Subgroups — C₆₇×#61414 #30740665

Order 17152.

1978 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3506	C₂	1	2	yes	maximal, complemented
2	C₆₇×#3401	C₂	1	8	yes	maximal, complemented
3	C₆₇×#3399	C₂	1	8	yes	maximal, complemented
4	C₆₇×#3155	C₂	1	2	yes	maximal, complemented
5	C₆₇×#3150	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#3045	C₂	1	4	yes	maximal, complemented
7	C₆₇×#3036	C₂	1	2	yes	maximal, complemented
8	C₆₇×#2961	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#2952	C₂	1	2	yes	maximal, complemented
10	C₂×#442×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×AGL(1,ℤ/8)×C₆₇	C₂²	1	4	yes	—
12	C₂×#135×C₆₇	C₂²	1	2	yes	—
13	C₂×#135×C₆₇	C₂²	1	2	yes	—
14	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
15	#559×C₆₇	C₂²	1	8	yes	complemented
16	#556×C₆₇	C₂²	1	4	yes	complemented
17	#555×C₆₇	—	2	2	no	—
18	#553×C₆₇	—	2	4	no	—
19	#550×C₆₇	C₂²	1	4	yes	complemented
20	#547×C₆₇	—	2	2	no	—
21	#538×C₆₇	—	2	4	no	—
22	#535×C₆₇	C₂²	1	4	yes	—
23	#533×C₆₇	C₂²	1	1	yes	characteristic
24	#532×C₆₇	C₂²	1	2	yes	—
25	#525×C₆₇	—	2	4	no	—
26	#520×C₆₇	C₂²	1	4	yes	—
27	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
28	#497×C₆₇	C₂²	1	4	yes	complemented
29	#496×C₆₇	C₂²	1	8	yes	—
30	#496×C₆₇	C₂²	1	4	yes	complemented
31	#495×C₆₇	C₂²	1	4	yes	—
32	#493×C₆₇	C₂²	1	4	yes	—
33	#464×C₆₇	C₂²	1	8	yes	complemented
34	#463×C₆₇	C₂²	1	8	yes	complemented
35	#463×C₆₇	C₂²	1	8	yes	complemented
36	#461×C₆₇	C₂²	1	4	yes	complemented
37	#460×C₆₇	C₂²	1	8	yes	complemented
38	#459×C₆₇	C₂²	1	4	yes	complemented
39	#459×C₆₇	C₂²	1	8	yes	complemented
40	#446×C₆₇	C₂²	1	8	yes	complemented
41	#442×C₆₇	C₂²	1	8	yes	—
42	#442×C₆₇	C₂²	1	4	yes	complemented
43	#441×C₆₇	C₂²	1	4	yes	complemented
44	C₄×SD₁₆×C₆₇	C₂²	1	8	yes	complemented
45	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic
46	#424×C₆₇	C₂²	1	2	yes	—
47	#423×C₆₇	C₂²	1	1	yes	characteristic
48	#417×C₆₇	C₂²	1	4	yes	—
49	C₂×#102×C₆₇	C₂²	1	2	yes	—
50	#405×C₆₇	C₂²	1	2	yes	—
51	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
52	C₂×#41×C₆₇	—	2	4	no	—
53	C₂×#41×C₆₇	C₂³	1	4	yes	—
54	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
55	C₂³×C₄×C₆₇	—	2	2	no	—
56	AGL(1,ℤ/8)×C₆₇	—	4	8	no	—
57	#135×C₆₇	—	2	8	no	—
58	#135×C₆₇	—	4	4	no	—
59	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
60	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
61	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
62	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
63	C₂×SD₁₆×C₆₇	C₂³	1	8	yes	—
64	C₂×D₈×C₆₇	—	2	4	no	—
65	C₂×D₈×C₆₇	C₂³	1	2	yes	—
66	C₂×D₈×C₆₇	C₂³	1	4	yes	—
67	C₂×D₈×C₆₇	C₂³	1	2	yes	—
68	C₂×D₈×C₆₇	C₂³	1	8	yes	—
69	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
70	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
71	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
72	C₄⋊Q₈×C₆₇	D₄	1	2	yes	—
73	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
74	#127×C₆₇	C₂³	1	2	yes	—
75	#127×C₆₇	C₂³	1	8	yes	—
76	#126×C₆₇	—	2	8	no	—
77	#125×C₆₇	D₄	1	2	yes	—
78	#124×C₆₇	—	2	8	no	—
79	#124×C₆₇	C₂³	1	4	yes	—
80	#124×C₆₇	C₂³	1	8	yes	—
81	#123×C₆₇	—	2	8	no	—
82	#123×C₆₇	—	2	8	no	—
83	C₂²⋊Q₈×C₆₇	—	2	4	no	—
84	C₂²⋊Q₈×C₆₇	D₄	1	4	yes	—
85	#121×C₆₇	—	2	2	no	—
86	#121×C₆₇	—	2	2	no	—
87	#121×C₆₇	—	2	4	no	—
88	#121×C₆₇	—	2	4	no	—
89	#121×C₆₇	—	2	4	no	—
90	#121×C₆₇	—	2	8	no	—
91	#120×C₆₇	—	2	8	no	—
92	C₄×Q₈×C₆₇	C₂³	1	8	yes	—
93	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
94	C₄×D₄×C₆₇	—	2	8	no	—
95	C₄×D₄×C₆₇	—	2	8	no	—
96	C₄×D₄×C₆₇	—	2	2	no	—
97	C₄×D₄×C₆₇	—	2	2	no	—
98	C₄×D₄×C₆₇	—	2	2	no	—
99	C₄×D₄×C₆₇	—	2	2	no	—
100	C₄×D₄×C₆₇	C₂³	1	4	yes	—
101	C₄×D₄×C₆₇	C₂³	1	8	yes	—
102	C₄×D₄×C₆₇	C₂³	1	4	yes	—
103	#117×C₆₇	—	2	4	no	—
104	#117×C₆₇	—	2	4	no	—
105	#117×C₆₇	C₂³	1	2	yes	—
106	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
107	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
108	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
109	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
110	#106×C₆₇	C₂³	1	4	yes	—
111	C₄⋊C₈×C₆₇	C₂³	1	8	yes	—
112	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
113	#103×C₆₇	C₂³	1	8	yes	—
114	#102×C₆₇	C₂³	1	8	yes	—
115	#102×C₆₇	C₂³	1	8	yes	—
116	#97×C₆₇	C₂³	1	4	yes	—
117	#97×C₆₇	C₂³	1	4	yes	—
118	C₄×C₈×C₆₇	C₂³	1	4	yes	—
119	C₂⁴×C₆₇	—	2	2	no	—
120	#41×C₆₇	—	2	8	no	—
121	#41×C₆₇	—	2	8	no	—
122	#41×C₆₇	—	8	8	no	—
123	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
124	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
125	C₂×D₄×C₆₇	—	2	4	no	—
126	C₂×D₄×C₆₇	—	2	2	no	—
127	C₂×D₄×C₆₇	—	2	2	no	—
128	C₂×D₄×C₆₇	—	2	8	no	—
129	C₂×D₄×C₆₇	—	2	2	no	—
130	C₂×D₄×C₆₇	—	2	8	no	—
131	C₂×D₄×C₆₇	—	2	4	no	—
132	C₂×D₄×C₆₇	—	2	4	no	—
133	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
134	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
135	C₂²×C₄×C₆₇	—	2	2	no	—
136	C₂²×C₄×C₆₇	—	2	4	no	—
137	C₂²×C₄×C₆₇	—	2	4	no	—
138	C₂²×C₄×C₆₇	—	2	8	no	—
139	C₂²×C₄×C₆₇	—	2	4	no	—
140	C₂²×C₄×C₆₇	—	8	4	no	—
141	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
142	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
143	Q₁₆×C₆₇	—	2	4	no	—
144	Q₁₆×C₆₇	—	4	2	no	—
145	SD₁₆×C₆₇	—	2	8	no	—
146	SD₁₆×C₆₇	—	4	4	no	—
147	SD₁₆×C₆₇	—	4	8	no	—
148	D₈×C₆₇	—	2	8	no	—
149	D₈×C₆₇	—	2	4	no	—
150	D₈×C₆₇	—	4	2	no	—
151	D₈×C₆₇	—	4	8	no	—
152	M₁₆×C₆₇	—	2	4	no	—
153	C₂×C₈×C₆₇	—	2	2	no	—
154	C₂×C₈×C₆₇	#41	1	4	yes	complemented
155	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
156	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
157	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
158	C₄⋊C₄×C₆₇	—	2	2	no	—
159	C₄⋊C₄×C₆₇	—	2	8	no	—
160	C₄⋊C₄×C₆₇	—	2	2	no	—
161	C₄⋊C₄×C₆₇	—	2	2	no	—
162	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
163	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
164	C₄⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
165	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
166	C₂²⋊C₄×C₆₇	—	2	4	no	—
167	C₂²⋊C₄×C₆₇	—	2	4	no	—
168	C₂²⋊C₄×C₆₇	—	2	8	no	—
169	C₂²⋊C₄×C₆₇	—	2	8	no	—
170	C₂²⋊C₄×C₆₇	—	2	4	no	—
171	C₂²⋊C₄×C₆₇	—	2	8	no	—
172	C₂²⋊C₄×C₆₇	—	2	4	no	—
173	C₂²⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
174	C₄²×C₆₇	—	2	8	no	—
175	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
176	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
177	C₄²×C₆₇	C₂⁴	1	2	yes	—
178	C₄²×C₆₇	C₂⁴	1	4	yes	—
179	C₂³×C₆₇	—	2	4	no	—
180	C₂³×C₆₇	—	2	4	no	—
181	C₂³×C₆₇	—	4	4	no	—
182	C₂³×C₆₇	—	8	2	no	—
183	C₂³×C₆₇	—	8	2	no	—
184	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
185	Q₈×C₆₇	—	2	4	no	—
186	Q₈×C₆₇	—	2	4	no	—
187	Q₈×C₆₇	—	4	4	no	—
188	D₄×C₆₇	—	2	4	no	—
189	D₄×C₆₇	—	2	4	no	—
190	D₄×C₆₇	—	2	4	no	—
191	D₄×C₆₇	—	2	4	no	—
192	D₄×C₆₇	—	4	4	no	—
193	D₄×C₆₇	—	4	8	no	—
194	D₄×C₆₇	—	4	4	no	—
195	D₄×C₆₇	—	8	2	no	—
196	D₄×C₆₇	—	8	8	no	—
197	D₄×C₆₇	—	8	8	no	—
198	C₂×C₄×C₆₇	—	2	4	no	—
199	C₂×C₄×C₆₇	—	2	4	no	—
200	C₂×C₄×C₆₇	—	4	2	no	—
201	C₂×C₄×C₆₇	—	4	8	no	—
202	C₂×C₄×C₆₇	—	4	8	no	—
203	C₂×C₄×C₆₇	—	8	4	no	—
204	C₂×C₄×C₆₇	—	8	8	no	—
205	C₂×C₄×C₆₇	—	8	2	no	—
206	C₂×C₄×C₆₇	C₂²×D₄	1	8	yes	—
207	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
208	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
209	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
210	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
211	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
212	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
213	C₈×C₆₇	—	2	2	no	—
214	C₈×C₆₇	—	2	4	no	—
215	C₈×C₆₇	C₂×#41	1	4	yes	—
216	C₂²×C₆₇	—	2	2	no	—
217	C₂²×C₆₇	—	4	4	no	—
218	C₂²×C₆₇	—	4	4	no	—
219	C₂²×C₆₇	—	4	4	no	—
220	C₂²×C₆₇	—	4	4	no	—
221	C₂²×C₆₇	—	4	4	no	—
222	C₂²×C₆₇	—	4	4	no	—
223	C₂²×C₆₇	—	8	8	no	—
224	C₂²×C₆₇	#550	1	2	yes	—
225	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
226	C₄×C₆₇	—	2	2	no	—
227	C₄×C₆₇	—	2	2	no	—
228	C₄×C₆₇	—	2	8	no	—
229	C₄×C₆₇	—	4	4	no	—
230	C₄×C₆₇	—	4	4	no	—
231	C₄×C₆₇	#550	1	2	yes	—
232	C₄×C₆₇	C₂²×#41	1	1	yes	characteristic
233	C₄×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
234	#61414	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
235	C₂×C₆₇	—	2	2	no	—
236	C₂×C₆₇	—	8	4	no	—
237	C₂×C₆₇	—	8	4	no	—
238	C₂×C₆₇	C₂×#550	1	1	yes	characteristic
239	C₂×C₆₇	#3585	1	2	yes	—
240	#3506	C₂×C₆₇	1	2	yes	complemented
241	#3401	C₂×C₆₇	1	8	yes	complemented
242	#3399	C₂×C₆₇	1	8	yes	complemented
243	#3155	C₂×C₆₇	1	2	yes	complemented
244	#3150	C₂×C₆₇	1	1	yes	characteristic, complemented
245	#3045	C₂×C₆₇	1	4	yes	complemented
246	#3036	C₂×C₆₇	1	2	yes	complemented
247	#2961	C₂×C₆₇	1	1	yes	characteristic, complemented
248	#2952	C₂×C₆₇	1	2	yes	complemented
249	C₂×#442	C₂×C₆₇	1	1	yes	characteristic, complemented
250	C₆₇	#61414	1	1	yes	characteristic, direct product factor, Sylow
251	C₂×AGL(1,ℤ/8)	C₂²×C₆₇	1	4	yes	—
252	C₂×#135	C₂²×C₆₇	1	2	yes	—
253	C₂×#135	C₂²×C₆₇	1	2	yes	—
254	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
255	#559	C₂²×C₆₇	1	8	yes	complemented
256	#556	C₂²×C₆₇	1	4	yes	complemented
257	#555	—	2	2	no	—
258	#553	—	2	4	no	—
259	#550	C₂²×C₆₇	1	4	yes	complemented
260	#547	—	2	2	no	—
261	#538	—	2	4	no	—
262	#535	C₂²×C₆₇	1	4	yes	—
263	#533	C₂²×C₆₇	1	1	yes	characteristic
264	#532	C₂²×C₆₇	1	2	yes	—
265	#525	—	2	4	no	—
266	#520	C₂²×C₆₇	1	4	yes	—
267	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
268	#497	C₂²×C₆₇	1	4	yes	complemented
269	#496	C₂²×C₆₇	1	8	yes	—
270	#496	C₂²×C₆₇	1	4	yes	complemented
271	#495	C₂²×C₆₇	1	4	yes	—
272	#493	C₂²×C₆₇	1	4	yes	—
273	#464	C₂²×C₆₇	1	8	yes	complemented
274	#463	C₂²×C₆₇	1	8	yes	complemented
275	#463	C₂²×C₆₇	1	8	yes	complemented
276	#461	C₂²×C₆₇	1	4	yes	complemented
277	#460	C₂²×C₆₇	1	8	yes	complemented
278	#459	C₂²×C₆₇	1	4	yes	complemented
279	#459	C₂²×C₆₇	1	8	yes	complemented
280	#446	C₂²×C₆₇	1	8	yes	complemented
281	#442	C₂²×C₆₇	1	8	yes	—
282	#442	C₂²×C₆₇	1	4	yes	complemented
283	#441	C₂²×C₆₇	1	4	yes	complemented
284	C₄×SD₁₆	C₂²×C₆₇	1	8	yes	complemented
285	C₂×#106	C₂²×C₆₇	1	1	yes	characteristic
286	#424	C₂²×C₆₇	1	2	yes	—
287	#423	C₂²×C₆₇	1	1	yes	characteristic
288	#417	C₂²×C₆₇	1	4	yes	—
289	C₂×#102	C₂²×C₆₇	1	2	yes	—
290	#405	C₂²×C₆₇	1	2	yes	—
291	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
292	C₂×#41	—	2	4	no	—
293	C₂×#41	C₂³×C₆₇	1	4	yes	—
294	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
295	C₂³×C₄	—	2	2	no	—
296	AGL(1,ℤ/8)	—	4	8	no	—
297	#135	—	2	8	no	—
298	#135	—	4	4	no	—
299	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
300	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
301	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
302	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
303	C₂×SD₁₆	C₂³×C₆₇	1	8	yes	—
304	C₂×D₈	—	2	4	no	—
305	C₂×D₈	C₂³×C₆₇	1	4	yes	—
306	C₂×D₈	C₂³×C₆₇	1	2	yes	—
307	C₂×D₈	C₂³×C₆₇	1	2	yes	—
308	C₂×D₈	C₂³×C₆₇	1	8	yes	—
309	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
310	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
311	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
312	C₄⋊Q₈	D₄×C₆₇	1	2	yes	—
313	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
314	#127	C₂³×C₆₇	1	2	yes	—
315	#127	C₂³×C₆₇	1	8	yes	—
316	#126	—	2	8	no	—
317	#125	D₄×C₆₇	1	2	yes	—
318	#124	—	2	8	no	—
319	#124	C₂³×C₆₇	1	4	yes	—
320	#124	C₂³×C₆₇	1	8	yes	—
321	#123	—	2	8	no	—
322	#123	—	2	8	no	—
323	C₂²⋊Q₈	—	2	4	no	—
324	C₂²⋊Q₈	D₄×C₆₇	1	4	yes	—
325	#121	—	2	2	no	—
326	#121	—	2	2	no	—
327	#121	—	2	4	no	—
328	#121	—	2	4	no	—
329	#121	—	2	4	no	—
330	#121	—	2	8	no	—
331	#120	—	2	8	no	—
332	C₄×Q₈	C₂³×C₆₇	1	8	yes	—
333	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
334	C₄×D₄	—	2	8	no	—
335	C₄×D₄	—	2	8	no	—
336	C₄×D₄	—	2	2	no	—
337	C₄×D₄	—	2	2	no	—
338	C₄×D₄	—	2	2	no	—
339	C₄×D₄	—	2	2	no	—
340	C₄×D₄	C₂³×C₆₇	1	4	yes	—
341	C₄×D₄	C₂³×C₆₇	1	4	yes	—
342	C₄×D₄	C₂³×C₆₇	1	8	yes	—
343	#117	—	2	4	no	—
344	#117	—	2	4	no	—
345	#117	C₂³×C₆₇	1	2	yes	—
346	C₂×C₄⋊C₄	—	2	4	no	—
347	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
348	C₂×C₂²⋊C₄	—	2	2	no	—
349	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
350	#106	C₂³×C₆₇	1	4	yes	—
351	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
352	C₄⋊C₈	C₂³×C₆₇	1	8	yes	—
353	#103	C₂³×C₆₇	1	8	yes	—
354	#102	C₂³×C₆₇	1	8	yes	—
355	#102	C₂³×C₆₇	1	8	yes	—
356	#97	C₂³×C₆₇	1	4	yes	—
357	#97	C₂³×C₆₇	1	4	yes	—
358	C₄×C₈	C₂³×C₆₇	1	4	yes	—
359	C₂⁴	—	2	2	no	—
360	#41	—	2	8	no	—
361	#41	—	2	8	no	—
362	#41	—	8	8	no	—
363	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
364	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
365	C₂×D₄	—	2	4	no	—
366	C₂×D₄	—	2	2	no	—
367	C₂×D₄	—	2	2	no	—
368	C₂×D₄	—	2	8	no	—
369	C₂×D₄	—	2	2	no	—
370	C₂×D₄	—	2	8	no	—
371	C₂×D₄	—	2	4	no	—
372	C₂×D₄	—	2	4	no	—
373	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
374	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
375	C₂²×C₄	—	2	2	no	—
376	C₂²×C₄	—	2	4	no	—
377	C₂²×C₄	—	2	4	no	—
378	C₂²×C₄	—	2	8	no	—
379	C₂²×C₄	—	2	4	no	—
380	C₂²×C₄	—	8	4	no	—
381	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
382	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
383	Q₁₆	—	2	4	no	—
384	Q₁₆	—	4	2	no	—
385	SD₁₆	—	2	8	no	—
386	SD₁₆	—	4	4	no	—
387	SD₁₆	—	4	8	no	—
388	D₈	—	2	8	no	—
389	D₈	—	2	4	no	—
390	D₈	—	4	2	no	—
391	D₈	—	4	8	no	—
392	M₁₆	—	2	4	no	—
393	C₂×C₈	—	2	2	no	—
394	C₂×C₈	#41×C₆₇	1	4	yes	complemented
395	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
396	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
397	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
398	C₄⋊C₄	—	2	2	no	—
399	C₄⋊C₄	—	2	8	no	—
400	C₄⋊C₄	—	2	2	no	—
401	C₄⋊C₄	—	2	2	no	—
402	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
403	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
404	C₄⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
405	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
406	C₂²⋊C₄	—	2	4	no	—
407	C₂²⋊C₄	—	2	4	no	—
408	C₂²⋊C₄	—	2	8	no	—
409	C₂²⋊C₄	—	2	8	no	—
410	C₂²⋊C₄	—	2	4	no	—
411	C₂²⋊C₄	—	2	8	no	—
412	C₂²⋊C₄	—	2	4	no	—
413	C₂²⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
414	C₄²	—	2	8	no	—
415	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
416	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
417	C₄²	C₂⁴×C₆₇	1	4	yes	—
418	C₄²	C₂⁴×C₆₇	1	2	yes	—
419	C₂³	—	2	4	no	—
420	C₂³	—	2	4	no	—
421	C₂³	—	4	4	no	—
422	C₂³	—	8	2	no	—
423	C₂³	—	8	2	no	—
424	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
425	Q₈	—	2	4	no	—
426	Q₈	—	2	4	no	—
427	Q₈	—	4	4	no	—
428	D₄	—	2	4	no	—
429	D₄	—	2	4	no	—
430	D₄	—	2	4	no	—
431	D₄	—	2	4	no	—
432	D₄	—	4	4	no	—
433	D₄	—	4	8	no	—
434	D₄	—	4	4	no	—
435	D₄	—	8	8	no	—
436	D₄	—	8	2	no	—
437	D₄	—	8	8	no	—
438	C₂×C₄	—	2	4	no	—
439	C₂×C₄	—	2	4	no	—
440	C₂×C₄	—	4	2	no	—
441	C₂×C₄	—	4	8	no	—
442	C₂×C₄	—	4	8	no	—
443	C₂×C₄	—	8	4	no	—
444	C₂×C₄	—	8	8	no	—
445	C₂×C₄	—	8	2	no	—
446	C₂×C₄	C₂²×D₄×C₆₇	1	8	yes	—
447	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
448	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
449	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
450	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
451	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
452	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
453	C₈	—	2	2	no	—
454	C₈	—	2	4	no	—
455	C₈	C₂×#41×C₆₇	1	4	yes	—
456	C₂²	—	2	2	no	—
457	C₂²	—	4	4	no	—
458	C₂²	—	4	4	no	—
459	C₂²	—	4	4	no	—
460	C₂²	—	4	4	no	—
461	C₂²	—	4	4	no	—
462	C₂²	—	4	4	no	—
463	C₂²	—	8	8	no	—
464	C₂²	#550×C₆₇	1	2	yes	—
465	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
466	C₄	—	2	2	no	—
467	C₄	—	2	2	no	—
468	C₄	—	2	8	no	—
469	C₄	—	4	4	no	—
470	C₄	—	4	4	no	—
471	C₄	#550×C₆₇	1	2	yes	—
472	C₄	C₂²×#41×C₆₇	1	1	yes	characteristic
473	C₄	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
474	C₂	—	2	2	no	—
475	C₂	—	8	4	no	—
476	C₂	—	8	4	no	—
477	C₂	C₂×#550×C₆₇	1	1	yes	characteristic
478	C₂	C₆₇×#3585	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ #533×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3506	C₂	semidirect
2. C₆₇×#3401	C₂	semidirect
3. C₆₇×#3399	C₂	semidirect
4. C₆₇×#3155	C₂	semidirect
5. C₆₇×#3150	C₂	semidirect
6. C₆₇×#3045	C₂	semidirect
7. C₆₇×#3036	C₂	semidirect
8. C₆₇×#2961	C₂	semidirect
9. C₆₇×#2952	C₂	semidirect
10. C₂×#442×C₆₇	C₂	semidirect
15. #559×C₆₇	C₂²	semidirect
16. #556×C₆₇	C₂²	semidirect
19. #550×C₆₇	C₂²	semidirect
28. #497×C₆₇	C₂²	semidirect
30. #496×C₆₇	C₂²	semidirect
33. #464×C₆₇	C₂²	semidirect
34. #463×C₆₇	C₂²	semidirect
35. #463×C₆₇	C₂²	semidirect
36. #461×C₆₇	C₂²	semidirect
37. #460×C₆₇	C₂²	semidirect
38. #459×C₆₇	C₂²	semidirect
39. #459×C₆₇	C₂²	semidirect
40. #446×C₆₇	C₂²	semidirect
42. #442×C₆₇	C₂²	semidirect
43. #441×C₆₇	C₂²	semidirect
44. C₄×SD₁₆×C₆₇	C₂²	semidirect
154. C₂×C₈×C₆₇	#41	semidirect
234. #61414	C₆₇	direct (N × H)
240. #3506	C₂×C₆₇	semidirect
241. #3401	C₂×C₆₇	semidirect
242. #3399	C₂×C₆₇	semidirect
243. #3155	C₂×C₆₇	semidirect
244. #3150	C₂×C₆₇	semidirect
245. #3045	C₂×C₆₇	semidirect
246. #3036	C₂×C₆₇	semidirect
247. #2961	C₂×C₆₇	semidirect
248. #2952	C₂×C₆₇	semidirect
249. C₂×#442	C₂×C₆₇	semidirect
250. C₆₇	#61414	direct (N × H)
255. #559	C₂²×C₆₇	semidirect
256. #556	C₂²×C₆₇	semidirect
259. #550	C₂²×C₆₇	semidirect
268. #497	C₂²×C₆₇	semidirect
270. #496	C₂²×C₆₇	semidirect
273. #464	C₂²×C₆₇	semidirect
274. #463	C₂²×C₆₇	semidirect
275. #463	C₂²×C₆₇	semidirect
276. #461	C₂²×C₆₇	semidirect
277. #460	C₂²×C₆₇	semidirect
278. #459	C₂²×C₆₇	semidirect
279. #459	C₂²×C₆₇	semidirect
280. #446	C₂²×C₆₇	semidirect
282. #442	C₂²×C₆₇	semidirect
283. #441	C₂²×C₆₇	semidirect
284. C₄×SD₁₆	C₂²×C₆₇	semidirect
394. C₂×C₈	#41×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3506	2
C₆₇×#3401	2
C₆₇×#3399	2
C₆₇×#3155	2
C₆₇×#3150	2
C₆₇×#3045	2
C₆₇×#3036	2
C₆₇×#2961	2
C₆₇×#2952	2
C₂×#442×C₆₇	2
#61414	67

Subgroups — C67×#61414 #30740665

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#61414 #30740665