	2	no	—
305	C₄×C₈	—	2	1	no	—
306	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
307	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
308	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
309	C₂⁴	—	4	2	no	—
310	C₂×Q₈	—	2	2	no	—
311	C₂×Q₈	C₂×D₄×C₆₇	1	1	yes	characteristic
312	C₂×D₄	—	2	2	no	—
313	C₂×D₄	—	2	2	no	—
314	C₂×D₄	—	2	2	no	—
315	C₂×D₄	—	4	2	no	—
316	C₂×D₄	—	4	2	no	—
317	C₂×D₄	—	4	4	no	—
318	C₂×D₄	—	4	4	no	—
319	C₂×D₄	—	8	2	no	—
320	C₂×D₄	—	8	2	no	—
321	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
322	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
323	C₂²×C₄	—	2	2	no	—
324	C₂²×C₄	—	2	1	no	—
325	Q₁₆	—	4	2	no	—
326	Q₁₆	—	8	1	no	—
327	SD₁₆	—	2	2	no	—
328	SD₁₆	—	8	2	no	—
329	SD₁₆	—	8	2	no	—
330	D₈	—	2	2	no	—
331	D₈	—	4	4	no	—
332	D₈	—	4	2	no	—
333	D₈	—	8	1	no	—
334	M₁₆	—	8	2	no	—
335	C₂×C₈	—	2	1	no	—
336	C₂×C₈	—	2	2	no	—
337	C₂×C₈	—	2	1	no	—
338	C₂×C₈	—	2	1	no	—
339	C₂×C₈	—	4	2	no	—
340	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
341	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
342	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
343	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
344	C₄⋊C₄	—	2	1	no	—
345	C₄⋊C₄	—	2	2	no	—
346	C₄⋊C₄	—	2	2	no	—
347	C₄⋊C₄	—	4	2	no	—
348	C₄⋊C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
349	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
350	C₄⋊C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
351	C₂²⋊C₄	—	2	1	no	—
352	C₂²⋊C₄	—	2	2	no	—
353	C₂²⋊C₄	—	4	2	no	—
354	C₂²⋊C₄	—	4	2	no	—
355	C₂²⋊C₄	—	4	2	no	—
356	C₄²	—	2	1	no	—
357	C₄²	—	4	2	no	—
358	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
359	C₂³	—	2	1	no	—
360	C₂³	—	2	2	no	—
361	C₂³	—	4	2	no	—
362	C₂³	—	8	4	no	—
363	C₂³	—	8	4	no	—
364	C₂³	—	16	2	no	—
365	Q₈	—	2	1	no	—
366	Q₈	—	2	1	no	—
367	Q₈	—	4	2	no	—
368	Q₈	—	4	2	no	—
369	D₄	—	2	2	no	—
370	D₄	—	2	1	no	—
371	D₄	—	2	2	no	—
372	D₄	—	4	2	no	—
373	D₄	—	4	2	no	—
374	D₄	—	4	2	no	—
375	D₄	—	4	2	no	—
376	D₄	—	8	2	no	—
377	D₄	—	8	4	no	—
378	D₄	—	8	4	no	—
379	D₄	#135×C₆₇	1	2	yes	—
380	C₂×C₄	—	2	1	no	—
381	C₂×C₄	—	2	1	no	—
382	C₂×C₄	—	2	2	no	—
383	C₂×C₄	—	2	1	no	—
384	C₂×C₄	—	4	2	no	—
385	C₂×C₄	—	4	1	no	—
386	C₂×C₄	—	4	2	no	—
387	C₂×C₄	—	4	1	no	—
388	C₂×C₄	—	4	2	no	—
389	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
390	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
391	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
392	C₈	—	2	1	no	—
393	C₈	—	2	1	no	—
394	C₈	—	2	1	no	—
395	C₈	—	4	1	no	—
396	C₈	—	4	1	no	—
397	C₈	—	4	2	no	—
398	C₈	—	4	1	no	—
399	C₈	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
400	C₂²	—	2	2	no	—
401	C₂²	—	4	1	no	—
402	C₂²	—	4	1	no	—
403	C₂²	—	4	2	no	—
404	C₂²	—	4	2	no	—
405	C₂²	—	4	2	no	—
406	C₂²	—	8	2	no	—
407	C₂²	—	8	2	no	—
408	C₂²	—	8	2	no	—
409	C₂²	—	16	8	no	—
410	C₂²	D₄²×C₆₇	1	1	yes	characteristic
411	C₄	—	2	1	no	—
412	C₄	—	4	1	no	—
413	C₄	—	4	2	no	—
414	C₄	—	4	1	no	—
415	C₄	—	4	1	no	—
416	C₄	—	8	2	no	—
417	C₄	C₂×#135×C₆₇	1	1	yes	characteristic
418	C₄	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
419	C₄	#576×C₆₇	1	1	yes	characteristic
420	C₄	#576×C₆₇	1	1	yes	characteristic
421	C₂	—	4	2	no	—
422	C₂	—	8	2	no	—
423	C₂	—	16	2	no	—
424	C₂	C₆₇×#3274	1	1	yes	characteristic
425	C₂	C₆₇×#3280	1	1	yes	characteristic
426	C₂	C₆₇×#3290	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₇
prime radical	D₄×D₈×C₆₇	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3393	C₂	semidirect
2. C₆₇×#3324	C₂	semidirect
4. C₆₇×#1709	C₂	semidirect
5. C₆₇×#1701	C₂	semidirect
6. C₆₇×#1669	C₂	semidirect
7. C₆₇×#1668	C₂	semidirect
8. C₆₇×#1639	C₂	semidirect
9. C₆₇×#1628	C₂	semidirect
10. C₆₇×#1589	C₂	semidirect
11. C₈×SD₁₆×C₆₇	C₂	semidirect
13. #556×C₆₇	C₂²	semidirect
16. #494×C₆₇	C₂²	semidirect
18. #486×C₆₇	C₂²	semidirect
19. #486×C₆₇	C₂²	semidirect
25. #461×C₆₇	C₂²	semidirect
26. #460×C₆₇	C₂²	semidirect
30. C₈×Q₈×C₆₇	C₂²	semidirect
34. #436×C₆₇	C₂²	semidirect
37. #335×C₆₇	C₂²	semidirect
38. #334×C₆₇	C₂²	semidirect
39. #332×C₆₇	C₂²	semidirect
40. #331×C₆₇	C₂²	semidirect
41. Q₈⋊C₈×C₆₇	C₂²	semidirect
43. D₄⋊C₈×C₆₇	C₂²	semidirect
44. C₈²×C₆₇	C₂²	semidirect
207. #19900	C₆₇	direct (N × H)
214. #3393	C₂×C₆₇	semidirect
215. #3324	C₂×C₆₇	semidirect
217. #1709	C₂×C₆₇	semidirect
218. #1701	C₂×C₆₇	semidirect
219. #1669	C₂×C₆₇	semidirect
220. #1668	C₂×C₆₇	semidirect
221. #1639	C₂×C₆₇	semidirect
222. #1628	C₂×C₆₇	semidirect
223. #1589	C₂×C₆₇	semidirect
224. C₈×SD₁₆	C₂×C₆₇	semidirect
225. C₆₇	#19900	direct (N × H)
227. #556	C₂²×C₆₇	semidirect
230. #494	C₂²×C₆₇	semidirect
232. #486	C₂²×C₆₇	semidirect
233. #486	C₂²×C₆₇	semidirect
239. #461	C₂²×C₆₇	semidirect
240. #460	C₂²×C₆₇	semidirect
244. C₈×Q₈	C₂²×C₆₇	semidirect
248. #436	C₂²×C₆₇	semidirect
251. #335	C₂²×C₆₇	semidirect
252. #334	C₂²×C₆₇	semidirect
253. #332	C₂²×C₆₇	semidirect
254. #331	C₂²×C₆₇	semidirect
255. Q₈⋊C₈	C₂²×C₆₇	semidirect
257. D₄⋊C₈	C₂²×C₆₇	semidirect
258. C₈²	C₂²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3393	2
C₆₇×#3324	2
D₄×D₈×C₆₇	2
C₆₇×#1709	2
C₆₇×#1701	2
C₆₇×#1669	2
C₆₇×#1668	2
C₆₇×#1639	2
C₆₇×#1628	2
C₆₇×#1589	2
C₈×SD₁₆×C₆₇	2
#19900	67

Subgroups — C67×#19900 #30699151

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#19900 #30699151