Subgroups — C₆₇×#32812 #30712063

Order 17152.

788 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2641	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₇×#2640	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2055	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2054	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2026	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#2025	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#2023	C₂	1	2	yes	maximal, complemented
8	C₆₇×#2014	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#2012	C₂	1	2	yes	maximal, complemented
10	C₆₇×#2011	C₂	1	1	yes	maximal, characteristic
11	C₆₇×#1832	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	—
13	C₂²×D₈×C₆₇	C₂²	1	2	yes	—
14	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#124×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
19	C₂×#103×C₆₇	C₂²	1	2	yes	—
20	C₂×#102×C₆₇	C₂²	1	2	yes	—
21	C₂×#102×C₆₇	C₂²	1	2	yes	—
22	C₂×#102×C₆₇	C₂²	1	2	yes	—
23	#397×C₆₇	—	2	2	no	—
24	#394×C₆₇	—	2	2	no	—
25	#394×C₆₇	—	2	2	no	—
26	#393×C₆₇	—	2	2	no	—
27	#391×C₆₇	C₂²	1	1	yes	characteristic
28	#391×C₆₇	C₂²	1	1	yes	characteristic
29	#390×C₆₇	C₂²	1	1	yes	characteristic
30	#390×C₆₇	C₂²	1	2	yes	—
31	#390×C₆₇	C₂²	1	1	yes	characteristic
32	#390×C₆₇	C₂²	1	2	yes	—
33	#376×C₆₇	C₂²	1	1	yes	characteristic
34	#340×C₆₇	C₂²	1	2	yes	—
35	#340×C₆₇	C₂²	1	2	yes	—
36	#336×C₆₇	C₂²	1	1	yes	characteristic
37	#336×C₆₇	C₂²	1	2	yes	—
38	#336×C₆₇	C₂²	1	1	yes	characteristic
39	C₂²×Q₈×C₆₇	—	2	1	no	—
40	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
41	C₂²×D₄×C₆₇	—	2	2	no	—
42	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
43	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
44	C₂×SD₁₆×C₆₇	—	4	2	no	—
45	C₂×SD₁₆×C₆₇	—	4	2	no	—
46	C₂×SD₁₆×C₆₇	—	4	2	no	—
47	C₂×D₈×C₆₇	—	4	2	no	—
48	C₂×D₈×C₆₇	—	4	2	no	—
49	C₂×D₈×C₆₇	—	4	2	no	—
50	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
51	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
52	C₄⋊Q₈×C₆₇	—	4	2	no	—
53	#127×C₆₇	—	4	2	no	—
54	#125×C₆₇	—	4	2	no	—
55	#124×C₆₇	—	4	2	no	—
56	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
57	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
58	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
59	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
60	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
61	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
62	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
63	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
64	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
65	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
66	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
67	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
68	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
69	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
70	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
71	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
72	C₄⋊C₈×C₆₇	—	4	2	no	—
73	#103×C₆₇	—	4	2	no	—
74	#102×C₆₇	—	4	2	no	—
75	#102×C₆₇	—	4	2	no	—
76	#102×C₆₇	—	4	2	no	—
77	#95×C₆₇	—	2	2	no	—
78	#95×C₆₇	—	2	2	no	—
79	#95×C₆₇	D₄	1	2	yes	—
80	#95×C₆₇	D₄	1	2	yes	—
81	C₂⁴×C₆₇	—	2	1	no	—
82	C₂⁴×C₆₇	—	2	2	no	—
83	C₂×Q₈×C₆₇	—	2	2	no	—
84	C₂×Q₈×C₆₇	—	2	2	no	—
85	C₂×Q₈×C₆₇	—	2	1	no	—
86	C₂×Q₈×C₆₇	—	2	1	no	—
87	C₂×Q₈×C₆₇	—	4	2	no	—
88	C₂×Q₈×C₆₇	—	8	2	no	—
89	C₂×D₄×C₆₇	—	2	2	no	—
90	C₂×D₄×C₆₇	—	2	2	no	—
91	C₂×D₄×C₆₇	—	2	2	no	—
92	C₂×D₄×C₆₇	—	2	2	no	—
93	C₂×D₄×C₆₇	—	2	2	no	—
94	C₂×D₄×C₆₇	—	2	2	no	—
95	C₂×D₄×C₆₇	—	2	1	no	—
96	C₂×D₄×C₆₇	—	2	2	no	—
97	C₂×D₄×C₆₇	—	2	1	no	—
98	C₂×D₄×C₆₇	—	2	2	no	—
99	C₂×D₄×C₆₇	—	4	4	no	—
100	C₂×D₄×C₆₇	—	8	2	no	—
101	C₂×D₄×C₆₇	—	8	2	no	—
102	C₂²×C₄×C₆₇	—	2	2	no	—
103	C₂²×C₄×C₆₇	—	2	1	no	—
104	C₂²×C₄×C₆₇	—	2	1	no	—
105	C₂²×C₄×C₆₇	—	2	1	no	—
106	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
109	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
110	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
111	SD₁₆×C₆₇	—	4	8	no	—
112	D₈×C₆₇	—	4	8	no	—
113	C₂×C₈×C₆₇	—	2	2	no	—
114	C₂×C₈×C₆₇	—	2	2	no	—
115	C₂×C₈×C₆₇	—	2	2	no	—
116	C₂×C₈×C₆₇	—	2	2	no	—
117	C₂×C₈×C₆₇	—	2	2	no	—
118	C₂×C₈×C₆₇	—	2	2	no	—
119	C₄⋊C₄×C₆₇	—	2	2	no	—
120	C₄⋊C₄×C₆₇	—	2	2	no	—
121	C₄⋊C₄×C₆₇	—	2	2	no	—
122	C₄⋊C₄×C₆₇	—	2	2	no	—
123	C₄⋊C₄×C₆₇	—	4	4	no	—
124	C₄⋊C₄×C₆₇	—	4	2	no	—
125	C₂²⋊C₄×C₆₇	—	4	2	no	—
126	C₂²⋊C₄×C₆₇	—	4	2	no	—
127	C₂²⋊C₄×C₆₇	—	8	2	no	—
128	C₂²⋊C₄×C₆₇	—	8	2	no	—
129	C₂²⋊C₄×C₆₇	—	8	2	no	—
130	C₂²⋊C₄×C₆₇	—	8	2	no	—
131	C₂²⋊C₄×C₆₇	—	8	2	no	—
132	C₂²⋊C₄×C₆₇	—	8	2	no	—
133	C₄²×C₆₇	—	2	1	no	—
134	C₄²×C₆₇	—	2	2	no	—
135	C₄²×C₆₇	—	2	2	no	—
136	C₄²×C₆₇	—	2	1	no	—
137	C₂³×C₆₇	—	4	1	no	—
138	C₂³×C₆₇	—	4	2	no	—
139	C₂³×C₆₇	—	4	2	no	—
140	C₂³×C₆₇	—	4	1	no	—
141	C₂³×C₆₇	—	4	1	no	—
142	C₂³×C₆₇	—	4	2	no	—
143	C₂³×C₆₇	—	4	2	no	—
144	C₂³×C₆₇	—	4	2	no	—
145	C₂³×C₆₇	—	4	2	no	—
146	C₂³×C₆₇	—	4	2	no	—
147	C₂³×C₆₇	—	4	2	no	—
148	C₂³×C₆₇	—	4	2	no	—
149	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
150	Q₈×C₆₇	—	4	2	no	—
151	Q₈×C₆₇	—	4	2	no	—
152	Q₈×C₆₇	—	8	4	no	—
153	D₄×C₆₇	—	4	2	no	—
154	D₄×C₆₇	—	4	2	no	—
155	D₄×C₆₇	—	4	2	no	—
156	D₄×C₆₇	—	4	2	no	—
157	D₄×C₆₇	—	4	2	no	—
158	D₄×C₆₇	—	4	2	no	—
159	D₄×C₆₇	—	8	8	no	—
160	C₂×C₄×C₆₇	—	2	1	no	—
161	C₂×C₄×C₆₇	—	2	1	no	—
162	C₂×C₄×C₆₇	—	2	2	no	—
163	C₂×C₄×C₆₇	—	2	2	no	—
164	C₂×C₄×C₆₇	—	2	2	no	—
165	C₂×C₄×C₆₇	—	2	2	no	—
166	C₂×C₄×C₆₇	—	2	2	no	—
167	C₂×C₄×C₆₇	—	2	2	no	—
168	C₂×C₄×C₆₇	—	2	2	no	—
169	C₂×C₄×C₆₇	—	2	2	no	—
170	C₂×C₄×C₆₇	—	4	1	no	—
171	C₂×C₄×C₆₇	—	4	2	no	—
172	C₂×C₄×C₆₇	—	4	1	no	—
173	C₂×C₄×C₆₇	—	4	1	no	—
174	C₂×C₄×C₆₇	—	4	1	no	—
175	C₂×C₄×C₆₇	—	4	1	no	—
176	C₂×C₄×C₆₇	—	4	2	no	—
177	C₂×C₄×C₆₇	—	4	2	no	—
178	C₂×C₄×C₆₇	—	4	2	no	—
179	C₂×C₄×C₆₇	—	4	2	no	—
180	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
181	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
182	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
183	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
184	C₈×C₆₇	—	4	2	no	—
185	C₈×C₆₇	—	4	2	no	—
186	C₂²×C₆₇	—	8	1	no	—
187	C₂²×C₆₇	—	8	2	no	—
188	C₂²×C₆₇	—	8	1	no	—
189	C₂²×C₆₇	—	8	2	no	—
190	C₂²×C₆₇	—	8	1	no	—
191	C₂²×C₆₇	—	8	2	no	—
192	C₂²×C₆₇	—	8	2	no	—
193	C₂²×C₆₇	—	8	2	no	—
194	C₂²×C₆₇	—	8	2	no	—
195	C₂²×C₆₇	—	8	2	no	—
196	C₂²×C₆₇	—	8	2	no	—
197	C₂²×C₆₇	—	8	2	no	—
198	C₂²×C₆₇	#535	1	1	yes	characteristic
199	C₂²×C₆₇	#535	1	1	yes	characteristic
200	C₂²×C₆₇	#537	1	1	yes	characteristic
201	C₂²×C₆₇	#576	1	2	yes	—
202	C₂²×C₆₇	#577	1	2	yes	—
203	C₄×C₆₇	—	4	1	no	—
204	C₄×C₆₇	—	4	1	no	—
205	C₄×C₆₇	—	4	2	no	—
206	C₄×C₆₇	—	4	2	no	—
207	C₄×C₆₇	—	8	1	no	—
208	C₄×C₆₇	—	8	1	no	—
209	C₄×C₆₇	—	8	1	no	—
210	C₄×C₆₇	—	8	2	no	—
211	C₄×C₆₇	H(3,𝔽₄)	1	2	yes	—
212	C₄×C₆₇	#562	1	2	yes	—
213	#32812	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
214	C₂×C₆₇	—	16	2	no	—
215	C₂×C₆₇	—	16	1	no	—
216	C₂×C₆₇	#2819	1	1	yes	characteristic
217	C₂×C₆₇	#3220	1	1	yes	characteristic
218	C₂×C₆₇	#3221	1	1	yes	characteristic
219	C₂×C₆₇	#3224	1	2	yes	—
220	C₂×C₆₇	#3257	1	2	yes	—
221	#2641	C₂×C₆₇	1	1	yes	characteristic, complemented
222	#2640	C₂×C₆₇	1	1	yes	characteristic, complemented
223	#2055	C₂×C₆₇	1	2	yes	complemented
224	#2054	C₂×C₆₇	1	2	yes	complemented
225	#2026	C₂×C₆₇	1	1	yes	characteristic, complemented
226	#2025	C₂×C₆₇	1	1	yes	characteristic, complemented
227	#2023	C₂×C₆₇	1	2	yes	complemented
228	#2014	C₂×C₆₇	1	1	yes	characteristic, complemented
229	#2012	C₂×C₆₇	1	2	yes	complemented
230	#2011	C₂×C₆₇	1	1	yes	characteristic
231	#1832	C₂×C₆₇	1	1	yes	characteristic, complemented
232	C₆₇	#32812	1	1	yes	characteristic, direct product factor, Sylow
233	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	—
234	C₂²×D₈	C₂²×C₆₇	1	2	yes	—
235	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
236	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
237	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic
238	C₂×#124	C₂²×C₆₇	1	1	yes	characteristic
239	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
240	C₂×#103	C₂²×C₆₇	1	2	yes	—
241	C₂×#102	C₂²×C₆₇	1	2	yes	—
242	C₂×#102	C₂²×C₆₇	1	2	yes	—
243	C₂×#102	C₂²×C₆₇	1	2	yes	—
244	#397	—	2	2	no	—
245	#394	—	2	2	no	—
246	#394	—	2	2	no	—
247	#393	—	2	2	no	—
248	#391	C₂²×C₆₇	1	1	yes	characteristic
249	#391	C₂²×C₆₇	1	1	yes	characteristic
250	#390	C₂²×C₆₇	1	1	yes	characteristic
251	#390	C₂²×C₆₇	1	2	yes	—
252	#390	C₂²×C₆₇	1	1	yes	characteristic
253	#390	C₂²×C₆₇	1	2	yes	—
254	#376	C₂²×C₆₇	1	1	yes	characteristic
255	#340	C₂²×C₆₇	1	2	yes	—
256	#340	C₂²×C₆₇	1	2	yes	—
257	#336	C₂²×C₆₇	1	1	yes	characteristic
258	#336	C₂²×C₆₇	1	2	yes	—
259	#336	C₂²×C₆₇	1	1	yes	characteristic
260	C₂²×Q₈	—	2	1	no	—
261	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
262	C₂²×D₄	—	2	2	no	—
263	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
264	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
265	C₂×SD₁₆	—	4	2	no	—
266	C₂×SD₁₆	—	4	2	no	—
267	C₂×SD₁₆	—	4	2	no	—
268	C₂×D₈	—	4	2	no	—
269	C₂×D₈	—	4	2	no	—
270	C₂×D₈	—	4	2	no	—
271	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
272	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
273	C₄⋊Q₈	—	4	2	no	—
274	#127	—	4	2	no	—
275	#125	—	4	2	no	—
276	#124	—	4	2	no	—
277	C₂×C₄⋊C₄	—	2	2	no	—
278	C₂×C₄⋊C₄	—	2	1	no	—
279	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
280	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
281	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
282	C₂×C₂²⋊C₄	—	2	1	no	—
283	C₂×C₂²⋊C₄	—	2	1	no	—
284	C₂×C₂²⋊C₄	—	2	2	no	—
285	C₂×C₂²⋊C₄	—	2	2	no	—
286	C₂×C₂²⋊C₄	—	2	2	no	—
287	C₂×C₂²⋊C₄	—	2	2	no	—
288	C₂×C₂²⋊C₄	—	2	2	no	—
289	C₂×C₂²⋊C₄	—	2	2	no	—
290	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
291	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
292	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
293	C₄⋊C₈	—	4	2	no	—
294	#103	—	4	2	no	—
295	#102	—	4	2	no	—
296	#102	—	4	2	no	—
297	#102	—	4	2	no	—
298	#95	—	2	2	no	—
299	#95	—	2	2	no	—
300	#95	D₄×C₆₇	1	2	yes	—
301	#95	D₄×C₆₇	1	2	yes	—
302	C₂⁴	—	2	1	no	—
303	C₂⁴	—	2	2	no	—
304	C₂×Q₈	—	2	2	no	—
305	C₂×Q₈	—	2	2	no	—
306	C₂×Q₈	—	2	1	no	—
307	C₂×Q₈	—	2	1	no	—
308	C₂×Q₈	—	4	2	no	—
309	C₂×Q₈	—	8	2	no	—
310	C₂×D₄	—	2	2	no	—
311	C₂×D₄	—	2	2	no	—
312	C₂×D₄	—	2	2	no	—
313	C₂×D₄	—	2	2	no	—
314	C₂×D₄	—	2	2	no	—
315	C₂×D₄	—	2	2	no	—
316	C₂×D₄	—	2	1	no	—
317	C₂×D₄	—	2	2	no	—
318	C₂×D₄	—	2	1	no	—
319	C₂×D₄	—	2	2	no	—
320	C₂×D₄	—	4	4	no	—
321	C₂×D₄	—	8	2	no	—
322	C₂×D₄	—	8	2	no	—
323	C₂²×C₄	—	2	1	no	—
324	C₂²×C₄	—	2	1	no	—
325	C₂²×C₄	—	2	1	no	—
326	C₂²×C₄	—	2	2	no	—
327	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
328	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
329	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
330	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
331	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
332	SD₁₆	—	4	8	no	—
333	D₈	—	4	8	no	—
334	C₂×C₈	—	2	2	no	—
335	C₂×C₈	—	2	2	no	—
336	C₂×C₈	—	2	2	no	—
337	C₂×C₈	—	2	2	no	—
338	C₂×C₈	—	2	2	no	—
339	C₂×C₈	—	2	2	no	—
340	C₄⋊C₄	—	2	2	no	—
341	C₄⋊C₄	—	2	2	no	—
342	C₄⋊C₄	—	2	2	no	—
343	C₄⋊C₄	—	2	2	no	—
344	C₄⋊C₄	—	4	4	no	—
345	C₄⋊C₄	—	4	2	no	—
346	C₂²⋊C₄	—	4	2	no	—
347	C₂²⋊C₄	—	4	2	no	—
348	C₂²⋊C₄	—	8	2	no	—
349	C₂²⋊C₄	—	8	2	no	—
350	C₂²⋊C₄	—	8	2	no	—
351	C₂²⋊C₄	—	8	2	no	—
352	C₂²⋊C₄	—	8	2	no	—
353	C₂²⋊C₄	—	8	2	no	—
354	C₄²	—	2	1	no	—
355	C₄²	—	2	2	no	—
356	C₄²	—	2	2	no	—
357	C₄²	—	2	1	no	—
358	C₂³	—	4	1	no	—
359	C₂³	—	4	2	no	—
360	C₂³	—	4	2	no	—
361	C₂³	—	4	1	no	—
362	C₂³	—	4	1	no	—
363	C₂³	—	4	2	no	—
364	C₂³	—	4	2	no	—
365	C₂³	—	4	2	no	—
366	C₂³	—	4	2	no	—
367	C₂³	—	4	2	no	—
368	C₂³	—	4	2	no	—
369	C₂³	—	4	2	no	—
370	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
371	Q₈	—	4	2	no	—
372	Q₈	—	4	2	no	—
373	Q₈	—	8	4	no	—
374	D₄	—	4	2	no	—
375	D₄	—	4	2	no	—
376	D₄	—	4	2	no	—
377	D₄	—	4	2	no	—
378	D₄	—	4	2	no	—
379	D₄	—	4	2	no	—
380	D₄	—	8	8	no	—
381	C₂×C₄	—	2	1	no	—
382	C₂×C₄	—	2	1	no	—
383	C₂×C₄	—	2	2	no	—
384	C₂×C₄	—	2	2	no	—
385	C₂×C₄	—	2	2	no	—
386	C₂×C₄	—	2	2	no	—
387	C₂×C₄	—	2	2	no	—
388	C₂×C₄	—	2	2	no	—
389	C₂×C₄	—	2	2	no	—
390	C₂×C₄	—	2	2	no	—
391	C₂×C₄	—	4	1	no	—
392	C₂×C₄	—	4	2	no	—
393	C₂×C₄	—	4	1	no	—
394	C₂×C₄	—	4	1	no	—
395	C₂×C₄	—	4	1	no	—
396	C₂×C₄	—	4	1	no	—
397	C₂×C₄	—	4	2	no	—
398	C₂×C₄	—	4	2	no	—
399	C₂×C₄	—	4	2	no	—
400	C₂×C₄	—	4	2	no	—
401	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
402	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
403	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
404	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
405	C₈	—	4	2	no	—
406	C₈	—	4	2	no	—
407	C₂²	—	8	1	no	—
408	C₂²	—	8	2	no	—
409	C₂²	—	8	1	no	—
410	C₂²	—	8	2	no	—
411	C₂²	—	8	1	no	—
412	C₂²	—	8	2	no	—
413	C₂²	—	8	2	no	—
414	C₂²	—	8	2	no	—
415	C₂²	—	8	2	no	—
416	C₂²	—	8	2	no	—
417	C₂²	—	8	2	no	—
418	C₂²	—	8	2	no	—
419	C₂²	#535×C₆₇	1	1	yes	characteristic
420	C₂²	#535×C₆₇	1	1	yes	characteristic
421	C₂²	#537×C₆₇	1	1	yes	characteristic
422	C₂²	#576×C₆₇	1	2	yes	—
423	C₂²	#577×C₆₇	1	2	yes	—
424	C₄	—	4	1	no	—
425	C₄	—	4	1	no	—
426	C₄	—	4	2	no	—
427	C₄	—	4	2	no	—
428	C₄	—	8	1	no	—
429	C₄	—	8	1	no	—
430	C₄	—	8	1	no	—
431	C₄	—	8	2	no	—
432	C₄	H(3,𝔽₄)×C₆₇	1	2	yes	—
433	C₄	#562×C₆₇	1	2	yes	—
434	C₂	—	16	2	no	—
435	C₂	—	16	1	no	—
436	C₂	C₆₇×#2819	1	1	yes	characteristic
437	C₂	C₆₇×#3220	1	1	yes	characteristic
438	C₂	C₆₇×#3221	1	1	yes	characteristic
439	C₂	C₆₇×#3224	1	2	yes	—
440	C₂	C₆₇×#3257	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₆₇×#2011	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ #376×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#2641	C₂	semidirect
2. C₆₇×#2640	C₂	semidirect
3. C₆₇×#2055	C₂	semidirect
4. C₆₇×#2054	C₂	semidirect
5. C₆₇×#2026	C₂	semidirect
6. C₆₇×#2025	C₂	semidirect
7. C₆₇×#2023	C₂	semidirect
8. C₆₇×#2014	C₂	semidirect
9. C₆₇×#2012	C₂	semidirect
11. C₆₇×#1832	C₂	semidirect
213. #32812	C₆₇	direct (N × H)
221. #2641	C₂×C₆₇	semidirect
222. #2640	C₂×C₆₇	semidirect
223. #2055	C₂×C₆₇	semidirect
224. #2054	C₂×C₆₇	semidirect
225. #2026	C₂×C₆₇	semidirect
226. #2025	C₂×C₆₇	semidirect
227. #2023	C₂×C₆₇	semidirect
228. #2014	C₂×C₆₇	semidirect
229. #2012	C₂×C₆₇	semidirect
231. #1832	C₂×C₆₇	semidirect
232. C₆₇	#32812	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₇×#2641	2
C₆₇×#2640	2
C₆₇×#2055	2
C₆₇×#2054	2
C₆₇×#2026	2
C₆₇×#2025	2
C₆₇×#2023	2
C₆₇×#2014	2
C₆₇×#2012	2
C₆₇×#2011	2
C₆₇×#1832	2
#32812	67

Subgroups — C67×#32812 #30712063

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#32812 #30712063